# RUN-003 — Round 02 (v0.2.2): the ledger, the scans and the exact values reproduce; the main report's numbers are its grid's, and the three-link family goes higher than its corrected curve **Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab **Date:** 2026-10-19 **Subject:** `Moser_Skew_Lab_v0.2.2.zip` (SHA-256 `4375eba4…`, pinned in [`corpus.py`](../code/corpus.py)): 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 **Tools:** [`run003_round02_phase_scan_chirality.py`](../code/run003_round02_phase_scan_chirality.py) — its own grid program with dual multipliers (HiGHS dual simplex, tolerances 10⁻¹⁰, as RUN-002, every scale re-evaluated at the solver's translation); exact minima over all phases by a 3,600-phase scan with every local minimum refined by golden section; [`gate.py`](../code/gate.py) for the drill **Logs:** [`run003-round02-phase-scan-chirality.json`](../data/gate-logs/run003-round02-phase-scan-chirality.json); citation snapshot [`round02-citations.json`](../data/external/round02-citations.json) **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. Reproduced: the best grid candidate's contact-pressure ledger to the digit; the 1,080 logged phase-scan rows of the three curves the documents define, with their minima, maxima, barriers, local-minimum counts and contact-signature counts (for the two undefined arcs, the summary agrees with the round's own rows); the exact three-normal formula and every exact value and derived difference the two addenda state, to 10⁻¹⁰; and both citations. Not as stated: the main report's numbers and its conclusion were computed on a 256-direction grid that misses the triangle's hypotenuse normal, where the program is a relaxation — the round's own exact addendum corrects them; its stated U and best-curve scales sit 2.9×10⁻⁹ and 7.2×10⁻⁹ above the grid program's minima, so their last printed digits are not the curves'. Findings: this run's search of the round's three-link family reaches a congruent critical scale of 0.998697792, 1.24×10⁻⁴ above the round's corrected 0.998573795; the round's grid candidate, which the addendum puts above 1, fits reflected at 0.954897; the two scanned arcs are defined nowhere, and the U-family surface's grid and the jump rule are stated only loosely — only the surface's corners rebuild. The drill planted 19 defects; every one was caught by exactly the checks named for it, and the 4 controls disturbed nothing. --- ## What Round 02 claims Round 02 keeps Round 01's scale program (RUN-002) but moves it to 256 support directions, and scans five curves — U_equal, arc_270, semicircle, segment and Round 01's adversarial candidate — through 360 rotation phases in the triangle T/1.0048. It reads the program's dual as a **contact-pressure ledger**: multipliers λ_j ≥ 0 with Σλ_j u(θ_j) = 0 and Σλ_j h_C(θ_j) = 1, so that s* = Σλ_j h_γ(θ_j − φ). It searches a three-link family — directions (α₁, 0, −α₃), lengths (l₁, l₂, l₃) summing to 1, U_equal being α₁ = α₃ = 90°, all l = 1/3 — over a symmetric grid and 120 asymmetric candidates, and reports a best curve at 0.9933108210 against U's 0.9512368779. Two addenda in the same package correct this. The **exact addendum** places T with its right angle at the origin and its legs on the positive axes, T = {x ≥ 0, y ≥ 0, x/A + y/B ≤ 1}. A rotated polyline with vertices (x_i, y_i), translated by t, lies in sT exactly when every vertex does, that is when x_i + t_x ≥ 0, y_i + t_y ≥ 0 and (x_i + t_x)/A + (y_i + t_y)/B ≤ s for all i; since A, B > 0 the least s comes from the least admissible translation, t_x = −min x_i and t_y = −min y_i, which puts the least x on the leg x = 0 and the least y on the leg y = 0, so s(φ) = max_i (x_i/A + y_i/B) − min_i x_i / A − min_i y_i / B, exactly, with no direction grid. On it the grid candidate is 1.000837369637, and a re-optimized three-link reaches 1.007315039573 — above the certified scale 1. The **chirality correction** resolves that: the certifying preprint covers by congruent copies, reflections allowed, so a curve's congruent critical scale is the lesser of its two handednesses, s_cong = min{s₊, s₋}; the 1.007315 curve's mirror image fits at 0.952938974740, and a search that maximizes s_cong ends at 0.998573794671 (s₊ = 0.998574374711, s₋ = 0.998573794671). Round 02 keeps the two conventions apart for good: orientation-preserving for the problem posed with rotations and translations, congruent when comparing with the 2026 preprint. ## The checks | id | guards | result | | --- | --- | --- | | Y1 | at its recorded phase, the grid candidate's 256-direction program gives its recorded scale and translation; its dual has weight on exactly the three recorded directions with the recorded weights, and balances; the container features there are the recorded ones; no phase gives less by more than 5×10⁻⁷ | **PASS** | | Y2 | each of the 1,080 logged rows of the three rebuildable curves is the 256-direction program's scale at its phase; for all five curves, the summary's minimum, minimum phase, maximum and barrier are the logged rows' | **PASS** | | Y3 | the 256-direction grid misses the hypotenuse's normal, so its value never exceeds the exact one; U's exact value lies above its grid value | **PASS** | | Y4 | the exact formula equals the three-normal linear program; with it, every exact value of the addendum and the correction reproduces, as a minimum over all phases, to 10⁻¹⁰; the stated differences, distances to 1 and chirality gaps are the differences they name | **PASS** | | Y5 | each logged three-link curve is its stated parameters: lengths summing to 1, directions (α₁, 0, −α₃) | **PASS** | | Y6 | the README, the report's phase and ledger tables, and the two addenda state the data's numbers | **PASS** | | Y7 | each curve's count of strict local minima over its 360 logged phases is the summary's | **PASS** | | Y8 | the two works the report cites are what it says; the preprint's convention is the congruent copy, reflections included | **PASS** | | Y9 | for the three rebuildable curves, the distinct sets of container features at the dual's active directions, over 360 phases, number as the summary's contact signatures | **PASS** | ## Findings ### 1. What reproduces, and how exactly The package's `src/contact_lp.py` is a docstring: it states the program and that "the negative HiGHS inequality marginals are nonnegative contact weights", and contains no code. Everything here is recomputed; read after the rebuild, the docstring describes the same program this tree solves. - **The ledger.** At phase 2.171084228853177 the grid candidate's 256-direction program gives 0.9933108209696215 and the recorded translation, both to 10⁻⁹. Its dual puts weight on exactly three directions, 29.53125°, 180° and 270°, with the recorded weights 2.093720944, 1.821719357 and 1.031991268, and balances to rounding. The container features are the recorded ones: the two legs' normals, and at 29.53125° — the grid direction nearest the hypotenuse's 30° normal — the vertex (A, 0) at the end of the short leg, not the hypotenuse (Y1). - **The scans.** The 360 logged rows of each of U_equal, the segment and Round 01's candidate are the 256-direction program's scale at their phases, to 10⁻⁹; the summary's minima, maxima, barriers (maximum minus minimum over the logged phases) and local-minimum counts are the logged rows' (Y2, Y7); and counting the distinct sets of container features the dual's active directions meet, over the 360 phases, gives the summary's contact-signature counts — 1, 4 and 2 (Y9). The report's table states the summary to nine places (Y6). - **The exact values.** The addendum's formula is the triangle's exact program for a polyline — on random polylines it equals the three-normal linear program to 10⁻¹⁰ — and with it, taken as a minimum over all phases, every value the two addenda state reproduces to 10⁻¹⁰: U 0.957829839129, the grid candidate 1.000837369637, the re-optimized three-link 1.007315039573 and its mirror image 0.952938974740, the corrected curve's 0.998574374711 and 0.998573794671 (Y4). The three logged curves are their stated parameters (Y5). U's exact value is the one RUN-002 found for Round 01's U, so Round 02 itself carries the correction RUN-002 made to Round 01's 0.95783224. - **The precision of two stated values.** The README and the report's §4 give U as 0.9512368779 and the best curve as 0.9933108210, as critical scales. On the same 256-direction program U's minimum is 0.9512368750, at phase 0 — the value the report's own §2 table prints, 0.951236875 — and the best curve's is 0.9933108137, reached 1.4×10⁻⁸ rad from the recorded phase. The stated values are 2.9×10⁻⁹ and 7.2×10⁻⁹ above them, so their last printed digits are not the curves' and the README's difference 4.2073943098×10⁻² is the minima's 4.207393878×10⁻² — the kind of finding RUN-002 §3 made for Round 01, at a hundredth of its size, and inside Y1's 5×10⁻⁷. ### 2. The main report's numbers are the 256-grid's The 256-direction grid is spaced 1.40625°, and the hypotenuse's normal at 30° falls between two of its directions. The grid directions nearest it, 29.53° and 30.94°, give support lines through T's two acute vertices, and the polygon the grid cuts out leaves a thin sliver beyond the hypotenuse: it contains T, so on it the program asks less than the triangle does and its value never exceeds the exact one (Y3). Round 01's 192-direction grid held all three normals and was exact for the triangle (RUN-002). The report's phase table, its U and its best curve are therefore relaxation values — U's exact value is 0.957830, the best curve's orientation-preserving 1.000837 — and its conclusion that the three-link family "尚未逼近 Wetzel 認證尺度 1" (has not yet approached the certified scale 1) does not hold as stated: under the congruent convention Round 02 itself prescribes for comparing with the preprint, the family reaches 0.998574 (the correction's curve) and 0.998698 (§3 below), within 1.4×10⁻³ and 1.3×10⁻³ of 1. The exact addendum says the grid underestimated ("可見斜邊法向近似確實造成了尺度低估", the hypotenuse-normal approximation did underestimate the scale) and gives the exact values; this run confirms them. A reader should take the addenda's numbers, not the main report's, wherever the two differ. ### 3. The three-link family goes higher than the corrected curve The correction's search maximizes s_cong over the three-link family and reports 0.998573794671; it does not claim a maximum. This run's own search — Nelder–Mead from four starts, each candidate's s_cong the lesser of its two exact minima over phases on a 360-phase scan — ends at equal thirds with equal turns, and a golden-section refinement of the turn there, on the 3,600-phase scan, gives s_cong = 0.998697792 at l₁ = l₂ = l₃ = 1/3, α₁ = α₃ = 82.435472°, 1.24×10⁻⁴ above the round's value. The curve is mirror-symmetric, so its two handednesses agree. The maximum there is a sharp kink, falling by 1.6×10⁻³ per degree on one side and 4.7×10⁻³ on the other, which is how a search can stop short of it; the round's curve, nearly symmetric with turns of 82.34°, is about 0.1° from it. This is a search, not a proof that the family's maximum is here. It matters for the next round, which measures its gains against a three-link baseline of 0.998574374711, the corrected curve's orientation-preserving value: 0.998697792 is above Round 03's best seven-link, 0.998674157956, and against it Round 03's five-link gain shrinks from 1.80×10⁻⁴ to 5.66×10⁻⁵. Whether Round 03's families contain this curve is RUN-004's question. ### 4. The grid candidate, reflected The exact addendum puts the round's grid candidate at 1.000837369637, above 1. The addendum's status note, written after the correction, says without computing it that asymmetric candidates above 1 are not counterexamples because the preprint allows reflection, and the correction computes the mirror image only of the re-optimized curve. The grid candidate's mirror image fits at 0.954897 (exact, minimum over phases), so its congruent value is 0.954897 and the note holds for it too. Round 01's candidate, scanned here, likewise has orientation-preserving value 0.918731 and congruent value 0.894545 (RUN-002 §4). The addendum's §3 body was not updated by its note: it still says the 1.007315 curve "已逼近認證 Wetzel 三角形的完整尺度" (has approached the certified triangle's full scale) and does not challenge 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. ### 5. What cannot be rebuilt - **The two arcs.** The documents do not define the scanned semicircle and arc_270. Every reading tried, and its largest difference from the logged scan over 360 phases, is in the gate log (`arc_readings`): 64, 128 and 256 chords of total length 1 — RUN-002's reading of Round 01's arcs is the first; the same chords inscribed in the unit arc; the unit arc itself (2,048 chords), centred on the x-axis; and the unit arc turned by a fitted angle or given a fitted radius, both fitted on every tenth phase. The closest unfitted readings still differ by 6.8×10⁻⁵ (the semicircle as the unit arc) and 1.01×10⁻⁴ (arc_270 as 256 chords of total length 1); the fitted ones, on the phases they were fitted to, by 5.1×10⁻⁵ and 7.3×10⁻⁵. Every curve the documents define reproduces to 10⁻⁹. The arcs' 720 rows are reported as the round's; only their summary statistics are checked against them (Y2, Y7). - **The U-family surface.** `u_family_surface.csv` is 14 × 21, and its grid is stated only as α from 65° to 115° and l from 0.18 to 0.42. Its four corners — rows running over l, columns over α — reproduce exactly on a 128-direction grid with 36 phases, and are off by 9×10⁻⁴ to 1.0×10⁻² on the round's 256-direction grid (36 or 360 phases); its interior does not reproduce on evenly spaced axes (off by 1.3×10⁻³ and 2.7×10⁻³ at the two interior cells probed). The surface is the output of a different program from the scans', and how its interior grid was laid out is not stated (`u_surface`, `u_surface_corners_on_256`). - **The jumps.** The report describes a jump as the dual's active directions changing ("當對偶活動方向改變時"), but gives no rule for counting. Counting changes of the active set between consecutive logged phases gives 0, 24 and 4 for U, the segment and Round 01's candidate, against the stated 27, 26 and 26 (`jumps_as_active_set_changes`); the stated indices come in runs of consecutive phases, which suggests another rule. - **The 120 asymmetric candidates** of the topology-guided search are not logged. ## At what strength each result stands | result | strength | | --- | --- | | the exact formula for the triangle | exact (the argument in "What Round 02 claims"; checked against the three-normal LP) | | the ledger, the 1,080 scan rows, the counts, the stated values and differences | numerical, floating-point linear programs and the formula in floating point, to the stated tolerances | | the exact minima over phases | numerical: a 3,600-phase scan, every local minimum refined by golden section; not proved global | | the three-link family's 0.998697792 | numerical: this run's search, refined; not proved maximal | | the two arcs' 720 rows, the U-family surface's interior, the jumps, the 120 candidates | the round's, unchecked | | T/1.0048 covers every unit arc, by congruent copies | cited theorem (preprint; the round cites it as arXiv:2606.14625) | ## The drill | | | | --- | ---: | | planted defects | 19 | | caught by exactly the checks named for them | **19** | | controls (the package unmodified, the report with CRLF endings, the scans copied, the README with trailing spaces) | 4 | | controls that disturbed nothing | **4** | | must-stay-green assertions held | **175/175** | Plants include a triangle leg, a 255-direction grid, a scan row moved and one dropped, a ledger weight, a ledger feature, a barrier, a local-minimum count, a mirror scale, a chirality gap, a three-link parameter, a curve point, the preprint's reflection convention, a digit of the report's table, a reference title, a README digit, the rotation's sign, a term of the exact formula, and a signature count. Each check, replaced in turn by one that always passes, was reported blind in exactly as many cases as plants name it — Y1 4, Y2 6, Y3 1, Y4 5, Y5 2, Y6 8, Y7 2, Y8 2, Y9 2 (nine runs; a blind run writes no log). Two runs of the drill made side by side wrote gate logs with the same SHA-256. The instrument was wrong before it was right, in four ways worth recording. Its first refinement of an exact minimum (SciPy's bounded Brent method, RUN-002's) stalled at kinks and left minima high (`brent_minus_golden`: from 0 to 5.2×10⁻⁹ over the twelve handedness minima); golden section, which cannot stall there, fixed it. RUN-002's two candidate minima, refined the old way, are high by 3.4×10⁻⁹ (orientation-preserving) and 9.6×10⁻¹⁰ (mirror) — below its tolerances and the precision it prints, so none of its findings move. Its first pattern for the preprint's convention matched a sentence about "reflected left-tail contacts", which would have let the check pass for a reason unrelated to the claim; it now looks for the convention itself ("Other orientations are obtained by rotations and reflections, in the congruent-copy convention used for the cover"). A first check of the U-family surface appeared to pass only because that run had replaced it with a blind one. And the drill harness's test for a vacuous plant compared dictionaries holding curves with `==`; it now compares at any depth — and a re-run of RUN-002 under the corrected harness, written to a separate file, matched its committed gate log byte for byte. An independent reading of this report before it was committed found the precision of §1, the verdict's label, the reliance of §2 on orientation-preserving values, and several readings that were measured but not logged; all are corrected here, and the readings are now in the gate log. ## What this run does not establish Any bound for Moser's problem. That the minima over phases, or this run's three-link value, are global. Anything about the two arcs, the surface's interior, the jumps or the 120 candidates. That the preprint's proof is correct. ## Re-running ``` python code/fetch_source.py python code/run003_round02_phase_scan_chirality.py --drill ``` The citation snapshot is committed with this run; `--fetch` rewrites it from arXiv (the records of both cited works and two phrases of the preprint's full text). The run is correct when the gate log is byte-identical to the committed one.