[ { "run": "RUN-001", "report_file": "RUN-001-CORPUS-IDENTITY.md", "source_document": "The Moser case as published on the AMRAL site (amral.evemisslab.com/moser/ and its English edition /en/moser/, site revision 127fb1c9; the case's two directories are trees eccb1c2b and 0590bafe, as at d7a8e7a4), against the fifteen round packages", "verdict_line": "**Verdict: the packages PASS; the pages FAIL five of seven checks, all on presentation.**", "items": [ { "claim": "The hub states the packages are the original bytes, unaltered and verified package by package by SHA-256 (「原始封包未經改動」; 「SHA-256 逐包核驗 · 原始位元組,不重新打包」); each package matches its pinned size, SHA-256 and member count.", "kind": "certificate", "numbers": [ "15 packages", "12,039,698 bytes", "282 members" ], "quote": "originals the site says they are: size, SHA-256 and member count agree with both manifests and with the bytes the", "report_finding": "PASS", "section": "C1", "strength": "exact-arithmetic certificate" }, { "claim": "Both manifests (moser/manifest.json, en/moser/manifest.json) list exactly the pinned packages in the programme's order with filename, size, SHA-256 and member count, and the served zips are those packages.", "kind": "specification", "numbers": [ "2 × 15 entries", "15 served packages" ], "quote": "both manifests list exactly the pinned packages in the programme's order, with filename, size, SHA-256, member count; the served zips match | **PASS** |", "report_finding": "PASS", "section": "C2", "strength": "exact-arithmetic certificate" }, { "claim": "The hub implies every file a zh round page serves is the package member at the same path; 267 of the 272 are byte-identical (the five that differ are recorded separately).", "kind": "specification", "numbers": [ "267/272", "272 served files" ], "quote": "| C3 | every file a zh page serves is byte-identical to the package member at the same path | **FAIL** — 267/272; 5 differ |", "report_finding": "PARTIAL", "section": "C3", "strength": "exact-arithmetic certificate", "audit": "finding moved from 'PASS' by the strength audit: the report's C3 is FAIL (267/272); the strength is that of the 267 byte-identical files, provenance by content hash, the closest fit" }, { "claim": "The round pages present five served documents (rounds 4, 8, 12) as the package's files.", "kind": "specification", "numbers": [ "5 of 272", "rounds 4, 8, 12", "equals_package_with_escapes_restored true for all five" ], "quote": "But they are not the package bytes. The page presents them as the package's files, says nothing about", "report_finding": "FAIL", "section": "C3; Findings 1", "strength": "corrected" }, { "claim": "Five package documents state TeX formulas (round 4 ROUND_04_CUSP_AUDIT.md; round 8 ROUND_08_DENSE_POINTCLOUD_CROSSCHECK.md, ROUND_08_FINITE_WIDTH_AUDIT.md, ROUND_08_PEAK_DENSE_POINTCLOUD_CROSSCHECK.md; round 12 ROUND_12_REPORT.md).", "kind": "reading", "numbers": [ "round 4: 0x07 ×1, 0x08 ×2, 0x0c ×1, one split by a newline", "round 8: 0x0b ×1; 0x08 ×1, 0x0b ×2; 0x0b ×1", "round 12: 0x08 ×9, 0x09 ×4, 0x0b ×1" ], "quote": "Five documents in three packages were written through a non-raw string literal, so TeX backslash sequences became", "report_finding": "NOT AS STATED", "section": "Findings 1", "strength": "corrected" }, { "claim": "Each files.json names its own package and lists exactly the served files with their sizes.", "kind": "specification", "numbers": [ "5 files", "1–14 bytes smaller" ], "quote": "the repair, and its `files.json` still gives the package sizes (1–14 bytes smaller than the files it serves).", "report_finding": "FAIL", "section": "C4", "strength": "corrected" }, { "claim": "Each round page shows its own round, version, date, SHA-256, package link, progress counter and previous/next links.", "kind": "specification", "numbers": [ "30 pages" ], "quote": "Everything else C5 guards holds on all thirty pages: round number, version, date, SHA-256, the package link, the", "report_finding": "PASS", "section": "C5; Findings 2", "strength": "label only" }, { "claim": "The English hub links every package for download.", "kind": "specification", "numbers": [ "15 English hub links", "404", "200" ], "quote": "English hub are dead links. Checked on the live site the same day by HTTP HEAD, outside the gate log", "report_finding": "FAIL", "section": "C5; Findings 2", "strength": "refuted" }, { "claim": "The round pages state each package's size (round 8 1575.6 KB, round 14 1409.4 KB, round 15 3943.7 KB, in both languages).", "kind": "specification", "numbers": [ "6 size labels", "round 8: 1575.6 KB vs 1,613,363 bytes, 1575.5 KB", "round 14: 1409.4 KB vs 1409.3", "round 15: 3943.7 KB vs 3942.8" ], "quote": "1,613,363 bytes, 1575.5 KB), round 14 says 1409.4 KB (1409.3), round 15 says 3943.7 KB (3942.8). The SHA-256 beside", "report_finding": "FAIL", "section": "C5; Findings 2", "strength": "corrected" }, { "claim": "Each round page's status line is the stage status the package's documents state of themselves, reproduced as written (「包內文件自陳的階段性狀態,原樣照登」).", "kind": "reading", "numbers": [ "15 status lines" ], "quote": "Every status line occurs in the round's main report. None is byte-identical: each source has full-width semicolons", "report_finding": "PASS", "section": "C6; Findings 3", "strength": "label only" }, { "claim": "Each quotation is attributed to a numbered, titled section of a named document and cites figures from it.", "kind": "reading", "numbers": [ "579 leaf boxes, round 12" ], "quote": "The attributions themselves all check out: every cited section exists, has the title the page", "report_finding": "PASS", "section": "C6; Findings 4", "strength": "label only" }, { "claim": "The quotations 「…」 on the round pages are the source's words, in order, in the cited sections (rounds 2–6, 8–15).", "kind": "reading", "numbers": [ "typographic: round 12", "markup: rounds 2, 3, 4, 6, 8, 9, 10, 11, 13, 15", "words: rounds 5, 14 (and 7)" ], "quote": "Quotations: thirteen hold, one is spliced, one ends early (C6)", "report_finding": "PASS", "section": "C6; Findings 4", "strength": "label only" }, { "claim": "Round 1's page quotes §4 「初步結構發現」 as 「固定相位下,平移項只占據 Fourier 的一階正弦與餘弦模態……真正壓迫容器的是平移後仍存在的高階峰值,對應曲線凸包的角、…」.", "kind": "reading", "numbers": [ "1 quotation, 2 parts" ], "quote": "sense is the source's; the words in quotation marks are not.", "report_finding": "FAIL", "section": "C6; Findings 4", "strength": "corrected" }, { "claim": "Round 7's page quotes the working proposition, closing 「…而是:曲率在曲線參數上的集中位置。」.", "kind": "reading", "numbers": [], "quote": "The words are the source's (so C6 passes), but the sentence is", "report_finding": "PARTIAL", "section": "C6; Findings 4", "strength": "corrected" }, { "claim": "Round 5's status note quotes 「真正成果不是小數點後的提升,而是把候選的生成原因完整寫出。」.", "kind": "reading", "numbers": [], "quote": "the source continues with a colon and the boxed result it announces; the sentence is complete in sense.", "report_finding": "PASS", "section": "Findings 4", "strength": "label only" }, { "claim": "The hub quotes 「不構成 Moser 問題的新上下界」 as something every round's README says of itself (holds verbatim for 12 rounds, in other words for rounds 1 and 8).", "kind": "reading", "numbers": [ "12/15 verbatim", "14 as paraphrase" ], "quote": "Read as a paraphrase, the hub's sentence holds for fourteen rounds; as a quotation, for twelve.", "report_finding": "PASS", "section": "C7; Findings 5", "strength": "label only" }, { "claim": "Round 12's README, like every round's, disclaims new upper or lower bounds for the Moser problem.", "kind": "reading", "numbers": [ "round 12" ], "quote": "For round 12 it does not hold in either reading.", "report_finding": "FAIL", "section": "C7; Findings 5", "strength": "refuted" }, { "claim": "The English round pages serve the packages' files under their original paths, listed at the originals' sizes, with the status note \"reproduced verbatim\".", "kind": "specification", "numbers": [ "38 English translations" ], "quote": "but the translations are not marked as translations.", "report_finding": "NOT AS STATED", "section": "Findings 6", "strength": "corrected" }, { "claim": "Each round page gives the package's file list.", "kind": "specification", "numbers": [ "10 of 282 members not served", "JSON over 30,000 bytes", "CSV over 1 MiB" ], "quote": "The pages do not say the lists are partial.", "report_finding": "PARTIAL", "section": "Not defects — What the file lists leave out", "strength": "label only" }, { "claim": "Package versions: four packages carry a sub-version in the filename (v0.2.2, v0.4.1, v0.6.1, v0.8.3) and a base version in their wrapper folder and READMEs (round 8's README says v0.8.3).", "kind": "specification", "numbers": [ "v0.2.2, v0.4.1, v0.6.1, v0.8.3" ], "quote": "The filename is the identifier used throughout this tree.", "report_finding": "PASS", "section": "Not defects — Version names", "strength": "label only" }, { "claim": "Each page's date tag is the package's own MANIFEST.json date (round 1's zip date).", "kind": "specification", "numbers": [ "15 dates" ], "quote": "date), and all fifteen agree.", "report_finding": "PASS", "section": "Not defects — Dates", "strength": "label only" }, { "claim": "The hub labels the zip timestamps UTC; the packages date round 12 2026-07-27 and rounds 13–15 2026-07-26.", "kind": "specification", "numbers": [ "round 12 declares 2026-07-27", "rounds 13–15 declare 2026-07-26", "round 12 files 2026-07-26 16:56", "2026-07-27 02:49–03:39" ], "quote": "these zips has an extended-timestamp field, so the packages do not support that label.", "report_finding": "NOT AS STATED", "section": "Not defects — Dates", "strength": "label only" }, { "claim": "Round pages name each round (rounds 2, 11 and 14 named differently on neighbouring pages; five English pages label links \"Previous\"/\"Next\").", "kind": "specification", "numbers": [ "rounds 2, 11, 14", "English rounds 6–10" ], "quote": "Neighbouring pages name rounds 2, 11 and 14 differently:", "report_finding": "PARTIAL", "section": "Not defects — Names", "strength": "label only" }, { "claim": "(Recomputation's instrument) The drill plants defects in the corpus and checks each is caught by exactly its named checks.", "kind": "method", "numbers": [ "65 planted", "65 caught", "4 controls", "411/411" ], "quote": "The drill planted 65 defects; every one was caught by", "report_finding": "PASS", "section": "The drill", "strength": "label only" } ], "corrections": [ { "what": "Five served documents (C3)", "source_says": "The round pages serve the package's files.", "report_finds": "Five documents in rounds 4, 8 and 12 are served as repaired copies, each exactly the package text with its TeX escapes put back, undisclosed.", "quote": "repaired copies, and each repaired copy is **exactly** the package text with those escapes put back" }, { "what": "TeX in five package documents", "source_says": "TeX formulas such as boxed, varepsilon, frac, ast, text and nabla commands.", "report_finds": "The backslash sequences became control characters (backspace, vertical tab, form feed, bell, TAB, newline) inside the package; restore_escapes restores them and is the restoration the site used.", "quote": "Five documents in three packages were written through a non-raw string literal, so TeX backslash sequences became" }, { "what": "files.json sizes (C4)", "source_says": "files.json lists the served files with their sizes.", "report_finds": "For the five repaired files it gives the package sizes, 1–14 bytes smaller than the files served.", "quote": "the repair, and its `files.json` still gives the package sizes (1–14 bytes smaller than the files it serves)." }, { "what": "English hub download links (C5)", "source_says": "Each package is downloadable from the English hub at ./packages/.", "report_finds": "/en/moser/packages/ does not exist; all fifteen buttons are dead; /moser/packages/ serves the zips, and the English round pages link there correctly.", "quote": "`/en/moser/packages/Moser_Skew_Lab_v0.1.zip` returns 404; `/moser/packages/Moser_Skew_Lab_v0.1.zip` returns 200." }, { "what": "Package size labels (C5)", "source_says": "Round 8 1575.6 KB, round 14 1409.4 KB, round 15 3943.7 KB.", "report_finds": "1575.5 KB (1,613,363 bytes), 1409.3 KB, 3942.8 KB; the SHA-256 beside each is correct.", "quote": "Three round pages state the package size wrongly, in both languages: round 8 says 1575.6 KB (the package is" }, { "what": "Round 1 quotation (C6)", "source_says": "A quotation of §4 「初步結構發現」.", "report_finds": "The subject 它 is replaced by its antecedent without brackets, and the §4.3 heading is joined to its bullet list, dropping the body's clause without an ellipsis; it should be marked as a paraphrase or quoted as written.", "quote": "The page replaced its subject 它 with the" }, { "what": "Round 7 quotation (C6)", "source_says": "The working proposition closes with a full stop after the boxed phrase.", "report_finds": "The source continues without a stop into the contacts the concentration controls; the page's sentence is narrower than the report's.", "quote": "and a list of the contacts it controls. The page's full stop makes the" }, { "what": "README disclaimer (C7)", "source_says": "Every round's README says 「不構成 Moser 問題的新上下界」.", "report_finds": "Verbatim in 12 READMEs, in other words in rounds 1 and 8; round 12's README (and package) makes no statement about bounds.", "quote": "Round 12's README is one line about its partial `mpmath.iv` replay, and makes no statement about bounds." }, { "what": "English edition files and status note", "source_says": "The English pages serve the package files, and the status note is \"reproduced verbatim\".", "report_finds": "38 files are English translations served under the original paths at the Chinese originals' sizes, unmarked; the status note is an English rendering.", "quote": "reads \"reproduced verbatim\" over an English rendering of a Chinese status line." } ], "open_items": [ "Whether any figure in any package is right.", "It should be marked as a paraphrase, or quoted as" ], "cross_refs": [ "RUN-002 (the runs from RUN-002 on check the packages' figures, one round at a time, from the package bytes; the five documents of rounds 4, 8 and 12 must be read with their escapes restored)" ], "unclear": [ "The 'source' here is the site's presentation layer, not a round document: items record what the pages say about the packages.", "C1, C2 and the 267 identical served files are byte/hash identity, labelled 'exact-arithmetic certificate' as the closest fit.", "The five repaired served copies and the files.json sizes are labelled 'corrected', not 'refuted': the repair is correct and the documents' role survives; the report calls it a presentation defect.", "The English hub's dead links are labelled 'refuted' (false as stated for every button); the live HTTP check was made outside the gate log.", "Status lines are recorded as PASS / label only although none is byte-identical (punctuation width differs), as the report rules.", "Round 7's quotation passes C6 but narrows the sentence; recorded PARTIAL / corrected.", "The hub's UTC label for zip timestamps is unsupported rather than shown false; recorded NOT AS STATED with 'label only'." ], "for_paper": "The fifteen Moser round packages served by the AMRAL site are the pinned originals: size, SHA-256 and member count agree with both manifests and with the served bytes, so every later run reads verified package bytes. The pages around them are inexact on five of seven checks, all presentation: five served documents (rounds 4, 8, 12) are undisclosed repaired copies whose TeX escapes, broken into control characters in the packages themselves, were restored; files.json gives package sizes for them; six size labels are wrong; the English hub's fifteen download links are dead; round 1's quotation is spliced and round 7's ends early. The hub's quoted README disclaimer holds verbatim for twelve rounds, as a paraphrase for fourteen, and not at all for round 12. The run settles nothing about any figure in any package." }, { "run": "RUN-002", "report_file": "RUN-002-ROUND01-SUPPORT-SKEW-LP.md", "source_document": "Round 01 (v0.1): Moser_Skew_Lab_v0.1.zip (SHA-256 d7a832b4…) — its README, reports/ROUND_01_REPORT.md, and its data (experiment_summary.json, benchmark_records.json, adversarial_history.json, adversarial_best_curve.csv, adversarial_skew_field.csv), read from the package bytes", "verdict_line": "**Verdict: PARTIAL** — what Round 01 computes for the curves its documents define reproduces, but several of the numbers it states do not hold as stated.", "items": [ { "claim": "The container T has legs (3+4√3)/18 and (4+√3)/6, Wetzel's 30-60-90 triangle, used scaled by 1/1.0048.", "kind": "theorem", "numbers": [ "(3+4√3)/18", "(4+√3)/6", "b = √3·a", "inscribed square side exactly 1/3" ], "quote": "| T's geometry: the right angle, b = √3·a, the inscribed square of side 1/3 | exact (symbolic) |", "report_finding": "PASS", "section": "W1; Findings 1 — The triangle", "strength": "analytic (proof checked)" }, { "claim": "The three containers' areas (disc of radius 1/2, 30° sector of radius 1, T/1.0048) are as the report gives them.", "kind": "computation", "numbers": [ "T/1.0048 area 0.26095623981…", "nine places", "summary to 10⁻¹⁵" ], "quote": "T/1.0048 has area 0.26095623981…, as the report says to nine places.", "report_finding": "PASS", "section": "W1", "strength": "numerical only" }, { "claim": "The candidate's recorded translation places T with its right angle at the origin and its legs on the axes (handedness not stated).", "kind": "reading", "numbers": [ "the other handedness gives 1.391" ], "quote": "placed that way, only the one with the short leg on the x-axis gives the recorded scale (the other gives 1.391).", "report_finding": "PASS", "section": "Findings 1 — The triangle", "strength": "numerical only" }, { "claim": "A curve's critical scale is found by a linear program over 192 support directions (least s with a translation t at fixed rotation), and its worst positive skew by a second program at fixed s and rotation.", "kind": "method", "numbers": [ "192 support directions", "feasibility tolerances 10⁻¹⁰" ], "quote": "| W2 | both programs as stated;", "report_finding": "PASS", "section": "W2", "strength": "numerical only" }, { "claim": "The library's largest critical scales in the disc and the sector are 0.99986613 and 0.99986614 (the unit segment).", "kind": "computation", "numbers": [ "0.99986613 (disc)", "0.99986614 (sector)", "cos(π/192) = 0.99986614", "5.8×10⁻⁹", "true maximum 1", "every other record ≤ 0.97142 (random_16_6s), true value < 0.972" ], "quote": "The true maximum is 1 in both.", "report_finding": "NOT AS STATED", "section": "W2; Findings 2", "strength": "corrected" }, { "claim": "The disc records of segment, V_60, V_90, V_120 and U_equal are those curves' critical scales in the disc (e.g. V_60 at 0.86590947).", "kind": "computation", "numbers": [ "V_60 0.86590947", "√3/2 = 0.86602540", "factor cos(π/192) to 2×10⁻¹²" ], "quote": "V_60 is recorded at 0.86590947, its true value is √3/2 = 0.86602540.", "report_finding": "NOT AS STATED", "section": "Findings 2", "strength": "corrected" }, { "claim": "The disc records of the semicircle and arc_270 are their critical scales in the disc.", "kind": "computation", "numbers": [ "factor 0.99990 (semicircle)", "factor 0.999991 (arc_270)", "semicircle within 7.3×10⁻⁹ of 2/π" ], "quote": "decimals — is the two effects nearly cancelling, not the program computing the arc.", "report_finding": "NOT AS STATED", "section": "Findings 2", "strength": "corrected" }, { "claim": "The triangle records are critical scales in T/1.0048 itself.", "kind": "reading", "numbers": [ "three edge normals on the 192-grid" ], "quote": "grid in the round's placement: its three edge normals are grid directions, so its relaxation is the triangle itself.", "report_finding": "PASS", "section": "Findings 2", "strength": "analytic (proof checked)" }, { "claim": "The 102 records give each curve's critical scale, the least over rotations from 60–120 starting phases (12 of the 21 rebuildable records hold within tolerance).", "kind": "computation", "numbers": [ "12 of 21", "within 5×10⁻⁷ above or 2×10⁻⁷ below", "V_90 sector 1.5×10⁻⁷ above", "segment sector 5.8×10⁻⁹ above", "seven disc records 4×10⁻⁹ to 1.0×10⁻⁷ below" ], "quote": "No record lies below its scanned minimum by more than its own logged residual (at most 1.0×10⁻⁷, all in the disc),", "report_finding": "PASS", "section": "W3; Findings 3", "strength": "numerical only" }, { "claim": "The records of segment (triangle), V_60 (sector), V_90, V_120, U_equal (triangle), semicircle (sector, triangle) and arc_270 (sector, triangle) are those curves' critical scales.", "kind": "computation", "numbers": [ "9 of 21", "5×10⁻⁷ to 4.4×10⁻⁶ above", "segment triangle 0.910863460 vs 0.910859678", "V_60 sector 0.866029837 vs 0.866025404", "V_90 triangle 0.879823896 vs 0.879822887", "V_120 triangle 0.910862299 vs 0.910859678", "semicircle sector 0.954941594 vs 0.954940311", "semicircle triangle 0.914631706 vs 0.914629508", "arc_270 sector 0.912880309 vs 0.912879782", "arc_270 triangle 0.925582665 vs 0.925581618" ], "quote": "for these 9 records the last printed digits are not the curve's", "report_finding": "FAIL", "section": "W3; Findings 3", "strength": "corrected" }, { "claim": "The triangle's headline maximum is U_equal at 0.95783224.", "kind": "computation", "numbers": [ "0.95783224", "0.95782984", "0.957829839128556", "Round 02 exact 0.957829839129" ], "quote": "from the sixth on: 0.95783224 against 0.95782984. Round 02's own exact three-normal value for U_equal is", "report_finding": "NOT AS STATED", "section": "W3; Findings 3", "strength": "corrected" }, { "claim": "The report's top-ten table for the triangle and its three maxima are the records'.", "kind": "reading", "numbers": [], "quote": "| W5 | the report's top-ten table and its three maxima are the records' | **PASS** |", "report_finding": "PASS", "section": "W5", "strength": "label only" }, { "claim": "The adversarial candidate (nine points, length 1) has critical scale 0.9187329185 at rotation 1.1024251219 with its recorded translation.", "kind": "computation", "numbers": [ "1.1024251219", "0.9187329184910361 vs recorded …0362", "translation (0.20923044773400137, 0.3573006223101446) within 10⁻⁹" ], "quote": "program gives 0.9187329184910361 against the recorded …0362, the recorded translation (0.20923044773400137,", "report_finding": "PARTIAL", "section": "W4; Findings 1 — The candidate", "strength": "numerical only", "audit": "finding moved from 'PASS' after the independent reading: the value at the recorded rotation reproduces, but W4 fails (one rotation gives 1.6×10⁻⁶ less) and it is one handedness's value" }, { "claim": "The candidate has \"about 3\" active directions (the summary: 3).", "kind": "computation", "numbers": [ "3 tight directions", "hypotenuse normal and the two legs' normals" ], "quote": "through the right angle at the origin and so carry no scale (the summary's count is 3; the report says \"about 3\").", "report_finding": "PASS", "section": "W4; Findings 1 — The candidate", "strength": "numerical only" }, { "claim": "With the container shrunk to 98 % of the candidate's scale, the residual worst skew is E = 3.6918923462×10⁻³, and the logged skew field is that placement's.", "kind": "computation", "numbers": [ "98 %", "E = 3.6918923462×10⁻³", "192 logged skew values" ], "quote": "At 98 % of that scale, the least worst skew is the recorded E, and the skew field at that placement equals all 192", "report_finding": "PASS", "section": "W4; Findings 1 — The candidate", "strength": "numerical only" }, { "claim": "The candidate's critical scale (least over rotations) is 0.9187329185.", "kind": "computation", "numbers": [ "1.6×10⁻⁶ above the scanned minimum", "rotation 3.2×10⁻⁶ rad from the recorded one" ], "quote": "recorded scale is likewise 1.6×10⁻⁶ above the scanned minimum of its own handedness, reached at a rotation 3.2×10⁻⁶", "report_finding": "FAIL", "section": "W4; Findings 3", "strength": "corrected" }, { "claim": "The candidate's critical scale 0.9187 is measured against the certified triangle T/1.0048.", "kind": "reading", "numbers": [ "0.918733 orientation-preserving", "0.894545 mirror image", "2.6 % below" ], "quote": "certified triangle, so by Round 02's own rule that comparison is 0.8945", "report_finding": "NOT AS STATED", "section": "Findings 4", "strength": "corrected" }, { "claim": "35 curves were tested (report \"基準與隨機曲線共測試 35 條\", README, experiment_summary.json benchmark_curve_count).", "kind": "specification", "numbers": [ "35 stated", "102 records", "34 curves × 3 containers" ], "quote": "(`benchmark_curve_count`: 35) all say 35. `benchmark_records.json` holds 102 records — 34 curves in each of the three", "report_finding": "FAIL", "section": "W6; Findings 5", "strength": "corrected" }, { "claim": "Translation occupies only the first Fourier mode of the support signal.", "kind": "theorem", "numbers": [ "modes 0–6 symbolic" ], "quote": "| translation is the first Fourier mode only | exact (symbolic for modes 0–6; the general statement is classical) |", "report_finding": "PASS", "section": "W7; Findings 1 — The structure", "strength": "analytic (proof checked)" }, { "claim": "Rotation is a phase shift of the support signal.", "kind": "theorem", "numbers": [ "10⁻¹⁴", "three rotations" ], "quote": "| rotation is a phase shift of the support signal | an identity; checked on the grid to 10⁻¹⁴ |", "report_finding": "PASS", "section": "W7; Findings 1 — The structure", "strength": "numerical only" }, { "claim": "§4.3–4.4: structural interpretation (higher-order peaks remaining after translation press the container).", "kind": "reading", "numbers": [], "quote": "interpretation and are not checked.", "report_finding": "UNMEASURED", "section": "Findings 1 — The structure", "strength": "not measured" }, { "claim": "T/1.0048 is certified (\"已認證\"), by 599 SOCP models the round did not redo; no source is cited.", "kind": "certificate", "numbers": [ "arXiv:2606.14625", "599 cone models", "certified lower endpoints exceed 1.0048", "area 0.260956…" ], "quote": "It is a preprint. This run records it as a cited theorem at that standing and checks", "report_finding": "PASS by cited theorem", "section": "W8; Findings 8", "strength": "cited theorem" }, { "claim": "Library names V_60, V_90, V_120 denote two halves of a unit segment (undefined in the package).", "kind": "reading", "numbers": [ "angle turned at the joint", "other reading misses by 0.1 to 0.3", "within 4.4×10⁻⁶" ], "quote": "The other reading misses V_60 and V_120 by 0.1 to 0.3; this one is within 4.4×10⁻⁶ everywhere.", "report_finding": "PASS", "section": "W9; Findings 6", "strength": "numerical only" }, { "claim": "Library names semicircle and arc_270 (64 segments in the documents) denote the arcs (undefined in the package).", "kind": "reading", "numbers": [ "64 chords of total length 1", "arc misses by 1.2×10⁻⁵ to 1.7×10⁻⁴", "chord polylines within 2.2×10⁻⁶", "arc_270 triangle: arc 0.925415, record 0.925583" ], "quote": "polylines, not of the arcs — for arc_270 in the triangle the arc's own value is 0.925415, the record 0.925583.", "report_finding": "PASS", "section": "W9; Findings 6", "strength": "numerical only" }, { "claim": "Records of zigzag_4, zigzag_8, almost_circle and 24 random curves.", "kind": "computation", "numbers": [ "27 curves", "81 records" ], "quote": "package, and their 81 records are reported here as the round's, not checked.", "report_finding": "UNMEASURED", "section": "Findings 6", "strength": "not measured" }, { "claim": "The package's src/moser_skew_lab.py stands for the program and points to the report.", "kind": "specification", "numbers": [], "quote": "The round ships no program: its `src/moser_skew_lab.py` is a three-line placeholder pointing to the report.", "report_finding": "PASS", "section": "Findings 1", "strength": "label only" }, { "claim": "The five-generation adversarial search ends in the reported candidate (critical scale 0.9187329185).", "kind": "computation", "numbers": [ "generations 0–4: 0.91889, 0.91889, 0.91889, 0.93326, 0.93451", "candidate 0.91873" ], "quote": "The package does not say which curve the history's values belong to, and this run cannot tell.", "report_finding": "PARTIAL", "section": "Findings 7", "strength": "open" }, { "claim": "README and §5: the round proposes no new bound for Moser's problem.", "kind": "reading", "numbers": [], "quote": "Its README and its §5 say it proposes no new", "report_finding": "PASS", "section": "What Round 01 claims", "strength": "label only" }, { "claim": "(Recomputation's instrument) The drill plants defects and checks each is caught by exactly its named checks.", "kind": "method", "numbers": [ "19 planted", "19 caught", "4 controls", "174/174" ], "quote": "| must-stay-green assertions held | **174/174** |", "report_finding": "PASS", "section": "The drill", "strength": "label only" } ], "corrections": [ { "what": "Disc and sector library maxima", "source_says": "0.99986613 (disc) and 0.99986614 (sector).", "report_finds": "Both are the 192-direction grid's cos(π/192) for the unit segment; the true maximum is exactly 1 in both, and no other curve exceeds 0.972.", "quote": "out at exactly cos(π/192) = 0.99986614 (W2). That is the sector's printed maximum; the disc's, 0.99986613, is the" }, { "what": "Disc records of the other symmetric curves and the arcs", "source_says": "e.g. V_60 at 0.86590947 in the disc.", "report_finds": "Grid values: the true value times cos(π/192) to 2×10⁻¹² (V_60 true value √3/2 = 0.86602540); the arc polylines low by factors 0.99990 and 0.999991.", "quote": "the disc's grid value is the true value times cos(π/192), to 2×10⁻¹²:" }, { "what": "Nine records above the scanned minimum", "source_says": "Each record is the curve's critical scale.", "report_finds": "9 of 21 rebuildable records lie 5×10⁻⁷ to 4.4×10⁻⁶ above this tree's scanned minimum of the same program; their last printed decimals are not the curve's.", "quote": "refined — is lower, by more than 5×10⁻⁷, than 9 of the 21 records it can rebuild (W3), and lower within that margin" }, { "what": "Triangle headline maximum", "source_says": "U_equal 0.95783224.", "report_finds": "0.95782984 (scanned 0.957829839128556; Round 02's exact 0.957829839129), wrong from the sixth decimal.", "quote": "from the sixth on: 0.95783224 against 0.95782984. Round 02's own exact three-normal value for U_equal is" }, { "what": "Candidate's critical scale", "source_says": "0.9187329185.", "report_finds": "1.6×10⁻⁶ above the scanned minimum of its handedness, reached 3.2×10⁻⁶ rad from the recorded rotation.", "quote": "recorded scale is likewise 1.6×10⁻⁶ above the scanned minimum of its own handedness, reached at a rotation 3.2×10⁻⁶" }, { "what": "Candidate against the certified triangle", "source_says": "0.9187, compared with the certified T/1.0048.", "report_finds": "0.9187 is the orientation-preserving value; the candidate is chiral and its mirror image fits at 0.894545, the value under the certificate's congruent convention.", "quote": "critical scale of 0.894545 in the same triangle, 2.6 % below the recorded 0.918733." }, { "what": "Number of curves", "source_says": "35 curves tested.", "report_finds": "The records hold 34 curves (102 records, 34 per container); which curve is missing is not said.", "quote": "(`benchmark_curve_count`: 35) all say 35. `benchmark_records.json` holds 102 records — 34 curves in each of the three" } ], "open_items": [ "The package does not say which curve is missing.", "The package does not say which curve the history's values belong to, and this run cannot tell.", "would then be one handedness's too, and they are unchecked (§6).", "That the minima of §3 are global — they come from scans refined locally, as the round's", "Anything about the 27 undefined curves. That the preprint's proof is correct.", "conventions of §4 the programme's own question is: Round 02 keeps both." ], "cross_refs": [ "RUN-003 (Round 02: its exact three-normal U_equal value 0.957829839129 and its two-convention rule are used here; RUN-003 also re-refines this run's two candidate minima, finding them high by 3.4×10⁻⁹ and 9.6×10⁻¹⁰, below this run's tolerances and printed precision)" ], "unclear": [ "The report names Round 02, not RUN-003; the RUN-003 cross-reference is inferred from RUN-003's subject.", "T's geometry and the Fourier statement are labelled 'analytic (proof checked)' because the report checks them symbolically (SymPy); the rotation phase shift is labelled 'numerical only' because the report calls it an identity but checks it on the grid.", "The certified triangle is labelled 'cited theorem': the round cites no source, the preprint (a preprint, not peer-reviewed) is the report's own identification, and nothing of its proof is checked.", "W5 (the report's table matches the records) is labelled 'label only' as a transcription check.", "The search-history mismatch (Findings 7) is labelled PARTIAL / open: the report cannot tell whether the history or the candidate is off.", "The arc names are recorded as a PASS reading; the report adds that the records are of 64-chord polylines, not of the arcs (arc_270 triangle 0.925415 vs record 0.925583), which is noted in numbers rather than as a correction because the documents themselves give 64 segments." ], "for_paper": "Round 01's two grid linear programs, its three containers, its 30-60-90 triangle (checked symbolically) and its adversarial candidate's recorded scale, translation, three tight directions, 98 % test and skew field reproduce, numerically; translation is analytically the first Fourier mode only, and T/1.0048 covering every unit arc holds by a cited preprint (arXiv:2606.14625) that the round itself does not cite. Several stated numbers do not hold as stated: the disc and sector library maxima are the 192-direction grid's cos(π/192), not the true 1; nine of the 21 rebuildable records are local-search values 5×10⁻⁷ to 4.4×10⁻⁶ above the scanned minimum (the triangle's headline U_equal 0.95783224 against 0.95782984); and the records hold 34 curves, not 35. The candidate's 0.9187 is its orientation-preserving value; under the certificate's congruent convention it fits at 0.8945. Twenty-seven of the 34 curves are defined nowhere, so their 81 records are unchecked, and the search history's best (0.93451) does not match the reported candidate. No bound for Moser's problem is claimed or established." }, { "run": "RUN-003", "report_file": "RUN-003-ROUND02-PHASE-SCAN-CHIRALITY.md", "source_document": "Round 02 (v0.2.2): Moser_Skew_Lab_v0.2.2.zip (SHA-256 4375eba4…) — its README, reports/ROUND_02_REPORT.md, ROUND_02_EXACT_ADDENDUM.md, ROUND_02_CHIRALITY_CORRECTION.md, and its data (round02_summary.json, five phase_scan_*.csv, best_contact_stress_ledger.json, exact_triangle_validation.json, chirality_corrected_summary.json, three logged curves, u_family_surface.csv), read from the package bytes", "verdict_line": "**Verdict: PARTIAL** — in RUN-002's sense: what the round computes for the curves its documents define reproduces, and some of the numbers it states do not hold as stated.", "items": [ { "claim": "Contact-pressure ledger: at phase 2.171084228853177 the grid candidate's 256-direction program gives 0.9933108210 with its recorded translation, and its dual puts weights 2.093720944, 1.821719357, 1.031991268 on three directions and balances.", "kind": "computation", "numbers": [ "phase 2.171084228853177", "0.9933108209696215", "10⁻⁹", "29.53125°, 180°, 270°", "2.093720944, 1.821719357, 1.031991268" ], "quote": "270°, with the recorded weights 2.093720944, 1.821719357 and 1.031991268, and balances to rounding.", "report_finding": "PASS", "section": "Y1; Findings 1 — The ledger", "strength": "numerical only" }, { "claim": "The ledger's container features at the active directions are the two legs' normals and, at 29.53125°, the vertex (A, 0).", "kind": "computation", "numbers": [ "29.53125°", "vertex (A, 0)" ], "quote": "the vertex (A, 0) at the end of the short leg, not the hypotenuse (Y1).", "report_finding": "PASS", "section": "Y1; Findings 1 — The ledger", "strength": "numerical only" }, { "claim": "The phase scans of U_equal, the segment and Round 01's candidate (360 phases each in T/1.0048, 256 directions) are logged row by row.", "kind": "computation", "numbers": [ "1,080 rows", "10⁻⁹" ], "quote": "program's scale at their phases, to 10⁻⁹; the summary's minima, maxima, barriers (maximum minus minimum over the", "report_finding": "PASS", "section": "Y2; Findings 1 — The scans", "strength": "numerical only" }, { "claim": "For all five scanned curves the summary's minimum, minimum phase, maximum and barrier are those of the logged rows.", "kind": "computation", "numbers": [ "5 curves" ], "quote": "for all five curves, the summary's minimum, minimum phase, maximum and barrier are the logged rows'", "report_finding": "PASS", "section": "Y2", "strength": "numerical only" }, { "claim": "Each curve's count of strict local minima over its 360 logged phases is as the summary states.", "kind": "computation", "numbers": [ "360 phases" ], "quote": "| Y7 | each curve's count of strict local minima over its 360 logged phases is the summary's | **PASS** |", "report_finding": "PASS", "section": "Y7", "strength": "numerical only" }, { "claim": "The contact-signature counts (distinct sets of container features at the dual's active directions over 360 phases) are 1, 4 and 2.", "kind": "computation", "numbers": [ "1, 4, 2" ], "quote": "gives the summary's contact-signature counts — 1,", "report_finding": "PASS", "section": "Y9", "strength": "numerical only" }, { "claim": "The README, the report's phase and ledger tables, and the two addenda state the data's numbers.", "kind": "reading", "numbers": [ "nine places" ], "quote": "| Y6 | the README, the report's phase and ledger tables, and the two addenda state the data's numbers | **PASS** |", "report_finding": "PASS", "section": "Y6", "strength": "label only" }, { "claim": "README and §4: U_equal's critical scale is 0.9512368779.", "kind": "computation", "numbers": [ "0.9512368779 stated", "0.9512368750 grid minimum, at phase 0", "0.951236875 (§2 table)", "2.9×10⁻⁹" ], "quote": "The stated values are 2.9×10⁻⁹ and 7.2×10⁻⁹ above them, so their last printed digits are", "report_finding": "NOT AS STATED", "section": "Findings 1 — The precision of two stated values", "strength": "corrected" }, { "claim": "README and §4: the best three-link curve's critical scale is 0.9933108210.", "kind": "computation", "numbers": [ "0.9933108210 stated", "0.9933108137 grid minimum", "1.4×10⁻⁸ rad from the recorded phase", "7.2×10⁻⁹" ], "quote": "and the best curve's is 0.9933108137, reached 1.4×10⁻⁸ rad", "report_finding": "NOT AS STATED", "section": "Findings 1 — The precision of two stated values", "strength": "corrected" }, { "claim": "README: the best curve exceeds U by 4.2073943098×10⁻².", "kind": "computation", "numbers": [ "4.2073943098×10⁻²", "4.207393878×10⁻²" ], "quote": "the README's difference 4.2073943098×10⁻² is the minima's 4.207393878×10⁻²", "report_finding": "NOT AS STATED", "section": "Findings 1 — The precision of two stated values", "strength": "corrected" }, { "claim": "Exact addendum: the hypotenuse-normal approximation of the 256-direction grid underestimated the scale (\"可見斜邊法向近似確實造成了尺度低估\").", "kind": "theorem", "numbers": [ "grid spacing 1.40625°", "nearest directions 29.53° and 30.94°" ], "quote": "triangle does and its value never exceeds the exact one (Y3).", "report_finding": "PASS", "section": "Y3; Findings 2", "strength": "analytic (proof checked)" }, { "claim": "Main report: the phase table, U (0.9512368779) and the best curve (0.9933108210) are critical scales in T/1.0048.", "kind": "computation", "numbers": [ "U exact 0.957830", "best curve orientation-preserving 1.000837" ], "quote": "values — U's exact value is 0.957830, the best curve's orientation-preserving 1.000837 — and its conclusion that the", "report_finding": "NOT AS STATED", "section": "Findings 2", "strength": "corrected" }, { "claim": "Main report's conclusion: the three-link family \"尚未逼近 Wetzel 認證尺度 1\" (has not yet approached the certified scale 1).", "kind": "reading", "numbers": [ "0.998574 (the correction's curve)", "0.998698 (this run's search)", "within 1.4×10⁻³ and 1.3×10⁻³ of 1" ], "quote": "(has not yet approached the certified scale 1) does not hold as stated:", "report_finding": "NOT AS STATED", "section": "Findings 2", "strength": "refuted" }, { "claim": "Exact addendum: with T = {x ≥ 0, y ≥ 0, x/A + y/B ≤ 1}, a rotated polyline's least scale is s(φ) = max_i (x_i/A + y_i/B) − min_i x_i / A − min_i y_i / B, with no direction grid.", "kind": "theorem", "numbers": [ "equals the three-normal LP to 10⁻¹⁰ on random polylines" ], "quote": "| the exact formula for the triangle | exact (the argument in \"What Round 02 claims\"; checked against the three-normal LP) |", "report_finding": "PASS", "section": "Y4; What Round 02 claims", "strength": "analytic (proof checked)" }, { "claim": "Exact addendum values: U 0.957829839129, the grid candidate 1.000837369637, the re-optimized three-link 1.007315039573 (above the certified scale 1).", "kind": "computation", "numbers": [ "0.957829839129", "1.000837369637", "1.007315039573", "10⁻¹⁰", "3,600-phase scan, golden-section refined" ], "quote": "the two addenda state reproduces to 10⁻¹⁰: U 0.957829839129, the grid candidate 1.000837369637, the re-optimized", "report_finding": "PASS", "section": "Y4; Findings 1 — The exact values", "strength": "numerical only" }, { "claim": "Chirality correction values: the 1.007315 curve's mirror image fits at 0.952938974740; the search maximizing s_cong ends at s₊ = 0.998574374711, s₋ = 0.998573794671.", "kind": "computation", "numbers": [ "0.952938974740", "0.998574374711", "0.998573794671", "10⁻¹⁰" ], "quote": "three-link 1.007315039573 and its mirror image 0.952938974740, the corrected curve's 0.998574374711 and", "report_finding": "PASS", "section": "Y4; Findings 1 — The exact values", "strength": "numerical only" }, { "claim": "The addenda's stated differences, distances to 1 and chirality gaps are the differences they name.", "kind": "computation", "numbers": [], "quote": "the stated differences, distances to 1 and chirality gaps are the differences they name", "report_finding": "PASS", "section": "Y4", "strength": "numerical only" }, { "claim": "Each logged three-link curve is its stated parameters: directions (α₁, 0, −α₃), lengths summing to 1.", "kind": "specification", "numbers": [ "3 logged curves" ], "quote": "| Y5 | each logged three-link curve is its stated parameters: lengths summing to 1, directions (α₁, 0, −α₃) | **PASS** |", "report_finding": "PASS", "section": "Y5", "strength": "numerical only" }, { "claim": "The certifying preprint covers by congruent copies, reflections allowed: T/1.0048 covers every unit arc.", "kind": "certificate", "numbers": [ "arXiv:2606.14625" ], "quote": "| T/1.0048 covers every unit arc, by congruent copies | cited theorem (preprint; the round cites it as arXiv:2606.14625) |", "report_finding": "PASS by cited theorem", "section": "Y8", "strength": "cited theorem" }, { "claim": "Chirality correction: a curve's congruent critical scale is s_cong = min{s₊, s₋}; orientation-preserving values serve the problem posed with rotations and translations, congruent values the comparison with the 2026 preprint.", "kind": "definition", "numbers": [], "quote": "handednesses, s_cong = min{s₊, s₋}; the 1.007315 curve's mirror image fits at 0.952938974740, and a search that", "report_finding": "PASS", "section": "What Round 02 claims; Y8", "strength": "label only" }, { "claim": "Chirality correction: the search maximizing s_cong over the three-link family reaches 0.998573794671 (no maximum claimed).", "kind": "computation", "numbers": [ "0.998697792", "l₁ = l₂ = l₃ = 1/3", "α₁ = α₃ = 82.435472°", "1.24×10⁻⁴ above", "kink slopes 1.6×10⁻³ and 4.7×10⁻³ per degree", "round's turns 82.34°, about 0.1° away" ], "quote": "1.24×10⁻⁴ above the round's value.", "report_finding": "PARTIAL", "section": "Findings 3", "strength": "corrected" }, { "claim": "Addendum status note: asymmetric candidates above 1 (the grid candidate at 1.000837369637) are not counterexamples, because the preprint allows reflection (stated without computing).", "kind": "reading", "numbers": [ "1.000837369637", "mirror image 0.954897" ], "quote": "candidate's mirror image fits at 0.954897 (exact, minimum over phases), so its congruent value is 0.954897", "report_finding": "PASS", "section": "Findings 4", "strength": "numerical only" }, { "claim": "Exact addendum §3 body: the 1.007315 curve \"已逼近認證 Wetzel 三角形的完整尺度\" and does not challenge the certificate \"因為結果仍小於 1\" (because the result is still below 1).", "kind": "reading", "numbers": [ "1.007315 > 1" ], "quote": "the certificate \"因為結果仍小於 1\" (because the result is still below 1) — while the number it reports is above 1.", "report_finding": "NOT AS STATED", "section": "Findings 4", "strength": "corrected" }, { "claim": "Phase scans of the semicircle and arc_270 (720 logged rows; the arcs are not defined in the documents).", "kind": "computation", "numbers": [ "720 rows", "closest unfitted readings off by 6.8×10⁻⁵ and 1.01×10⁻⁴", "fitted readings off by 5.1×10⁻⁵ and 7.3×10⁻⁵" ], "quote": "are reported as the round's; only their summary statistics are checked against them (Y2, Y7).", "report_finding": "UNMEASURED", "section": "Findings 5 — The two arcs", "strength": "not measured" }, { "claim": "The U-family surface u_family_surface.csv (14 × 21, α from 65° to 115°, l from 0.18 to 0.42): its corners.", "kind": "computation", "numbers": [ "14 × 21", "exact on a 128-direction grid with 36 phases", "off by 9×10⁻⁴ to 1.0×10⁻² on the 256-direction grid" ], "quote": "Its four corners — rows running over l, columns over α — reproduce exactly on a 128-direction", "report_finding": "PARTIAL", "section": "Findings 5 — The U-family surface", "strength": "numerical only" }, { "claim": "The U-family surface's interior values.", "kind": "computation", "numbers": [ "off by 1.3×10⁻³ and 2.7×10⁻³ at two probed cells on evenly spaced axes" ], "quote": "different program from the scans', and how its interior grid was laid out is not stated", "report_finding": "UNMEASURED", "section": "Findings 5 — The U-family surface", "strength": "not measured" }, { "claim": "Jump counts (a jump being a change of the dual's active directions, \"當對偶活動方向改變時\"): 27, 26 and 26 for U, the segment and Round 01's candidate.", "kind": "computation", "numbers": [ "stated 27, 26, 26", "active-set changes 0, 24, 4" ], "quote": "gives no rule for counting. Counting changes of the active set between consecutive logged phases gives 0, 24 and 4", "report_finding": "UNMEASURED", "section": "Findings 5 — The jumps", "strength": "not measured" }, { "claim": "A topology-guided search over 120 asymmetric three-link candidates.", "kind": "computation", "numbers": [ "120 candidates" ], "quote": "of the topology-guided search are not logged.", "report_finding": "UNMEASURED", "section": "Findings 5", "strength": "not measured" }, { "claim": "The package's src/contact_lp.py states the program and that the negative HiGHS inequality marginals are nonnegative contact weights.", "kind": "specification", "numbers": [], "quote": "rebuild, the docstring describes the same program this tree solves.", "report_finding": "PASS", "section": "Findings 1", "strength": "label only" }, { "claim": "(Recomputation's instrument) The drill plants defects and checks each is caught by exactly its named checks.", "kind": "method", "numbers": [ "19 planted", "19 caught", "4 controls", "175/175" ], "quote": "| must-stay-green assertions held | **175/175** |", "report_finding": "PASS", "section": "The drill", "strength": "label only" } ], "corrections": [ { "what": "Precision of U and the best curve (README, §4)", "source_says": "U 0.9512368779; best curve 0.9933108210.", "report_finds": "On the same 256-direction program the minima are 0.9512368750 and 0.9933108137, 2.9×10⁻⁹ and 7.2×10⁻⁹ lower; the last printed digits are not the curves'.", "quote": "The stated values are 2.9×10⁻⁹ and 7.2×10⁻⁹ above them, so their last printed digits are" }, { "what": "README difference", "source_says": "4.2073943098×10⁻².", "report_finds": "The minima's difference is 4.207393878×10⁻².", "quote": "the README's difference 4.2073943098×10⁻² is the minima's 4.207393878×10⁻²" }, { "what": "Main report's numbers are the 256-grid's", "source_says": "Phase table, U and best curve as critical scales in T/1.0048.", "report_finds": "The grid misses the hypotenuse normal and is a relaxation; U's exact value is 0.957830, the best curve's orientation-preserving 1.000837 (as the round's own exact addendum gives).", "quote": "The report's phase table, its U and its best curve are therefore relaxation" }, { "what": "Conclusion on approaching the certified scale", "source_says": "The three-link family has not yet approached the certified scale 1.", "report_finds": "Under the congruent convention the round prescribes, the family reaches 0.998574 and 0.998698, within 1.4×10⁻³ and 1.3×10⁻³ of 1.", "quote": "0.998574 (the correction's curve) and 0.998698 (§3 below), within 1.4×10⁻³ and 1.3×10⁻³ of 1." }, { "what": "Best congruent three-link", "source_says": "The search maximizing s_cong ends at 0.998573794671.", "report_finds": "This run's search reaches 0.998697792 at equal thirds, α₁ = α₃ = 82.435472°, 1.24×10⁻⁴ higher (a search, not a proof of maximality).", "quote": "s_cong = 0.998697792 at l₁ = l₂ = l₃ = 1/3, α₁ = α₃ = 82.435472°," }, { "what": "Exact addendum §3 body", "source_says": "The 1.007315 curve does not challenge the certificate because the result is still below 1.", "report_finds": "1.007315 is above 1; the body was not updated by the status note, and the reason that holds is chirality (its mirror image fits at 0.952938974740).", "quote": "The addendum's §3 body was not updated by its note:" } ], "open_items": [ "numerical: this run's search, refined; not proved maximal", "That the minima over phases, or this run's three-link value, are global.", "the two arcs, the surface's interior, the jumps or the 120 candidates. That the preprint's proof is correct.", "Round 03's families contain this curve is RUN-004's question.", "different program from the scans', and how its interior grid was laid out is not stated", "stated indices come in runs of consecutive phases, which suggests another rule." ], "cross_refs": [ "RUN-002 (same grid program and tolerances; Round 02's exact U carries the correction RUN-002 made to Round 01's 0.95783224; Round 01's candidate scanned here at 0.918731 orientation-preserving and 0.894545 congruent; RUN-002's two candidate minima, refined by Brent, are high by 3.4×10⁻⁹ and 9.6×10⁻¹⁰, below its tolerances)", "RUN-004 (whether Round 03's five- and seven-link families contain the equal-thirds three-link at 0.998697792; Round 03's three-link baseline 0.998574374711 and best seven-link 0.998674157956)" ], "unclear": [ "The best congruent three-link (Findings 3) is labelled 'corrected' rather than 'refuted' because the round's correction does not claim a maximum; the report's higher value is a search, not proved maximal.", "The exact addendum's §3 body ('because the result is still below 1') is labelled 'corrected': the statement is false as written, but the round's own status note and chirality correction repair the conclusion.", "The U-family surface's corners are labelled PARTIAL / numerical only: they reproduce on a 128-direction, 36-phase program, not the scans' 256-direction one, and the round does not state its grid; whether that is a correction of a source statement is not clear from the report.", "The jump counts are labelled 'not measured': the report tested one reading (0, 24, 4 vs 27, 26, 26) but says the round gives no rule, so the stated counts are recorded as unchecked.", "The open item 'Round 03's families contain this curve is RUN-004's question.' is answered in RUN-004 (both families contain it)." ], "for_paper": "Round 02's 256-direction grid misses the hypotenuse normal and is therefore a relaxation, so the main report's numbers (U 0.9512368779, best curve 0.9933108210, themselves 2.9×10⁻⁹ and 7.2×10⁻⁹ above the grid minima) and its conclusion that the three-link family has not approached the certified scale 1 do not hold as stated. The round's own exact addendum supplies the exact three-normal formula, which checks analytically, and every exact value of the addendum and the chirality correction (U 0.957829839129; grid candidate 1.000837369637; re-optimized three-link 1.007315039573 and its mirror 0.952938974740; corrected curve 0.998574374711 / 0.998573794671) reproduces to 10⁻¹⁰ as numerical minima over phases; the contact-pressure ledger and the 1,080 rebuildable scan rows reproduce to 10⁻⁹. Under the congruent convention the three-link family reaches 0.998697792 at equal thirds with α = 82.435472° (a numerical search, not proved maximal), 1.24×10⁻⁴ above the round's corrected curve, and the grid candidate fits reflected at 0.954897. The two arcs, the U-family surface's interior, the jump counts and the 120 asymmetric candidates are not rebuilt." }, { "run": "RUN-004", "report_file": "RUN-004-ROUND03-SYMMETRIC-LINKS.md", "source_document": "Round 03 (v0.3): Moser_Skew_Lab_v0.3.zip (SHA-256 9f4a3644…) — its README, reports/ROUND_03_REPORT.md, src/METHOD.md, and its data (round03_summary.json, resolution_stability.json, best_5link_curve.csv, best_7link_curve.csv), with Round 02's best_congruence_three_link_curve.csv for the baseline, read from the package bytes", "verdict_line": "**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces, and one of its two conclusions does not hold as stated.", "items": [ { "claim": "§2 and METHOD.md: Round 03 keeps Round 02's exact formula for T/1.0048, s(φ) = max_i (x_i/A + y_i/B) − min_i x_i / A − min_i y_i / B, and maximizes over curves the least scale over rotations.", "kind": "theorem", "numbers": [], "quote": "| the exact formula for the triangle | exact (RUN-003) |", "report_finding": "PASS", "section": "Z9; At what strength", "strength": "label only", "audit": "relabelled from 'analytic (proof checked)' by the strength audit: the exact formula is derived and checked in RUN-003 (Y4) and counted there; RUN-004 uses it" }, { "claim": "The logged five-link (α, β, 0, −β, −α; l₁, l₂, l₃, l₂, l₁) and seven-link (α, β, γ, 0, −γ, −β, −α; l₁, l₂, l₃, l₄, l₃, l₂, l₁) curves are their stated parameters, of length 1, and each equals its mirror up to parameter reversal (METHOD.md).", "kind": "specification", "numbers": [ "length 1 to 10⁻¹²" ], "quote": "10⁻¹², and each is its own mirror image traversed backwards, as METHOD.md says (Z9)", "report_finding": "PASS", "section": "Z1; Z9", "strength": "numerical only" }, { "claim": "Mirror-symmetric families exclude the chirality question at the level of the parameters (\"手性假警報因此在參數層被排除\").", "kind": "theorem", "numbers": [ "mirror images give the same scales (Z1)" ], "quote": "A mirror-symmetric curve is congruent to its own reflection, so its two handednesses agree and", "report_finding": "PASS", "section": "What Round 03 claims; Z1", "strength": "analytic (proof checked)" }, { "claim": "The three-link baseline (Round 02's corrected curve, re-evaluated on phase grids up to 65,536 phases) is s₃ = 0.998574374711.", "kind": "computation", "numbers": [ "0.9985743747111", "1.4×10⁻¹³", "phase to 5.3×10⁻¹³", "summary's s₃ is the 32,768-phase row, 1.3×10⁻¹³ from the 65,536 row" ], "quote": "The least exact scale over all phases is 0.9985743747111 for the baseline, 0.9987543716677 for the", "report_finding": "PASS", "section": "Z1; Findings 1 — The values", "strength": "numerical only" }, { "claim": "The best five-link has s₅ = 0.998754371668 at its stated phase.", "kind": "computation", "numbers": [ "0.9987543716677", "6.4×10⁻¹⁴", "phase to 2.4×10⁻¹³" ], "quote": "The least exact scale over all phases is 0.9985743747111 for the baseline, 0.9987543716677 for the", "report_finding": "PASS", "section": "Z1; Findings 1 — The values", "strength": "numerical only" }, { "claim": "The best seven-link has s₇ = 0.998674157956 at its stated phase.", "kind": "computation", "numbers": [ "0.9986741579563", "3.4×10⁻¹⁴", "phase to 1.2×10⁻¹³" ], "quote": "five-link and 0.9986741579563 for the seven-link (`exact_minima`) — the stated values to 1.4×10⁻¹³, 6.4×10⁻¹⁴ and", "report_finding": "PASS", "section": "Z1; Findings 1 — The values", "strength": "numerical only" }, { "claim": "Contact ledgers and skeleton: both best curves touch the triangle at four points, both endpoints on the hypotenuse and one interior vertex on each leg.", "kind": "computation", "numbers": [ "baseline [2], [1], [0, 3]", "five-link [3], [2], [0, 5]", "seven-link [4], [3], [0, 7]", "multiplicity 4" ], "quote": "seven-link — multiplicity 4 each, and for the two symmetric curves this is the stated skeleton (Z2).", "report_finding": "PASS", "section": "Z2; Findings 1 — The contacts", "strength": "numerical only" }, { "claim": "The stated differences and distances to 1.", "kind": "computation", "numbers": [], "quote": "The stated differences and distances to 1 are the differences of the stated values (Z3).", "report_finding": "PASS", "section": "Z3", "strength": "numerical only" }, { "claim": "Resolution table: on six phase grids (2,048 to 65,536) the three curves have 7, 8 and 7 strict local minima, with the refined scale and phase per row.", "kind": "computation", "numbers": [ "18 rows", "2,048 to 65,536 phases", "7, 8, 7", "scale to 5.5×10⁻¹²", "phase to 2.1×10⁻¹¹" ], "quote": "On every grid from 2,048 to 65,536 phases the exact scale has 7, 8 and 7 strict local", "report_finding": "PASS", "section": "Z4; Findings 1 — The resolution table", "strength": "numerical only" }, { "claim": "Each curve's \"metrics\" block (minimum, maximum, barrier, local minima and count) in the summary.", "kind": "computation", "numbers": [ "8,192 phases, unrefined", "10⁻¹²" ], "quote": "exact scale sampled on 8,192 phases, unrefined, to 10⁻¹² (Z5).", "report_finding": "PASS", "section": "Z5; Findings 1 — The metrics", "strength": "numerical only" }, { "claim": "The summary's metrics.minimum and best_local_minima are labelled as the curves' minima.", "kind": "reading", "numbers": [ "1.37×10⁻⁴ to 1.38×10⁻⁴ above the refined minimum", "about three quarters of the five-link's stated gain" ], "quote": "a reader of the summary should not take `metrics.minimum` or `best_local_minima` for critical scales.", "report_finding": "NOT AS STATED", "section": "Z5; Findings 1 — The metrics", "strength": "corrected" }, { "claim": "The container's contact-pressure entropy is H_λ = 0.963370649827, the normalized entropy of the balance weights of the triangle's three normals.", "kind": "computation", "numbers": [ "0.963370649827", "weights (1/A, 1/B, 1)" ], "quote": "value 0.963370649827 is the entropy of the weights (1/A, 1/B, 1), which balance the three constraint rows as the", "report_finding": "PASS", "section": "Z7; Findings 5", "strength": "numerical only" }, { "claim": "H_λ is \"容器法向幾何決定的基準\" (a baseline determined by the container's normal geometry).", "kind": "reading", "numbers": [ "unit normals: weights (B, A, √(A² + B²)), entropy 0.965163253185", "T undivided: 0.963701869" ], "quote": "Either number belongs to the triangle and a choice of scaling; neither depends on any", "report_finding": "NOT AS STATED", "section": "Findings 5", "strength": "corrected" }, { "claim": "The README and the report state the data's values, gains, phases, lengths, angles and contact ledgers (including N₃ = 7 and four active contacts).", "kind": "reading", "numbers": [], "quote": "| Z8 | the README and the report state the data's values, gains, phases, lengths, angles and contact ledgers | **PASS** |", "report_finding": "PASS", "section": "Z8", "strength": "label only" }, { "claim": "README, §6 (boxed) and §12: going to seven links brought no further improvement (\"增加到七段並未繼續改善\"); s₇ < s₅ is \"更重要的限制\" and the way forward cannot be reduced to increasing the curve's complexity.", "kind": "reading", "numbers": [ "round's five-link as a seven-link: 0.998754371668", "s₇ 0.998674157956", "8.0×10⁻⁵ below" ], "quote": "So of the three causes §11 item 7 allows, the family itself is excluded, and the comparison is a statement about the", "report_finding": "FAIL", "section": "Z6; Findings 2", "strength": "refuted" }, { "claim": "README and §6: the seven-link shortfall shows the added parameters largely degenerate into near-parallel sub-segments (\"新增參數大量退化為近平行子段\").", "kind": "reading", "numbers": [], "quote": "cannot lower a family's optimum: a seven-link curve whose added links are exactly parallel is a five-link curve, with", "report_finding": "NOT AS STATED", "section": "Findings 2", "strength": "refuted" }, { "claim": "§5: the seven-link family did not surpass the five-link family in this round's search (\"七段族在本輪搜尋中沒有超越五段族\").", "kind": "reading", "numbers": [], "quote": "The report's §5 puts it that way", "report_finding": "PASS", "section": "Findings 2", "strength": "numerical only" }, { "claim": "§11 item 7: seven below five may be due to the family itself, the parameterization or insufficient search, and cannot yet be read as a general negative.", "kind": "reduction", "numbers": [ "γ = β, (l₁, l₂/2, l₂/2, l₃)" ], "quote": "the supremum is still at least s₅, since the least scale over rotations depends", "report_finding": "PARTIAL", "section": "Findings 2", "strength": "analytic (proof checked)" }, { "claim": "The round's best seven-link s₇ = 0.998674157956 is its family's best curve found, above the three-link baseline.", "kind": "computation", "numbers": [ "equal-thirds three-link 0.998697791610 in both families, same value to 10⁻¹²", "s₇ 2.36×10⁻⁵ below it", "this run's seven-link 0.998904534 (2.30×10⁻⁴ above s₇)", "from the round's own seven-link: 0.998827199" ], "quote": "to 10⁻¹² (Z6). The reported seven-link is 2.36×10⁻⁵ below it; the reported five-link is 5.66×10⁻⁵ above it.", "report_finding": "FAIL", "section": "Z6; Findings 2", "strength": "corrected" }, { "claim": "The round's best five-link s₅ = 0.998754371668 is its family's best curve found (§11 item 2: global optimality not proved).", "kind": "computation", "numbers": [ "nudge (M15 parameters) +2.67×10⁻⁵", "this run's five-link 0.998903093 at (0.194587, 0.141402, 0.328021), α = 83.4458°, β = 80.6281°", "1.49×10⁻⁴ above s₅", "from the equal-thirds three-link: 0.998832412", "Round v0.4.1: 0.998903750476" ], "quote": "α = 83.4458°, β = 80.6281° — 1.49×10⁻⁴ above the round's s₅; from the equal-thirds three-link, 0.998832412;", "report_finding": "PARTIAL", "section": "Findings 2", "strength": "corrected" }, { "claim": "§8 and §10: the next round should level the lowest phase branches.", "kind": "method", "numbers": [ "round's five-link: one branch at φ = 2π/3, next 0.999238612 (4.8×10⁻⁴ higher)", "round's seven-link: next 3.6×10⁻⁴ higher", "this run's five-link: three branches level to 1.2×10⁻¹¹", "this run's seven-link: three branches level to 10⁻¹⁴" ], "quote": "branches is what the report's §8 and §10 propose for the next round.", "report_finding": "PASS", "section": "Findings 2", "strength": "numerical only" }, { "claim": "The five- and seven-link curves' least scales fall at their stated phases (2π/3).", "kind": "computation", "numbers": [ "2π/3 to the last bit of a double", "stated phases to 2.4×10⁻¹³", "endpoints agree to 2×10⁻¹⁶", "one-sided slopes −0.269 and +0.615 / +0.616", "baseline 1.66×10⁻⁶ rad from 2π/3" ], "quote": "| the five-link and seven-link curves' minima fall at φ = 2π/3 | numerical; the endpoints' tie at that phase is exact, by symmetry |", "report_finding": "PASS", "section": "Findings 4", "strength": "numerical only" }, { "claim": "§7: a curve can saturate the same hypotenuse constraint with both endpoints at once — \"這是第 3 輪最重要的結構發現\" (round 3's most important structural finding).", "kind": "reading", "numbers": [ "φ = 2π/3" ], "quote": "is the part of the skeleton that follows from the construction rather than from the search.", "report_finding": "PARTIAL", "section": "Findings 4", "strength": "corrected", "audit": "finding moved from 'NOT AS STATED' after the independent reading: the double hypotenuse contact reproduces (Z2); what does not hold is reading it as a finding of the search, since the imposed symmetry forces it" }, { "claim": "§6: five links give a small but stable rise in pressure over three; \"三連桿不是本搜尋架構中的終點\" (the three-link is not the end point of this search architecture).", "kind": "reading", "numbers": [ "highest congruent three-link found 0.998697792" ], "quote": "architecture) still holds — the five-link is higher than every congruent three-link value Round 02 or RUN-003 found", "report_finding": "PASS", "section": "Findings 3", "strength": "numerical only" }, { "claim": "The five-link's gain over the three-link baseline s₃ is 1.79997×10⁻⁴.", "kind": "computation", "numbers": [ "1.79997×10⁻⁴ stated", "1.80577×10⁻⁴ against the congruent baseline 0.998573794671" ], "quote": "one; against the congruent baseline the five-link gain is 1.80577×10⁻⁴ instead of 1.79997×10⁻⁴. The difference is", "report_finding": "NOT AS STATED", "section": "Findings 3", "strength": "corrected" }, { "claim": "The five-link improves on the three-link family by about 1.80×10⁻⁴ (and the seven-link improves on it too).", "kind": "computation", "numbers": [ "5.66×10⁻⁵ against the equal-thirds three-link", "seven-link's gain negative" ], "quote": "and against the equal-thirds three-link the five-link's gain is 5.66×10⁻⁵, about a third of the stated 1.80×10⁻⁴,", "report_finding": "NOT AS STATED", "section": "Findings 3", "strength": "corrected" }, { "claim": "No bound for Moser's problem is claimed.", "kind": "reading", "numbers": [], "quote": "No bound is claimed.", "report_finding": "PASS", "section": "What Round 03 claims", "strength": "label only" }, { "claim": "The package's five figures.", "kind": "reading", "numbers": [ "5 figures" ], "quote": "figures, which were not examined.", "report_finding": "UNMEASURED", "section": "What this run does not establish", "strength": "not measured" }, { "claim": "(Recomputation's instrument) The drill plants defects and checks each is caught by exactly its named checks.", "kind": "method", "numbers": [ "16 planted", "16 caught", "4 controls", "146/146" ], "quote": "| must-stay-green assertions held | **146/146** |", "report_finding": "PASS", "section": "The drill", "strength": "label only" } ], "corrections": [ { "what": "metrics blocks in the summary", "source_says": "metrics.minimum and best_local_minima per curve.", "report_finds": "They are an unrefined 8,192-phase grid's values; the sampled minimum lies 1.37×10⁻⁴ to 1.38×10⁻⁴ above the true kink minimum, so they are not critical scales.", "quote": "the true minimum is a sharp kink between grid points, and the sampled minimum lies 1.37×10⁻⁴ to 1.38×10⁻⁴ above it" }, { "what": "Reading of the entropy", "source_says": "H_λ = 0.963370649827, a baseline determined by the container's normal geometry.", "report_finds": "It is the entropy of the weights (1/A, 1/B, 1) of the formula's rows; with unit normals the weights are the side lengths and the entropy is 0.965163253185; the round's reading changes with the triangle's size (0.963701869 for T undivided).", "quote": "lengths (B, A, √(A² + B²)), and the entropy is 0.965163253185. The round's reading also changes with the triangle's" }, { "what": "Seven links do not improve on five (README, §6 boxed, §12)", "source_says": "Seven links brought no further improvement; the added parameters degenerate into near-parallel sub-segments; s₇ < s₅ is the more important limitation.", "report_finds": "Setting γ = β and splitting the second side, every five-link curve is a seven-link curve, so the seven-link optimum is at least s₅; s₇ < s₅ is where the search stopped, and degeneration cannot lower a family's optimum.", "quote": "seven-link curve, with every length positive, and the seven-link family's optimum is at least the five-link family's." }, { "what": "s₇ against the equal-thirds three-link", "source_says": "s₇ = 0.998674157956 is the best seven-link.", "report_finds": "Both families contain the equal-thirds three-link of RUN-003 (0.998697791610); s₇ is 2.36×10⁻⁵ below it; this run's seven-link search reaches 0.998904534.", "quote": "to 10⁻¹² (Z6). The reported seven-link is 2.36×10⁻⁵ below it; the reported five-link is 5.66×10⁻⁵ above it." }, { "what": "Best five-link", "source_says": "s₅ = 0.998754371668.", "report_finds": "A small nudge raises it by 2.67×10⁻⁵; this run's search reaches 0.998903093, 1.49×10⁻⁴ higher (numerical, not proved maximal).", "quote": "α = 83.4458°, β = 80.6281° — 1.49×10⁻⁴ above the round's s₅; from the equal-thirds three-link, 0.998832412;" }, { "what": "§7's most important structural finding", "source_says": "Both endpoints saturating the hypotenuse is round 3's most important structural finding.", "report_finds": "At φ = 2π/3 the mirror axis is parallel to the hypotenuse normal, so the endpoints' tie follows from the imposed symmetry; what the search found is that the minimum falls there, with one interior vertex on each leg.", "quote": "the imposed symmetry gives at the phase where the minimum falls; what the search found is that the minimum falls" }, { "what": "Baseline convention for the gains", "source_says": "Five-link gain 1.79997×10⁻⁴ over s₃ = 0.998574374711.", "report_finds": "s₃ is the orientation-preserving value; against the congruent baseline 0.998573794671 the gain is 1.80577×10⁻⁴.", "quote": "one; against the congruent baseline the five-link gain is 1.80577×10⁻⁴ instead of 1.79997×10⁻⁴. The difference is" }, { "what": "Gain against the best three-link", "source_says": "Five links improve on three by about 1.80×10⁻⁴.", "report_finds": "Against the equal-thirds three-link (0.998697792) the five-link's gain is 5.66×10⁻⁵ and the seven-link's is negative.", "quote": "and against the equal-thirds three-link the five-link's gain is 5.66×10⁻⁵, about a third of the stated 1.80×10⁻⁴," } ], "open_items": [ "Neither search is proved maximal.", "Any bound for Moser's problem. That the minima over phases, the round's five-link value or this run's searches are", "That seven links beat five in general: this run's 1.44×10⁻⁶ is the difference of two searches, and the", "seven-link search stopped where it did: the package has no code and no search log.", "report a five-link at 0.998903750476, 6.6×10⁻⁷ above this run's; RUN-005 will check it.", "figures, which were not examined." ], "cross_refs": [ "RUN-003 (the exact three-normal formula and the minima-over-phases method; the equal-thirds three-link at 0.998697792 that both Round 03 families contain; Round 02's corrected curve as the baseline)", "RUN-002 (the sense of PARTIAL used here)", "RUN-005 (Round v0.4.1 reports a five-link at 0.998903750476, 6.6×10⁻⁷ above this run's; to be checked there)" ], "unclear": [ "The chirality exclusion is labelled 'analytic (proof checked)': the report gives the one-line reason (a mirror-symmetric curve is congruent to its reflection) and checks the mirror images numerically (Z1).", "The round's best five- and seven-link curves are labelled 'corrected', not 'refuted': §11 item 2 itself disclaims global optimality, and the report's higher values are searches, not proofs. The seven-link conclusion (README, §6, §12) is labelled 'refuted' because the nesting is exact (the construction in §2).", "The summary's metrics blocks are labelled 'corrected' though the round's report does not use them; the correction is to the summary's labelling.", "The entropy value reproduces under the round's row reading; only its reading as a property of the container's normal geometry is corrected.", "Two item quotes (s₃ and s₅) share one report line, which states both values.", "The drill note says one of Z6's two catches (plant M15) is weaker: it reprints an existing failure with a new value rather than finding a new one." ], "for_paper": "Round 03's three critical scales (three-link baseline 0.998574374711, five-link 0.998754371668, seven-link 0.998674157956) reproduce to 1.4×10⁻¹³ under the exact three-normal formula, with minima over phases found numerically by a refined 3,600-phase scan; the contact ledgers and four-contact skeleton, the 18-row resolution table, the metrics blocks (as an unrefined 8,192-phase grid) and the entropy (under the formula's row scaling) reproduce. The seven-link family contains every five-link curve (set γ = β and split the second side), so the reported s₇ < s₅ is where the search stopped, not a property of the family, and the README's and §6/§12's degeneration explanation is refuted; s₇ also lies 2.36×10⁻⁵ below the equal-thirds three-link that both families contain. This run's own numerical searches reach 0.998903093 (five links) and 0.998904534 (seven links), neither proved maximal. Against the best known three-link the five-link gain is 5.66×10⁻⁵, not 1.80×10⁻⁴, and the double hypotenuse contact that §7 calls the round's most important structural finding follows from the imposed mirror symmetry at φ = 2π/3. No bound for Moser's problem is claimed." }, { "run": "RUN-005", "report_file": "RUN-005-ROUND04-BRANCH-EQUALIZATION.md", "source_document": "Round 04 (v0.4.1) - Moser_Skew_Lab_v0.4.1.zip (SHA-256 acc4235c…): its README, reports/ROUND_04_REPORT.md, reports/ROUND_04_CUSP_AUDIT.md, src/METHOD.md, and its data (round04_summary.json, branch_sensitivity_matrix.csv, equalized_5link_curve.csv), with Round 03's summary for the starting curve and round v0.5's README for one cross-check", "verdict_line": "**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces, and two of the things it states do not hold as stated.", "items": [ { "claim": "The equalized five-link has (l₁, l₂, l₃) = (0.1943225812, 0.1415865167, 0.3281818042) and (β, α) = (80.68621471°, 83.41615042°), with l₃ = 1 − 2l₁ − 2l₂, length 1 and mirror symmetry.", "kind": "specification", "numbers": [ "(0.1943225812, 0.1415865167, 0.3281818042)", "(80.68621471°, 83.41615042°)", "10⁻¹⁷" ], "quote": "The logged curve is the after parameters to 10⁻¹⁷, with l₃ = 1 − 2l₁ − 2l₂ exactly, of length 1, and", "report_finding": "PASS", "section": "E1; Findings 1 — The curves", "strength": "numerical only" }, { "claim": "Round 04 starts from Round 03's five-link (RUN-004), whose least scale is Round 03's.", "kind": "specification", "numbers": [ "0.998754371668" ], "quote": "least scale is Round 03's, 0.998754371668 (E2)", "report_finding": "PASS", "section": "E1, E2; Findings 1 — The curves", "strength": "numerical only" }, { "claim": "A full rescan of phases gives the critical scale s₅⁽⁴⁾ = 0.998903750476; the cusp audit puts the true global scale at 0.998903750476358, at φ = 4.712388980384675, close to the exact 3π/2.", "kind": "computation", "numbers": [ "0.998903750476358", "0.9989037504763532", "5×10⁻¹⁵", "φ = 3π/2", "1.5×10⁻¹⁴" ], "quote": "0.9989037504763583 to 5×10⁻¹⁵, reached at φ = 3π/2 to the last bit of a double (the audit's phase to 1.5×10⁻¹⁴)", "report_finding": "PASS", "section": "E2; Findings 1 — The global scale", "strength": "numerical only" }, { "claim": "The critical scale is 1.493788082036×10⁻⁴ above Round 03; the stated distance to 1 and the README's tracker overestimate 1.301910645957×10⁻⁸ are differences of the stated values.", "kind": "computation", "numbers": [ "1.493788082036×10⁻⁴", "1.301910645957×10⁻⁸" ], "quote": "The stated improvement, distance to 1 and tracker overestimate are the differences they name (E6).", "report_finding": "PASS", "section": "E6; Findings 1 — The global scale", "strength": "numerical only" }, { "claim": "Cusp audit: the equalized curve has nine local minima over phases (grids up to 65,536 phases, every local minimum refined), the four lowest being B1 to B4.", "kind": "computation", "numbers": [ "nine", "8.6×10⁻¹⁴", "4.9×10⁻¹³" ], "quote": "Its nine local minima over phases are the stated nine, values to 8.6×10⁻¹⁴ and phases to", "report_finding": "PASS", "section": "E3; Findings 1 — The landscape", "strength": "numerical only" }, { "claim": "Cusp audit resolution table: on grids of 2,048 to 65,536 phases the landscape has 8, 9, 9, 9, 9 and 9 strict local minima, each row with its global scale.", "kind": "computation", "numbers": [ "2,048 to 65,536", "8, 9, 9, 9, 9 and 9", "6.0×10⁻¹⁵" ], "quote": "On grids of 2,048 to 65,536 phases the landscape has 8, 9, 9, 9, 9 and 9 strict local", "report_finding": "PASS", "section": "E4; Findings 1 — The resolution table", "strength": "numerical only" }, { "claim": "Cusp audit: the audit refined around every local minimum ('在所有局部極小附近細化'), and its resolution table's global-scale column records the true global scale at each resolution.", "kind": "reading", "numbers": [ "3.3×10⁻¹⁵", "2.7×10⁻¹⁵ to 6.0×10⁻¹⁵" ], "quote": "so the round did refine, as the audit says", "report_finding": "PASS", "section": "E4; Findings 1 — The resolution table", "strength": "numerical only" }, { "claim": "The tracker follows the four lowest local minima (B1 and B2 at low phases, B3 at 120°, B4 at 270°) as functions m_r(η), and the stated spread is that of the four tracked values.", "kind": "computation", "numbers": [ "3.0×10⁻⁸ rad", "3.2×10⁻¹¹ to 1.30×10⁻⁸", "2.6×10⁻¹¹ to 1.28×10⁻⁸" ], "quote": "Each tracked phase (after) lies within 3.0×10⁻⁸ rad of a true local minimum", "report_finding": "PASS", "section": "E5; Findings 1 — The tracker", "strength": "numerical only" }, { "claim": "README ('四分支等高誤差') and main report §2 and §10: the four branches are level to 2.734812376559×10⁻¹².", "kind": "computation", "numbers": [ "2.734812376559×10⁻¹²", "1.30×10⁻⁸", "0.99890376346193 (B1)", "0.99890376346325 (B2)", "0.99890375813127 (B3)", "0.99890375047635 (B4)" ], "quote": "level to 2.73×10⁻¹². That is the spread of the tracker's four values. The four true branch minima are", "report_finding": "NOT AS STATED", "section": "§2 — The level claim is the tracker's", "strength": "corrected" }, { "claim": "Cusp audit: the level claim is withdrawn; the tracker's four epigraph constraints are level to 2.73×10⁻¹², the four true heights lie in a band of about 10⁻⁸, and the true global scale is 0.998903750476358.", "kind": "reading", "numbers": [ "2.73×10⁻¹²", "about 10⁻⁸", "1.30×10⁻⁸", "0.998903750476358" ], "quote": "the audit puts them in a band of about 10⁻⁸ and withdraws the claim, and the README carries", "report_finding": "PASS", "section": "§2 — The level claim is the tracker's", "strength": "numerical only" }, { "claim": "Cusp audit: the gap between tracker and true global scale comes from the 270° branch, a cusp where the active vertex switches.", "kind": "reading", "numbers": [ "1.3×10⁻⁸" ], "quote": "The audit names one cusp, the 270° branch.", "report_finding": "PASS", "section": "§2 — The level claim is the tracker's", "strength": "numerical only" }, { "claim": "(Report's own measurement) Every one of the four branch minima is a kink with one-sided slopes of opposite sign, not only the 270° one; the 120° branch's tracked value is also above its minimum.", "kind": "computation", "numbers": [ "5.4×10⁻⁹", "−0.330 and +0.024 at B1", "−0.024 and +0.234 at B2", "−0.268 and +0.613 at B3", "−0.330 and +0.432 at B4" ], "quote": "The 120° branch's tracked value is also 5.4×10⁻⁹ above its minimum, and every one of the four minima is a kink, with", "report_finding": "PASS", "section": "§2 — The level claim is the tracker's", "strength": "numerical only" }, { "claim": "Finite-difference sensitivity matrix G_rj = ∂m_r/∂η_j, η = (l₁, l₂, β, δ), printed to eight decimals in the summary and branch_sensitivity_matrix.csv.", "kind": "computation", "numbers": [ "2.3×10⁻⁵", "2.4×10⁻⁵", "1.3×10⁻⁴", "1.5×10⁻⁴", "10⁻⁶", "3.7×10⁻⁹" ], "quote": "and its rows B3 and B4 to 1.3×10⁻⁴ and 1.5×10⁻⁴, about the third (`sensitivity_row_max_difference`).", "report_finding": "FAIL", "section": "E7; §4 — The sensitivity matrix and the pressure vector", "strength": "corrected", "audit": "quotation replaced after the independent reading: the earlier one began after the report's 'No', which ends the line before, and so read as a pass" }, { "claim": "From G, the branches respond to the same parameter in conflicting directions ('反應方向互相衝突').", "kind": "reading", "numbers": [], "quote": "The sign pattern the report reads — \"反應方向互相衝突\" (conflicting directions) — is the", "report_finding": "PASS", "section": "§4 — The sensitivity matrix and the pressure vector", "strength": "numerical only" }, { "claim": "Second-level branch pressure μ on the simplex with Σ_r μ_r ∇_η m_r ≈ 0, residual 1.840159765877×10⁻⁷.", "kind": "computation", "numbers": [ "1.840159765877×10⁻⁷", "(0.02802, 0.02871, 0.73655, 0.20672)", "2.9×10⁻⁴", "8.3×10⁻⁶", "2.0×10⁻⁴" ], "quote": "The pressure vector's arithmetic holds exactly (E8).", "report_finding": "PASS", "section": "E8; §4 — The sensitivity matrix and the pressure vector", "strength": "numerical only" }, { "claim": "No local direction raises all four branches, so a max-min level point forms ('形成 max-min 等高點') at the round's point.", "kind": "reading", "numbers": [ "of order 10⁻⁵", "2.0×10⁻⁴", "6.5×10⁻⁸" ], "quote": "direction raises all four branches at once, at a rate of order 10⁻⁵ — and the round's point is not a level point", "report_finding": "PARTIAL", "section": "§4 — The sensitivity matrix and the pressure vector", "strength": "corrected" }, { "claim": "Per-branch contact ledgers: B1 has vertices 0 and 1 on the leg x = 0, B2 vertices 1 and 2, B3 the two endpoints on the hypotenuse, B4 the two endpoints on the leg x = 0; the four branches are different contact topologies.", "kind": "computation", "numbers": [], "quote": "says, the four branches are different contact topologies, not rotated copies of one placement.", "report_finding": "PASS", "section": "E9; Findings 1 — The landscape", "strength": "numerical only" }, { "claim": "The README, the report and the audit state the data's values, parameters, tables and ledgers; METHOD.md states the epigraph problem max z s.t. m_r(η) ≥ z and the pressure condition.", "kind": "specification", "numbers": [], "quote": "METHOD.md states the report's epigraph problem and pressure condition | **PASS** |", "report_finding": "PASS", "section": "E10; Findings 1 — The documents", "strength": "label only" }, { "claim": "Cusp audit: the next round must not use only ∇_η m_r; the cusps need one-sided derivatives, Clarke subdifferentials, switching equations, exact special phases and a piecewise contact-event system.", "kind": "method", "numbers": [ "1.5×10⁻⁹" ], "quote": "audit describes is in φ, not in η, and its methodological conclusion — that ∇_η m_r alone will not do — does not", "report_finding": "PARTIAL", "section": "§3 — The four branch minima sit at exact phases", "strength": "corrected" }, { "claim": "The four tracked branches are the four lowest local minima of the phase landscape: B1 and B2 at low phases, B3 at 120°, B4 at 270°.", "kind": "computation", "numbers": [ "π/2 − α", "π/2 − β", "2π/3", "3π/2", "2.2×10⁻¹⁵", "2.8×10⁻¹³" ], "quote": "Each branch minimum is where the curve's contact with the triangle changes identity, and each such phase is exact", "report_finding": "PASS", "section": "§3 — The four branch minima sit at exact phases", "strength": "numerical only" }, { "claim": "(Report's own computation, the audit's 'exact special phases' item carried out) The epigraph problem solved on the four closed-form branch values m_r(η) = s(φ_r(η); η) levels the branches above the round's value.", "kind": "computation", "numbers": [ "1.1×10⁻¹⁵", "0.998903757133", "6.7×10⁻⁹", "(0.194313, 0.141593, 0.328188)", "α = 83.4151°", "β = 80.6883°", "status 8", "5.68, 0.314, 0.0407 and 2.7×10⁻¹¹", "(0.0280, 0.0287, 0.7365, 0.2067)", "6.5×10⁻⁸", "2.04×10⁻⁹" ], "quote": "So the point is a local maximum of the four-branch model, by these probes; it is not proved", "report_finding": "PASS", "section": "§3 — The four branch minima sit at exact phases", "strength": "numerical only" }, { "claim": "Round v0.5's README gives a common event scale ('事件共同尺度') of 0.998903757132509.", "kind": "computation", "numbers": [ "0.998903757132509", "1.2×10⁻¹⁵" ], "quote": "of 0.998903757132509, this run's value to 1.2×10⁻¹⁵; RUN-006 will check it.", "report_finding": "PASS", "section": "§3 — The four branch minima sit at exact phases", "strength": "numerical only" }, { "claim": "§7, §10: without adding links, levelling only the weakest phase branches, the five-link family's critical scale can still be raised ('仍能繼續提高'); a curve's difficulty is a balance of several ledgers among competing placement branches.", "kind": "reading", "numbers": [ "1.493788082036×10⁻⁴", "0.998903093", "6.6×10⁻⁷" ], "quote": "The round's headline — levelling the weakest branches raises the five-link's scale without adding links — holds. But", "report_finding": "PASS", "section": "§3 — The four branch minima sit at exact phases", "strength": "numerical only" }, { "claim": "§8 item 5 asks whether a new contact identity can break the current 0.9989 plateau ('可突破目前的 $0.9989$ 平台').", "kind": "reading", "numbers": [ "0.998904534", "7.84×10⁻⁷", "7.77×10⁻⁷" ], "quote": "That is above the round's five-link by less than 10⁻⁶, and RUN-004's search, like this one, is not proved maximal.", "report_finding": "PARTIAL", "section": "§5 — The seven-link family", "strength": "open" }, { "claim": "§9 limits: floating-point one-dimensional minimization, finite differences, no global guarantee for SLSQP, no interval certificate, no proof that the active identities are stable in a whole neighbourhood; no bound is claimed.", "kind": "reading", "numbers": [], "quote": "the five-link family, or that the contact identities are stable in a whole neighbourhood.", "report_finding": "PASS", "section": "What this run does not establish", "strength": "open" }, { "claim": "(Recomputation's instrument) Drill of planted defects and controls against checks E1-E10.", "kind": "method", "numbers": [ "17", "4", "166/166" ], "quote": "The drill planted 17 defects; each tripped exactly the checks named for it, and the 4", "report_finding": "PASS", "section": "The drill", "strength": "label only" } ], "corrections": [ { "what": "Four-branch level error (README '四分支等高誤差', main report §2 and §10)", "source_says": "The four branches are level to 2.734812376559×10⁻¹².", "report_finds": "That is the spread of the tracker's four values; the true branch minima 0.99890376346193, 0.99890376346325, 0.99890375813127 and 0.99890375047635 span 1.30×10⁻⁸, and the round's own cusp audit withdraws the claim.", "quote": "The README's \"四分支等高誤差：2.734812376559e-12\" and the main report's §2 and §10 state that the four branches are" }, { "what": "Sensitivity matrix G (E7)", "source_says": "G_rj = ∂m_r/∂η_j printed to eight decimals.", "report_finds": "Against central differences of the true branch minima, rows B1 and B2 agree only to 2.3×10⁻⁵ and 2.4×10⁻⁵, rows B3 and B4 to 1.3×10⁻⁴ and 1.5×10⁻⁴; the sign pattern is the same; the cause is not established (the package has no code).", "quote": "its rows B3 and B4 to 1.3×10⁻⁴ and 1.5×10⁻⁴, about the third" }, { "what": "Pressure reading at the round's point (§4)", "source_says": "Σμ_r∇_η m_r ≈ 0 (residual 1.840159765877×10⁻⁷): no local direction raises all four branches, so a max-min level point forms.", "report_finds": "On this run's G at the round's point the least-norm simplex combination leaves 8.3×10⁻⁶ (the round's μ leaves 2.0×10⁻⁴), so some direction raises all four at a rate of order 10⁻⁵ and only B4 is active there; the multiplier reading holds at the levelled point of §3, residual 6.5×10⁻⁸.", "quote": "(only B4 is active there). The multiplier reading belongs at the levelled point of §3, where it holds: μ* > 0 and the" }, { "what": "Cusp audit's methodological conclusion", "source_says": "The next round must not use only ∇_η m_r at the cusps.", "report_finds": "The non-smoothness is in φ, not η: each branch value is a closed form m_r(η) = s(φ_r(η); η) at an exact phase, smooth in η where the contact identities hold, and its gradient equals central differences of the true minima to 1.5×10⁻⁹.", "quote": "follow at this point: ∇_η m_r is well defined here, and computed from the closed forms it is exact to that precision." } ], "open_items": [ "Any bound for Moser's problem. That the minima over phases, the round's equalized curve or this run's are maximal in", "the five-link family, or that the contact identities are stable in a whole neighbourhood.", "rows differ from the derivatives: the package has no code. Anything about the package's eight figures, which were not", "So the point is a local maximum of the four-branch model, by these probes; it is not proved", "That is above the round's five-link by less than 10⁻⁶, and RUN-004's search, like this one, is not proved maximal." ], "cross_refs": [ "RUN-001 (restore_escapes applied to the package documents)", "RUN-002 (the sense of PARTIAL)", "RUN-003 (exact formula for the triangle and minima over phases)", "RUN-004 (Round 03's five-link start; five-link family; symmetry ties §4; direct max-min search 0.998903093; seven-link search 0.998904534)", "RUN-006 (checks round v0.5's common event scale 0.998903757132509)" ], "unclear": [ "verdict_line joins the report's lines 15-16 (one sentence across a hard wrap).", "The level claim is labelled 'corrected' rather than 'refuted': the round's own cusp audit already withdraws it, and the report's §3 repairs the levelling (1.1×10⁻¹⁵ at the exact phases); it could be read as refuted.", "The pressure reading at the round's point and the cusp audit's methodological conclusion are labelled PARTIAL/corrected with corrections rows, although the report's verdict counts only two not-as-stated items (the level claim and the sensitivity matrix).", "The branch-minima item is labelled numerical only: the report calls the phases' identities exact (a link parallel to a leg, a symmetry tie), but that the minima sit there is numerical.", "Resolution-table reading: the report says the global-scale column cannot tell refinement from coincidence (3π/2 is a grid point under φ_k = 2πk/n, a convention the package does not state); recorded PASS because the scatter shows the round did refine.", "§8 item 5 is a question, not a claim; recorded PARTIAL/open on RUN-004's seven-link data.", "E7's first part (the summary's matrix equals the csv's) passes; only the derivative comparison fails. The reading η = (l₁, l₂, β, δ) with δ = α − β is the report's." ], "for_paper": "Round 04's equalized five-link reproduces in floating point as a curve, with least scale 0.998903750476 at φ = 3π/2 (to 5×10⁻¹⁵), its nine local minima, the resolution counts, the four contact ledgers and the pressure arithmetic. Its headline level error of 2.73×10⁻¹² is the tracker's spread: the true branch minima span 1.30×10⁻⁸, as the round's own cusp audit concedes, and its sensitivity matrix, printed to eight decimals, is right only to the third or fourth. The report finds the four branch minima at exact phases (π/2 − α, π/2 − β, 2π/3, 3π/2), so each branch value is a closed form smooth in η, contrary to the audit's methodological worry. Re-equalized on these closed forms the branches level to 1.1×10⁻¹⁵ at 0.998903757133, 6.7×10⁻⁹ higher, a local maximum of the four-branch model by probes only. No bound for Moser's problem is claimed or established." }, { "run": "RUN-006", "report_file": "RUN-006-ROUND05-CONTACT-EVENTS.md", "source_document": "Round 05 (v0.5) - Moser_Skew_Lab_v0.5.zip (SHA-256 2a86975f…): its README, reports/ROUND_05_REPORT.md, src/event_system.py (a docstring, no code), and its data (round05_summary.json, event_candidate_curve.csv, event_branch_gradients.csv, event_kkt_jacobian.csv, perturbation_convergence.json)", "verdict_line": "**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces, and two of its printed figures are not right to the digits printed.", "items": [ { "claim": "The triangle's legs A and B are the certified legs divided by 1.0048.", "kind": "specification", "numbers": [ "1.0048", "1.1×10⁻¹⁶" ], "quote": "A and B are the certified legs divided by 1.0048, to 1.1×10⁻¹⁶ (V1, `precision`).", "report_finding": "PASS", "section": "V1; Findings 1 — The setup", "strength": "numerical only" }, { "claim": "φ₃ = 2π/3 comes from the Wetzel triangle, with B/A = √3.", "kind": "theorem", "numbers": [ "B/A = √3", "3(4 + √3) = √3(3 + 4√3)", "2.2×10⁻¹⁶" ], "quote": "exactly — 3(4 + √3) = √3(3 + 4√3) — and in floating point to 2.2×10⁻¹⁶.", "report_finding": "PASS", "section": "V1; Findings 1 — The setup", "strength": "analytic (proof checked)" }, { "claim": "The candidate curve is the event parameters, with α = β + δ and l₃ = 1 − 2(l₁ + l₂), of length 1 and mirror-symmetric.", "kind": "specification", "numbers": [ "5.6×10⁻¹⁷" ], "quote": "the event parameters to 5.6×10⁻¹⁷, of length 1, its own mirror image (V1); the event phases are π/2 − α, π/2 − β,", "report_finding": "PASS", "section": "V1; Findings 1 — The setup", "strength": "numerical only" }, { "claim": "The four branch phases follow from the active contact identities: φ₁ = π/2 − α (q₀, q₁ on the left leg), φ₂ = π/2 − β (q₁, q₂ on it), φ₃ = 2π/3 (both endpoints on the hypotenuse), φ₄ = 3π/2 (both endpoints on the left leg).", "kind": "reduction", "numbers": [ "π/2 − α", "π/2 − β", "2π/3", "3π/2" ], "quote": "2π/3 and 3π/2 exactly (V2).", "report_finding": "PASS", "section": "V2; Findings 1 — The setup", "strength": "analytic (proof checked)" }, { "claim": "The four branch values are closed forms in p = (l₁, l₂, β, δ): m₁ = H_φ₁(q₃), m₂ = H_φ₂(q₃) − x_φ₂(q₁)/A, m₃ = −x_φ₃(q₃)/A − y_φ₃(q₂)/B, m₄ = y₂/A + (D − x₂)/B.", "kind": "theorem", "numbers": [ "3.3×10⁻¹⁶", "10⁻¹⁴" ], "quote": "event phases to 3.3×10⁻¹⁶ and give the stated branch values (V3) — so the report's formulas are the right ones for", "report_finding": "PASS", "section": "V3; Findings 1 — The closed forms", "strength": "numerical only" }, { "claim": "s_event = 0.998903757132509, with the four branch values level to 2.220446049250×10⁻¹⁶.", "kind": "computation", "numbers": [ "0.998903757132509", "2.220446049250×10⁻¹⁶" ], "quote": "the common scale and the spread are theirs | **PASS** |", "report_finding": "PASS", "section": "V3", "strength": "numerical only" }, { "claim": "The event scale is the curve's least scale over all phases, each of the four events being a local (kink) minimum.", "kind": "computation", "numbers": [ "0.9989037571325088", "0.0301" ], "quote": "The least exact scale over all phases is the common scale, 0.9989037571325088; all four events", "report_finding": "PASS", "section": "V4; Findings 1 — The landscape", "strength": "numerical only" }, { "claim": "§11: a 262,144-phase audit has grid minimum equal to the event scale, at 3π/2 (the other event phases not being grid points).", "kind": "computation", "numbers": [ "262,144 = 2¹⁸" ], "quote": "The 262,144-phase audit reproduces: 262,144 = 2¹⁸", "report_finding": "PASS", "section": "V4; Findings 1 — The landscape", "strength": "numerical only" }, { "claim": "The logged branch gradients ∇_p m_r (event_branch_gradients.csv) at the root.", "kind": "computation", "numbers": [ "8.9×10⁻¹⁶", "10⁻¹²" ], "quote": "The logged gradients are the complex-step derivatives of the closed forms to 8.9×10⁻¹⁶ (V5).", "report_finding": "PASS", "section": "V5; Findings 1 — The equations", "strength": "numerical only" }, { "claim": "m₁ = m₂ = m₃ = m₄ alone is three equations in four unknowns and in general forms a one-parameter family of candidates ('一般形成一維候選族').", "kind": "reduction", "numbers": [ "5.68, 0.314, 0.0407 and 1.7×10⁻¹⁷" ], "quote": "level set m₁ = m₂ = m₃ = m₄ is a smooth curve there — the report's one-parameter family — with tangent v; along it the", "report_finding": "PASS", "section": "§2 — The root is a strict local maximum", "strength": "numerical only" }, { "claim": "Adding the branch pressures (μ_r ≥ 0, Σμ_r = 1, Σμ_r∇_p m_r = 0) gives nine equations in nine unknowns (p, s, μ₁, …, μ₄), which the stated solution satisfies.", "kind": "reduction", "numbers": [ "5.7×10⁻¹⁶", "10⁻¹⁴" ], "quote": "At the stated solution the nine equations leave 5.7×10⁻¹⁶ (V6)", "report_finding": "PASS", "section": "V6; Findings 1 — The equations", "strength": "numerical only" }, { "claim": "The residual at the solution is ‖F‖ = 2.777480200316×10⁻¹⁶.", "kind": "computation", "numbers": [ "2.777480200316×10⁻¹⁶", "5.7×10⁻¹⁶" ], "quote": "the round's 2.8×10⁻¹⁶ is reproduced in order, not", "report_finding": "PASS", "section": "V6; Findings 1 — The equations", "strength": "corrected" }, { "claim": "Pressures μ = (0.0280, 0.0288, 0.7365, 0.2067) lie on the simplex; the gain over Round 04's audited value is 6.656150519646×10⁻⁹, with the stated distance to 1.", "kind": "computation", "numbers": [ "(0.0280, 0.0288, 0.7365, 0.2067)", "6.656150519646×10⁻⁹" ], "quote": "the improvement over Round 04 and the distance to 1 are the differences they name", "report_finding": "PASS", "section": "V6", "strength": "numerical only" }, { "claim": "§5 glosses the gain over Round 04 as '約 $10^{-9}$' (about 10⁻⁹).", "kind": "reading", "numbers": [ "6.656150519646×10⁻⁹" ], "quote": "6.656150519646×10⁻⁹, nearer 10⁻⁸ on a log scale; a loose gloss, not counted.", "report_finding": "PASS", "section": "§4 — Printed digits that are not the quantities'", "strength": "label only" }, { "claim": "The 9×9 KKT Jacobian (event_kkt_jacobian.csv) has numerical rank 9.", "kind": "computation", "numbers": [ "9×9", "rank 9", "2.65×10⁻⁹", "1.06×10⁻⁹", "1.000000000372, 1.0000000026549 and 0.99999999986574" ], "quote": "Its rank 9 is not in doubt", "report_finding": "PASS", "section": "V7; Findings 1 — The equations", "strength": "numerical only" }, { "claim": "The Jacobian's smallest singular value is 3.607219048175×10⁻² and its condition number 162.781733366.", "kind": "computation", "numbers": [ "3.607219048175×10⁻²", "3.607219010513×10⁻²", "3.8×10⁻¹⁰", "162.781733366", "162.781735059", "1.7×10⁻⁶", "4.2×10⁻¹⁰" ], "quote": "are the logged difference Jacobian's. The exact Jacobian", "report_finding": "NOT AS STATED", "section": "§4 — Printed digits that are not the quantities'", "strength": "corrected" }, { "claim": "40 of 40 perturbed starts return to the same root, at most 8.3×10⁻¹⁵ away.", "kind": "computation", "numbers": [ "40 of 40", "8.3×10⁻¹⁵", "10⁻³ or 10⁻²", "32 of 40 at 3×10⁻²" ], "quote": "The 40 trials all succeed, at most 8.3×10⁻¹⁵ from the root (V8).", "report_finding": "PASS", "section": "V8; §3 — The root is reached from the earlier rounds", "strength": "numerical only" }, { "claim": "At all four events D₋s(φ_r) < 0 < D₊s(φ_r), so each is a kink minimum, with the recorded contact ledgers.", "kind": "computation", "numbers": [ "0.0238" ], "quote": "one-sided derivatives have opposite signs, the smallest in size 0.0238 (V9).", "report_finding": "PASS", "section": "V9; Findings 1 — The cusps", "strength": "numerical only" }, { "claim": "§9 table of one-sided derivatives at the four events, printed to nine decimals.", "kind": "computation", "numbers": [ "3.0×10⁻⁹ to 3.9×10⁻⁸", "4×10⁻⁸", "−0.267655431", "−0.2676554550", "step 10⁻⁸" ], "quote": "but by the standard RUN-002 and RUN-003 applied to printed last digits, these are not as stated.", "report_finding": "NOT AS STATED", "section": "§4 — Printed digits that are not the quantities'", "strength": "corrected" }, { "claim": "The README and the report state the data's numbers and tables; src/event_system.py, a docstring with no code, states the nine equations and unknowns.", "kind": "specification", "numbers": [], "quote": "slope tables to the digits printed, and the docstring states the nine equations (V10).", "report_finding": "PASS", "section": "V10; Findings 1 — The documents", "strength": "label only" }, { "claim": "The candidate is a numerically isolated stationary point in this fixed contact topology ('目前候選是此固定接觸拓撲中的數值孤立駐定點').", "kind": "theorem", "numbers": [ "−0.1227", "5.68, 0.314, 0.0407 and 1.7×10⁻¹⁷" ], "quote": "condition holds: the root is a strict local maximum of the four-branch max-min min_r m_r(p).", "report_finding": "PASS", "section": "§2 — The root is a strict local maximum", "strength": "numerical only" }, { "claim": "(Report's finding) Since every contact identity holds with a strict margin, the curve's least scale S(p) equals the four-branch minimum near the root, so the root is a strict local maximum of S on a neighbourhood of unestablished size.", "kind": "theorem", "numbers": [ "0.0067", "|D±| ≥ 0.0238", "0.030", "2.04×10⁻⁹, 2.04×10⁻⁷ and 2.04×10⁻⁵", "6.1×10⁻⁸, 5.5×10⁻⁷ and 6.1×10⁻⁶", "2.2×10⁻¹⁶", "5.6×10⁻¹⁶" ], "quote": "the root is a strict local maximum of S. That holds at the strength of the margins, which are double-precision", "report_finding": "PASS", "section": "§2 — From the model to the curve", "strength": "numerical only" }, { "claim": "The round plans a draft computational certificate that no higher solution exists locally ('局部不存在更高解的計算證書草案') and the active identities' neighbourhood; §14: no interval Newton or Krawczyk certificate.", "kind": "certificate", "numbers": [], "quote": "\"活動身份鄰域\" (the active identities' neighbourhood). It is not attempted here.", "report_finding": "UNMEASURED", "section": "§2 — From the model to the curve", "strength": "open" }, { "claim": "(Report's own measurement) Newton's method on the nine equations reaches the stated root from Round 04's equalized point and from RUN-005's SLSQP point.", "kind": "computation", "numbers": [ "2.2×10⁻¹⁶", "8.0×10⁻¹⁶", "3.1×10⁻¹⁵", "6×10⁻⁹", "1.0×10⁻¹⁵" ], "quote": "Jacobian — returns to the stated root to 2.2×10⁻¹⁶ from the root itself, and reaches it from Round 04's equalized point", "report_finding": "PASS", "section": "§3 — The root is reached from the earlier rounds", "strength": "numerical only" }, { "claim": "§14 and the reading's hedge: not a proof of analytic uniqueness ('這不構成解析唯一性證明'); perturbation return is not uniqueness; no proof this topology is optimal among five-links; no bound.", "kind": "reading", "numbers": [], "quote": "uniqueness of the root, optimality of this topology among five-links | not established (as the round says) |", "report_finding": "PASS", "section": "At what strength each result stands", "strength": "open" }, { "claim": "(Recomputation's instrument) Drill of planted defects and controls against checks V1-V10.", "kind": "method", "numbers": [ "19", "4", "177/177" ], "quote": "The drill planted 19 defects; each tripped exactly the checks named for it, and the 4 controls disturbed", "report_finding": "PASS", "section": "The drill", "strength": "label only" } ], "corrections": [ { "what": "One-sided derivatives at the events (§9)", "source_says": "One-sided derivatives printed to nine decimals, e.g. B3's left slope −0.267655431.", "report_finds": "They are one-sided differences of the exact scale at step 10⁻⁸; the exact derivatives differ by 3.0×10⁻⁹ to 3.9×10⁻⁸ (B3's left slope is −0.2676554550); the signs D₋ < 0 < D₊ stand.", "quote": "(B3's left slope prints −0.267655431; the derivative is −0.2676554550)." }, { "what": "Smallest singular value and condition number of the KKT Jacobian", "source_says": "σ_min = 3.607219048175×10⁻², condition number 162.781733366.", "report_finds": "These are the logged difference Jacobian's; the exact Jacobian gives σ_min = 3.607219010513×10⁻² (3.8×10⁻¹⁰ lower) and κ = 162.781735059 (1.7×10⁻⁶ higher); rank 9 and isolation unaffected.", "quote": "gives σ_min = 3.607219010513×10⁻², 3.8×10⁻¹⁰ below the printed 3.607219048175×10⁻² — right to eight significant" }, { "what": "Residual of the nine equations", "source_says": "‖F‖ = 2.777480200316×10⁻¹⁶.", "report_finds": "The nine equations leave 5.7×10⁻¹⁶ at the stated solution; the stated residual is reproduced in order, not in digits.", "quote": "the round's 2.8×10⁻¹⁶ is reproduced in order, not" } ], "open_items": [ "Any bound for Moser's problem. That the root is the five-link family's maximum, or the only root, or that this contact", "The size of the neighbourhood on which the root is a local maximum of the curve's least", "Any interval certificate. Anything about the package's eight figures, which were not examined.", "The round does not say how far its 40 starts were perturbed.", "\"活動身份鄰域\" (the active identities' neighbourhood). It is not attempted here." ], "cross_refs": [ "RUN-001 (restore_escapes applied to the package documents)", "RUN-002 (the sense of PARTIAL; the standard for printed last digits)", "RUN-003 (exact formula and minima over phases; the standard for printed last digits)", "RUN-005 (Round 04's branches; the event phases found exact in its §3; its SLSQP point 0.998903757133; E7's 10⁻⁶ tolerance)" ], "unclear": [ "verdict_line joins the report's lines 15-16 (one sentence across a hard wrap).", "The residual 2.777480200316×10⁻¹⁶ is labelled corrected (reproduced in order, not in digits) following the brief's rule on figures printed past their digits; the report's verdict counts only the one-sided derivatives and σ_min/κ as not as stated.", "The event phases are labelled analytic (proof checked): the report calls them exact via the contact identities (RUN-005 §3); the V2 check itself compares stated values.", "The closed forms are labelled numerical only: the strength table calls them exact for these identities, but the check is agreement with the exact scale to 3.3×10⁻¹⁶ in floating point.", "The round's reading that black-box search in this topology 'has largely converged' ('已大致收束') is not tested as such; the report's Newton returns (§3) bear on it; not itemised.", "The '約 10⁻⁹' gloss is recorded PASS/label only because the report calls it a loose gloss and does not count it.", "V9 passes at its 10⁻⁶ tolerance while the printed nine-decimal slopes are not as stated; recorded as one PASS item (signs, ledgers) and one NOT AS STATED item (values)." ], "for_paper": "Round 05's contact-event system reproduces in double precision: the four event phases are the exact contact-identity phases, the report's closed forms equal the exact scale there to 3.3×10⁻¹⁶, the nine equations hold at the stated root to 5.7×10⁻¹⁶, and the common scale 0.998903757132509 is the curve's least scale over all phases, with the next local minimum 0.030 higher. Two printed figures are not right to their digits: the one-sided derivatives are step-10⁻⁸ differences (off by up to 3.9×10⁻⁸), and σ_min and κ are a difference Jacobian's (the exact values lie 3.8×10⁻¹⁰ and 1.7×10⁻⁶ away). The report shows that the root satisfies the second-order sufficient condition, with curvature −0.1227 along the single critical direction, so it is a strict local maximum of the four-branch model, and, by continuity from strict double-precision margins, of the curve's least scale on a neighbourhood of unestablished size. Newton's method reaches the same root from Round 04's and RUN-005's points. No interval certificate, no uniqueness or optimality of the topology, and no bound for Moser's problem is established." }, { "run": "RUN-007", "report_file": "RUN-007-ROUND06-CHIRALITY-ESCAPE.md", "source_document": "Round 06 (v0.6.1) - Moser_Skew_Lab_v0.6.1.zip (SHA-256 6fddf556…): its README, reports/ROUND_06_REPORT.md, ROUND_06_INTERPRETATION.md, ROUND_06_LOCAL_ESCAPE_AUDIT.md, src/METHOD.md, and its data (round06_summary.json, antisymmetric_mode_profiles.json, topology_census.json, local_escape_audit.json, topology_escape_curve.csv), with Round 05's summary for the root", "verdict_line": "**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces, and two of the things it states do not hold as stated.", "items": [ { "claim": "§1: the general five-link near the root joins four antisymmetric parameters (d_l₁, d_l₂, d_α, d_β) to (l₁, l₂, β, δ), and every curve is scored by the congruent scale s_cong = min{s₊, s₋}.", "kind": "definition", "numbers": [], "quote": "The logged escape curve is the stated escape parameters built as §1 defines the general five-link,", "report_finding": "PASS", "section": "A1; Findings 1 — The family", "strength": "label only" }, { "claim": "(Report's finding) The eight parameters are a chart of every unit five-link near the root up to rigid motion, and s₋(p, d) = s₊(p, −d).", "kind": "theorem", "numbers": [ "4.4×10⁻¹⁶" ], "quote": "The eight parameters are a chart of every unit five-link near the root, up to rigid motion: the map from (p, d) to the", "report_finding": "PASS", "section": "§2 — The event root is a strict local maximum among all nearby five-links", "strength": "analytic (proof checked)" }, { "claim": "The reference is Round 05's event solution, its two handednesses both at the event scale, with no chirality gap.", "kind": "computation", "numbers": [ "0.998903757132509", "2.2×10⁻¹⁶", "nine local minima", "131,072 phases" ], "quote": "least scale equal to the event scale 0.998903757132509 to 2.2×10⁻¹⁶ (`reference_handednesses`), with nine local", "report_finding": "PASS", "section": "A1; Findings 1 — The reference", "strength": "numerical only" }, { "claim": "Four antisymmetric modes, each scanned alone over 101 values: 404 rows of plus, minus and congruent values and chirality gaps, with per-mode summaries.", "kind": "computation", "numbers": [ "404", "101", "2,048-phase", "6.7×10⁻¹⁶" ], "quote": "All 404 rows are the 2,048-phase grid minima of the curve and of its mirror image, to 6.7×10⁻¹⁶,", "report_finding": "PASS", "section": "A2; Findings 1 — The mode profiles", "strength": "numerical only" }, { "claim": "The mode tables are consistent with the mirror symmetry: the plus value at t is the minus value at −t.", "kind": "computation", "numbers": [], "quote": "+t, so the plus value at t must be the minus value at −t; the rows satisfy it to rounding (A3).", "report_finding": "PASS", "section": "A3; Findings 1 — The mode profiles", "strength": "numerical only" }, { "claim": "In all four modes the best value is zero perturbation, with no gain over the centre.", "kind": "computation", "numbers": [ "±1.052, ±1.137, ±0.198 and ±0.251", "5.6×10⁻⁴" ], "quote": "modes peak strictly at zero, with one-sided slopes ±1.052, ±1.137, ±0.198 and ±0.251 (`mode_profiles_true`) —", "report_finding": "PASS", "section": "§3 — What the single-mode scans can and cannot show", "strength": "numerical only" }, { "claim": "The modes table's largest chirality gaps in the scan ('掃描內最大手性差') are 1.686×10⁻³, 2.448×10⁻³, 1.329×10⁻² and 5.370×10⁻³.", "kind": "computation", "numbers": [ "1.686×10⁻³", "2.448×10⁻³", "1.329×10⁻²", "5.370×10⁻³", "1.743×10⁻³", "2.083×10⁻³", "1.330×10⁻²", "5.069×10⁻³", "3.7×10⁻⁴" ], "quote": "least scales the largest gaps are 1.743×10⁻³, 2.083×10⁻³, 1.330×10⁻² and 5.069×10⁻³, against the printed 1.686×10⁻³,", "report_finding": "NOT AS STATED", "section": "§3 — What the single-mode scans can and cannot show", "strength": "corrected" }, { "claim": "From the single-mode peaks near zero: mirror symmetry is no arbitrary aesthetic restriction but the natural balance point of the congruent minimum between the chirality branches.", "kind": "reading", "numbers": [], "quote": "four peaks along four axes do not exclude a gain along a combination of modes", "report_finding": "PARTIAL", "section": "§3 — What the single-mode scans can and cannot show", "strength": "numerical only" }, { "claim": "The eight-dimensional search's best candidate, checked on 131,072 phases: congruent 0.998557207698881, 3.47×10⁻⁴ below the event scale, chirality gap 3.30×10⁻⁵, with the stated improvement and distance to 1.", "kind": "computation", "numbers": [ "0.998557207699", "0.998590245470", "9.8×10⁻¹⁴", "3.7×10⁻¹³", "8 and 9", "3.47×10⁻⁴", "3.30×10⁻⁵" ], "quote": "Its two handednesses, 0.998557207699 and 0.998590245470, reproduce to 9.8×10⁻¹⁴, their phases to", "report_finding": "PASS", "section": "A4; Findings 1 — The escape", "strength": "numerical only" }, { "claim": "Interpretation note: the eight-dimensional search's best candidate re-converged to near mirror symmetry ('最佳候選重新收斂到近鏡像對稱').", "kind": "reading", "numbers": [ "d_l₁ = 0.0111", "d_l₂ = −0.0111", "5.6 %", "8.1 %", "−0.357°", "+0.368°", "3.47×10⁻⁴", "3.3×10⁻⁵" ], "quote": "So the interpretation note's \"最佳候選重新收斂到近鏡像對稱\" does not hold", "report_finding": "NOT AS STATED", "section": "§4 — The eight-dimensional search did not return to the platform", "strength": "refuted" }, { "claim": "Interpretation note: the search produced no reproducible positive escape.", "kind": "reading", "numbers": [], "quote": "That it found no escape is consistent with §2, but it is not evidence for", "report_finding": "PASS", "section": "§4 — The eight-dimensional search did not return to the platform", "strength": "numerical only" }, { "claim": "Census of 1,600 general five-links near the root, the top twenty re-checked at high resolution, five distinct winning contact signatures among them.", "kind": "computation", "numbers": [ "1,600", "twenty", "4.5×10⁻¹³", "1.6×10⁻¹²", "five" ], "quote": "For each of the twenty logged rows the coarse values are the 2,048-phase grid minima, the refined", "report_finding": "PASS", "section": "A5; Findings 1 — The census", "strength": "numerical only" }, { "claim": "The census table's '排名' (rank) column orders the candidates.", "kind": "reading", "numbers": [ "5.7×10⁻⁵ to 5.6×10⁻⁴" ], "quote": "so the report's \"排名\" (rank) column is the coarse rank, as its procedure says", "report_finding": "PASS", "section": "Findings 1 — The census", "strength": "label only" }, { "claim": "Local escape audit: 6,300 general five-links at seven radii (7 × 900), the top thirty re-checked, none above the root, the best non-base 1.13×10⁻⁴ below it.", "kind": "computation", "numbers": [ "6,300", "7 × 900", "1.13×10⁻⁴", "4.6×10⁻¹³", "1.7×10⁻⁴ to 6.2×10⁻⁴", "+1.1×10⁻¹⁶" ], "quote": "The fifteen logged candidates' handednesses reproduce to 4.6×10⁻¹³, and their gains, norms, the best", "report_finding": "PASS", "section": "A6; Findings 1 — The audit", "strength": "numerical only" }, { "claim": "The census's 1,600 samples and the audit's 6,300 samples (and the audit's other fifteen re-checked candidates) bear out the stated rankings, signatures and counts.", "kind": "computation", "numbers": [ "1,600", "6,300" ], "quote": "| the census and the audit beyond their logged rows | the round's, unchecked (the samples are not logged) |", "report_finding": "UNMEASURED", "section": "At what strength each result stands", "strength": "not measured" }, { "claim": "Local stability draft: the root's inactive-vertex margins, and 500 random samples in each of twelve symmetric parameter boxes from 10⁻⁷ to 5×10⁻⁴ all keeping the four signatures ('局部數值證書草案，不是區間證明').", "kind": "computation", "numbers": [ "1.1×10⁻¹⁶", "100 in each of the twelve boxes", "10⁻⁷ to 5×10⁻⁴" ], "quote": "this run's own box samples, 100 in each of the twelve boxes, keep the four signatures, and each stated box minimum", "report_finding": "PASS", "section": "A7; Findings 1 — The stability draft", "strength": "numerical only" }, { "claim": "The stability draft re-prints Round 05's Jacobian figures and least one-sided slope (2.377371632889×10⁻²) as the event root's existing diagnostics.", "kind": "reading", "numbers": [ "2.377371632889×10⁻²", "0.0237736992" ], "quote": "the last digits RUN-006 §4 found not right; that is charged to Round 05 there and not counted again.", "report_finding": "PASS", "section": "A7; Findings 1 — The stability draft", "strength": "label only" }, { "claim": "The README, the report, the audit note and METHOD.md state the data's numbers, tables and the congruent objective; the interpretation note carries its reading.", "kind": "specification", "numbers": [], "quote": "The README, the report, the audit note and METHOD.md state the data's numbers, tables and the", "report_finding": "PASS", "section": "A8; Findings 1 — The documents", "strength": "label only" }, { "claim": "(Report's finding) Round 05's four branch identities give closed forms m_r(p, d) in all eight parameters, and near the root s_cong(p, d) = min over r of {m_r(p, d), m_r(p, −d)}.", "kind": "computation", "numbers": [ "3.3×10⁻¹⁶", "4.4×10⁻¹⁶", "1.1×10⁻¹⁵", "4.1×10⁻⁶", "2.5×10⁻⁵" ], "quote": "value is a closed form m_r(p, d) in the eight parameters; at the root they are the event scale to 3.3×10⁻¹⁶, and their", "report_finding": "PASS", "section": "§2 — The event root is a strict local maximum among all nearby five-links", "strength": "numerical only" }, { "claim": "(Report's finding) G_d is nonsingular, so the critical cone is the single line (v, 0) and the curvature along it is −0.1227: the event root is a strict local maximum of the eight-dimensional model.", "kind": "theorem", "numbers": [ "2.52, 0.239, 0.135 and 0.0394", "−0.1227" ], "quote": "strict local maximum of the eight-dimensional model", "report_finding": "PASS", "section": "§2 — The event root is a strict local maximum among all nearby five-links", "strength": "numerical only" }, { "claim": "(Report's finding) The congruent least scale equals the eight-parameter model on some neighbourhood of the root, so there no unit five-link, chiral or not, does better.", "kind": "theorem", "numbers": [], "quote": "congruent least scale equals the model on some neighbourhood of the root, whose size is not established.", "report_finding": "PASS", "section": "§2 — The event root is a strict local maximum among all nearby five-links", "strength": "numerical only" }, { "claim": "Interpretation note: the present data support the event root's local stability against small chirality breaking ('因此目前資料支持五連桿事件根對小型手性破缺具有局部穩定性').", "kind": "reading", "numbers": [ "−0.1227" ], "quote": "root's local stability against small chirality breaking) — at the strength of a double-precision second-order", "report_finding": "PASS", "section": "§2 — The event root is a strict local maximum among all nearby five-links", "strength": "numerical only" }, { "claim": "The stability draft is 'not an interval proof' and the escape audit 'is not a proof of local optimality among general five-links' ('這不是一般五連桿局部最優證明'); no bound is claimed.", "kind": "reading", "numbers": [], "quote": "the neighbourhood's size not established; no interval certificate", "report_finding": "PASS", "section": "At what strength each result stands", "strength": "open" }, { "claim": "(Recomputation's instrument) Drill of planted defects and controls against checks A1-A8.", "kind": "method", "numbers": [ "21", "4", "161/161" ], "quote": "The drill planted 21 defects; each tripped exactly the", "report_finding": "PASS", "section": "The drill", "strength": "label only" } ], "corrections": [ { "what": "Largest chirality gaps in the mode scans ('掃描內最大手性差')", "source_says": "1.686×10⁻³, 2.448×10⁻³, 1.329×10⁻² and 5.370×10⁻³ (METHOD.md defines s± as minimized over rotation).", "report_finds": "These are differences of 2,048-phase grid samples, which lie above the least scales by up to 5.6×10⁻⁴; on the least scales the largest gaps are 1.743×10⁻³, 2.083×10⁻³, 1.330×10⁻² and 5.069×10⁻³, up to 3.7×10⁻⁴ different. The modes' best values are right a fortiori.", "quote": "best values are therefore right a fortiori, but the table's \"掃描內最大手性差\" are differences of grid samples; on the" }, { "what": "Interpretation note: the best eight-dimensional candidate 're-converged to near mirror symmetry'", "source_says": "The search's best candidate re-converged to near mirror symmetry, returning to the platform.", "report_finds": "Its antisymmetric lengths d_l₁ = 0.0111 and d_l₂ = −0.0111 are 5.6 % and 8.1 % of l₁ and l₂, its angles −0.357° and +0.368°, and it lies 3.47×10⁻⁴ below the event root, itself the member d = 0 of the searched family: by the round's own test it did not return to the platform; its small chirality gap is what a max-min over the two handednesses balances toward.", "quote": "not near zero, by the report's own test." } ], "open_items": [ "Any bound for Moser's problem. That the root is the general five-link family's global maximum, or that no other", "The size of the neighbourhood of §2.", "unlogged samples, or the package's seven figures, which were not examined.", "the other fifteen, like the 6,300 samples, are not logged.", "congruent least scale equals the model on some neighbourhood of the root, whose size is not established." ], "cross_refs": [ "RUN-001 (restore_escapes applied to the package documents)", "RUN-002 (the sense of PARTIAL)", "RUN-003 (exact formula and minima over phases, both handednesses)", "RUN-006 (Round 05's root; its identity margins and second-order argument §2, carried over here; its §4 finding on Round 05's printed Jacobian figures and least slope, re-printed in Round 06)" ], "unclear": [ "verdict_line joins the report's lines 16-17 (one sentence across a hard wrap).", "The 're-converged to near mirror symmetry' reading is labelled refuted (false as stated and used to support stability) rather than corrected; the stability conclusion itself survives via the report's §2 and is recorded separately as PASS.", "The single-mode reading is labelled PARTIAL/numerical only without a corrections row: the report says the peaks at zero are forced by the symmetry and could not show the conclusion alone, but the conclusion holds by G_d's nonsingularity; it is not among the verdict's two not-as-stated items.", "The re-printed Round 05 Jacobian figures and least slope are recorded PASS/label only because the report charges their last digits to Round 05 (RUN-006) and does not count them again.", "The audit's seven radii (0.02 to 1.0) are in units the documents do not state.", "The census rank column is recorded label only: it is the coarse rank, as the round's procedure says, not the refined order.", "The chart and mirror identity are labelled analytic (proof checked): the report gives the linear invertible map and the reflection argument, and also confirms the identity to 4.4×10⁻¹⁶ numerically." ], "for_paper": "Round 06's reference, mode profiles, escape candidate, logged census rows and logged audit candidates reproduce in double precision; the census's 1,600 and the audit's 6,300 samples are not logged and were not checked. Two statements do not hold as stated: the modes table's largest chirality gaps are differences of 2,048-phase grid samples, off from the least scales' gaps by up to 3.7×10⁻⁴, and the interpretation note's claim that the eight-dimensional search re-converged to near mirror symmetry is false, its best candidate having antisymmetric lengths 5.6 % and 8.1 % of the symmetric ones and lying 3.47×10⁻⁴ below the event root. The report writes the four branches as closed forms in all eight parameters, finds the antisymmetric gradient block G_d nonsingular, and so carries Round 05's second-order condition (curvature −0.1227) over: the event root is a strict local maximum of the eight-parameter model and, in double precision, of the congruent least scale among nearby unit five-links on a neighbourhood of unestablished size. The single-mode peaks at zero are forced by mirror symmetry and could not show this alone; no interval certificate and no bound is established." }, { "run": "RUN-008", "report_file": "RUN-008-ROUND07-CURVATURE-WINGS.md", "source_document": "Round 07 (v0.7) - Moser_Skew_Lab_v0.7.zip (SHA-256 61b02f8d…): its README, MANIFEST, reports/ROUND_07_REPORT.md, ROUND_07_INTERPRETATION.md, ROUND_07_SEARCH_AUDIT.md, and its data (round07_summary.json, segmented_results.json, circular_result.json, circular_equalized_result.json, the five segmented_m*.csv, event_five_link_control.csv, continuous_circular_wing_sample.csv), with Round 05's summary for the control", "verdict_line": "**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces, and one of its readings does not hold as stated.", "items": [ { "claim": "Round 05's event five-link, 0.998903757132509, is the explicit control, with nine local minima over phases.", "kind": "specification", "numbers": [ "0.998903757132509", "4.4×10⁻¹⁶", "4.5×10⁻¹⁴", "nine" ], "quote": "its least scale is the event scale to 4.4×10⁻¹⁶; its nine stated local minima reproduce to 4.5×10⁻¹⁴", "report_finding": "PASS", "section": "C1; Findings 1 — The control", "strength": "numerical only" }, { "claim": "Segmented wings of m = 2, 3, 4, 6, 8 pieces per side, mirror-symmetric with a horizontal centre segment, directions passing monotonically from α to β, piece lengths set by an exponential bias and turning by a curvature power (no formulas given).", "kind": "specification", "numbers": [ "exp(skew·k/(m − 1))", "β + δ·(1 − k/(m − 1))^power", "1.1×10⁻¹⁶" ], "quote": "Every logged curve is its parameters under this", "report_finding": "PASS", "section": "C2; Findings 1 — The segmented family", "strength": "numerical only" }, { "claim": "Search audit: the m = 2 family contains the event five-link.", "kind": "reduction", "numbers": [], "quote": "For m = 2 the rule gives directions (α, β) and any length ratio, so the family is the", "report_finding": "PASS", "section": "C2; Findings 1 — The segmented family", "strength": "analytic (proof checked)" }, { "claim": "Each segmented candidate's least scale, phase and local-minimum count on 32,768 phases, found by one heuristic search each; the best, m = 6, at 0.998839177646286.", "kind": "computation", "numbers": [ "32,768", "4.1×10⁻¹³", "0.998839177646286" ], "quote": "Each segmented candidate's least scale, phase and local-minimum count reproduce, the scales to", "report_finding": "PASS", "section": "C3; Findings 1 — The candidates", "strength": "numerical only" }, { "claim": "The m = 6 candidate's contact signature at its least phase, with a resolution audit on grids of 4,096 to 131,072 phases.", "kind": "computation", "numbers": [ "4,096 to 131,072", "six rows", "9.7×10⁻¹⁴" ], "quote": "The m = 6 candidate's signature is its contacts at its least phase, and its resolution audit's six", "report_finding": "PASS", "section": "C4; Findings 1 — The candidates", "strength": "numerical only" }, { "claim": "The m = 6 signature differs from Round 05's four-event skeleton ('此簽章不同於第 5 輪的四事件骨架').", "kind": "reading", "numbers": [ "seven local minima", "7.957°" ], "quote": "So the round's first clause is literally true — the candidate's least is a single event, not Round 05's", "report_finding": "PASS", "section": "§4 — The m = 6 signature is one of Round 05's four events", "strength": "numerical only" }, { "claim": "The m = 6 signature shows that multi-piece curvature wings really enter another kind of local placement mechanism ('另一種局部放置機制'), though the pressure is still lower.", "kind": "reading", "numbers": [ "7.957°", "91.55°, 91.06°, 90.54° and 89.99999999996°", "0.998854829779", "0.998854830818", "1.57×10⁻⁵", "1.028" ], "quote": "(four-event skeleton) of four level branches — but the inference fails: its events are the B2, B3 and B4 that Round 04", "report_finding": "NOT AS STATED", "section": "§4 — The m = 6 signature is one of Round 05's four events", "strength": "refuted" }, { "claim": "The final arc scales are an analytic support computation over the arcs' end points, their tangent points and the centre segment's ends ('最終固定相位尺度是圓弧模型內的解析支撐計算').", "kind": "method", "numbers": [], "quote": "For an arc, the extreme of each of the triangle's three linear functionals lies at a junction or at", "report_finding": "PASS", "section": "C5; Findings 1 — The arcs", "strength": "analytic (proof checked)" }, { "claim": "Constant-curvature arc wings (each side an arc of length w whose tangent turns linearly from α to β, centre 1 − 2w), equalized on three branches (a low phase, 120°, 270°) to 9.99×10⁻¹⁶ at s_arc = 0.998862397495093.", "kind": "computation", "numbers": [ "0.998860080009", "0.998862397495", "6.7×10⁻¹⁶", "3.2×10⁻¹³", "9.99×10⁻¹⁶", "curvature −δ/w", "seven local minima" ], "quote": "(0.998860080009) and the equalized one (0.998862397495) — their least scales to 6.7×10⁻¹⁶, their seven local minima", "report_finding": "PASS", "section": "C5; Findings 1 — The arcs", "strength": "numerical only" }, { "claim": "The reliable comparison: the equalized arc lies 4.135963741603×10⁻⁵ below the event control.", "kind": "computation", "numbers": [ "4.135963741603×10⁻⁵", "4.14×10⁻⁵", "1.1×10⁻¹⁰", "3.7×10⁻⁶ to 1.1×10⁻⁵", "eight starts" ], "quote": "deficit, 4.14×10⁻⁵, is the gap between the event five-link and a local maximum of the arc family, at the strength of a", "report_finding": "PASS", "section": "§2 — The arc wing's deficit stands", "strength": "numerical only" }, { "claim": "Provisional working proposition, limited to this round's families: constant-curvature smoothing slightly lowers the congruent containment pressure ('常曲率平滑化略微降低合同容納壓力').", "kind": "reading", "numbers": [], "quote": "proposition is, as it says, provisional and limited to this family.", "report_finding": "PASS", "section": "§2 — The arc wing's deficit stands", "strength": "open" }, { "claim": "The package's continuous circular wing sample (and the summary's continuous_circular_wing entry) is a polyline sample of the arc wing.", "kind": "computation", "numbers": [ "179 equal-arc chords per wing", "31 equal steps", "0.9999999961", "1 + 3.9×10⁻⁹", "1.1×10⁻¹⁵", "1.6×10⁻⁹", "3.7×10⁻⁴" ], "quote": "rescaled by 1 + 3.9×10⁻⁹ to length 1: it matches that to 1.1×10⁻¹⁵", "report_finding": "PASS", "section": "C6; Findings 1 — The sample", "strength": "numerical only" }, { "claim": "The gains, distances, the ranking and the MANIFEST's two scales.", "kind": "computation", "numbers": [], "quote": "The gains, distances, the ranking and the manifest's two scales are the values they name (C7)", "report_finding": "PASS", "section": "C7; Findings 1 — The figures", "strength": "numerical only" }, { "claim": "The README, the report, the interpretation note and the search audit state the data's numbers, tables and signature.", "kind": "specification", "numbers": [], "quote": "the README, the report, the interpretation note and the search audit state the data's numbers, tables and signature | **PASS** |", "report_finding": "PASS", "section": "C8; Findings 1 — The figures", "strength": "label only" }, { "claim": "Search audit: the m = 2 search found 0.998530736905809, 3.73×10⁻⁴ below the event five-link its family contains, which 'can only be explained as insufficient search, not as a mathematical disadvantage'; so the m > 2 results are only current best candidates.", "kind": "reading", "numbers": [ "0.998530736905809", "3.73×10⁻⁴" ], "quote": "The round's audit — the m = 2 family contains the event five-link, so its candidate's deficit is the search's —", "report_finding": "PASS", "section": "§3 — Every segmented family reaches the event five-link", "strength": "analytic (proof checked)" }, { "claim": "§8: it cannot be inferred that the m > 2 candidates are their families' optima.", "kind": "reading", "numbers": [ "6.5×10⁻⁵", "3.7×10⁻⁴", "−0.98 to 0.21", "0.82 to 2.09" ], "quote": "Each segmented family's supremum is therefore at least the event", "report_finding": "PASS", "section": "§3 — Every segmented family reaches the event five-link", "strength": "analytic (proof checked)" }, { "claim": "§10 asks whether a family θ(s) = β + (α − β)F(s) with the curvature concentrated in a narrowing layer tends to the event scale as the layer shrinks.", "kind": "theorem", "numbers": [ "power 160", "1.3×10⁻¹³", "7×10⁻¹⁵ (m = 3)", "1.5×10⁻⁵ (m = 8)", "−1.43 to −6.04" ], "quote": "This is the round's own §10.2 question — does a narrowing curvature layer tend to the event", "report_finding": "PASS", "section": "§3 — Every segmented family reaches the event five-link", "strength": "analytic (proof checked)" }, { "claim": "The ranking places the equalized arc above this round's best multi-piece candidate ('本輪最佳多段曲率候選'), under the README heading '可靠控制排序' (reliable control ranking).", "kind": "reading", "numbers": [], "quote": "candidates that is accurate, and the round says its m > 2 results are only candidates, so it is not counted here — unlike", "report_finding": "PASS", "section": "§3 — Every segmented family reaches the event five-link", "strength": "numerical only" }, { "claim": "(Recomputation's instrument) Drill of planted defects and controls against checks C1-C8.", "kind": "method", "numbers": [ "20", "4", "154/154" ], "quote": "The drill planted 20 defects; each tripped exactly the checks named for it, and the 4 controls disturbed", "report_finding": "PASS", "section": "The drill", "strength": "label only" } ], "corrections": [ { "what": "Reading of the m = 6 contact signature", "source_says": "The signature differs from Round 05's four-event skeleton, showing that multi-piece curvature wings really enter another kind of local placement mechanism.", "report_finds": "At the least phase (7.957°) wing link 3 of six is parallel to the leg x = 0 (to 4×10⁻¹¹ degrees), with vertices 3 and 4 on that leg, vertex 0 on the leg y = 0 and vertex 7 on the hypotenuse: Round 05's branch B2; the other low minima are B3's (120°) and B4's (270°), 1.57×10⁻⁵ above. These are among the four placement mechanisms Round 04 named, not another kind.", "quote": "local placement mechanism); at its least phase one wing link is exactly parallel to the leg x = 0, with the endpoint on" } ], "open_items": [ "Any bound for Moser's problem. That the equalized arc is the arc family's global maximum. That any segmented family's", "supremum exceeds the event scale (only that it reaches it), or that the round's search domain contained the skews and", "powers the limit needs. Anything about the package's five figures, which were not examined.", "a higher arc elsewhere in the family is not excluded" ], "cross_refs": [ "RUN-001 (restore_escapes applied to the package documents)", "RUN-002 (the sense of PARTIAL)", "RUN-003 (exact formula and minima over phases)", "RUN-004 (the same proviso about angle ordering in its §2; its seven-link item, where the round drew a family conclusion, contrasted with this round's ranking)", "RUN-005 (Round 04/05 branch B2's contact ledger, §1, and its phase π/2 − β, §3)", "RUN-007 (vectorized phase grid reused)" ], "unclear": [ "verdict_line joins the report's lines 16-17 (one sentence across a hard wrap).", "The segmented rule (length ∝ exp(skew·k/(m − 1)), direction β + δ(1 − k/(m − 1))^power) is inferred by the report from the five logged curves; the documents give no formula and no search box; recorded PASS numerical only.", "The m = 6 mechanism reading is labelled refuted (the inference fails); its literal first clause, that the least is a single event rather than four level branches, is recorded separately as PASS.", "The logged arc sample and the summary's continuous_circular_wing entry are the earlier, unequalized arc, not the equalized one the README and the report quote; the report does not count this as a misstatement.", "The README heading '可靠控制排序' (reliable control ranking) 'sits uneasily' with the search audit, which names only the arc and the event as reliable; the report does not count it, so it is folded into the PASS ranking item.", "The m = 2 audit, the §8 item and the §10.2 limit are labelled analytic (proof checked): the report calls the limit exact with skew and power unrestricted (numerical at power 160 to 10⁻¹³); whether the round's search domain contained those skews and powers is not established.", "The provisional working proposition is recorded PASS/open: it rests only on a local search of the arc family." ], "for_paper": "Round 07's control, its five segmented-wing candidates (the best, m = 6, at 0.998839177646286) and both constant-curvature arc wings reproduce in double precision, the arcs through an exact support computation. The equalized arc's deficit of 4.14×10⁻⁵ below the event five-link stands at the strength of a derivative-free local search (no move from the arc, none of eight multistarts higher), so the round's smoothing proposition stays provisional. The report extends the round's own m = 2 search audit to every m: with skew and power unrestricted, each segmented family contains the event five-link as the power tends to infinity (to 10⁻¹³ at power 160), so every segmented candidate's deficit is the search's, answering the round's §10.2 question yes for the segmented rule. The round's reading that the m = 6 signature is another kind of placement mechanism fails: its least is Round 05's B2 event, a wing link exactly parallel to the leg x = 0, and its other low minima are B3 and B4. No bound for Moser's problem, and no global maximum of the arc family, is established." }, { "run": "RUN-009", "report_file": "RUN-009-ROUND08-SMOOTH-LAYER.md", "source_document": "Round 08 (v0.8.3): Moser_Skew_Lab_v0.8.3.zip (SHA-256 26ea8d85…) — README, MANIFEST, src/METHOD.md, reports/ROUND_08_REPORT.md, ROUND_08_FINITE_WIDTH_AUDIT.md, ROUND_08_DENSE_POINTCLOUD_CROSSCHECK.md, ROUND_08_PEAK_DENSE_POINTCLOUD_CROSSCHECK.md, and its data (round08_summary.json, round08_focused_equalization.json, round08_peak_scan.json, the two point-cloud records, the eleven smooth_eps_*.csv and best_finite_width_smooth_curve.csv), with Round 05's summary for the event five-link", "verdict_line": "**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces, including its headline, and four things it states do not hold as stated.", "items": [ { "claim": "Round 08 replaces each wing's inner corner of the event five-link with an endpoint-normalized tanh tangent layer θ_ε(u) = β + (α − β) F_ε(u; c) of width ε, with a horizontal centre of length 1 − 2w and a mirrored wing, tending to the event five-link as ε → 0 (the documents give F only in words).", "kind": "definition", "numbers": [], "quote": "documents give F only in words — \"Endpoint-normalized tanh tangent layer\" — and the curves and the curvature ledger fix", "report_finding": "PASS", "section": "What Round 08 claims; At what strength each result stands", "strength": "numerical only" }, { "claim": "The eleven sweep curve files (smooth_eps_*.csv) are the tanh-layer curves of their stated parameters, mirror-symmetric and of length 1.", "kind": "computation", "numbers": [ "2,400 equal wing steps, 49 centre steps", "10⁻⁷", "5.8×10⁻¹⁰ at ε = 0.2", "5.7×10⁻⁸ at ε = 0.002" ], "quote": "The eleven sweep curve files, by contrast, are their parameters' curves, symmetric and of length 1, sampled from", "report_finding": "PASS", "section": "L1", "strength": "numerical only" }, { "claim": "The published best_finite_width_smooth_curve.csv is the best candidate's curve (ε = 0.037), the curve published as beating the event five-link.", "kind": "specification", "numbers": [ "8,034 points", "99 %", "0.0033 chord at −0.96°", "u = 1 − (k + ⅔)/4,000", "2.8×10⁻⁵", "5.5×10⁻⁵", "0.99992566", "0.998794830", "1.09×10⁻⁴ below the event five-link", "1.20×10⁻⁴ below the candidate", "0.99891433908463 with corner and endpoint added", "0.998873602 (corner alone)", "0.998823385 (endpoint alone)" ], "quote": "congruent least scale is 0.998794830 — 1.09×10⁻⁴ below the event five-link, 1.20×10⁻⁴ below the candidate it is meant to", "report_finding": "FAIL", "section": "L2; 2. The best candidate's curve file is not the candidate", "strength": "refuted" }, { "claim": "For each of the eleven sweep widths the round's full-audit least scale and its fixed-event-parameters audit are the curve's least scales, with the stated local-minimum counts and distances.", "kind": "computation", "numbers": [ "2×10⁻⁸", "1.1×10⁻⁸", "1.1×10⁻⁹ to 4.9×10⁻⁹", "6.7° to 8.0° (ε ≥ 0.02)", "1.8×10⁻¹² (ε ≤ 0.012)" ], "quote": "- **The sweep.** The eleven-width sweep's full audits and fixed-event-parameter audits are this run's least scales to", "report_finding": "PASS", "section": "L3; 4. Discretizations — the sweep", "strength": "numerical only" }, { "claim": "Re-equalizing four branches over (w, β, δ, c) at nine widths from 0.008 to 0.05 and six more from 0.034 to 0.046 gives the stated best scales, audit rows, counts, gains and tracked spreads, the summary holding the same records.", "kind": "computation", "numbers": [ "fifteen widths", "10⁻¹⁰", "6.3×10⁻¹³" ], "quote": "The fifteen re-equalized widths' scales are this run's to 6.3×10⁻¹³ (L4, `equalized_minus_exact`), and their peak —", "report_finding": "PASS", "section": "L4", "strength": "numerical only" }, { "claim": "The scale rises with layer width and falls back, with best tested width ε* = 0.037.", "kind": "computation", "numbers": [ "ε* = 0.037" ], "quote": "0.037 of the widths tried — is as stated.", "report_finding": "PASS", "section": "1. The headline reproduces, in independent code and in high precision", "strength": "numerical only" }, { "claim": "§4/§12: the best candidate (ε = 0.037) has least scale s* = 0.998914339084632 (stationary-support value) at the 270° cusp, matching the peak scan's ε = 0.037 record and the manifest.", "kind": "computation", "numbers": [ "0.998914339084632", "3π/2", "10⁻¹³", "0.9989143390846327172529644…", "7.2×10⁻¹⁶", "2,000 panels", "3,600-phase scan" ], "quote": "stated stationary-support value, 0.998914339084632 (L5), reached at 3π/2, where the two endpoints tie on the leg x = 0.", "report_finding": "PASS", "section": "L5; 1. The headline reproduces", "strength": "numerical only" }, { "claim": "The gain over the event five-link (s₀ = 0.998903757132509) is s* − s₀ = 1.058195212322×10⁻⁵.", "kind": "computation", "numbers": [ "1.058195212322×10⁻⁵", "s₀ = 0.998903757132509", "1.0581952124×10⁻⁵", "2.3×10⁻¹⁶" ], "quote": "2.3×10⁻¹⁶ (`event_root_branches_40_digits`) — so the gain is s* − s₀ = 1.0581952124×10⁻⁵. Both curves are their own mirror", "report_finding": "PASS", "section": "L5; 1. The headline reproduces", "strength": "numerical only" }, { "claim": "Of the candidate's branches the true least is at the 270° cusp.", "kind": "computation", "numbers": [ "0.998914339085 at 3π/2", "0.998914340723 at 2π/3", "0.998914384874", "0.998914385987", "four above 1.029" ], "quote": "branches, and four above 1.029 (`best_minima`). So the least is the 270° cusp, and the gain over the event five-link holds", "report_finding": "PASS", "section": "1. The headline reproduces", "strength": "numerical only" }, { "claim": "The package publishes the gain twice: 1.058195209325×10⁻⁵ from the point cloud (README, summary, §7, peak note) and 1.058195212322×10⁻⁵ from the stationary support (§4, §12, manifest).", "kind": "computation", "numbers": [ "1.058195209325×10⁻⁵", "1.058195212322×10⁻⁵", "3.0×10⁻¹⁴ apart" ], "quote": "figures in §4 and §12, the manifest), 3.0×10⁻¹⁴ apart.", "report_finding": "PASS", "section": "1. The headline reproduces; L8", "strength": "numerical only" }, { "claim": "The dense point cloud of 240,017 points, which does not use the stationary-point inversion, gives 0.998914339084602 at the exact 270° cusp, 3.0×10⁻¹⁴ below; the cloud notes' stationary values, differences and gains.", "kind": "computation", "numbers": [ "240,017 points", "0.998914339084602", "3.0×10⁻¹⁴", "10⁻¹¹", "4.1×10⁻¹² to 4.7×10⁻¹²", "1.2×10⁻¹⁴ to 3.5×10⁻¹⁴" ], "quote": "phases (L6): at the smooth low-phase branches by 4.1×10⁻¹² to 4.7×10⁻¹² — a cloud of points misses a smooth extreme by a", "report_finding": "PASS", "section": "L6", "strength": "numerical only" }, { "claim": "A tracking local minimizer beside a non-smooth cusp is off by about 10⁻⁸ (tracked branches B3 at 120°, B4 at 270°).", "kind": "computation", "numbers": [ "1.4×10⁻¹⁵", "2.1×10⁻⁸ and 1.7×10⁻⁸ rad" ], "quote": "- **The tracked branches.** At the candidate the tracker's B3 and B4 values are the exact scale at their own phases,", "report_finding": "PASS", "section": "4. Discretizations — the tracked branches", "strength": "numerical only" }, { "claim": "At the candidate the tracked branches B1–B4 level to 3.3×10⁻⁸ (the tracked branch spread).", "kind": "computation", "numbers": [ "3.3×10⁻⁸", "6.5×10⁻⁹", "4.7×10⁻⁸" ], "quote": "3.3×10⁻⁸, mixes the two; the true four span 4.7×10⁻⁸ and are not level.", "report_finding": "NOT AS STATED", "section": "4. Discretizations — the tracked branches", "strength": "corrected" }, { "claim": "Each curvature ledger gives its curve's peak curvature, 5–95 % width, turning δ and centroid.", "kind": "computation", "numbers": [ "2,400 steps (sweep)", "12,000 steps (candidate)", "2.7×10⁻¹² relative", "1,306 steps of w/12,000", "10⁻⁸ (turning)", "10⁻⁶ (centroid)" ], "quote": "fitting and now tested by L7: each peak is, to 2.7×10⁻¹² relative, the largest central difference δ|F′|/w on an", "report_finding": "PASS", "section": "L7; 4. Discretizations — the curvature ledgers", "strength": "numerical only" }, { "claim": "The report's curvature ledger for the candidate: ‖κ‖∞ = 2.092618923824 and 5–95 % width 0.036579511462, printed to twelve digits as the curve's.", "kind": "computation", "numbers": [ "2.092618923824", "0.036579511462", "2.092622664354", "0.036616788565", "3.7×10⁻⁶", "3.7×10⁻⁵", "1.9×10⁻⁶ at ε = 0.2", "1.9 % at ε = 0.002" ], "quote": "forms give 2.092622664354 and 0.036616788565 (`best_ledger_exact`) — the printed twelve digits are not the curve's.", "report_finding": "NOT AS STATED", "section": "4. Discretizations — the curvature ledgers", "strength": "corrected" }, { "claim": "The finite-width audit says the gain is only about 10⁻⁶.", "kind": "reading", "numbers": [ "10⁻⁶", "1.045806040811×10⁻⁵", "10⁻⁵ (main report §10)" ], "quote": "says \"提升量只有約 $10^{-6}$\" (the gain is only about 10⁻⁶); its own figure, above it, is", "report_finding": "NOT AS STATED", "section": "4. Discretizations — the finite-width audit", "strength": "corrected" }, { "claim": "The ε = 0.04 cloud note gives the difference between the two methods (兩種方法差) as 7.703293336547×10⁻⁹ and the gain as 1.046576370145×10⁻⁵.", "kind": "computation", "numbers": [ "7.703293336547×10⁻⁹", "1.046576370145×10⁻⁵", "9.0×10⁻⁹ rad", "2.2×10⁻⁹", "5.5×10⁻⁹", "−2.0×10⁻¹⁴", "1.0458060388×10⁻⁵" ], "quote": "least by 7.7033×10⁻⁹ (`cloud_eps_0.04_note`). The figure is a phase gap — 2.2×10⁻⁹ of it the 120° branch's own", "report_finding": "NOT AS STATED", "section": "4. Discretizations — the ε = 0.04 cloud note", "strength": "corrected" }, { "claim": "§2: the event five-link is a numerically isolated polyline candidate in a fixed contact topology, not a proof of local optimality in the full curve space; the fixed-event-parameter audits show a finite-width layer at the event's own parameters above the event root.", "kind": "computation", "numbers": [ "1.6×10⁻¹¹", "1.6×10⁻⁶ at ε = 0.008", "1.0×10⁻⁶ at 0.005", "5.0×10⁻⁸ at 0.00025", "1.9997×10⁻⁴ ε" ], "quote": "proof of local optimality among curves, and its own fixed-event-parameter audits show why; this run confirms them (to", "report_finding": "PASS", "section": "3. The event five-link is not a local maximum among curves", "strength": "numerical only" }, { "claim": "Left open by the round's caveat: whether the event five-link is locally optimal among curves.", "kind": "theorem", "numbers": [ "2.0×10⁻⁴ ε", "layers 0.00025 to 0.008", "1.06×10⁻⁵ at ε = 0.037" ], "quote": "to the event five-link uniformly (in Hausdorff distance), it is not a local maximum among curves in that topology;", "report_finding": "PASS", "section": "3. The event five-link is not a local maximum among curves", "strength": "numerical only" }, { "claim": "Round 08's finding: the five-link polyline platform can be slightly surpassed by a finite-width smooth curvature layer.", "kind": "reading", "numbers": [ "1.06×10⁻⁵", "0.998904534 (RUN-004's seven-link)", "7.77×10⁻⁷", "9.8×10⁻⁶" ], "quote": "A polyline had in fact passed the five-link \"平台\" (platform) before any smoothing: RUN-004's best seven-link, 0.998904534,", "report_finding": "PASS", "section": "3. The event five-link is not a local maximum among curves", "strength": "numerical only" }, { "claim": "Limitation: the round's two support paths still share the numerical integration of the curve's coordinates.", "kind": "reading", "numbers": [], "quote": "The round's two support paths share its integration of the curve; this run shares neither. It builds the curve from", "report_finding": "PASS", "section": "1. The headline reproduces", "strength": "label only" }, { "claim": "README: the result is a candidate numerical breakthrough, not yet confirmed by arbitrary precision or an interval certificate.", "kind": "reading", "numbers": [], "quote": "— at the strength of a double-precision scan with every local minimum refined and a 40-digit value at the least phase, not", "report_finding": "PASS", "section": "1. The headline reproduces", "strength": "label only" }, { "claim": "No bound for Moser's problem is claimed; the candidate is below 1.", "kind": "reading", "numbers": [], "quote": "Any bound for Moser's problem: the candidate is below 1, as the round says.", "report_finding": "PASS", "section": "What this run does not establish", "strength": "label only" }, { "claim": "The README's figures, the report's §3–§8 tables and numbers, the finite-width audit's ε*, scale, gain and spread, and both cloud notes' scales, differences and gains state the data's figures.", "kind": "specification", "numbers": [], "quote": "the README's figures, the report's §3–§8 tables and numbers, the finite-width audit's ε*, scale, gain and spread", "report_finding": "PASS", "section": "L8", "strength": "label only" }, { "claim": "The recomputation's own drill (17 planted defects, 4 controls) — instrument, not source.", "kind": "method", "numbers": [ "17", "4", "140/140" ], "quote": "each tripped exactly the checks named for it, and the 4 controls disturbed nothing.", "report_finding": "PASS", "section": "The drill", "strength": "label only" } ], "corrections": [ { "what": "The best candidate's curve file", "source_says": "best_finite_width_smooth_curve.csv is the ε = 0.037 candidate, published as beating the event five-link.", "report_finds": "Its centre stops at 99 % of 1 − 2w, a 0.0033 chord cuts the corner by 5.5×10⁻⁵ to a right wing sampled two-thirds of a step off and missing its endpoint; length 0.99992566, not mirror-symmetric; read as a curve its congruent least scale is 0.998794830, 1.09×10⁻⁴ below the event five-link.", "quote": "`best_finite_width_smooth_curve.csv` has 8,034 points. Its left wing is the candidate's, 4,000 equal steps. Its centre" }, { "what": "Tracked branch spread at the candidate", "source_says": "The tracked branches level to 3.3×10⁻⁸.", "report_finds": "B3 and B4 are the exact scale at phases beside the cusps (the stated mechanism); B1 and B2 are 6.5×10⁻⁹ below the exact scale at their phases, from some other evaluation; the true four span 4.7×10⁻⁸ and are not level.", "quote": "round's own stated mechanism. Its B1 and B2 values are 6.5×10⁻⁹ below the exact scale at their phases, which no" }, { "what": "The candidate's curvature ledger", "source_says": "‖κ‖∞ = 2.092618923824 and 5–95 % width W = 0.036579511462.", "report_finds": "These are a 12,000-step grid's largest central difference and quantized band width; the closed forms give 2.092622664354 and 0.036616788565 (3.7×10⁻⁶ and 3.7×10⁻⁵ off).", "quote": "steps of w/12,000. For the candidate the report prints ‖κ‖∞ = 2.092618923824 and W = 0.036579511462; the closed" }, { "what": "Finite-width audit's size of the gain", "source_says": "The gain is only about 10⁻⁶.", "report_finds": "Its own figure is 1.045806040811×10⁻⁵, and the main report's §10 says about 10⁻⁵.", "quote": "1.045806040811×10⁻⁵, and the main report's §10 says \"提升量只有約 $10^{-5}$\"." }, { "what": "ε = 0.04 cloud note's method difference and gain", "source_says": "The difference between the two methods is 7.703293336547×10⁻⁹; the gain 1.046576370145×10⁻⁵.", "report_finds": "The figure is a phase gap (the cloud's value at the tracked B3 phase, 9.0×10⁻⁹ rad beside the 120° cusp); the record's own method difference at the exact cusp is −2.0×10⁻¹⁴ and its gain 1.0458060388×10⁻⁵.", "quote": "height above the least, 5.5×10⁻⁹ the 9.0×10⁻⁹ rad offset (`exact_at_2pi_3_minus_least`); the record's own method" } ], "open_items": [ "maximum — its branches are not level — or that the least over phases is certified: the 40-digit value is at the least", "phase, the phase found by a double-precision scan. Anything about the package's six figures, which were not examined.", "The mechanism is not shown; the pattern fits", "offset from a minimum can produce: they come from some other evaluation.", "among nearby five-links, symmetric or not, on a neighbourhood of unestablished size." ], "cross_refs": [ "RUN-001 (restore_escapes restores the audit notes' TeX)", "RUN-002 (the sense of PARTIAL)", "RUN-004 (best seven-link 0.998904534, already above the event five-link)", "RUN-006 (event five-link numerically a strict local maximum among nearby five-links)", "RUN-007 (the same, symmetric or not)", "RUN-010 (builds on this run's layer F and relies on its double-precision scan for the least's location)", "RUN-011 (this run's golden refinement had already found the interval-9 smooth minimum)", "RUN-012 (uses this run's exact-support wing in double precision)" ], "unclear": [ "The tanh layer's form is inferred by the report from the curve files and the ledger (the documents give it in words); labelled numerical only on the strength of L1's match to 10⁻⁷.", "The source prints the gain as 1.058195212322×10⁻⁵; the report gives 1.0581952124×10⁻⁵ because s₀ is known only in double precision, but does not call the source's figure wrong, so the gain is recorded as PASS, not a correction.", "Round 08's 'platform surpassed' finding is recorded PASS; the report adds that a polyline (RUN-004's seven-link) had already passed the platform and that polylines can approach the candidate arbitrarily closely.", "The ledger's ∫|κ| ds ≈ 0.052047349063 is not discussed beyond L7's turning test; no separate item.", "The resolution table (3,001 to 12,001 integration points) is covered only by L8's table check; no separate item.", "The best curve file is labelled refuted: the report calls its construction exact as to the file's points and its least scale 0.998794830 numerical." ], "for_paper": "Round 08's headline reproduces in independent code: the ε = 0.037 tanh-layer candidate has least scale 0.998914339084632 at 3π/2 (0.99891433908463271725… in 40 digits at that phase), a gain of 1.0581952124×10⁻⁵ over the event five-link, at the strength of a double-precision scan with every local minimum refined, not an interval certificate. The sweep, the fifteen re-equalized widths, the point clouds and the curvature ledgers (read as grid central differences and quantized widths) reproduce. Not as stated: the published best curve file is not the candidate (read as a curve its least scale is 0.998794830, below the event five-link), the candidate's twelve-digit curvature figures are grid values, the finite-width audit's 'about 10⁻⁶' is 10⁻⁵, and the ε = 0.04 note's method difference is a phase gap (the true method difference is −2.0×10⁻¹⁴). The tracked branches are not level (true spread 4.7×10⁻⁸), and the event five-link is, numerically, not a local maximum among curves: a layer at its own parameters raises the scale by about 2.0×10⁻⁴ ε." }, { "run": "RUN-010", "report_file": "RUN-010-ROUND09-PRECISION-CUSP.md", "source_document": "Round 09 (v0.9): Moser_Skew_Lab_v0.9.zip (SHA-256 2cc4d8b3…) — README, MANIFEST, reports/ROUND_09_REPORT.md, ROUND_09_ARBITRARY_PRECISION_AUDIT.md, ROUND_09_INTERVAL_CERTIFICATE_DRAFT.md, src/high_precision_verify.py, and its data (round09_summary.json, branch_audit.json, darboux_cusp_enclosure.csv, precision_convergence.csv, epsilon_peak_fit.json), with Round 08's summary and peak scan and Round 05's event root", "verdict_line": "**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces, and three things it prints do not hold to their digits.", "items": [ { "claim": "§2: Round 08's best candidate is fixed as a curve defined by explicit decimal constants ε = 0.037, w = 0.3361057714712081, β = 1.405382794839393, δ = 0.05204734906280986, c = 0.5801781668857768, with θ(u) = β + δ (g(u) − g(1))/(g(0) − g(1)), g(u) = ½[1 − tanh((u − c)/ε)].", "kind": "specification", "numbers": [ "ε = 0.037", "w = 0.3361057714712081", "β = 1.405382794839393", "δ = 0.05204734906280986", "c = 0.5801781668857768", "10⁻⁴⁰" ], "quote": "the decimal constants are Round 08's best parameters and Round 05's event root, as shortest decimals", "report_finding": "PASS", "section": "P1", "strength": "numerical only" }, { "claim": "§3: at φ = 3π/2 the scale reduces to s₂₇₀ = y₁/A + (x₁ + l₀)/B, so the key candidate value needs only two one-dimensional integrals.", "kind": "reduction", "numbers": [], "quote": "tie on the leg and the wing's end p₁ carries the hypotenuse — derived here, not checked in interval arithmetic.", "report_finding": "PASS", "section": "3. The Darboux bound is sound", "strength": "analytic (proof checked)" }, { "claim": "§4: s₂₇₀ by tanh–sinh and Gauss–Legendre quadrature; the summary, the audit note and the manifest give it to 120 digits.", "kind": "computation", "numbers": [ "0.998914339084632717252964418536702246180192119314833519984798128351274681109990178989484072997947224021575344613070601820…", "120 digits", "200 digits" ], "quote": "(`s270_120_decimals_cut`) holds the summary's, the audit note's and the manifest's 120 digits (P2). Each convergence", "report_finding": "PASS", "section": "P2; 1. The cusp value, the event control and their difference reproduce", "strength": "numerical only" }, { "claim": "§4: at 40, 60, 80 and 120 digits the two quadratures agree in every saved digit (the convergence rows).", "kind": "computation", "numbers": [ "41, 61 and 81 decimals", "120", "125 printed" ], "quote": "row is right to its working precision — the 40-, 60- and 80-digit rows to 41, 61 and 81 decimals, the 120-digit rows to", "report_finding": "PASS", "section": "P2", "strength": "numerical only" }, { "claim": "§5: the event five-link's four branches, recomputed in high precision, differ at 10⁻¹⁶ because Round 05 kept only a double-precision root; s₀ is their minimum.", "kind": "computation", "numbers": [ "8.2×10⁻¹⁷", "2.3×10⁻¹⁶", "5.7×10⁻¹⁷", "200 digits" ], "quote": "120 digits (P3); B1 is the least, and B2, B3, B4 lie 8.2×10⁻¹⁷, 2.3×10⁻¹⁶ and 5.7×10⁻¹⁷ above it", "report_finding": "PASS", "section": "P3", "strength": "numerical only" }, { "claim": "s₂₇₀ − s₀ = 1.0581952124042×10⁻⁵, printed to 124 decimals in the summary, the manifest and the audit note.", "kind": "computation", "numbers": [ "1.05819521240422978345936361457659629238926026414161953594484…×10⁻⁵", "124 decimals printed", "121 held" ], "quote": "which the summary, the manifest and the audit note print to 124 decimals; a difference of two 120-digit values, it holds", "report_finding": "PASS", "section": "P3", "strength": "numerical only" }, { "claim": "§4 adopts the 120-digit result and boxes an 80-digit s₂₇₀, which the README repeats.", "kind": "computation", "numbers": [ "80 digits", "72 decimals held" ], "quote": "- **The adopted value.** §4 says \"採用 $120$ 位結果\" and boxes an 80-digit s₂₇₀, which the README repeats; it holds to 72", "report_finding": "NOT AS STATED", "section": "P9; 2. Printed digits", "strength": "corrected" }, { "claim": "§5 and the README print an 80-digit event minimum s₀.", "kind": "computation", "numbers": [ "80 digits", "71 decimals held" ], "quote": "decimals, and the 80-digit event minimum of §5 and the README to 71 (`long_figures_digits_held`). The 70-digit s₂₇₀", "report_finding": "NOT AS STATED", "section": "P9; 2. Printed digits", "strength": "corrected" }, { "claim": "§13's 70-digit s₂₇₀ and s₀, and the 64- and 49-digit differences of §5, §13 and the README.", "kind": "computation", "numbers": [ "70 digits", "64 digits", "49 digits" ], "quote": "and s₀ of §13, and the 64- and 49-digit differences of §5, §13 and the README, hold to every digit.", "report_finding": "PASS", "section": "P9; 2. Printed digits", "strength": "numerical only" }, { "claim": "§6: θ decreases inside (0, π/2), so sin θ decreases and cos θ increases, and the left and right sums bound the two integrals (positivity from the integrands' monotone order).", "kind": "theorem", "numbers": [ "α = 1.4574", "β = 1.4054" ], "quote": "θ decreases (δ > 0) from α = 1.4574 to β = 1.4054, inside (0, π/2), so sin θ decreases and cos θ increases: on each part", "report_finding": "PASS", "section": "P4; 3. The Darboux bound is sound", "strength": "analytic (proof checked)" }, { "claim": "darboux_cusp_enclosure.csv and the audit note: each Darboux row's lower and upper bounds, width and margin.", "kind": "computation", "numbers": [ "100 to 102 decimals held of 104 to 110 printed", "99 decimals asked" ], "quote": "row's bounds, width and margin hold to 100 to 102 decimals of the 104 to 110 printed (`darboux_rows_digits_held`); P4", "report_finding": "PASS", "section": "P4", "strength": "numerical only" }, { "claim": "§6: at 4,096 parts the Darboux lower bound for s(3π/2) exceeds s₀ by more than 7.8×10⁻⁶.", "kind": "certificate", "numbers": [ "4,096 parts", "0.9989115747096505504998971", "8.8×10⁻²⁹", "7.8175771418×10⁻⁶" ], "quote": "statement that s(3π/2) exceeds s₀ by more than 7.8×10⁻⁶ is certified.", "report_finding": "PASS", "section": "3. The Darboux bound is sound", "strength": "exact-arithmetic certificate" }, { "claim": "§6 limit: the Darboux envelope proves only that the exact 270° cusp value is above the event control, not that it is the global minimum over the phase circle.", "kind": "reading", "numbers": [], "quote": "It concerns one phase, as the round says.", "report_finding": "PASS", "section": "3. The Darboux bound is sound", "strength": "label only" }, { "claim": "§7: the branch audit's four branches — two low-phase stationary minima and the cusps at 2π/3 and 3π/2 — with their contact labels, the nearest competitor 1.6383×10⁻⁹ above s₂₇₀ at 120°.", "kind": "computation", "numbers": [ "10⁻⁶⁸", "10⁻⁷ rad", "1.6383×10⁻⁹", "0.154699127028335487754398589663", "0.124080806550179655854701882269", "0.998914384873626650041301807…", "0.998914385986533194245715981…" ], "quote": "functional ties them; the cusps are those ties, and the labels name one point of each tied pair (`branches.*.active`).", "report_finding": "PASS", "section": "P5; 5. The branches and the peak fits", "strength": "numerical only" }, { "claim": "branch_audit.json gives the four branches' values to 75 digits; §7's table presents the two low ones as the branch minima.", "kind": "computation", "numbers": [ "75 digits printed", "70 or 71 decimals held", "0.15469913367438023", "0.12408080074251457", "6.6×10⁻⁹ and −5.8×10⁻⁹ rad", "7.3×10⁻¹⁸ and 5.6×10⁻¹⁸", "17 decimals" ], "quote": "table presents them, they hold to 17 decimals.", "report_finding": "NOT AS STATED", "section": "P9; 2. Printed digits — the branch audit", "strength": "corrected" }, { "claim": "§8 estimates the cusps' one-sided slopes as difference quotients with h = 10⁻⁹.", "kind": "computation", "numbers": [ "h = 10⁻⁹", "10⁻⁴⁵" ], "quote": "P6 reproduces every quotient in the branch audit to 10⁻⁴⁵ — and the exact one-sided derivatives, from the active", "report_finding": "PASS", "section": "P6", "strength": "numerical only" }, { "claim": "§8 prints the cusps' one-sided derivatives to 50 digits.", "kind": "computation", "numbers": [ "50 digits", "0.4995 h", "9 of 50 decimals", "120°: D₋ = −0.267658287627918161991503299…, D₊ = 0.612416405839174998855469498…", "270°: D₋ = −0.329480283321469717992727949…, D₊ = 0.432686758448835705462640528…" ], "quote": "Every quotient lies 0.4995 h from its derivative, at every h (`quotient_minus_exact`): §8's figures hold to 9 of their", "report_finding": "NOT AS STATED", "section": "P9; 2. Printed digits — the slopes", "strength": "corrected" }, { "claim": "§8 concludes D₋ < 0 < D₊ at both cusps.", "kind": "computation", "numbers": [], "quote": "50 decimals. The signs D₋ < 0 < D₊, which is what §8 concludes, hold.", "report_finding": "PASS", "section": "P6; 2. Printed digits — the slopes", "strength": "numerical only" }, { "claim": "§9 fits Round 08's peak scan, placing ε* between 0.0367 and 0.0371.", "kind": "computation", "numbers": [ "0.0367", "0.0371", "3.9×10⁻¹²", "2.2×10⁻¹⁶", "5.2×10⁻¹¹" ], "quote": "The peak fits reproduce: ε* to 3.9×10⁻¹², the", "report_finding": "PASS", "section": "P7; 5. The branches and the peak fits", "strength": "numerical only" }, { "claim": "The interval-certificate draft's target for the comparison, s(φ) ≥ s₂₇₀ − η for every φ with η < s₂₇₀ − s₀ (the full phase-interval exclusion listed in §10 as not done).", "kind": "certificate", "numbers": [ "u = j/65,536", "M₂ = 15.129", "5.9×10⁻¹⁰ by 2.0×10⁻¹⁰", "radius at most 0.2674", "Lip = 1.3306", "42,838 intervals", "0.99891425946916933623", "0.998903757132508674955129824903", "1.0502336×10⁻⁵", "99 %" ], "quote": "a certified margin of more than 1.0502336×10⁻⁵, 99 % of the round's gain (`certificate`); every figure of a bound is cut", "report_finding": "PASS", "section": "4. The whole phase circle, certified for the comparison", "strength": "exact-arithmetic certificate" }, { "claim": "§11: the open step is the 10⁻⁹-level gap between 120° and 270°, which with a global Lipschitz constant and a uniform grid would force an extremely fine grid.", "kind": "reading", "numbers": [ "10⁻⁹", "1.6×10⁻⁹", "1.06×10⁻⁵", "about 6,500 times wider" ], "quote": "with η < s₂₇₀ − s₀, a margin of 1.06×10⁻⁵, about 6,500 times wider. This run certifies it.", "report_finding": "PARTIAL", "section": "4. The whole phase circle, certified for the comparison", "strength": "corrected" }, { "claim": "§11's target: s(φ) ≥ s₂₇₀ for every φ, i.e. the candidate's least is reached at 3π/2 (left open by the round).", "kind": "theorem", "numbers": [ "0.99891425946", "8.0×10⁻⁸" ], "quote": "s₂₇₀, within 8.0×10⁻⁸. That it is reached at 3π/2 — §11's target, \"s(φ) ≥ s₂₇₀\" for every φ — is not certified here.", "report_finding": "PARTIAL", "section": "4. The whole phase circle, certified for the comparison", "strength": "numerical only", "audit": "relabelled from 'open' after the independent reading: the report measures the least at 3π/2 by a double-precision scan (certified only to within 8.0×10⁻⁸ here); RUN-011 certifies it" }, { "claim": "The report's tables (§4–§10) and short figures, the README's lists and Darboux margin, the draft's open target, the manifest and the summary's copies of the data files state the data's figures.", "kind": "specification", "numbers": [], "quote": "the report's tables (§4–§10) and short figures, the README's lists and Darboux margin, the draft's open target, the manifest", "report_finding": "PASS", "section": "P8", "strength": "label only" }, { "claim": "README: the package does not constitute a new upper or lower bound for the Moser problem.", "kind": "reading", "numbers": [ "about 0.99891" ], "quote": "Any bound for Moser's problem: the candidate's least scale is about 0.99891, below 1, and the README says the package", "report_finding": "PASS", "section": "What this run does not establish", "strength": "label only" }, { "claim": "The recomputation's own drill (19 planted defects, 4 controls) — instrument, not source.", "kind": "method", "numbers": [ "19", "4", "180/180" ], "quote": "certified here. The drill planted 19 defects; each tripped exactly the checks named for it, and the 4 controls disturbed", "report_finding": "PASS", "section": "The drill", "strength": "label only" } ], "corrections": [ { "what": "The adopted 80-digit s₂₇₀ (§4, README)", "source_says": "The 120-digit result is adopted; §4 boxes an 80-digit s₂₇₀.", "report_finds": "The 80-digit figure holds to 72 decimals; the 120-digit value is reproduced to every digit.", "quote": "Not as stated: §4 says \"採用 $120$ 位結果\" (the" }, { "what": "The 80-digit event minimum s₀ (§5, README)", "source_says": "s₀ printed to 80 digits.", "report_finds": "It holds to 71 decimals.", "quote": "README and §5 an 80-digit event minimum that holds to 71; the branch audit's 75-digit values hold to 70 or 71 decimals," }, { "what": "The branch audit's values and the low branches as minima (§7)", "source_says": "75-digit branch values; the low branches' values are the branch minima.", "report_finds": "The 75-digit values hold to 70 or 71 decimals; the low branches' values are at minimizer phases 6.6×10⁻⁹ and −5.8×10⁻⁹ rad from the stationary points, exceeding the minima by 7.3×10⁻¹⁸ and 5.6×10⁻¹⁸, so as minima they hold to 17 decimals.", "quote": "and the two low branches' are values at the phases a minimizer returned, 6.6×10⁻⁹ and 5.8×10⁻⁹ rad from the stationary" }, { "what": "§8's 50-digit one-sided derivatives", "source_says": "The cusps' one-sided slopes to 50 digits.", "report_finds": "They are difference quotients at h = 10⁻⁹, each 0.4995 h from the exact derivative, so 9 of their 50 decimals are the derivative's; the signs D₋ < 0 < D₊ hold.", "quote": "difference quotients at h = 10⁻⁹, 5.0×10⁻¹⁰ from the derivatives, so 9 of their digits are the derivative's." }, { "what": "What the round's comparison still needs (§11)", "source_says": "The open step is the 10⁻⁹-level gap between the 120° and 270° branches.", "report_finds": "The comparison with the event five-link needs only the draft's target, with a margin of 1.06×10⁻⁵ (about 6,500 times wider), which this run certifies over the whole phase circle; the 10⁻⁹ gap matters only for locating the least at 3π/2, left open here.", "quote": "draft's target, with a margin of 1.06×10⁻⁵, not the 10⁻⁹-level gap its §11 cites; this run certifies it in interval" } ], "open_items": [ "That the least is reached at 3π/2: the certificate bounds the least from below, within", "8.0×10⁻⁸ of the value there, and the location rests on RUN-009's double-precision scan.", "family's best, or a local maximum among curves.", "which a displacement bound would carry far inside the certified margin, but the transfer was not carried out.", "package's five figures, which were not examined.", "derived here, not checked in interval arithmetic." ], "cross_refs": [ "RUN-002 (the sense of PARTIAL)", "RUN-009 (the layer F; its double-precision scan locates the least at 3π/2)", "RUN-011 (certifies the least at 3π/2, using this run's event five-link upper bound and trapezoid boxes)", "RUN-012 (uses this run's high-precision candidate)" ], "unclear": [ "Convergence rows (125 decimals printed, held to 41/61/81/120), the difference (124 printed, 121 held) and the Darboux rows (104-110 printed, 100-102 held) are figures printed past their working precision; the report reads them at that precision and counts them as holding, separately from the three P9 groups, so they are recorded PASS, not as corrections.", "The §11 framing is recorded as corrected with a correction row; the report states it as a finding, not among the three not-as-stated groups.", "The 7.8×10⁻⁶ Darboux certificate and the whole-circle certificate rest on the support identity at 3π/2, which the report derives but does not check in interval arithmetic (RUN-011 later checks its premises in interval arithmetic).", "P1 (decimal constants) is an exact string identity plus a 10⁻⁴⁰ high-precision match of the g-form to F; labelled numerical only.", "The whole-circle certificate is for Round 09's decimal curves; the transfer to Round 08's double-precision curves was not carried out." ], "for_paper": "Round 09's high-precision values reproduce: s₂₇₀ to 120 digits by two independent quadratures at 200 digits, the event five-link's four branches, and s₂₇₀ − s₀ = 1.0581952124042297834593636×10⁻⁵. The Darboux premise is checked analytically, and the 4,096-part bound s(3π/2) − s₀ > 7.8×10⁻⁶ is certified in interval arithmetic, given the support identity at 3π/2. Over the whole phase circle the run certifies, in interval arithmetic and for Round 09's decimal curves, that the candidate's least scale is at least 0.99891425946916933623, above the event five-link's least (at most 0.998903757132508674955129824903): a margin of more than 1.0502336×10⁻⁵, 99 % of the gain; that the least is reached at 3π/2 is not certified here. Not as stated: the adopted 80-digit s₂₇₀ (holds to 72 decimals) and s₀ (71), the branch audit's low-branch values read as minima (17 decimals), and §8's 50-digit slopes (difference quotients, 9 digits of the derivative)." }, { "run": "RUN-011", "report_file": "RUN-011-ROUND10-PHASE-MAP.md", "source_document": "Round 10 (v0.10): Moser_Skew_Lab_v0.10.zip (SHA-256 pinned in corpus.py) — README, MANIFEST, reports/ROUND_10_REPORT.md, ROUND_10_GLOBAL_CERTIFICATE_INTERFACE.md, src/phase_contact_map.py (a docstring), and its data (round10_summary.json, round10_global_exclusion_audit.json, phase_contact_intervals.csv), with Round 08's best candidate, Round 09's decimal constants and Round 05's event root", "verdict_line": "**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces, and two things it states do not hold.", "items": [ { "claim": "The curve is Round 08's best candidate with Round 09's decimal constants; the source note states the support labels (p0–p3, L, R) and the envelope-theorem derivative on a fixed signature.", "kind": "specification", "numbers": [], "quote": "the curve parameters are Round 08's best and Round 09's decimal constants; the source note states the labels and the envelope formula", "report_finding": "PASS", "section": "S1", "strength": "label only" }, { "claim": "§3: the phase circle splits into 18 intervals of fixed active support identity with 18 distinct signatures (I_min x, I_min y, I_max d) and 17 boundaries; the CSV and the summary hold the same intervals.", "kind": "computation", "numbers": [ "18 intervals", "18 signatures", "17 boundaries", "10⁻¹²", "60 digits" ], "quote": "every interval's midpoint the active supports are the stated signature (S2). Each of the 17 switches, located by", "report_finding": "PASS", "section": "S2; 1. The map, the switches and the stated stationary points reproduce", "strength": "numerical only" }, { "claim": "§6: the 17 switches' phases and scales (and the special phases' values).", "kind": "computation", "numbers": [ "5.22×10⁻¹⁴ in phase", "3.58×10⁻¹⁴ to 1.04×10⁻¹³ in scale", "2×10⁻¹³", "8.38×10⁻¹⁴ (double value at 3π/2 below exact)", "8.37×10⁻¹⁴ (§7, against a double reference)" ], "quote": "bisection on the active labels, lies within 5.22×10⁻¹⁴ of its stated phase, and its stated scale is the exact one less", "report_finding": "PASS", "section": "S2, S3; 1. The map, the switches and the stated stationary points reproduce", "strength": "numerical only" }, { "claim": "The number of switches is 17 in the manifest and the §6 table and 18 in the README and §4.", "kind": "reading", "numbers": [ "17", "18", "φ = 0", "−0.654 before, −0.214 after" ], "quote": "the one omitted from §6 is the switch at φ = 0, where the two ends tie in y, and it is not a minimum (the derivative is", "report_finding": "PASS", "section": "1. The map, the switches and the stated stationary points reproduce", "strength": "label only" }, { "claim": "§4: the map is the same from 32,768 to 262,144 phases — no new small interval or new local minimum as the resolution grows.", "kind": "computation", "numbers": [ "32,768", "65,536", "131,072", "262,144", "18 cyclic switches", "18 signatures", "8 sampled local minima" ], "quote": "32,768, 65,536, 131,072 and 262,144 phases give 18 cyclic switches, 18 signatures, 8 sampled local minima and the least", "report_finding": "PASS", "section": "S7", "strength": "numerical only" }, { "claim": "§5: the 12 listed smooth stationary points, with their kinds and signatures.", "kind": "computation", "numbers": [ "12", "7.88×10⁻¹³ in phase", "9.58×10⁻¹⁴ in scale", "40 digits" ], "quote": "stated stationary points, found here at 40 digits, are exact ones of their kinds and signatures (S4). This run's scans at", "report_finding": "PASS", "section": "S4; 1. The map, the switches and the stated stationary points reproduce", "strength": "numerical only" }, { "claim": "§5: the 12 smooth stationary points are all of them; no extra sign change of the derivative hidden between coarse grid points has been found.", "kind": "computation", "numbers": [ "12", "13", "95 points an interval", "15 geometric steps to 10⁻¹⁵ of the width" ], "quote": "list has 12 points where this run finds 13 (`stationary_count`)", "report_finding": "NOT AS STATED", "section": "S4; 3. The minimum at 3.028 rad is smooth, and missed", "strength": "corrected", "audit": "kept by the strength audit; one finding (the smooth minimum hidden in interval 9, 5.44×10⁻⁹ rad before switch 9) surfacing in four of the round's statements (S3-S6); not four independent errors" }, { "claim": "§5: interval 9 is strictly decreasing, its derivative's root count being zero on 257, 1,025 and 4,097 samples.", "kind": "computation", "numbers": [ "257", "1,025", "4,097", "−0.006893 (largest sampled derivative)", "−0.3817 to +0.1732 (exact derivative over interval 9)" ], "quote": "its guards find no root on 257, 1,025 or 4,097 samples (the largest derivative they see is −0.006893), the stationary", "report_finding": "NOT AS STATED", "section": "S5; 3. The minimum at 3.028 rad is smooth, and missed", "strength": "corrected", "audit": "kept by the strength audit; one finding (the smooth minimum hidden in interval 9, 5.44×10⁻⁹ rad before switch 9) surfacing in four of the round's statements (S3-S6); not four independent errors" }, { "claim": "§6: four contact switches form cusp local minima, among them switch 9 at 3.028226470697 rad (min x passing from the right wing's interior to p₃), with one-sided derivatives −0.006893 and +0.173166.", "kind": "computation", "numbers": [ "3.028226470697", "−0.006893", "+0.173166", "φ* = 3.028226465255688362712441…", "s(φ*) = 1.0998102257068941255…", "5.44×10⁻⁹ rad", "6.16×10⁻¹¹" ], "quote": "as a cusp minimum with derivatives −0.006893 and +0.173166. The exact derivative is +0.173166 on both sides, so the switch", "report_finding": "NOT AS STATED", "section": "S3; 3. The minimum at 3.028 rad is smooth, and missed", "strength": "corrected", "audit": "kept by the strength audit; one finding (the smooth minimum hidden in interval 9, 5.44×10⁻⁹ rad before switch 9) surfacing in four of the round's statements (S3-S6); not four independent errors" }, { "claim": "§6: at the twelve switches where a wing's interior support reaches or leaves a corner, the table gives two different one-sided derivatives.", "kind": "computation", "numbers": [ "twelve wing switches", "0.047 to 0.21 apart", "4.6×10⁻⁹", "1.2×10⁻⁸", "1.4×10⁻⁸ rad of α up to u = 0.3", "2.5×10⁻⁸ of β from u = 0.85", "−0.3292, −0.3239, −0.2087, −0.1305, −0.0523" ], "quote": "them), while the round's two stated values are apart by 0.047 to 0.21 (`switches`; S3). One of each pair is the exact", "report_finding": "NOT AS STATED", "section": "S3; 2. At a wing switch the derivative is continuous, and steep", "strength": "corrected" }, { "claim": "§6: one-sided derivatives at the five switches between two corners (π/2, 2π/3, π, 3π/2, 5π/3) and at the two special phases.", "kind": "computation", "numbers": [ "1.11×10⁻⁸", "10⁻⁷" ], "quote": "special phases, the stated one-sided derivatives are the exact ones to 1.11×10⁻⁸ (difference quotients).", "report_finding": "PASS", "section": "S3; 2. At a wing switch the derivative is continuous, and steep", "strength": "numerical only" }, { "claim": "§3/§7: the global exclusion ledger gives each interval's minimum, and §7 ranks the 8 local minima, four of them cusp minima.", "kind": "computation", "numbers": [ "8 local minima", "18 ledger rows", "2×10⁻¹³", "three cusp minima, five smooth" ], "quote": "and the ledger gives interval 9 the switch's value (S3–S6). So there are three cusp minima, not four, and five smooth", "report_finding": "NOT AS STATED", "section": "S6; 3. The minimum at 3.028 rad is smooth, and missed", "strength": "corrected", "audit": "kept by the strength audit; one finding (the smooth minimum hidden in interval 9, 5.44×10⁻⁹ rad before switch 9) surfacing in four of the round's statements (S3-S6); not four independent errors" }, { "claim": "(Report's own argument, bearing on §5's count) On the twelve intervals whose three supports are all corners there is at most one stationary point, a maximum.", "kind": "theorem", "numbers": [ "s″ = −s < 0" ], "quote": "ones. On the twelve intervals whose three supports are all corners, s is a fixed sum of sinusoids with s″ = −s < 0, so", "report_finding": "PASS", "section": "3. The minimum at 3.028 rad is smooth, and missed", "strength": "analytic (proof checked)" }, { "claim": "§7: the 120° cusp is 1.6383×10⁻⁹ above s(3π/2).", "kind": "certificate", "numbers": [ "1.6383×10⁻⁹", "1.630111×10⁻⁹", "64 sub-intervals", "0.05 rad" ], "quote": "box enters once, is at least 1.630111×10⁻⁹ on either side (`s_2pi_3_corners_minus_s270_at_least`) — the 120° margin of", "report_finding": "PASS", "section": "4. The least at 3π/2, certified", "strength": "exact-arithmetic certificate" }, { "claim": "The interface note's final target, s(φ) ≥ s(3π/2) for every φ, which the round does not yet claim (its conclusion that no branch lies below 270° rests on floating-point sampling).", "kind": "certificate", "numbers": [ "[0.998914339066915, 0.998914339102351]", "262,144 steps a wing", "3.7×10⁻¹¹ in x, 1.3×10⁻¹¹ in y", "1,558 intervals", "Lip = 1.3306", "1.855678×10⁻⁹", "4.69×10⁻⁸", "30 digits" ], "quote": "So the candidate's least scale is s(3π/2), reached at 3π/2 alone, in [0.998914339066915, 0.998914339102351]; against", "report_finding": "PASS", "section": "4. The least at 3π/2, certified", "strength": "exact-arithmetic certificate" }, { "claim": "(Report's own measurement) The candidate's least scale exceeds the event five-link's least.", "kind": "certificate", "numbers": [ "0.998903757132508674955129824903", "1.0581934×10⁻⁵" ], "quote": "the event five-link's least, at most 0.998903757132508674955129824903 (RUN-010), the gain is at least 1.0581934×10⁻⁵", "report_finding": "PASS", "section": "4. The least at 3π/2, certified", "strength": "exact-arithmetic certificate" }, { "claim": "Limitation: that the derivative has no hidden root rests on stability across grids, not on a rigorous bound; the conclusion cannot be replaced by a rigorous theorem.", "kind": "reading", "numbers": [], "quote": "ranking and the ledger all miss: the hidden root its own limitation names. The global conclusion is untouched.", "report_finding": "PASS", "section": "Verdict; 3. The minimum at 3.028 rad is smooth, and missed", "strength": "label only" }, { "claim": "The report's tables (§3–§7) and figures, the README's counts and lines, the manifest and the interface note state the data's figures.", "kind": "specification", "numbers": [], "quote": "the report's tables (§3–§7) and figures, the README's counts and lines, the manifest and the interface note state the data's", "report_finding": "PASS", "section": "S8", "strength": "label only" }, { "claim": "§7 and §11: the long figures of s₂₇₀, s₀ and their difference.", "kind": "computation", "numbers": [], "quote": "the report's long figures of s₂₇₀, s₀ and their difference (§7, §11) hold to their digits", "report_finding": "PASS", "section": "S9", "strength": "numerical only" }, { "claim": "README: the package does not constitute a new upper or lower bound for the Moser problem.", "kind": "reading", "numbers": [ "about 0.99891" ], "quote": "Any bound for Moser's problem: the candidate's least scale is about 0.99891, below 1, and the README says the package", "report_finding": "PASS", "section": "What this run does not establish", "strength": "label only" }, { "claim": "The recomputation's own drill (18 planted defects, 4 controls) — instrument, not source.", "kind": "method", "numbers": [ "18", "4", "174/174" ], "quote": "The drill planted 18 defects; each tripped exactly the checks named for it, and", "report_finding": "PASS", "section": "The drill", "strength": "label only" } ], "corrections": [ { "what": "One-sided derivatives at the twelve wing switches (§6)", "source_says": "Two one-sided derivatives at each wing switch, apart by 0.047 to 0.21.", "report_finds": "s is continuously differentiable there (the support coincides with a corner, the others are single corners); this run's one-sided values agree to 4.6×10⁻⁹; one stated value is the derivative to 1.2×10⁻⁸, the other the derivative at a sample inside the interval, where the derivative is steep.", "quote": "differentiable, but the round states two one-sided derivatives there, apart by up to 0.21 — one of them the derivative at" }, { "what": "Switch 9 as a cusp minimum (§6)", "source_says": "Switch 9 at 3.028226470697 rad is a cusp local minimum with derivatives −0.006893 and +0.173166; four switches are cusp minima.", "report_finds": "The exact derivative is +0.173166 on both sides, so the switch is not a minimum; the minimum is a smooth stationary point φ* = 3.028226465255688362712441…, 5.44×10⁻⁹ rad before the switch and 6.16×10⁻¹¹ lower; three cusp minima, five smooth.", "quote": "a sample inside, not at the switch; and the cusp minimum it places at 3.028 rad is not one — the derivative is +0.173 on" }, { "what": "Completeness of the stationary list (§5)", "source_says": "12 smooth stationary points; no hidden sign change found.", "report_finds": "There are 13; the thirteenth is the smooth minimum in interval 9, the hidden root the round's own limitation names.", "quote": "which the stationary list (13 points, not 12), interval 9's classification and root guards at all three densities, the" }, { "what": "Interval 9's classification and root guards (§5)", "source_says": "Interval 9 is strictly decreasing; no root on 257, 1,025 or 4,097 samples.", "report_finds": "The exact derivative runs from −0.3817 to +0.1732 and crosses zero 5.44×10⁻⁹ rad before the switch; the guards' largest sampled value is −0.006893.", "quote": "1.0998102257069 at 3.0282264653). The round's sampling reaches neither: interval 9 is classified \"strictly decreasing\"," }, { "what": "Ranking and ledger for interval 9 (§3, §7)", "source_says": "The ranking lists the minimum as a cusp at the switch; the ledger gives interval 9 the switch's value.", "report_finds": "The interval's minimum is the smooth stationary value 1.0998102257068941255…, 6.16×10⁻¹¹ below the switch's; the global minimum is unaffected (it is 0.10 above the least).", "quote": "the ranking lists the minimum as a cusp at the switch," } ], "open_items": [ "The stationary points as an interval enclosure: on the corner-only intervals their number is", "the round's but not a proof — the certificate of §4 does not need it. An upper bound for the gain. Anything about the", "curve Round 08 stored in double precision, as against Round 09's decimal one. Nor the package's six figures, which were", "certified: a lower bound; its upper bound would need the event five-link's least from below, not done" ], "cross_refs": [ "RUN-002 (the sense of PARTIAL)", "RUN-009 (its golden refinement had found the interval-9 minimum, 1.0998102257069 at 3.0282264653)", "RUN-010 (high-precision candidate and exact support, trapezoid boxes, the event five-link's upper bound 0.998903757132508674955129824903)", "RUN-012 (Round 11's root boxes miss this run's thirteenth stationary point; cites this certificate for the round's conclusions)" ], "unclear": [ "The hidden root in interval 9 is one finding that surfaces in S3 (switch 9's cusp), S4 (list completeness), S5 (classification and guards) and S6 (ranking and ledger); recorded as four corrected items with four correction rows, but a census should not count four independent errors.", "The switch count (17 in the manifest and §6, 18 in the README and §4) is recorded as label only and PASS: the report explains it (the omitted switch at φ = 0) without calling it a defect.", "The certified 120° margin is a lower bound, 1.630111×10⁻⁹, not the stated 1.6383×10⁻⁹ itself; labelled certificate since the stated ordering is what is certified.", "The 13th stationary point and φ* rest on a dense, geometrically refined scan of the exact derivative on the wing intervals (high precision, not an interval enclosure); the corrected items carry that strength.", "The gain lower bound 1.0581934×10⁻⁵ uses RUN-010's interval upper bound for the event five-link." ], "for_paper": "Round 10's phase-contact map reproduces at 60 digits: 18 intervals and signatures, the 17 switches to 5.22×10⁻¹⁴ in phase and 1.04×10⁻¹³ in scale, the 12 listed stationary points, and the counts at every resolution. Not as stated: at the twelve wing switches s is continuously differentiable, yet the round prints two one-sided derivatives up to 0.21 apart, one of them a sample inside the interval; and its cusp minimum at 3.028 rad is not one — the minimum is a smooth stationary point 5.44×10⁻⁹ rad before the switch, 6.16×10⁻¹¹ lower, missed by the stationary list (13 points, not 12), the classification, the guards, the ranking and the ledger (three cusp minima, five smooth). The global conclusion is untouched and is certified here in interval arithmetic for Round 09's decimal curve: the least scale is s(3π/2), reached there alone, in [0.998914339066915, 0.998914339102351], the 120° cusp at least 1.630111×10⁻⁹ above it, and the gain over the event five-link at least 1.0581934×10⁻⁵ (a lower bound only)." }, { "run": "RUN-012", "report_file": "RUN-012-ROUND11-BOX-CERTIFICATE.md", "source_document": "Round 11 (v0.11): Moser_Skew_Lab_v0.11.zip (SHA-256 pinned in corpus.py) — README, MANIFEST, reports/ROUND_11_REPORT.md, ROUND_11_CERTIFICATE_AUDIT.md, ROUND_12_BALL_ARITHMETIC_HANDOFF.md, src/METHOD.md, src/special_branch_verify.py (read, not run), and its data (round11_summary.json, exact_contact_boundaries.json, special_branch_interval_certificate.json, phase_derivative_box_certificate.json, stationary_root_boxes.json, boundary_neighborhood_audit.json and the two CSVs), with Round 10's map, Round 09's decimal constants and event branches and Round 05's event root", "verdict_line": "**Verdict: PARTIAL** — in RUN-002's sense: what the round computes mostly holds, and two things it states do not.", "items": [ { "claim": "§2: Round 10's switches are exact angles (π/2 − α, π/2 − β, β − π/3, α − π/3, π/2, π − α, π − β, 2π/3, π/2 + β, π/2 + α, π, 5π/3 − α, 5π/3 − β, π + β, π + α, 3π/2, 5π/3), so the phase partition rests on finitely many exact angle events, not bisected decimals.", "kind": "theorem", "numbers": [ "19 boundary formulas", "60 digits", "6.0×10⁻⁶⁰", "5.240×10⁻¹⁴", "10⁻¹⁵", "10⁻⁴⁰" ], "quote": "Every boundary formula, evaluated at 60 digits, satisfies its switch's own condition — two corners level under the", "report_finding": "PASS", "section": "U1; 1. What holds", "strength": "numerical only" }, { "claim": "§3: s₂₇₀ ∈ [0.9989143390757518, 0.9989143390935133], by a composite midpoint rule on 262,144 parts with a global second-derivative bound and a float padding.", "kind": "certificate", "numbers": [ "[0.9989143390757518, 0.9989143390935133]", "262,144 parts", "8.88×10⁻¹²" ], "quote": "special enclosures contain the exact values: s₂₇₀ lies 8.88×10⁻¹² inside each end of its interval, and s₁₂₀ − s₂₇₀ =", "report_finding": "PASS", "section": "U2; 1. What holds", "strength": "numerical only" }, { "claim": "§3: s₁₂₀ − s₂₇₀ ∈ [1.634956007484312×10⁻⁹, 1.641712345829326×10⁻⁹], the series' first explicit error box for the 10⁻⁹-level 120°/270° competition.", "kind": "certificate", "numbers": [ "[1.634956007484312×10⁻⁹, 1.641712345829326×10⁻⁹]", "1.63833406919315×10⁻⁹", "1.634956×10⁻⁹ vs RUN-011's 1.630111×10⁻⁹" ], "quote": "1.63833406919315×10⁻⁹ inside [1.634956×10⁻⁹, 1.641712×10⁻⁹] (`s270_minus_interval_ends`, `s120_minus_s270_exact`); the", "report_finding": "PASS", "section": "U2; 1. What holds", "strength": "numerical only" }, { "claim": "The enclosures' error bounds are the composite-midpoint bound M₂/(24N²) with the global second-derivative bound M₂ = |θ′|² + |θ″|, plus a float padding.", "kind": "method", "numbers": [ "M₂/(24N²) + 2×10⁻¹⁵", "M_q·M₂/(24N²) + 2×10⁻¹⁵", "M₂ = 15.128024568137338" ], "quote": "bounds are M₂/(24N²) + 2×10⁻¹⁵ for the coordinates and M_q·M₂/(24N²) + 2×10⁻¹⁵ for the difference, with M₂ =", "report_finding": "PASS", "section": "U2, U9; 1. What holds", "strength": "numerical only" }, { "claim": "§4: the gain interval is the s₂₇₀ enclosure less the event five-link's first branch B1; its lower end is positive.", "kind": "computation", "numbers": [ "84 decimals" ], "quote": "bounds, as shortest decimals, less Round 09's B1, cut to 84 decimals (U3). Each of the 579 derivative boxes contains", "report_finding": "PASS", "section": "U3; 1. What holds", "strength": "numerical only" }, { "claim": "§5: 579 adaptive derivative boxes, none unresolved, cover the 18 intervals' interiors, so that apart from the root boxes and boundary boxes every derivative sign is excluded by boxes.", "kind": "certificate", "numbers": [ "579 boxes", "13 points a box", "8.9×10⁻¹²", "guards 2×10⁻¹⁰ or 2×10⁻⁸" ], "quote": "the exact derivative at nine equally spaced points and four near its ends, by at least 8.9×10⁻¹² (`boxes_least_slack`),", "report_finding": "PASS", "section": "U4; 1. What holds", "strength": "numerical only" }, { "claim": "§6: the smooth stationary points lie in root boxes, each holding one stationary point, with second-derivative boxes that exclude zero and scale intervals that contain s.", "kind": "certificate", "numbers": [ "12 root boxes", "1.4×10⁻⁸", "60 digits" ], "quote": "Each root box holds one stationary point of its kind; its second-derivative interval contains s″ at four points inside", "report_finding": "PASS", "section": "U5; 1. What holds", "strength": "numerical only" }, { "claim": "§6 counts 12 smooth stationary points in all, §11 speaks of 12 unique stationary-point root boxes, and the summary gives 4 smooth minima.", "kind": "computation", "numbers": [ "12", "13", "5.44×10⁻⁹ rad before π/2 + α", "4 smooth minima", "5" ], "quote": "The 12 root boxes are 12 of the 13 stationary points RUN-011 found; the thirteenth, the smooth minimum 5.44×10⁻⁹ rad", "report_finding": "NOT AS STATED", "section": "U5; 3. Thirteen stationary points, not twelve", "strength": "corrected" }, { "claim": "§7: the neighbourhood of π/2 + α, left unresolved by its derivative intervals, is excluded by the scale's lower bound (a smooth support point leaving the wing at its end, with a wider derivative box).", "kind": "computation", "numbers": [ "[−0.06691, 0.03742]", "+0.17317", "7.39×10⁻²" ], "quote": "above s₂₇₀. So the exclusion holds, and the report's account of the unresolved switch — a smooth support point leaving", "report_finding": "PASS", "section": "3. Thirteen stationary points, not twelve", "strength": "numerical only" }, { "claim": "§7: each switch neighbourhood (radius 2×10⁻⁶, or 2×10⁻⁸ where both supports are corners) has scale intervals containing s, and its classification is the derivative's sign pattern.", "kind": "computation", "numbers": [ "2×10⁻⁶", "2×10⁻⁸" ], "quote": "contain s on either side, the switch included, and every classification is the sign pattern of the derivative over the", "report_finding": "PASS", "section": "U6; 1. What holds", "strength": "numerical only" }, { "claim": "§7: each neighbourhood's one-sided derivative intervals enclose the derivative on [φ, φ ± r] on each side of the switch.", "kind": "certificate", "numbers": [ "[φ ± 2×10⁻¹⁰, φ ± r]", "14 of the 17 switches", "0.022 to 0.157 (12 wing switches)", "1.9×10⁻¹⁰ to 2.1×10⁻¹⁰ (π/2, π, one side of 5π/3 − α)", "padding 2.2×10⁻⁶ to 3.0×10⁻⁶", "[−0.19350, −0.08917] at π/2 − α", "−0.193496 vs −0.329242", "0.136, 0.157, 0.078, 0.038, 0.044, 0.022", "padding 0.9×10⁻¹¹ to 2.0×10⁻¹¹" ], "quote": "[φ ± 2×10⁻¹⁰, φ ± r], not [φ, φ ± r]. The derivative boxes' guards are 2×10⁻¹⁰ on the corner intervals and 2×10⁻⁸ on", "report_finding": "NOT AS STATED", "section": "U6; 2. The 2×10⁻¹⁰ rad beside each switch", "strength": "refuted" }, { "claim": "Audit note: every boundary passes by the derivative classification, the neighbourhood's scale lower bound or a special-branch certificate; the neighbourhoods of π/2 − α, π/2 − β and β − π/3 pass by derivative classification.", "kind": "certificate", "numbers": [ "2.69×10⁻²", "2.30×10⁻²", "7.7×10⁻⁴" ], "quote": "The conclusion is true there — the derivative keeps its sign — but the certificate does not show it.", "report_finding": "NOT AS STATED", "section": "2. The 2×10⁻¹⁰ rad beside each switch", "strength": "open" }, { "claim": "§7: the other 11 neighbourhoods whose derivative intervals fail to enclose are also excluded by the scale's lower bound.", "kind": "computation", "numbers": [ "11" ], "quote": "classifications pass — and at the 11 failing switches where the scale bound also excludes, nothing rests on them. At", "report_finding": "PASS", "section": "2. The 2×10⁻¹⁰ rad beside each switch", "strength": "numerical only" }, { "claim": "The summary counts 14 neighbourhoods as passing by the scale's lower bound.", "kind": "computation", "numbers": [ "14", "13", "−1.8×10⁻¹¹ (scale margin at 3π/2)" ], "quote": "so of the 14 the summary counts as passing by scale, 13 pass by their margins.)", "report_finding": "NOT AS STATED", "section": "U6; 2. The 2×10⁻¹⁰ rad beside each switch", "strength": "corrected" }, { "claim": "Round 11 calls its result a semi-validated computational certificate, since NumPy's basic operations are not directed-rounding throughout, and its handoff asks the next round to recompute everything in ball arithmetic.", "kind": "reading", "numbers": [], "quote": "round's own handoff asks for these objects to be recomputed in ball arithmetic; this is where it matters. (At 3π/2 the", "report_finding": "PASS", "section": "2. The 2×10⁻¹⁰ rad beside each switch", "strength": "label only" }, { "claim": "Round 11's conclusions: the least at 3π/2, s₁₂₀ above s₂₇₀, and the candidate above the event five-link.", "kind": "theorem", "numbers": [ "s₁₂₀ − s₂₇₀ ≥ 1.630111×10⁻⁹", "gain ≥ 1.0581934×10⁻⁵" ], "quote": "the round's qualitative conclusions (the least at 3π/2; s₁₂₀ > s₂₇₀; the candidate above the event five-link) | certified by RUN-011, with its own bounds", "report_finding": "PASS", "section": "Verdict; At what strength each result stands", "strength": "label only", "audit": "relabelled from 'exact-arithmetic certificate' by the strength audit: the conclusions are certified by RUN-011 with its own bounds and counted there; RUN-012's replay does not certify them" }, { "claim": "The summary, the CSVs and the global candidate hold the data files' figures.", "kind": "specification", "numbers": [], "quote": "the summary, the CSVs and the global candidate hold the data files' figures", "report_finding": "PASS", "section": "U7", "strength": "label only" }, { "claim": "The report's tables (§2, §5–§7) and numbers (§3, §4), the README, the audit note, the handoff and the manifest state the data's figures.", "kind": "specification", "numbers": [], "quote": "the report's tables (§2, §5–§7) and numbers (§3, §4), the README, the audit note, the handoff and the manifest", "report_finding": "PASS", "section": "U8", "strength": "label only" }, { "claim": "The verifier's constants are Round 09's decimals and its bounds are |θ′| and |θ′|² + |θ″|, as the method note states.", "kind": "reading", "numbers": [], "quote": "the verifier's constants are Round 09's decimals; its bounds are |θ′| and |θ′|² + |θ″|; the method note states them", "report_finding": "PASS", "section": "U9", "strength": "label only" }, { "claim": "README: the package does not constitute a new upper or lower bound for the Moser problem.", "kind": "reading", "numbers": [], "quote": "Any bound for Moser's problem; the README says the package", "report_finding": "PASS", "section": "What this run does not establish", "strength": "label only" }, { "claim": "The recomputation's own drill (17 planted defects, 4 controls) — instrument, not source.", "kind": "method", "numbers": [ "17", "4", "168/168" ], "quote": "The drill planted 17 defects; each tripped exactly the checks named for", "report_finding": "PASS", "section": "The drill", "strength": "label only" } ], "corrections": [ { "what": "Count of smooth stationary points (§6, §11, summary)", "source_says": "12 smooth stationary points in all, 12 unique root boxes; 4 smooth minima.", "report_finds": "There are 13 (RUN-011); the thirteenth, a smooth minimum 5.44×10⁻⁹ rad before π/2 + α, lies inside that switch's neighbourhood, which is excluded by scale; the smooth minima are 5. The scheme survives there; the count does not.", "quote": "points and §11's 12 \"個唯一駐點根盒\" (unique stationary-point root boxes), as the full count, are not, and the summary's 4" }, { "what": "Switch neighbourhoods' one-sided derivative intervals (§7) and the three exclusions resting on them", "source_says": "Each neighbourhood's derivative intervals enclose the derivative on each side of the switch; switches π/2 − α, π/2 − β, β − π/3 are excluded by their derivative signs.", "report_finds": "The intervals cover [φ ± 2×10⁻¹⁰, φ ± r], not [φ, φ ± r]; the 2×10⁻¹⁰ rad beside each switch is enclosed by nothing, and at 14 of 17 switches the derivative there lies outside (by 0.022 to 0.157 at the 12 wing switches, 1.9×10⁻¹⁰ to 2.1×10⁻¹⁰ at π/2, π and one side of 5π/3 − α). At switches 1 to 3 the scale bounds do not exclude (2.69×10⁻², 2.30×10⁻², 7.7×10⁻⁴ below s₂₇₀), so the exclusion rests on these intervals: the sign is right but the certificate does not show it.", "quote": "neighbourhood's one-sided derivative intervals cover [φ ± 2×10⁻¹⁰, φ ± r], not [φ, φ ± r] — the 2×10⁻¹⁰ rad beside" }, { "what": "Neighbourhoods passing by scale (summary)", "source_says": "14 neighbourhoods pass by the scale's lower bound.", "report_finds": "At 3π/2 the scale margin is −1.8×10⁻¹¹ (its lower bound is s₂₇₀'s own) and that neighbourhood is excluded by the special certificate; 13 pass by their margins.", "quote": "scale margin is −1.8×10⁻¹¹, its lower bound being s₂₇₀'s own; that neighbourhood is excluded by the special certificate," } ], "open_items": [ "That the 579 boxes and 12 root boxes are valid enclosures: they contain the derivative wherever this run looked, which is", "evidence, not a proof. A repair of the three neighbourhood exclusions inside Round 11's scheme: RUN-011's certificate", "their own rounding model unverified, as the round says", "These are sampled checks — the derivative at points of each box — not a re-verification of the", "The conclusion is true there — the derivative keeps its sign — but the certificate does not show it.", "Nor the package's four figures, which were not examined." ], "cross_refs": [ "RUN-002 (the sense of PARTIAL)", "RUN-009 (the exact-support wing in double precision used for sampling)", "RUN-010 (the high-precision candidate)", "RUN-011 (13 stationary points; its certificate covers the round's conclusions and the three neighbourhood exclusions by another route; Round 11's s₂₇₀ interval lies inside RUN-011's)" ], "unclear": [ "The round's qualitative conclusions are labelled exact-arithmetic certificate because the report says they are certified by RUN-011; the certificate was computed there, not in this run, so a census should not count it twice.", "The three exclusions at π/2 − α, π/2 − β and β − π/3 are labelled open (not refuted): the derivative keeps its sign there (60 digits) and RUN-011 covers them by another route; the defect itself is recorded as refuted in the derivative-interval item, with one correction row covering both.", "Round 11's s₂₇₀ and s₁₂₀ − s₂₇₀ enclosures are labelled numerical only: the report checks that they contain the exact values at high precision, but the enclosures are the round's float-padded midpoint rule, not replayed in interval arithmetic.", "The gain interval is an exact decimal subtraction of the round's figures; labelled numerical only because the s₂₇₀ enclosure's own rounding model is unverified.", "The error-bound formula and M₂ are recomputed (U2, U9); labelled numerical only, as the report does not set out the derivation of M₂ as a supremum." ], "for_paper": "Round 11's exact switch formulas hold (each satisfies its switch condition at 60 digits, to 6.0×10⁻⁶⁰), its s₂₇₀ and s₁₂₀ − s₂₇₀ enclosures contain the exact values, and its 579 derivative boxes and 12 root boxes contain the derivative, s″ and s wherever sampled — evidence, not an interval re-verification. Not as stated: every switch neighbourhood's one-sided derivative intervals cover [φ ± 2×10⁻¹⁰, φ ± r] rather than [φ, φ ± r], so the 2×10⁻¹⁰ rad beside each switch is enclosed by nothing, and at 14 of 17 switches the derivative there lies outside the intervals (by 0.022 to 0.157 at the wing switches); the exclusions at π/2 − α, π/2 − β and β − π/3 rest on these intervals, so the certificate does not show them, although the derivative keeps its sign. The count of 12 smooth stationary points is 13 (the thirteenth inside the π/2 + α neighbourhood, excluded by scale), and 13, not 14, neighbourhoods pass by their scale margins. The round's qualitative conclusions hold, as RUN-011 certified with bounds of its own." }, { "run": "RUN-013", "report_file": "RUN-013-ROUND12-IV-REPLAY.md", "source_document": "Round 12 (v0.12), Moser_Skew_Lab_v0.12.zip: README, MANIFEST, reports/ROUND_12_REPORT.md, ROUND_12_IV_REPLAY_AUDIT.md, ROUND_13_SMOOTH_KKT_HANDOFF.md, the two replay scripts (read, not run) and its data (round12_summary.json, special_branch_iv_certificate.json, exact_boundary_balls.json, root_interval_newton_replay.json, boundary_iv_replay.json), with Round 11's boundaries, neighbourhoods and root boxes, Round 10's map, Round 09's constants and Round 05's event root", "verdict_line": "**Verdict: PARTIAL** — in RUN-002's sense: what the round computes holds, and four things it states do not.", "items": [ { "claim": "§2: in mpmath's iv (directed interval arithmetic) s₂₇₀ is enclosed in [0.99891433894289…, 0.99891433922637…].", "kind": "certificate", "numbers": [ "[0.99891433894289…, 0.99891433922637…]", "1.4174×10⁻¹⁰" ], "quote": "The special intervals contain the exact values: s₂₇₀ lies 1.4174×10⁻¹⁰ inside each end of its interval, and s₁₂₀ − s₂₇₀", "report_finding": "PASS", "section": "W1; Findings 1 — What holds", "strength": "exact-arithmetic certificate" }, { "claim": "§2: s₁₂₀ − s₂₇₀ is enclosed in [1.5846×10⁻⁹, 1.6920×10⁻⁹], so its lower bound is strictly positive (\"下界仍嚴格為正\").", "kind": "certificate", "numbers": [ "[1.5846×10⁻⁹, 1.6920×10⁻⁹]", "5.3711×10⁻¹¹" ], "quote": "written anew and not running the round's script — gives the stated intervals to 3×10⁻⁸¹ at", "report_finding": "PASS", "section": "W1; Findings 1 — What holds", "strength": "exact-arithmetic certificate", "audit": "quotation replaced after the independent reading: the replay, not the containment, carries the label" }, { "claim": "§2: the two special intervals are the output of the stated method — midpoint sums over the wing in interval arithmetic with the analytic midpoint error.", "kind": "method", "numbers": [ "65,536 parts", "75 digits", "M/(24N²)", "3×10⁻⁸¹" ], "quote": "either end (`replay_minus_stated`): the round computed what it says.", "report_finding": "PASS", "section": "W1; Findings 1 — What holds", "strength": "exact-arithmetic certificate" }, { "claim": "§2: the candidate's margin over the event control's first branch lies in [1.0581810×10⁻⁵, 1.0582094×10⁻⁵], keeping a strictly positive lower bound (\"同樣保持嚴格正下界\").", "kind": "computation", "numbers": [ "[1.0581810×10⁻⁵, 1.0582094×10⁻⁵]" ], "quote": "interval holds Round 09's B1, and the gain interval is the s₂₇₀ interval less it (W1). This run's own replay of the", "report_finding": "PASS", "section": "W1; Findings 1 — What holds", "strength": "numerical only", "audit": "kept 'numerical only' after the independent reading: the gain interval is the replayed s₂₇₀ interval less the event interval, whose containment of B1 was checked at high precision, not replayed" }, { "claim": "§3: the 19 boundary formulas are recomputed as balls, each enclosing its formula's value.", "kind": "computation", "numbers": [ "19", "2.2×10⁻⁷⁰", "90 digits" ], "quote": "The 19 balls each hold their formula's exact value,", "report_finding": "PASS", "section": "W2; Findings 1 — What holds", "strength": "numerical only" }, { "claim": "§3 (repeated by the audit note's \"解析邊界：19/19\" and the summary's and manifest's statuses): all 19 balls contain Round 11's saved floats (\"全部包含第 11 輪保存的浮點值\").", "kind": "reading", "numbers": [ "19/19 stated", "1 of 19", "3.8×10⁻¹⁶", "eight of them not the double nearest the angle" ], "quote": "nearest the angle. So no ball but the one at 0 can hold its double, and the data's flags say exactly that: one ball of 19", "report_finding": "NOT AS STATED", "section": "W6; Findings 2 — The boundary balls: one in 19", "strength": "corrected" }, { "claim": "Data: the balls' containment flags and the summary's all_contain flag record which balls hold Round 11's floats.", "kind": "computation", "numbers": [ "all_contain: false", "1 of 19" ], "quote": "holds Round 11's float, and the summary's `all_contain` is false (`balls_containing_round11_float`). The report's §3", "report_finding": "PASS", "section": "W2; Findings 2", "strength": "numerical only" }, { "claim": "§4: interval Newton on Round 11's 12 root boxes: 12/12 satisfy the interval Newton inclusion condition, each box holding one stationary point of its kind.", "kind": "computation", "numbers": [ "12/12", "60 digits" ], "quote": "Each root box holds one stationary point of its kind, the Newton image lies inside it and", "report_finding": "PASS", "section": "W3; Findings 1 — What holds", "strength": "numerical only" }, { "claim": "§4: the end-point derivative signs all fit the minimum/maximum classification; the second-derivative intervals exclude zero and the scale intervals hold s at the root.", "kind": "computation", "numbers": [ "four points per box", "60 digits, to 10⁻¹⁰" ], "quote": "holds the root, and the second-derivative test (at four points of the box), the end signs and the scale interval hold", "report_finding": "PASS", "section": "W3; Findings 1 — What holds", "strength": "numerical only" }, { "claim": "§1: the root replay runs in mpmath's directed interval arithmetic.", "kind": "method", "numbers": [ "10⁻⁵⁵", "80 digits" ], "quote": "and every Newton inclusion this run checks holds. The result is high-precision evidence, not the directed interval", "report_finding": "NOT AS STATED", "section": "W7; Findings 4 — The root replay's inputs are estimates", "strength": "corrected" }, { "claim": "§7: all smooth stationary points are reproduced by another interval arithmetic (\"所有光滑駐點，均被另一套區間運算重現\").", "kind": "reading", "numbers": [ "12 of 13", "5.44×10⁻⁹ rad before π/2 + α" ], "quote": "So \"所有光滑駐點，均被另一套區間運算重現\" is not so; 12 of 13", "report_finding": "NOT AS STATED", "section": "W3; Findings 3 — Twelve stationary points of thirteen", "strength": "corrected" }, { "claim": "§5: of the 17 switches' sides, 5 are resolved directly by the interval replay.", "kind": "computation", "numbers": [ "17", "5", "π/2, 2π/3, π, 3π/2 and 5π/3" ], "quote": "The five direct passes among the switches are π/2, 2π/3, π, 3π/2 and 5π/3, the five between two corners (W4).", "report_finding": "PASS", "section": "W4; Findings 1 — What holds", "strength": "label only" }, { "claim": "§5: at the switches not resolved directly, no counterexample was found (\"不是發現反例\").", "kind": "computation", "numbers": [ "eleven of the twelve", "60 digits" ], "quote": "At eleven of the twelve switches the round could not resolve directly, there is indeed no counterexample: the derivative", "report_finding": "PASS", "section": "W4; Findings 5 — \"Dependency inflation, not a counterexample\"", "strength": "numerical only" }, { "claim": "§5: what the fallback library cannot resolve at the other twelve switches is \"dependency inflation\".", "kind": "reading", "numbers": [ "2×10⁻¹⁰ rad", "0.022 to 0.157", "π/2 + α", "7.39×10⁻²" ], "quote": "But it is not the whole reason Round 11's narrow boxes could not be reproduced", "report_finding": "NOT AS STATED", "section": "Findings 5 — \"Dependency inflation, not a counterexample\"", "strength": "corrected" }, { "claim": "§5 (implicit): the round's own interval extensions through the inverse tanh near atanh(±1) are too wide to fix the signs.", "kind": "computation", "numbers": [], "quote": "may well be why the round's own intervals could not fix the signs; this run did not compute", "report_finding": "UNMEASURED", "section": "Findings 5 — \"Dependency inflation, not a counterexample\"", "strength": "not measured" }, { "claim": "§5: at the unresolved switches Round 11's analytic error bounds and scale exclusions stand (\"第 11 輪的解析誤差界與尺度排除仍保留\").", "kind": "reading", "numbers": [ "π/2 − α, π/2 − β, β − π/3" ], "quote": "twelve — π/2 − α, π/2 − β, β − π/3 — Round 11 has no scale exclusion, so \"第 11 輪的解析誤差界與尺度排除仍保留\" leaves those three", "report_finding": "NOT AS STATED", "section": "Findings 5 — \"Dependency inflation, not a counterexample\"", "strength": "refuted" }, { "claim": "Conclusion: the most important low-value competition (s₁₂₀ above s₂₇₀, the candidate above the event five-link, the least at 3π/2) is reproduced — support, not a full Arb certificate.", "kind": "reading", "numbers": [], "quote": "conclusions its figures support — s₁₂₀ above s₂₇₀, the candidate above the event five-link, the least at 3π/2 — hold, as", "report_finding": "PASS", "section": "Verdict", "strength": "label only", "audit": "relabelled from 'cited theorem' by the strength audit: the conclusions hold by RUN-011's interval certificate, an earlier run of this recomputation, not a published theorem; counted in RUN-011" }, { "claim": "The round does not finish the 579 leaf boxes, and finds no python-flint, Arb or Sage in its environment.", "kind": "reading", "numbers": [ "579" ], "quote": "It does not finish the 579 leaf boxes", "report_finding": "PASS", "section": "W5; What Round 12 claims", "strength": "open" }, { "claim": "The summary holds the data files' records and counts.", "kind": "specification", "numbers": [], "quote": "the summary holds the data files' records and counts; the leaf replay is not completed; the environment", "report_finding": "PASS", "section": "W5", "strength": "label only" }, { "claim": "The two shipped replay scripts use Round 09's decimals as constants and the methods the report states.", "kind": "specification", "numbers": [], "quote": "the replay scripts' constants are Round 09's decimals and their methods are those stated", "report_finding": "PASS", "section": "W7", "strength": "label only" }, { "claim": "The package's replay scripts produce its data files.", "kind": "specification", "numbers": [ "60-digit ends printed vs 80-digit ends in the data" ], "quote": "scripts do contain; the rest of the data comes from code the package does not ship.", "report_finding": "PARTIAL", "section": "Findings 6 — The scripts do not make all the data", "strength": "label only" }, { "claim": "The report's intervals (§2), tables (§4, §5), counts, the README, the audit note, the manifest and the handoff restate the data.", "kind": "reading", "numbers": [], "quote": "the report's intervals (§2) and tables (§4, §5), its counts, statuses, the README, the audit note, the manifest and the handoff", "report_finding": "PASS", "section": "W6", "strength": "label only" } ], "corrections": [ { "what": "§3 boundary balls and Round 11's saved floats", "source_says": "19/19 balls contain Round 11's saved floats; the audit note repeats 19/19 and the statuses call the balls passed.", "report_finds": "Each ball encloses the exact angle (about 10⁻⁷⁰ wide), but Round 11's phases are doubles up to 3.8×10⁻¹⁶ from the exact angles, so only the ball at 0 holds its double (1 of 19); the summary's own all_contain is false.", "quote": "nevertheless boxes 19/19 and says \"全部包含第 11 輪保存的浮點值\", the audit note repeats \"解析邊界：19/19\", and the summary's and" }, { "what": "§1 arithmetic of the root replay", "source_says": "The interval Newton replay runs in directed interval arithmetic (mpmath iv).", "report_finds": "Wing positions are boxed by the gap between two quadratures plus 10⁻⁵⁵ (an estimate, not a bound); the inverse tangent angle, the bounds on θ′ and the phase constant are computed in point arithmetic; A and B enter as decimal strings. The Newton inclusions hold at 60 digits, so the result is high-precision evidence.", "quote": "centre — widened by their difference and 10⁻⁵⁵, an estimate of the error, not a bound on it (W7 reads the line); it" }, { "what": "§7 all smooth stationary points reproduced", "source_says": "All smooth stationary points are reproduced by another interval arithmetic.", "report_finds": "12 of 13: the thirteenth, the smooth minimum 5.44×10⁻⁹ rad before π/2 + α, lies in no root box (it is inside the neighbourhood Round 11 excludes by scale).", "quote": "The 12 root boxes are Round 11's, and the thirteenth stationary point, the smooth minimum 5.44×10⁻⁹ rad before π/2 + α" }, { "what": "\"Dependency inflation, not a counterexample\" at the twelve unresolved switches", "source_says": "The unresolved sides are dependency inflation of the fallback library.", "report_finds": "Round 11's one-sided intervals leave out the 2×10⁻¹⁰ rad beside each switch, where the derivative moves by 0.022 to 0.157, so they are not enclosures and even an exact enclosure could not reproduce them; at π/2 + α the derivative changes sign on the left, at the thirteenth stationary point.", "quote": "every one of these twelve switches Round 11's one-sided interval leaves out the 2×10⁻¹⁰ rad beside the switch, where the" }, { "what": "Round 11's scale exclusions at three switches", "source_says": "Round 11's analytic error bounds and scale exclusions stand.", "report_finds": "At π/2 − α, π/2 − β and β − π/3 Round 11 has no scale exclusion, so those three rest on intervals that do not enclose; the conclusion (least at 3π/2) holds only by RUN-011's separate certificate.", "quote": "resting on intervals that do not enclose. The conclusion there is true — RUN-011 certified the least at 3π/2 by another" } ], "open_items": [ "That the round's root boxes or Round 11's boxes are certified: their inputs rest on estimates, and this run's checks of", "How the data the scripts do not make were made.", "π/2 − β and β − π/3 inside either round's scheme: RUN-011's certificate covers them by another route.", "may well be why the round's own intervals could not fix the signs; this run did not compute", "It does not finish the 579 leaf boxes", "Moser's problem. Nor the package's three figures, which were not examined." ], "cross_refs": [ "RUN-002 (sense of the PARTIAL verdict)", "RUN-010 (high-precision candidate, 40 to 200 digits, used for exact values, stationary points and derivatives beside switches)", "RUN-011 (13 stationary points; certified the least at 3π/2 and the conclusions by its own bounds)", "RUN-012 (thirteenth stationary point; Round 11's one-sided intervals miss the 2×10⁻¹⁰ rad beside each switch)" ], "unclear": [ "The 19/19 ball claim is labelled 'corrected', not 'refuted': the balls do pass as enclosures of the exact angles, and only §3's own criterion (holding Round 11's doubles) fails, 1 of 19.", "The event-branch/gain interval is labelled 'numerical only': this run's own interval replay covers the s₂₇₀ and difference intervals; the event interval's containment of Round 09's B1 is checked against high-precision values, not replayed.", "The conclusions (least at 3π/2 etc.) rest on RUN-011's certificate, an earlier run of this programme, not on a published theorem; 'cited theorem' / 'PASS by cited theorem' is the closest fit, chosen so this run is not credited with a certificate it did not compute.", "The five direct passes are labelled 'label only': the report identifies them as the five corner-to-corner switches but does not recompute the round's interval passes.", "The three switches without scale exclusion are labelled 'refuted' (within either round's scheme nothing excludes them); the conclusion survives only by RUN-011's other route.", "The 'documents restate the data' item is inferred PASS: W6 fails, and the report names only the boundary-ball account as the failure.", "Whether the source claims its scripts make the data is not stated in the report; the scripts item records the report's provenance finding as PARTIAL / label only." ], "for_paper": "Round 12's mpmath-iv enclosures of s₂₇₀ and of s₁₂₀ − s₂₇₀ are reproduced by this run's own interval replay (65,536 parts at 75 digits, analytic midpoint error) to 3×10⁻⁸¹ and contain the exact values with margins 1.4174×10⁻¹⁰ and 5.3711×10⁻¹¹, so s₁₂₀ > s₂₇₀ stands as an interval result. The 19 boundary balls enclose the exact angles, but only one holds Round 11's saved double, not 19 as §3 says. The 12 root boxes pass their Newton, second-derivative, end-sign and scale tests at 60 digits, but the round's replay rests on estimated wing positions, not directed interval arithmetic, and the thirteenth stationary point lies in no box (12 of 13). 'Dependency inflation' does not account for the twelve unresolved switches: Round 11's one-sided intervals are not enclosures, the derivative changes sign beside π/2 + α, and three switches have no scale exclusion. The least at 3π/2 holds only by RUN-011's separate certificate; no Moser bound is claimed." }, { "run": "RUN-014", "report_file": "RUN-014-ROUND13-SMOOTH-KKT.md", "source_document": "Round 13 (v0.13), Moser_Skew_Lab_v0.13.zip: README, MANIFEST, reports/ROUND_13_REPORT.md, ROUND_13_KKT_AUDIT.md, ROUND_14_CURVATURE_DISTRIBUTION_HANDOFF.md, the replay script (read, not run) and its data (round13_summary.json, smooth_kkt_candidate.json, full_phase_minima.json, branch_gradient_matrix.csv, basin_restarts.csv), with Round 09's constants and Round 05's event root", "verdict_line": "**Verdict: PARTIAL** — in RUN-002's sense: what the round computes holds, and four things it states do not.", "items": [ { "claim": "Round 08's reference (the base of the gain) is Round 09's decimal curve with its scale s(3π/2); the candidate's centre length, α, degrees and parameter shifts are its own arithmetic.", "kind": "specification", "numbers": [ "200 digits" ], "quote": "Round 08's reference is Round 09's decimal curve, with its s(3π/2) to", "report_finding": "PASS", "section": "X1; Findings 1 — What holds", "strength": "numerical only" }, { "claim": "§2: at the stated curve the four branches (B1, B2, the 120° and 270° cusps) take the stated values, and the two low phases are smooth minima with supports L|p0|p2.", "kind": "computation", "numbers": [ "0.998914343297489195 (B1)", "0.998914343297485032 (B2)", "0.998914343297542111 (120°)", "0.998914343297542440 (270°)", "7×10⁻¹⁷", "5×10⁻¹⁸" ], "quote": "stated values are within 7×10⁻¹⁷ of them, the low phases within 5×10⁻¹⁸ of the two smooth minima, both with the supports", "report_finding": "PASS", "section": "X2; Findings 1 — What holds", "strength": "numerical only" }, { "claim": "§2: with ε = 0.036823344353076 the common scale is 0.998914343297485, 4.2128524×10⁻⁹ above Round 08's (with the stated spread and distance to one).", "kind": "computation", "numbers": [ "0.998914343297485", "4.2128523149×10⁻⁹", "4.2128524003×10⁻⁹", "9×10⁻¹⁷", "spread 5.741×10⁻¹⁴" ], "quote": "B2, and its excess over Round 08's s(3π/2) is 4.2128523149×10⁻⁹ (`improvement_exact`), which the stated", "report_finding": "PASS", "section": "X2; Findings 1 — What holds", "strength": "numerical only" }, { "claim": "The five-parameter curve's least exceeds Round 08's, by about 4.21×10⁻⁹ (the round gives no interval certificate).", "kind": "certificate", "numbers": [ "4.0451×10⁻⁹", "4.2486×10⁻⁹", "s₁₃ − 1.5×10⁻¹⁰", "262,144 steps", "1,587 cover intervals", "0.998914343182500", "s(3π/2) ≤ 0.998914343315417", "Round 08 in [0.998914339066915, 0.998914339102351]" ], "quote": "exceeds Round 08's by at least 4.0451×10⁻⁹ and at most 4.2486×10⁻⁹ — the improvement is real, and the stated 4.2129×10⁻⁹", "report_finding": "PASS", "section": "Findings 8 — The certificate: the gain over Round 08", "strength": "exact-arithmetic certificate" }, { "claim": "§4: the 4×5 matrix of the branches' gradients in the five parameters.", "kind": "computation", "numbers": [ "1.6×10⁻⁷", "central differences, step 10⁻²⁰" ], "quote": "the gradients hold to 1.6×10⁻⁷, and the residual is the stated pressures times them.", "report_finding": "PASS", "section": "X3", "strength": "numerical only" }, { "claim": "§4: the weighted stationarity residual, of norm 4.025×10⁻⁸, is the stated pressures times the gradients; the pressures sum to one.", "kind": "computation", "numbers": [ "4.025×10⁻⁸" ], "quote": "1.6×10⁻⁷ (X3), and the stated residual is the stated pressures times the stated gradients, to the rounding (X4). The", "report_finding": "PASS", "section": "X4", "strength": "numerical only" }, { "claim": "§4: the residual is limited by finite differences and curve-integration interpolation and \"cannot be taken as an interval KKT proof\".", "kind": "reading", "numbers": [ "4.025×10⁻⁸", "2.9×10⁻⁸" ], "quote": "The stated point does not solve the twelve equations exactly: with exact gradients its pressures leave a residual of up", "report_finding": "PASS", "section": "Findings 2 — The KKT point exists, near the stated one, and is a maximum", "strength": "open" }, { "claim": "§1–3: the stated point (ε = 0.036823344353076, pressures 0.0282523300, 0.0283757600, 0.7360431600, 0.2073287400) solves the twelve-equation event-KKT system in (w, β, δ, c, ε, φ₁, φ₂, s, μ₁…μ₄).", "kind": "computation", "numbers": [ "|F| = 9.5×10⁻³⁹ in six steps", "ε = 0.0368234056750", "6.1×10⁻⁸", "6.8×10⁻⁸", "pressures (0.0282538923, 0.0283741595, 0.7360431486, 0.2073287995)", "1.6×10⁻⁶", "scale 0.99891434329753911", "5.4×10⁻¹⁴" ], "quote": "(`kkt_stated_minus_solution`). So the stated point is the system's solution to about 6×10⁻⁸ in its parameters; the", "report_finding": "PASS", "section": "X4; Findings 2 — The KKT point exists, near the stated one, and is a maximum", "strength": "numerical only" }, { "claim": "Verdict: the point is a minimax candidate (\"極小極大候選\") — the round does not test the second-order condition.", "kind": "theorem", "numbers": [ "gradient rank 3 (4.03, 0.312, 0.0132, 5×10⁻³¹)", "reduced Hessian eigenvalues −0.113, −0.023", "m_φφ = 0.344" ], "quote": "−0.113 and −0.023 (`reduced_hessian_eigenvalues`): negative definite. The solution is a strict local maximum of the", "report_finding": "PASS", "section": "X5; Findings 2 — The KKT point exists, near the stated one, and is a maximum", "strength": "numerical only" }, { "claim": "§7: the common gradient's projection onto the two-dimensional tangent space of the equal-height set is about 2.7×10⁻⁹.", "kind": "computation", "numbers": [ "2.7×10⁻⁹ stated", "5.985×10⁻⁹ (stated gradients)", "1.651×10⁻⁹ (exact gradients at the stated point)", "0 at the solution" ], "quote": "(`projection_exact_gradients_at_stated_point`); at this run's solution it is zero. Neither is §7's 2.7×10⁻⁹, and this", "report_finding": "NOT AS STATED", "section": "X9; Findings 4 — Section 7's projection", "strength": "corrected" }, { "claim": "§7: there is at present no identifiable first-order ascent direction (\"目前沒有可辨識的一階上升方向\").", "kind": "reading", "numbers": [], "quote": "That is the second-order condition the round does not test; its \"沒有可辨識的一階上升方向\" is true at", "report_finding": "PASS", "section": "Findings 2; Findings 4", "strength": "numerical only" }, { "claim": "§8: the 12×12 Jacobian has numerical rank 12.", "kind": "computation", "numbers": [ "rank 12", "steps 10⁻¹⁵ and 10⁻¹²" ], "quote": "(`jacobian_steps_relative_difference`). Rank 12 holds.", "report_finding": "PASS", "section": "X5; Findings 3 — The Jacobian", "strength": "numerical only" }, { "claim": "§8: the Jacobian's singular values run from 4.29×10⁸ down to 11.4, smallest 11.4, condition 3.77×10⁷.", "kind": "computation", "numbers": [ "stated 4.29×10⁸, 1.36×10⁸, 2.81×10⁷, …, 12.7, 11.4", "stated condition 3.77×10⁷", "exact 5.40 … 0.00254", "exact condition 2.13×10³", "Frobenius norm 7.5" ], "quote": "11.4 — are not these: the largest is 8×10⁷ times larger, the smallest 4,500 times larger, and the condition, 3.77×10⁷", "report_finding": "NOT AS STATED", "section": "X5; Findings 3 — The Jacobian", "strength": "corrected" }, { "claim": "§8, §13, §15 and the README: the system is of full rank but highly ill-conditioned (\"系統滿秩，但高度病態\").", "kind": "reading", "numbers": [ "condition 2.13×10³" ], "quote": "did not find what the stated values are the spectrum of. A condition of 2×10³ is not, on any usual reading, a high one,", "report_finding": "NOT AS STATED", "section": "X5; Findings 3 — The Jacobian", "strength": "corrected" }, { "claim": "§8: rank 12 \"supports numerical isolation, not a robust conclusion of interval invertibility\".", "kind": "reading", "numbers": [], "quote": "rank 12 in double precision is not an interval proof of invertibility (X5).", "report_finding": "PASS", "section": "Findings 3 — The Jacobian", "strength": "open" }, { "claim": "§9: twenty restarts with scales in [0.998914343297213, 0.998914343297252] (count, range, largest distance in the files); the manifest's figures and files.", "kind": "specification", "numbers": [ "20", "[0.998914343297213, 0.998914343297252]" ], "quote": "the data files hold the summary's records; the restarts' count, range and largest distance; the manifest's figures and files", "report_finding": "PASS", "section": "X6", "strength": "label only" }, { "claim": "§9: all twenty restarts return to the same scale basin (\"全部返回同一尺度盆地\").", "kind": "computation", "numbers": [ "restart scales 2.33×10⁻¹³ to 2.72×10⁻¹³ below the candidate's" ], "quote": "cannot say how their scales were evaluated or what \"返回同一尺度盆地\" (return to the same scale basin) meant in practice; the", "report_finding": "UNMEASURED", "section": "Findings 5 — The restarts", "strength": "not measured" }, { "claim": "§10: curvature ledger — total turn δ, peak 2.101101621113, 5–95 % width.", "kind": "computation", "numbers": [ "2.101101621113", "width to 1.5×10⁻¹¹" ], "quote": "peak δ/(ε(T₀ − T₁)w) to the double, and the width is the exact width to 1.5×10⁻¹¹ (X7). The full phase circle has 8", "report_finding": "PASS", "section": "X7", "strength": "numerical only" }, { "claim": "§11: the full phase circle has 8 local minima, four of them the controlling branches.", "kind": "computation", "numbers": [ "8", "2¹⁸-phase scan" ], "quote": "the count of 8 local minima and the four controlling ones", "report_finding": "PASS", "section": "X8; Findings 6 — The full phase audit", "strength": "numerical only" }, { "claim": "§11: outside the four controlling branches the lowest minimum is about 1.02935; no new low branch is introduced by the five-parameter correction.", "kind": "computation", "numbers": [ "1.02935", "1.0293495" ], "quote": "round's conclusion: the lowest minimum outside the four is the curve's 1.0293495, which §11's 1.02935 holds, and none is", "report_finding": "PASS", "section": "X8; Findings 6 — The full phase audit", "strength": "numerical only" }, { "claim": "§11: another local minimum at phase 3.77896, scale 1.1275893354.", "kind": "computation", "numbers": [ "3.77896", "1.1275893354", "1.3×10⁻⁶ in phase", "2×10⁻¹¹ in scale" ], "quote": "curve's minimum, to 1.3×10⁻⁶ in phase and 2×10⁻¹¹ in scale. The other three are not:", "report_finding": "PASS", "section": "X8; Findings 6 — The full phase audit", "strength": "numerical only" }, { "claim": "§11: another local minimum at phase 1.73608, scale 1.029349885, smooth.", "kind": "computation", "numbers": [ "1.73608", "1.029349885", "minimum 1.7361562900, 1.0293495204", "7.6×10⁻⁵ in phase", "3.6×10⁻⁷ above the minimum" ], "quote": "1.7361562900, 1.0293495204, smooth, 2.9×10⁻⁵ rad before the switch at π − β", "report_finding": "NOT AS STATED", "section": "X8; Findings 6 — The full phase audit", "strength": "corrected" }, { "claim": "§11: another local minimum at phase 3.02827, scale 1.0998191738, a contact switch.", "kind": "computation", "numbers": [ "3.02827", "1.0998191738", "minimum 3.0282125272, 1.0998092540", "5.7×10⁻⁵ in phase", "9.9×10⁻⁶ above the minimum", "5.0×10⁻⁹ rad before π/2 + α" ], "quote": "And the minimum near 3.028 is not a contact switch", "report_finding": "NOT AS STATED", "section": "X8; Findings 6 — The full phase audit", "strength": "corrected" }, { "claim": "§11: another local minimum at phase 5.23599, scale 1.0814431618, a contact switch.", "kind": "computation", "numbers": [ "5.23599", "1.0814431618", "5π/3 = 5.2359877560", "cusp scale 1.0814272247", "1.6×10⁻⁵ above the cusp's", "1.5×10⁻⁵ above s at its own phase" ], "quote": "five decimals, but its scale is 1.6×10⁻⁵ above the cusp's and 1.5×10⁻⁵ above s at its own phase, and this run did not", "report_finding": "NOT AS STATED", "section": "X8; Findings 6 — The full phase audit", "strength": "corrected" }, { "claim": "§3: the pressures, printed to ten decimals (0.0282523300, 0.0283757600, 0.7360431600, 0.2073287400).", "kind": "computation", "numbers": [ "ten decimals printed", "eight in the data", "five decimals hold for B1 and B2", "seven for the others" ], "quote": "The report prints the pressures to ten decimals (\"0.0282523300\") from data with eight; against the solution's pressures", "report_finding": "NOT AS STATED", "section": "X9; Findings 7 — Printed digits", "strength": "corrected" }, { "claim": "§4: the gradients, printed to nine decimals.", "kind": "computation", "numbers": [ "nine decimals printed", "six to nine hold" ], "quote": "It prints the gradients to nine decimals; they hold six", "report_finding": "NOT AS STATED", "section": "X9; Findings 7 — Printed digits", "strength": "corrected" }, { "claim": "§10: the 5–95 % width, printed as 0.036441801352.", "kind": "computation", "numbers": [ "0.036441801352 printed", "0.036441801337 exact", "ten decimals hold" ], "quote": "It prints the width to twelve decimals, 0.036441801352", "report_finding": "NOT AS STATED", "section": "X9; Findings 7 — Printed digits", "strength": "corrected" }, { "claim": "The README, the report's figures and tables (§2–§6, §8–§12), the audit note and the handoff restate the data.", "kind": "reading", "numbers": [], "quote": "the README, the report's figures and tables (§2–§6, §8–§12), the audit note and the handoff state the data's", "report_finding": "PASS", "section": "X9", "strength": "label only" }, { "claim": "The package's replay script (src/smooth_kkt_replay.py) replays the candidate.", "kind": "specification", "numbers": [], "quote": "`src/smooth_kkt_replay.py` prints the summary's candidate, rank and residual norm; it computes nothing. The system's", "report_finding": "PARTIAL", "section": "Findings 9 — The script makes none of the data", "strength": "label only" }, { "claim": "Verdict: a numerically isolated five-parameter event-KKT candidate, not an interval KKT or Arb certificate, and no new Moser bound.", "kind": "reading", "numbers": [], "quote": "ill-conditioned minimax candidate) — not an interval KKT or Arb certificate, and no new Moser bound.", "report_finding": "PASS", "section": "What Round 13 claims", "strength": "open" } ], "corrections": [ { "what": "§7 projection of the common gradient", "source_says": "About 2.7×10⁻⁹.", "report_finds": "5.985×10⁻⁹ with the stated gradients, 1.651×10⁻⁹ with exact gradients at the stated point, zero at this run's KKT solution; the computation giving 2.7×10⁻⁹ was not found. The conclusion drawn from it holds.", "quote": "The projection of the common gradient onto the plane on which the four branch gradients agree is 5.985×10⁻⁹ with the" }, { "what": "§8 Jacobian singular values and condition", "source_says": "Singular values 4.29×10⁸ down to 11.4, condition 3.77×10⁷.", "report_finds": "The exact Jacobian of the §1 system has singular values 5.40, 4.29, 2.15, 1.36, 1.18, 0.861, 0.433, 0.0887, 0.0663, 0.0236, 0.0135, 0.00254, condition 2.13×10³; its Frobenius norm 7.5 bounds every singular value.", "quote": "solution from second differences at 60 digits, has singular values 5.40, 4.29, 2.15, 1.36, 1.18, 0.861, 0.433, 0.0887," }, { "what": "\"Highly ill-conditioned\" (§8, §13, §15, README)", "source_says": "The system is of full rank but highly ill-conditioned.", "report_finds": "Condition 2.13×10³ is not high on any usual reading; the characterization rests on the stated numbers, not on the system. The round's caution about interval invertibility stands for another reason.", "quote": "and the round's \"高度病態\" (highly ill-conditioned), repeated in §15 and the README, with §13's \"系統條件數高\" (the system's" }, { "what": "§11 other minimum near 1.736", "source_says": "1.73608, 1.029349885, smooth.", "report_finds": "The minimum is at 1.7361562900, scale 1.0293495204; the stated phase is 7.6×10⁻⁵ away and its scale 3.6×10⁻⁷ above the minimum — a sample near the minimum.", "quote": "Two stated phases are 7.6×10⁻⁵ and 5.7×10⁻⁵ from the minima, with scales within 3.1×10⁻⁸ of s at those phases but" }, { "what": "§11 other minimum near 3.028", "source_says": "3.02827, 1.0998191738, contact switch.", "report_finds": "The minimum is smooth, at 3.0282125272 with scale 1.0998092540, 5.0×10⁻⁹ rad before the switch at π/2 + α, where s is continuously differentiable; the stated scale is 9.9×10⁻⁶ above it.", "quote": "α the support of min_x passes from the right wing's interior to its end p₃, where s is continuously differentiable, and" }, { "what": "§11 other minimum at 5π/3", "source_says": "5.23599, 1.0814431618, contact switch.", "report_finds": "It is the cusp at 5π/3 = 5.2359877560 with scale 1.0814272247; the stated scale is 1.6×10⁻⁵ too high, and its source was not found here.", "quote": "5π/3 = 5.2359877560, 1.0814272247, the cusp" }, { "what": "Printed digits of the pressures", "source_says": "Pressures to ten decimals: 0.0282523300, 0.0283757600, 0.7360431600, 0.2073287400.", "report_finds": "The data carry eight decimals; against the solution's pressures they hold five decimals for B1 and B2 and seven for the others.", "quote": "they hold five decimals for B1 and B2 and seven for the others." }, { "what": "Printed digits of the gradients", "source_says": "Gradients to nine decimals.", "report_finds": "They hold six to nine decimals; figures printed past what they were computed to.", "quote": "these are figures printed past what they were computed to, not claims of more (X9)." }, { "what": "Printed digits of the 5–95 % width", "source_says": "0.036441801352 (twelve decimals).", "report_finds": "The width is 0.036441801337; the printed figure holds ten decimals.", "quote": "the width is 0.036441801337, and the figure holds ten" } ], "open_items": [ "An interval enclosure of the KKT solution, of its isolation or of its second-order condition: they rest on 60-digit", "Interval enclosures of the eight minima's places: the certificate bounds", "How the stated singular values, §7's projection, the 5π/3 scale or the restarts'", "on a neighbourhood of unestablished size", "the round's numbers can be checked, as here, but not reproduced.", "Any bound for Moser's problem. Nor the package's six figures," ], "cross_refs": [ "RUN-002 (sense of the PARTIAL verdict)", "RUN-009 (wing used for values, derivatives and ledger)", "RUN-010 (high-precision candidate at 60 and 200 digits)", "RUN-011 (interval certificate machinery reused with a target; Round 08's least is its s(3π/2); smooth minimum before π/2 + α on Round 08's curve)", "RUN-012 (smooth minimum just before π/2 + α on Round 08's curve)" ], "unclear": [ "The curvature ledger is checked against an exact formula evaluated in double precision; labelled 'numerical only'.", "'Highly ill-conditioned' is labelled 'corrected' rather than 'refuted': the stated spectrum is wrong, but the round's caution (rank 12 in double precision is no interval proof) survives for another reason.", "The strict local maximum is this run's second-order test at 60 digits, which the round did not perform; recorded against the round's 'minimax candidate' wording as PASS / numerical only, on a neighbourhood of unestablished size.", "The gain certificate is this run's own interval computation (RUN-011's machinery) and uses RUN-011's finding that Round 08's least is its s(3π/2); labelled 'exact-arithmetic certificate'.", "The 'documents restate the data' item is inferred PASS: X9 fails, and the report names §7's projection and the printed digits as its failures.", "The replay-script item records a provenance finding (the script prints, computes nothing) as PARTIAL / label only; the report does not quote what the package says the script does." ], "for_paper": "Round 13's five-parameter event-KKT candidate holds at high precision: this run's own 60-digit solution of the twelve equations (|F| = 9.5×10⁻³⁹) lies within 6.1×10⁻⁸ of the stated parameters, with all pressures positive. On the plane the four gradients leave free, the Lagrangian's Hessian is negative definite (−0.113, −0.023), so the solution is a strict local maximum of the least branch on a neighbourhood of unestablished size. In interval arithmetic the stated curve's least is certified at no less than s₁₃ − 1.5×10⁻¹⁰, which puts the gain over Round 08 in [4.0451×10⁻⁹, 4.2486×10⁻⁹], around the stated 4.2129×10⁻⁹. The stated Jacobian spectrum (condition 3.77×10⁷) is not the system's (5.40 down to 2.54×10⁻³, condition 2.13×10³), so 'highly ill-conditioned' does not hold. §7's 2.7×10⁻⁹ projection is not reproduced, three of the four 'other' minima are phase samples or misclassified, and the pressures, gradients and width are printed past their digits. The round's conclusions — a four-way tie, a gain of about 4.21×10⁻⁹, no low branch outside the four — hold; no Moser bound is claimed." }, { "run": "RUN-015", "report_file": "RUN-015-ROUND14-DOUBLE-PEAK.md", "source_document": "Round 14 (v0.14), Moser_Skew_Lab_v0.14.zip: README, MANIFEST, reports/ROUND_14_REPORT.md, ROUND_14_SEARCH_AUDIT.md, ROUND_15_CURVATURE_MODE_HANDOFF.md, the model file (read, not run: a docstring) and its data (round14_summary.json, double_peak_search.json, double_peak_high_resolution_audit.json, asymmetry_congruence_audit.json, the three profile CSVs and the two curve files), with Round 13's package for the reference curve", "verdict_line": "**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces where it can be rebuilt, and its central conclusion does not hold.", "items": [ { "claim": "The two curve files are Round 13's curve and the search candidate's mixture (polylines centred at their mean); the reference scale is Round 13's common scale; the mixture's centres, widths and weight follow from its parameters.", "kind": "specification", "numbers": [ "12,000 steps per wing", "160 across the centre", "1.0×10⁻¹⁵", "2.0×10⁻¹⁵" ], "quote": "The reference scale is Round 13's common scale. The base curve file is RUN-009's polyline of Round 13's curve — 12,000", "report_finding": "PASS", "section": "Y1; Findings 1 — What holds", "strength": "numerical only" }, { "claim": "§3: the search candidate's four branch values under the high-resolution evaluator.", "kind": "computation", "numbers": [ "0.9989143567270", "0.9989143562161", "0.9989143424378", "0.9989143424543", "cusps exact to 3.4×10⁻¹⁵", "low values 2.09×10⁻¹⁰ and 2.17×10⁻¹⁰ below", "least the 120° cusp, 8.597×10⁻¹⁰ below Round 13's least" ], "quote": "0.9989143424543 (the cusps) (`search_exact`): the high-resolution cusp values are exact to 3.4×10⁻¹⁵, its low values", "report_finding": "PASS", "section": "Y2; Findings 1 — What holds", "strength": "numerical only" }, { "claim": "§3: the best double peak lies 6.55×10⁻¹⁰ below the single peak \"under the same high-resolution evaluator\" (\"同一高解析求值器下\").", "kind": "computation", "numbers": [ "−6.55×10⁻¹⁰ stated", "−8.597×10⁻¹⁰ between the curves", "single peak's polyline low branch 2.05×10⁻¹⁰ low" ], "quote": "−6.55×10⁻¹⁰ compares a polyline's low branch with an exact cusp (the curves' difference is −8.60×10⁻¹⁰)", "report_finding": "NOT AS STATED", "section": "Y2; Findings 4 — Why the round did not see it", "strength": "corrected" }, { "claim": "§3: the search's own evaluation of its candidate's branches (double_peak_search.json).", "kind": "computation", "numbers": [ "low branches 1.22×10⁻⁸ and 1.32×10⁻⁸ below exact", "cusps 6×10⁻¹⁰ above" ], "quote": "search's low branches are 1.22×10⁻⁸ and 1.32×10⁻⁸ below the exact values and its cusps 6×10⁻¹⁰ above", "report_finding": "NOT AS STATED", "section": "Findings 4 — Why the round did not see it", "strength": "corrected" }, { "claim": "§3: the best eight-parameter mixture has peak separation 3.3255×10⁻⁶, \"collapsed back to nearly one peak\" (\"重新塌縮為近單峰\").", "kind": "computation", "numbers": [ "3.3255×10⁻⁶", "split curve at d = 0.03 at least 1.1924×10⁻⁷ higher" ], "quote": "errors change with d. The search stopped at a separation of 3.3×10⁻⁶.", "report_finding": "PARTIAL", "section": "Findings 3; Findings 4 — Why the round did not see it", "strength": "refuted", "audit": "kept by the strength audit; one cause (the single peak is a saddle against splitting; the split curve is certified higher at d = 0.03) behind three of the round's statements; not three independent errors; finding moved from 'NOT AS STATED' after the independent reading: the separation is where the search stopped and holds; what is refuted is reading it as the family's best collapsing to one peak, since the family holds the split curve certified higher" }, { "claim": "§4: the separation profile is Round 13's curve split into two equal halves at fixed parameters, with the stated values and losses.", "kind": "computation", "numbers": [ "cusps exact to 1.5×10⁻¹⁴", "low values 1.75 to 2.70×10⁻⁹ low", "21 points with d ≤ 0.01" ], "quote": "its low values 1.75 to 2.70×10⁻⁹ low (`profile_stated_minus_exact`); its losses are its scales less the first, and its", "report_finding": "PASS", "section": "Y3; Findings 1 — What holds", "strength": "numerical only" }, { "claim": "§4: the profile falls as Δs(d) ≈ −0.033794468d²; the quadratic coefficient is negative (\"二次係數為負\").", "kind": "computation", "numbers": [ "−0.033794468", "−0.0337945", "m_dd −0.067619 (B1)", "m_dd −0.067561 (B2)" ], "quote": "coefficient −0.0337945 is what it measures: the low branches' second derivatives along d at fixed parameters are", "report_finding": "PASS", "section": "Y4; Findings 1 — What holds", "strength": "numerical only" }, { "claim": "Boxed verdict under \"有限數據支持：\": the single-peak curvature layer is locally stable against peak splitting (\"單峰曲率層對峰分裂方向局部穩定\"); the summary records single_peak_local_stability_supported: true and double_peak_escape: false.", "kind": "theorem", "numbers": [ "Lagrangian eigenvalues +1.5867×10⁻⁴ and −2.90×10⁻⁶", "m_dd: B1 −0.067619, B2 −0.067561, 120° +0.0042856, 270° +0.0040120", "pressures (0.028, 0.028, 0.736, 0.207)" ], "quote": "has curvature +1.587×10⁻⁴ along the separation, so the single peak is a saddle; split into two equal halves at", "report_finding": "NOT AS STATED", "section": "Y4; Findings 2 — The single peak is a saddle against splitting", "strength": "refuted", "audit": "kept by the strength audit; one cause (the single peak is a saddle against splitting; the split curve is certified higher at d = 0.03) behind three of the round's statements; not three independent errors" }, { "claim": "Conclusion: within the tested curvature-splitting and chirality-breaking families, Round 13's single-peak mirror-symmetric candidate has no identifiable ascent direction (\"沒有可辨識的上升方向\").", "kind": "reading", "numbers": [ "d = 0.004: +1.2790×10⁻⁹", "d = 0.008: +5.2352×10⁻⁹", "d = 0.016: +2.3051×10⁻⁸", "d = 0.03: +1.1942×10⁻⁷", "residuals 7×10⁻³⁰ to 2×10⁻²⁸ at 40 digits" ], "quote": "separation d and with the five parameters re-solved, the four branches tie 1.28×10⁻⁹ higher at d = 0.004 and 1.19×10⁻⁷", "report_finding": "NOT AS STATED", "section": "Findings 3 — The ascent, solved and certified", "strength": "refuted", "audit": "kept by the strength audit; one cause (the single peak is a saddle against splitting; the split curve is certified higher at d = 0.03) behind three of the round's statements; not three independent errors" }, { "claim": "(Report's own measurement, beyond the round's claims) The split curve at separation d = 0.03 with the five parameters re-solved, inside Round 14's double-peak family.", "kind": "certificate", "numbers": [ "s ≥ 0.998914462565054", "1,595 cover intervals, worst 0.998914462589249", "Round 13's least ≤ 0.998914343315417", "gain ≥ 1.1924×10⁻⁷" ], "quote": "its own curve. So the split curve's least exceeds Round 13's candidate's by at least 1.1924×10⁻⁷ — about twenty-eight", "report_finding": "PASS", "section": "Findings 3 — The ascent, solved and certified", "strength": "exact-arithmetic certificate" }, { "claim": "§4: the separation profile's low phases and scales at the larger separations (d ≥ 0.028).", "kind": "computation", "numbers": [ "0.14500000", "0.13699999", "0.13676", "merged minimum at 0.14085 and 0.13939", "derivative up to 8.9×10⁻⁴", "least 0.9988420155 vs stated 0.9988424470", "4.3×10⁻⁷" ], "quote": "stated scale 0.9988424470, 4.3×10⁻⁷ higher (`profile_merged_low_minimum`) (Y3). It changes nothing about the", "report_finding": "NOT AS STATED", "section": "Y3; Findings 5 — The profile at larger separations", "strength": "corrected" }, { "claim": "§6: both chirality modes peak at zero offset; the mirror symmetry is locally stable for the tested centre and width chirality modes (\"鏡像對稱對所測中心與寬度手性模態局部穩定\").", "kind": "theorem", "numbers": [ "centre mode (−0.00673, −0.02511, −0.00200, +0.01761)", "width mode (−0.000286, −0.000286, −0.0000068, −0.0000033)", "singular values 0.0315 and 0.000284", "cusp phase rates −0.0116 and −0.0000396" ], "quote": "The round's chirality verdict holds for the scale it scores, the congruent one, at first order. At RUN-014's solution", "report_finding": "PASS", "section": "Y5; Findings 6 — Chirality", "strength": "numerical only" }, { "claim": "§6: a curve is scored by the weaker of its two handednesses (the congruent scale), a curve being allowed to sit in the triangle reflected.", "kind": "definition", "numbers": [ "width-mode rise of one handedness 3.2×10⁻⁶ at η_ε = −0.45" ], "quote": "round's objective, a curve being allowed to sit in the triangle reflected; were placements rotations and translations", "report_finding": "PASS", "section": "Findings 6 — Chirality", "strength": "label only" }, { "claim": "§6: the congruent scale near zero is described by quadratic coefficients (\"零點附近二次係數\") −3.674929940 (centre) and −0.003092904 (width).", "kind": "computation", "numbers": [ "−3.674929940", "−0.003092904", "one-sided slope −0.0251 (centre)", "one-sided slope −0.000286 (width)", "−5.18×10⁻⁵ at η_c = ±0.002" ], "quote": "one-sided slope −0.0251 in the centre mode and −0.000286 in the width mode (`congruent_one_sided_slope`) — the profile's", "report_finding": "NOT AS STATED", "section": "Y5; Findings 6 — Chirality", "strength": "corrected" }, { "claim": "§6: the width-mode coefficient −0.003092904 is the fit of the shipped width profile.", "kind": "computation", "numbers": [ "−0.003092904" ], "quote": "handedness, reflection exchanges the two — and the width-mode fit is its window's fit of the shipped profile (Y5); the", "report_finding": "PASS", "section": "Y5; Findings 1 — What holds", "strength": "numerical only" }, { "claim": "§6: the centre-mode coefficient −3.674929940 is the fit of the shipped centre profile.", "kind": "computation", "numbers": [ "−3.6749 stated", "−3.6255 from the shipped profile", "13 points", "window 0.012" ], "quote": "the stated window 0.012 give −3.6255, not −3.6749 (`mode_fits_from_shipped_profiles`) (Y5). The congruent scale is the", "report_finding": "NOT AS STATED", "section": "Y5; Findings 6 — Chirality", "strength": "corrected" }, { "claim": "§6: the census of 80 random two-dimensional offsets (best 2.81×10⁻⁵ below zero) and the refined candidates; reflection exchanges the two handednesses.", "kind": "computation", "numbers": [ "80", "2.81×10⁻⁵" ], "quote": "chirality profiles, the census and the refined candidates are consistent — the congruent scale is the weaker", "report_finding": "PASS", "section": "Y5; Findings 1 — What holds", "strength": "numerical only" }, { "claim": "§7: the full-phase audit lists eight local minima of the base curve, and the double-peak curve's list.", "kind": "computation", "numbers": [ "six of eight are samples", "phases off by up to 1.28×10⁻⁴", "scales s at those phases to 3.5×10⁻¹¹", "up to 3.42×10⁻⁵ above the minima", "120° entry 2.0942673, 0.9989485504" ], "quote": "Six of the base curve's eight listed minima are the curve's values at phases near its minima, not the minima: the", "report_finding": "NOT AS STATED", "section": "Y6; Findings 7 — The full-phase lists are phase samples", "strength": "corrected" }, { "claim": "§7: the 270° cusp entry and the minimum near 3.7790 in the base list.", "kind": "computation", "numbers": [ "4×10⁻¹²" ], "quote": "3.7790 are minima to 4×10⁻¹² (`base_minima_against_exact`) (Y6). The double-peak list is evaluated at the base list's", "report_finding": "PASS", "section": "Y6; Findings 7 — The full-phase lists are phase samples", "strength": "numerical only" }, { "claim": "§7: the sampled global minimum is the 270° cusp, and no new low branch appears.", "kind": "computation", "numbers": [], "quote": "The sampled global minimum is the 270° cusp, and no low branch is new.", "report_finding": "PASS", "section": "Y6; Findings 7 — The full-phase lists are phase samples", "strength": "numerical only" }, { "claim": "The summary holds the data files' records; the CSVs are the JSON profiles; the manifest.", "kind": "specification", "numbers": [], "quote": "the summary holds the data files' records; the CSVs are the JSON profiles; the manifest", "report_finding": "PASS", "section": "Y7", "strength": "label only" }, { "claim": "The README, the report's figures and tables (§3, §4, §6–§8), the search audit and the handoff restate the data.", "kind": "reading", "numbers": [], "quote": "the README, the report's figures and tables (§3, §4, §6–§8), the search audit and the handoff state the data's", "report_finding": "PASS", "section": "Y8", "strength": "label only" }, { "claim": "The package's model file (src/curvature_distribution_models.py) defines the curvature-distribution models.", "kind": "specification", "numbers": [], "quote": "is a docstring: the model definitions, no code.", "report_finding": "PARTIAL", "section": "Findings 8 — No code", "strength": "label only" }, { "claim": "Stated limits: the round cannot prove that the single peak is best among all curvature densities, or that all double-peak models are worse.", "kind": "reading", "numbers": [], "quote": "single peak is best among all curvature densities, that all double-peak models are worse", "report_finding": "PASS", "section": "What Round 14 claims", "strength": "open" } ], "corrections": [ { "what": "§3 double-minus-single difference", "source_says": "The double peak lies 6.55×10⁻¹⁰ below the single peak under the same high-resolution evaluator.", "report_finds": "The figure sets the evaluator's single peak (least a polyline low branch, 2.05×10⁻¹⁰ low) against the double peak's exact cusp; the curves differ by −8.597×10⁻¹⁰.", "quote": "\"同一高解析求值器下\", sets the evaluator's single peak — whose least is a low branch, a polyline's, 2.05×10⁻¹⁰ low" }, { "what": "The search evaluator's branch values", "source_says": "The search's values for its candidate.", "report_finds": "Its low branches are 1.22×10⁻⁸ and 1.32×10⁻⁸ below the exact values and its cusps 6×10⁻¹⁰ above — errors ten times the whole gain at d = 0.004.", "quote": "errors ten times the whole gain at d = 0.004, 1.3×10⁻⁹" }, { "what": "§3 search result: collapse to one peak", "source_says": "The best eight-parameter mixture has separation 3.3255×10⁻⁶, collapsed back to nearly one peak.", "report_finds": "The search stopped at 3.3×10⁻⁶; inside the same family the split curve at d = 0.03 with re-solved parameters is certified at least 1.1924×10⁻⁷ above Round 13's candidate.", "quote": "times Round 13's own gain over Round 08 — inside the double-peak family Round 14 searched." }, { "what": "Single peak locally stable against splitting", "source_says": "單峰曲率層對峰分裂方向局部穩定; single_peak_local_stability_supported: true; double_peak_escape: false.", "report_finds": "At RUN-014's solution the Lagrangian's block in (d, r) has eigenvalues +1.5867×10⁻⁴ and −2.90×10⁻⁶: the single peak is a saddle, not a local maximum of the least scale in the double-peak family.", "quote": "(`splitting_lagrangian`, `splitting_eigenvalues`) (Y4). The positive one means the single peak is not a local maximum of" }, { "what": "No identifiable ascent direction in the tested families", "source_says": "Round 13's candidate has no identifiable ascent direction within the tested curvature-splitting and chirality-breaking families.", "report_finds": "Splitting the peak and re-solving the five parameters raises all four branches together: +1.279×10⁻⁹ at d = 0.004 up to +1.1942×10⁻⁷ at d = 0.03, the last certified in interval arithmetic.", "quote": "the least scale in the double-peak family: splitting the peak and re-balancing the five parameters raises all four" }, { "what": "§4 profile at d ≥ 0.028", "source_says": "The profile's low phases are minima and its scale is the least.", "report_finds": "From d = 0.028 the first low phase is 0.14500000 and from d = 0.032 the second is 0.13699999, ends of the search interval; the two minima have merged; at d = 0.04 the least is 0.9988420155, the stated scale 4.3×10⁻⁷ higher.", "quote": "From d = 0.028 on, the profile's first low phase is 0.14500000, and from d = 0.032 its second is 0.13699999, at every" }, { "what": "§6 quadratic coefficients near zero", "source_says": "Quadratic coefficients −3.674929940 (centre) and −0.003092904 (width) near zero.", "report_finds": "The congruent scale has a corner at zero, one-sided slopes −0.0251 (centre) and −0.000286 (width); the coefficients are fits of even polynomials to a function with a corner.", "quote": "coefficients near zero) are fits of even polynomials to a function with a corner." }, { "what": "§6 centre-mode fit", "source_says": "−3.674929940.", "report_finds": "The shipped profile's 13 points in the stated window 0.012 give −3.6255.", "quote": "profile's fit in its window; the centre-mode fit is not reproduced by the same fit — the shipped profile's 13 points in" }, { "what": "§7 listed minima", "source_says": "Eight local minima of the base curve (and a double-peak list).", "report_finds": "Six of the eight are values at phases near the minima (phases off by up to 1.28×10⁻⁴, scales up to 3.42×10⁻⁵ above the minima); the double-peak list is evaluated at the base list's phases.", "quote": "phases are off by up to 1.28×10⁻⁴, and the scales are s at those phases to 3.5×10⁻¹¹ but above the minima by up to" } ], "open_items": [ "The double-peak family's maximum, or where the four-branch tie stops being the least as d grows: this run solved four", "Any interval enclosure of the saddle's curvature or of the split solutions.", "wider family — Round 15's curvature modes — goes further. Any bound for Moser's problem.", "1.3×10⁻⁹; this run did not measure how the", "single peak is best among all curvature densities" ], "cross_refs": [ "RUN-002 (sense of the PARTIAL verdict)", "RUN-006 (the maximin test is the Lagrangian's curvature on the free directions)", "RUN-007 (chirality: the result holds with the five parameters re-solved or not)", "RUN-009 (polyline used for the curve files)", "RUN-014 (solution of Round 13's twelve equations, phase map and interval certificate reused; Round 13's 'other minima' traced here to four list entries, its 5π/3 scale to s at 5.235859924)" ], "unclear": [ "The verdict sentence spans two lines of the report; verdict_line joins them with one space.", "Three items (search collapse, boxed stability verdict, no ascent direction) are labelled 'refuted' and rest on one underlying finding (the saddle and the certified ascent at d = 0.03); they are distinct statements in the source.", "The d = 0.03 certificate is this run's own result beyond the round's claims; recorded as a PASS item with the claim marked as the report's own measurement.", "The fit window d ≤ 0.01 for the separation coefficient is inferred by the report, not stated by the round.", "Curve-file reproduction is labelled 'numerical only' though the report's strength table says 'exact (double precision for the polylines)'.", "The chirality verdict is PASS for the congruent scale only; the scope (reflected placements allowed) is recorded as a separate label-only definition item.", "The source's stated limit 'cannot prove all double-peak models are worse' is recorded as 'open', although the report's certified split curve shows that statement to be false." ], "for_paper": "Round 14's computations rebuild — the curve files, the search candidate's exact cusps (to 3.4×10⁻¹⁵), the fixed-parameter separation profile and its coefficient −0.0338 — but its central conclusion does not. At Round 13's KKT solution the Lagrangian of the four branches has curvature +1.587×10⁻⁴ along peak separation, so the single peak is a saddle, not locally stable against splitting. Splitting the peak with the five parameters re-solved raises the four-branch tie by 1.28×10⁻⁹ at d = 0.004 and 1.19×10⁻⁷ at d = 0.03 (high precision), and the d = 0.03 curve's least is certified in interval arithmetic at least 1.1924×10⁻⁷ above Round 13's candidate, inside the family the round searched. The round missed it because its profile held the parameters fixed and its search evaluator put the low branches about 10⁻⁸ low. The mirror symmetry is a strict local maximum of the congruent scale in both chirality modes at first order, as stated; but the quadratic coefficients describe a function with a corner, the centre-mode fit is not reproduced, and six of the eight listed minima are phase samples." }, { "run": "RUN-016", "report_file": "RUN-016-ROUND15-FUNCTIONAL-MODES.md", "source_document": "Round 15 (v0.15), Moser_Skew_Lab_v0.15.zip: README, MANIFEST, reports/ROUND_15_REPORT.md, ROUND_15_BRANCH_OPENING_AUDIT.md, ROUND_16_FIVE_BRANCH_HANDOFF.md, the model file (read, not run: a docstring) and its data (round15_summary.json, final_candidate.json, final_candidate_curve.csv, mode_profiles.json, newton_line_full_phase.csv, pressure_hessian.csv, pressure_hessian_eigenvectors.csv, reoptimized_multiplier_ledger.csv), with Round 13's package for the mode coordinate and the reference", "verdict_line": "**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces where it can be rebuilt, and some of what it states does not hold as stated.", "items": [ { "claim": "The modes are ψₙ = √(2n+1) Pₙ(t), n = 1…8, orthonormal for ρ₀ du = dt/2, with Gram error 2.220×10⁻¹⁶.", "kind": "definition", "numbers": [ "n = 1…8", "Gram error 2.220×10⁻¹⁶", "means 0, norms 1, maxima √(2n+1)" ], "quote": "the eight modes named; their weighted means, norms and maxima are the Legendre modes' (0, 1, √(2n+1)); the Gram matrix and its stated error", "report_finding": "PASS", "section": "Z1", "strength": "numerical only" }, { "claim": "The final candidate's curve file is the mode-deformed curve in the base curvature coordinate (\"基準曲率累積座標\").", "kind": "specification", "numbers": [ "96,000 steps", "320 centre steps", "4.3×10⁻¹² (t from Round 13's curve)", "5.0×10⁻¹¹ and 3.9×10⁻¹¹ (t from the base or RUN-014's solution)", "1.06×10⁻⁷ (t from the candidate's own layer)" ], "quote": "reproduced to 4.3×10⁻¹² (`curve_file_largest_difference`); with t from the round's base or from RUN-014's solution,", "report_finding": "PASS", "section": "Z2; Findings 1 — The candidate at 40 digits", "strength": "numerical only" }, { "claim": "§6–7: at m = 3.2275 the candidate's four branches and its latent fifth branch (highest-resolution audit), the least being the 120° cusp.", "kind": "computation", "numbers": [ "120° 0.99891448071696448", "270° 0.99891448071696486", "B1 0.99891448112086535", "B2 0.99891448173927110", "latent 0.99891448372155293", "1.84×10⁻¹⁴" ], "quote": "The stated values hold to 1.84×10⁻¹⁴; the least is 1.3741947945×10⁻⁷ above Round 13's (`gain_over_round13_exact`). The", "report_finding": "PASS", "section": "Z7; Findings 1 — The candidate at 40 digits", "strength": "numerical only" }, { "claim": "§7: the candidate's least exceeds Round 13's by about 1.374×10⁻⁷ (the round gives no interval or Arb certificate).", "kind": "certificate", "numbers": [ "gain in [1.3725×10⁻⁷, 1.3758×10⁻⁷]", "s ≥ 0.998914480566964", "1,598 cover intervals, worst 0.998914480841346", "s(3π/2) in [0.998914480714660, 0.998914480719270]", "Round 13's least in [0.998914343147485, 0.998914343315417]", "boxes at most 4.8×10⁻¹² wide", "262,144 steps", "1.78×10⁻⁸ above RUN-015's split curve" ], "quote": "recomputed here; so the candidate's least exceeds Round 13's by at least 1.3725×10⁻⁷ and at most 1.3758×10⁻⁷. With", "report_finding": "PASS", "section": "Findings 2 — The certificate", "strength": "exact-arithmetic certificate" }, { "claim": "§7: the latent branch lies about 3.0046×10⁻⁹ above the least.", "kind": "computation", "numbers": [ "3.0045884×10⁻⁹" ], "quote": "the latent branch 3.0045884×10⁻⁹ above it", "report_finding": "PASS", "section": "Z7; Verdict", "strength": "numerical only" }, { "claim": "§7 and the README: the boxed least s₁₅ = 0.998914480716946 (15 decimals).", "kind": "computation", "numbers": [ "0.998914480716946 stated", "0.998914480716964 exact", "holds 13 decimals" ], "quote": "The report boxes s₁₅ = 0.998914480716946 (the curve's least is 0.998914480716964), s₁₅ − s₁₃ = 1.374194610326×10⁻⁷", "report_finding": "NOT AS STATED", "section": "Z10; Findings 7 — Printed digits", "strength": "corrected" }, { "claim": "§7 and the README: the boxed gain s₁₅ − s₁₃ = 1.374194610326×10⁻⁷ (13 significant digits).", "kind": "computation", "numbers": [ "1.374194610326×10⁻⁷ stated", "1.374194794×10⁻⁷ exact", "holds 7 significant digits" ], "quote": "the gain and the latent margin are printed to 15 decimals, 13 and 13 significant digits where they hold 13 decimals, 7", "report_finding": "NOT AS STATED", "section": "Z10; Findings 7 — Printed digits", "strength": "corrected" }, { "claim": "§7 and the README: the boxed latent margin 3.004601123457×10⁻⁹ (13 significant digits).", "kind": "computation", "numbers": [ "3.004601123457×10⁻⁹ stated", "3.0045884×10⁻⁹ exact", "holds 5 significant digits" ], "quote": "(1.374194794×10⁻⁷) and the latent margin 3.004601123457×10⁻⁹ (3.0045884×10⁻⁹): they hold 13 decimals, and 7 and 5", "report_finding": "NOT AS STATED", "section": "Z10; Findings 7 — Printed digits", "strength": "corrected" }, { "claim": "§7: the candidate has nine local minima and passes the full-phase numerical audit.", "kind": "computation", "numbers": [ "9", "smooth minima to 4×10⁻⁹ in phase", "all to 1.84×10⁻¹⁴ in scale", "dense scan at 65,536 phases" ], "quote": "nine listed minima are the curve's: the smooth ones to 4×10⁻⁹ in phase, all to 1.84×10⁻¹⁴ in scale — the one at 3.0282", "report_finding": "PASS", "section": "Z7; Findings 1 — The candidate at 40 digits", "strength": "numerical only" }, { "claim": "§1: the re-solved base's least is the stated base scale (and the Newton line's first row).", "kind": "computation", "numbers": [ "cusps to 3.7×10⁻¹⁶", "least within 2.84×10⁻¹³", "base parameters up to 6.53×10⁻⁸ from RUN-014's solution", "first two pressures 1.13×10⁻⁵ off" ], "quote": "60 digits its cusps are the stated ones to 3.7×10⁻¹⁶, and its least is the cusps', within 2.84×10⁻¹³ of the stated base", "report_finding": "PASS", "section": "Z3; Findings 3 — The base and its gradients", "strength": "numerical only" }, { "claim": "§1: at the re-solved base the four branches are tied (the single-mode profiles at amplitude 0).", "kind": "computation", "numbers": [ "low branches 9.66×10⁻¹⁰ above the cusps", "profiles 9.7×10⁻¹⁰ below the curve's own" ], "quote": "scale (`base_exact`, `profile_zero_row_minus_base_exact`) (Z3). But its low branches lie 9.66×10⁻¹⁰ above the cusps,", "report_finding": "NOT AS STATED", "section": "Z3; Findings 3 — The base and its gradients", "strength": "corrected" }, { "claim": "§1: the base's parameter gradients, cusp rows.", "kind": "computation", "numbers": [ "3.0×10⁻¹⁰" ], "quote": "the cusp rows hold to 3.0×10⁻¹⁰", "report_finding": "PASS", "section": "Z3; Findings 3 — The base and its gradients", "strength": "numerical only" }, { "claim": "§1: the base's parameter gradients, low-branch rows.", "kind": "computation", "numbers": [ "1.02×10⁻⁵", "pressure offset 1.13×10⁻⁵" ], "quote": "the low rows only to 1.02×10⁻⁵", "report_finding": "NOT AS STATED", "section": "Z3; Findings 3 — The base and its gradients", "strength": "corrected" }, { "claim": "§1: the mode gradients and the pressure-projected first derivative of each mode at the base, nonzero (−1.978316×10⁻⁶ for P4).", "kind": "computation", "numbers": [ "−1.978316×10⁻⁶ (P4)", "mode gradients to 8.1×10⁻⁸", "weighted to 2.1×10⁻⁹" ], "quote": "from RUN-014's, which they would explain. The mode gradients are better: they hold to 8.1×10⁻⁸, the weighted ones to", "report_finding": "PASS", "section": "Z4; Findings 3 — The base and its gradients", "strength": "numerical only" }, { "claim": "§2: each mode is compensated by the five parameters so that the four branches move equally to first order.", "kind": "method", "numbers": [], "quote": "the minimum-norm solution of the equal-height conditions", "report_finding": "PASS", "section": "Z4; Findings 3 — The base and its gradients", "strength": "numerical only" }, { "claim": "The single-mode profiles (mode_profiles.json) along the compensated paths.", "kind": "computation", "numbers": [ "cusps to 1.7×10⁻¹⁵" ], "quote": "paths: their cusp values are the curve's there to 1.7×10⁻¹⁵ (`profile_cusps_stated_minus_exact`), and differ from the", "report_finding": "PASS", "section": "Z9; Findings 3 — The base and its gradients", "strength": "numerical only" }, { "claim": "§3: the pressure-weighted Lagrangian's finite-difference Hessian along the compensated paths has seven negative eigenvalues and one positive, 1.329×10⁻⁶.", "kind": "computation", "numbers": [ "1.329×10⁻⁶", "seven negative" ], "quote": "The Hessian is symmetric, its file is the summary's, its eigenpairs reproduce it, and one eigenvalue is positive (Z5).", "report_finding": "PASS", "section": "Z5; Findings 5 — The Hessian, the Newton direction and the Newton line", "strength": "label only", "audit": "relabelled from 'numerical only' after the independent reading: Z5 recomputes the shipped matrix's own arithmetic (symmetry, eigenpairs, the summary); the Hessian is not rebuilt from the curve" }, { "claim": "§3: from the positive eigenvalue, Round 13's single-peak candidate is not a strict local maximum in this eight-mode space (\"並非此八模態空間中的嚴格局部極大\").", "kind": "theorem", "numbers": [ "weighted mode gradients at RUN-014's solution: P2 4.81×10⁻⁷, P4 −1.979×10⁻⁶, P6 5.42×10⁻⁷, P8 3.95×10⁻⁷", "first-order term 1.558×10⁻⁷", "quadratic term −1.09×10⁻⁸", "model total 1.45×10⁻⁷ vs gain 1.3747×10⁻⁷" ], "quote": "round's conclusion, not a strict local maximum, is true for this first-order reason; its §3 draws it from the Hessian's", "report_finding": "NOT AS STATED", "section": "Z6; Findings 4 — Not stationary, so the Hessian decides nothing", "strength": "corrected" }, { "claim": "§5: the ascent follows a pressure-projected Newton direction (\"壓力投影 Newton 方向\"), its construction unstated.", "kind": "method", "numbers": [ "cosine with −H⁻¹g 0.634", "seven negative directions 0.99993", "six most negative 0.99999996", "ratios 0.0433 to 0.04353" ], "quote": "It is not −H⁻¹g (cosine 0.634), nor the step on the seven negative directions (0.99993): it is the Newton step on the", "report_finding": "PASS", "section": "Z5; Findings 5 — The Hessian, the Newton direction and the Newton line", "strength": "numerical only" }, { "claim": "The candidate's mode coefficients are the Newton direction at multiplier m = 3.2275.", "kind": "computation", "numbers": [ "3.2275" ], "quote": "The candidate's coefficients are exactly 3.2275 times it (Z7).", "report_finding": "PASS", "section": "Z7; Findings 5", "strength": "numerical only" }, { "claim": "§5: along the Newton direction the five parameters are re-equalized at each multiplier m (the Newton-line file).", "kind": "method", "numbers": [ "rebuilt at m = 1 and 2 to 1.1×10⁻¹⁵ and 5×10⁻¹⁶" ], "quote": "is not the re-equalized walk §5 describes but the first-order path p₀ + Q·m·a_N", "report_finding": "NOT AS STATED", "section": "Z8; Findings 5 — The Hessian, the Newton direction and the Newton line", "strength": "corrected" }, { "claim": "Audit note: a true ascent window exists on the Newton line (\"Newton 線存在真上升窗\").", "kind": "reading", "numbers": [ "5.3×10⁻⁸ above the base at most", "4.76×10⁻⁸ at m = 3.2" ], "quote": "to 5.3×10⁻⁸ above the base and is still 4.76×10⁻⁸ above it at m = 3.2", "report_finding": "PASS", "section": "Findings 5 — The Hessian, the Newton direction and the Newton line", "strength": "numerical only" }, { "claim": "§5: a ninth local minimum appears after m ≈ 2.2.", "kind": "computation", "numbers": [ "m = 2.2" ], "quote": "On the Newton line, a ninth local minimum first appears at m = 2.2, as stated (Z8).", "report_finding": "PASS", "section": "Z8; Findings 6 — The window", "strength": "numerical only" }, { "claim": "§5–6: near m ≈ 3.228 the new branch begins to take over (\"新分支開始接管\"), so a finite ascent window is bounded by branch opening.", "kind": "computation", "numbers": [ "below at m = 3.2235 (by 1.0×10⁻⁷), 3.224, 3.2244, 3.2245 (by 1.5×10⁻⁹)", "agrees from 3.2246 to 3.2275", "below for good from 3.228" ], "quote": "from 3.2246 to 3.2275, and falls below for good from 3.228 (`ledger_full_minus_four_branch`) (Z8). The latent branch", "report_finding": "NOT AS STATED", "section": "Z8; Findings 6 — The window", "strength": "corrected" }, { "claim": "The re-optimized multiplier ledger: norms 0.04 m, gains the scales less the base.", "kind": "specification", "numbers": [ "0.04 m" ], "quote": "scales — so its rows are checked for their own arithmetic and against each other, not rebuilt.", "report_finding": "PASS", "section": "Z8; Findings 8 — What cannot be checked", "strength": "numerical only" }, { "claim": "§4: a four-branch optimum at 0.998914692247305 is excluded by a hidden middle branch at φ ≈ 0.1390815.", "kind": "computation", "numbers": [ "0.998914692247305", "hidden branch 0.9989115500", "φ ≈ 0.1390815" ], "quote": "coefficients, so neither its 0.998914692247305 nor its hidden branch at 0.9989115500 can be rebuilt; this run checks", "report_finding": "UNMEASURED", "section": "Findings 8 — What cannot be checked", "strength": "not measured" }, { "claim": "The copies, the manifest, the README, the report's tables (§1, §3, §6–§8), the audit note and the handoff restate the data.", "kind": "reading", "numbers": [], "quote": "the copies, the manifest, the README, the report's tables (§1, §3, §6–§8), the audit note and the handoff state the data's", "report_finding": "PASS", "section": "Z10", "strength": "label only" }, { "claim": "The package's model file defines the mode deformation.", "kind": "specification", "numbers": [], "quote": "docstring: none of the numbers can be reproduced from the package, only checked.", "report_finding": "PARTIAL", "section": "Findings 8 — What cannot be checked", "strength": "label only" } ], "corrections": [ { "what": "Printed digits of the boxed least", "source_says": "s₁₅ = 0.998914480716946.", "report_finds": "The curve's least is 0.998914480716964; the figure holds 13 decimals.", "quote": "(the curve's least is 0.998914480716964)" }, { "what": "Printed digits of the boxed gain", "source_says": "s₁₅ − s₁₃ = 1.374194610326×10⁻⁷.", "report_finds": "The gain is 1.374194794×10⁻⁷ (certified in [1.3725×10⁻⁷, 1.3758×10⁻⁷]); the figure holds 7 significant digits.", "quote": "(1.374194794×10⁻⁷) and the latent margin" }, { "what": "Printed digits of the latent margin", "source_says": "3.004601123457×10⁻⁹.", "report_finds": "The latent margin is 3.0045884×10⁻⁹; the figure holds 5 significant digits.", "quote": "they hold 13 decimals, and 7 and 5" }, { "what": "Four-way tie at the base", "source_says": "At the re-solved base the four branches are tied (single-mode profiles at amplitude 0).", "report_finds": "The base's low branches lie 9.66×10⁻¹⁰ above the cusps; the round's evaluator puts the low branches about 9.7×10⁻¹⁰ low, so the tie holds only in that evaluator.", "quote": "while the round's single-mode profiles at amplitude 0 have them tied with the cusps, 9.7×10⁻¹⁰ below the curve's own:" }, { "what": "Low-branch parameter gradients at the base", "source_says": "The base's parameter gradients (residual, pressures).", "report_finds": "The cusp rows hold to 3.0×10⁻¹⁰ but the low rows only to 1.02×10⁻⁵, errors of the size of the pressures' 1.13×10⁻⁵ offset from RUN-014's.", "quote": "(`base_parameter_gradients_stated_minus_exact_largest`) (Z3) — errors of the size of the pressures' 1.13×10⁻⁵ offset" }, { "what": "§3 inference from the Hessian's positive eigenvalue", "source_says": "One positive eigenvalue, so Round 13's candidate is not a strict local maximum in the eight-mode space.", "report_finds": "At RUN-014's solution the pressure-weighted mode gradients are nonzero (up to 1.98×10⁻⁶ for P4), so the point is not stationary in the modes; the conclusion holds for this first-order reason and the Hessian decides nothing there.", "quote": "one positive eigenvalue, which at a non-stationary point decides nothing. The candidate bears this out: its coefficients" }, { "what": "§5 Newton line", "source_says": "Along the Newton direction the five parameters are re-equalized at each multiplier.", "report_finds": "The Newton-line file is the first-order path p₀ + Q·m·a_N, rebuilt at m = 1 and 2 to 1.1×10⁻¹⁵ and 5×10⁻¹⁶; on it the least rises only to 5.3×10⁻⁸ above the base.", "quote": "cusp, is the curve's there to 1.1×10⁻¹⁵ and 5×10⁻¹⁶ (`newton_line_rows_rebuilt`) (Z8)." }, { "what": "Takeover window near m ≈ 3.228", "source_says": "Near m ≈ 3.228 the new branch begins to take over; a finite ascent window bounded by branch opening.", "report_finds": "In the ledger the full-phase scale falls below the four branches' at m = 3.2235 to 3.2245, agrees again from 3.2246 to 3.2275, and falls below for good from 3.228; the candidate at 3.2275 is unaffected.", "quote": "below the four branches' at m = 3.2235 (by 1.0×10⁻⁷), 3.224, 3.2244 and 3.2245 (by 1.5×10⁻⁹), agrees with them again" } ], "open_items": [ "An interval enclosure of the mode gradients or of the latent branch's margin by itself (the certificate bounds the least", "Whether the eight-mode family, or a wider one, goes further; the five-branch", "The excluded candidate. Any bound for Moser's problem.", "That its rows are not one smooth family is an inference —" ], "cross_refs": [ "RUN-002 (sense of the PARTIAL verdict)", "RUN-010 (candidate at 60 digits)", "RUN-011 (smooth minimum just before the switch at π/2 + α, as on Round 08's curve)", "RUN-014 (solution of Round 13's system, branches, interval machinery and certified lower bound on Round 13's least)", "RUN-015 (split curve's certified least, an upper bound used for the 1.78×10⁻⁸ comparison; Round 14's evaluator also put low branches low)" ], "unclear": [ "The verdict sentence spans two lines of the report; verdict_line joins them with one space.", "The mode coordinate t is not pinned uniquely by the curve file: Round 13's layer or any layer within 10⁻⁷ of it reproduces the file; the report takes Round 13's.", "The identification of the Newton direction (Newton step on the six most negative eigen-directions) is the report's inference; the round does not state how it is formed.", "That the ledger's rows are not one smooth family is an inference from rates; the ledger's parameters are not shipped, so its rows are checked only for their own arithmetic.", "'Not a strict local maximum' is labelled 'corrected', not 'refuted': the conclusion is true, but for a first-order reason, not the Hessian reason the round gives.", "The Hessian, Newton direction, compensation and ledger checks are, in the report's words, 'the data's own arithmetic, recomputed'; labelled 'numerical only'.", "The certificate's upper end uses RUN-014's certified lower bound on Round 13's least, recomputed here; labelled 'exact-arithmetic certificate'.", "The round's own 'no interval or Arb certificate yet' is not recorded as an open item, because this run supplies that certificate." ], "for_paper": "Round 15's eight-mode candidate stands. At 40 digits its four branches and its latent fifth reproduce to 2×10⁻¹⁴: the least is the 120° cusp, 0.99891448071696448, with nine local minima and a latent margin of 3.0045884×10⁻⁹. This run certifies in interval arithmetic (Simpson boxes with interval Taylor bounds, RUN-014's cover) that the candidate's least exceeds Round 13's by at least 1.3725×10⁻⁷ and at most 1.3758×10⁻⁷. Round 13's point is not stationary in the modes (pressure-weighted mode gradients up to 1.98×10⁻⁶ at RUN-014's solution), so 'not a strict local maximum' holds for a first-order reason, and the Hessian's positive eigenvalue decides nothing there. The base's low branches tie with the cusps only in the round's evaluator (they sit 9.66×10⁻¹⁰ above them), and their parameter gradients hold only to 1.02×10⁻⁵. The ledger's takeover opens and closes at m = 3.2235 to 3.2245, before the stated 3.228, and the headline least, gain and margin are printed past their digits; no Moser bound is claimed." } ]