# RUN-040 — Round 37: the three depth-32 certificates are this tree's own, leaf for leaf; cell 33's continuation to depth 42 is this tree's, every checkpoint and all 166 segment files byte for byte; the merge never checks the premise its theorem rests on **Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab **Date:** 2026-09-27 **Subject:** `AMRAL_Lebesgue_Finite_Closure_Round_37_Depth32_DFRONT_Shards_v0.1.md` (AMRAL-LUC-FC-R37, dated 2026-09-20) and its package, read from the archive: - the three shard files, `COVERAGE_DASHBOARD.json`, `DEPTH32_BATCH_STATUS.json`, `CELL33_DFRONT_CHAIN.json`, `COMPOSITIONAL_DFRONT_MERGE_SPEC.json`, `NEXT_DEPTH32_SHARDS.json`, `VALIDATION.json`, `State_Crystal`; - cells 49, 64 and 66's certificate packages — `cell49.rhcert` (41,150,609 bytes), `cell64.rhcert` (44,278,537 bytes) and `cell66.rhcert` (44,674,727 bytes), their indexes, manifests and replay records; - cell 33's DFRONT chain — the four checkpoints (the depth-36 one Round 36's file) and their manifests, the 166 segment files with their records, and the merge spec again; - the reference bundle's ten scripts (the merge script shipped twice, identical); Round 29's markers and Round 26's ledger again. **Tools:** [`run040_round37_depth32.py`](../code/run040_round37_depth32.py) — RUN-037's decoder and replay, carried to depth 32; RUN-031's migration from the start bands on RUN-030's lifts; RUN-030's continuation of cell 33, one depth at a time from RUN-039's depth-36 frontier; RUN-039's DFRONT decoder and encoder; RUN-032's and RUN-034's outer polygons for markers 49, 64 and 66 **Logs:** [`run040-round37-depth32.json`](../data/gate-logs/run040-round37-depth32.json) **Verdict: PASS for eleven of Round 37's twelve judgments — the three trees, the three lower bounds and three byte replays, the promotion, the conservative geometry status and the two readings of the global claim. UNMEASURED for the MPFR marker replay, whose records are not shipped and whose arithmetic is not here.** - **The three certificates.** - Decoded here: the headers, the leaves using up each file, and the indexes of L + 1 offsets. - Each cell is lifted here to depth 32 and every leaf certified on its intended box from the band of its slack. On every one of the 9,986, 10,539 and 10,636 leaves, the certificate's density and hull count are this tree's, and the margins agree to 2.3×10⁻¹⁶. The density counts are the manifests' — and, for cell 49, §2's. - Every one of the 2,441,597, 2,624,186 and 2,646,674 stored vertices is in its tagged core under this tree's integer arithmetic. Every polygon is strictly convex, and every area is above 167/200. - The least margins, 3.153126719767511×10⁻⁸, 1.506934987941895×10⁻⁸ and 6.860894579634212×10⁻⁹, lie on the manifests' leaves and are the replay records', bit for bit, as is every chunk's. - One caution, caught before anything ran: RUN-035 to RUN-038 build a leaf's box by zipping its path against a split sequence of length 30. At depth 32 that would silently drop the last two splits and certify each deep leaf on its depth-30 ancestor's box. RUN-040 carries the sequence to 32 and asserts the lengths. - **The markers.** - The shard files' independent and MPFR margins are Round 29's. - The full polygons under the package's normals and ceiling have 5,156, 2,597 and 1,312 active constraints and the files' areas. - This tree's certificates need 256, 128 and 128 normals. The exact margins are 4.18×10⁻⁵, 6.55×10⁻⁵ and 1.71×10⁻⁴. - The MPFR replays' records are not shipped. - **The audits.** 1,659, 1,702 and 1,710 unique witness-core states on this tree's lifts, as the shard files say. Their MPFR results are recorded, not recomputed; the audits' own records are not shipped. - **Cell 33, depth 36 to 42.** - Carried here one depth at a time from this tree's depth-36 frontier (RUN-039), under the package's pad. - At 38, 40 and 42 the open boxes are the DFRONT files', code for code: 27,042, 40,934 and 78,507. - Each checkpoint file, written here from the spec with its parent checkpoint's digest in the header, is the shipped file byte for byte. The depth-36 checkpoint is Round 36's file. - Between checkpoints, the nodes evaluated and newly certified — (63,980, 18,975), (118,666, 45,441) and (205,928, 65,391) — are the manifests'. The least new slacks agree to 10⁻¹⁵, and the volumes and ratios (0.481963, 0.378430, 0.479473, and 0.087451 over six depths) follow. - 78,507 boxes are still open at 42, so the cell stays PARTIAL, as §11 says. - **The 166 segments.** Each of the 29, 55 and 82 is this tree's continuation of its parent slice: - the slices tile the parent checkpoint in canonical order; - each segment file, written here from this tree's descendants of its slice, is the shipped file byte for byte; - its evaluated and certified node counts follow from this tree's counts under the slice, and so does its volume; - the segments' boxes together are the child checkpoint's. The parent-slice hashes (a slice's codes as text) and the depth-42 bundle of segment digests are recomputed here. - **§9, the compositional merge.** - The theorem is right as stated: slices of a disjoint frontier, each refined only inside itself, give disjoint descendants. - Its premise — every segment box inside its own slice — holds for all 146,483 shipped boxes, checked here directly on the files. - The merge spec's seven requirements do not include that premise, and the shipped merge does not test it: it reads neither the slice hashes nor the parent boxes, only their number. - Two segments of one box each, the second lying in the first segment's parent, meet all seven requirements and overlap. - So containment comes from the refinement code's construction, not from the merge. A merge meant to stand on its own needs one ancestor lookup per box. - The shipped merge writes exactly the depth-42 manifest's fields. The depth-38 and depth-40 manifests have other fields, so their merges were not this script. - **The prepared trees.** - Cells 12, 59 and 57, lifted here, have 21,789 / 10,895, 22,009 / 11,005 and 22,261 / 11,131 nodes and leaves, the ledger's counts. Their topology and axis digests are the preparation file's. - Cells 49, 64 and 66's trees match their shard files the same way. - **The stages.** - Candidates {47, 48, 49, 64, 66, 68, 69, 70, 75, 76}: 10/77 and 10/66, with 56 below. - The depth-32 wave: 49, 64 and 66 are candidates; 12, 57 and 59 prepared; 45, 46, 58, 60, 65 and 67 remaining. - Geometry 66 / 11 with no new closure; cell 33 PARTIAL. - **The scripts,** read after the rebuild and never run (W8). - The mpmath emitter's start bands are RUN-035's, band for band. The byte emitter starts at the mpmath ledger's density, and the assembler's index counts L + 1 offsets. - The verifier quantizes the cores outward to a 10⁴⁰ grid and demands strict convexity. But it first rounds each rational endpoint to a 140-digit decimal, to nearest — the step RUN-034 found in the MPFR wrapper. This tree's replay has no such step. - The dominance audit keys a state by both backends' rotation and half-widths, which is the count made here. - Both marker replays prune the full cyclic hull at Round 29's density. They take the largest candidate over every sector the angle's interval meets: a valid bound, looser at a sector boundary. - The refinements take each box's floats straight from its code rather than by the lift's halving. The least new slacks move by 10⁻¹⁵, and no decision moves. - Not in the archive: the mpmath leaf ledgers and reference leaves the emitters read, the three trees' bits (their digests are shipped), and the marker and dominance scripts' outputs. Drill: 17 / 17 plants caught by exactly the named checks, 4 / 4 controls undisturbed, 231 / 231 must-stay-green assertions held. Each of the twelve checks was blinded in turn, and each was reported blind in exactly as many cases as plants name it. --- ## What Round 37 claims 1. **Cells 49, 64 and 66** (§0–4). - Trees rebuilt from durable sources: 19,971 / 9,986, 21,077 / 10,539 and 21,271 / 10,636 nodes and leaves, the ledger's counts. - Lower bounds 9,986/9,986, 10,539/10,539 and 10,636/10,636, least margins 3.15312672×10⁻⁸, 1.50693499×10⁻⁸ and 6.86089458×10⁻⁹; cell 49's density counts. - Marker uppers 0.8349813779061498, 0.8349716675759491 and 0.8348951300993619, independent and MPFR. - MPFR dominance over 1,659, 1,702 and 1,710 states. - RHCERT with 2,441,597, 2,624,186 and 2,646,674 vertices, replayed from the bytes alone. - All three PUBLICATION-CANDIDATE-SHARD. 2. **Coverage** (§5, §13). 10/77 ≈ 12.99 % and 10/66 ≈ 15.15 %, with 56 below; depth-32 batch 3/12 candidates and 3/12 prepared. 3. **Cell 33** (§6–8, §11). - Depth 36 → 38 → 40 → 42 in 29, 55 and 82 segments: 14,027 → 27,042 → 40,934 → 78,507 pending. - V from 2.9748×10⁻⁸ to 2.6015×10⁻⁹, V₄₂/V₃₆ = 0.087451, with no plateau. - Still PARTIAL; geometry 66/77, residual 11, 51/77 at depth 36. 4. **The compositional merge** (§9–10). Q = ∪Qᵢ is interior-disjoint from the parent's disjointness, each Qᵢ ⊆ ∪Pᵢ, and segment-level disjointness. It is merged by hashes and slice coverage, without a global scan. 5. **The next shards** (§12). Cells 12, 59 and 57: 21,789 / 10,895, 22,009 / 11,005 and 22,261 / 11,131, node counts matching. ## A — sources **A0.** 49 anchors. Round 37 has no reference list. ## W — rebuilt | check | Round 37 | rebuilt | result | | --- | --- | --- | --- | | W1 | §0–4 | the three certificates decoded; every vertex in this tree's arithmetic; each leaf against this tree's certificate from its start band | PASS, 9,986, 10,539 and 10,636 | | W2 | §2–4 | markers 49, 64 and 66 | PASS; the MPFR replays' records not shipped | | W3 | §2–4 | the audits' state counts on this tree's lifts | PASS | | W4 | §6–8, §10 | cell 33 continued here to 42; the four checkpoints box for box and byte for byte | PASS | | W5 | §6–9 | the 166 segments against this tree's continuation; the merge's premise; its requirements | PASS; the requirements do not imply the premise | | W6 | §1, §12 | the six trees lifted, encoded and hashed | PASS | | W7 | §5, §11, §13 | the stages; the geometry unchanged | PASS | | W8 | — | the ten scripts | read; the merge never tests containment | ## P — the package **P1.** - Each certificate's manifest has its file's digests and sizes. The replay record's chunks tile the leaves, each with its own count and no failure, and the shard file carries the manifest's figures and the package's size and digest. The main archive holds a second copy of cell 49's package, the same file. - Every segment record names its file's digest, its parent checkpoint's digest, the dependency hash and a slice as long as its range. Each checkpoint's manifest has its segments' number and totals — pending, evaluated, certified, the least new slack — and the slices' hashes and the bundle recompute. - The chain record, the merge spec (both copies identical) and the checkpoint manifests agree, and the depth-36 checkpoint is Round 36's file. - The depth-32 batch and the next shards have the Round 26 ledger's paths, node counts, margins and closure depths. - The crystal has the four checkpoints, the ratio, the three cells and their audits, 10/77 and the prepared shards. **P2.** `VALIDATION.json` describes the document and the crystal correctly. Its eight file digests are the archive's, its four package digests are the package's, and its seventeen checks follow from the data. ## Round 37, read | Round 37 | statement | verdict | note | | --- | --- | --- | --- | | §1 | node-count match = PASS | PASS | the six trees here (W6) | | §2–4 | 9,986/9,986, 10,539/10,539, 10,636/10,636 lower | PASS | this tree's certificates from the start bands (W1, W6) | | §2–4 | the same, bytes-only replay | PASS | this tree's arithmetic, this tree's own polygons (W1) | | §2 | MPFR marker replay: PASS | UNMEASURED | no record shipped; no MPFR here (W2) | | §0 | 49, 64, 66 → PUBLICATION-CANDIDATE-SHARD | PASS | the bytes (W1), the markers (W2); the MPFR dominance not recomputed | | §9 | the compositional merge theorem | PASS | a reading, not a registry judgment: right as stated; its premise holds here (W5), but the merge's requirements omit it | | §11 | geometry status remains conservative (PARTIAL) | PASS | 78,507 open at 42 here too (W4) | | §0, §15 | a_Leb ≥ 0.835 NOT CERTIFIED | PASS | reading | ## The status registry Round 37's 12 judgments are reviewed. It uses no phrase for the first time. | verdict | judgments | | --- | ---: | | PASS | 11 | | UNMEASURED | 1 | ## The drill | kind | cases | named checks | | --- | ---: | --- | | the document's 78,507 | 1 | A0 | | a leaf refused | 1 | W1 | | an index offset | 1 | W1 (also P1) | | a marker margin | 1 | W2 | | an audit's state count | 1 | W3 | | a checkpoint box | 1 | W4 (also W5, P1) | | a checkpoint volume | 1 | W4 | | a segment's volume | 1 | W5 | | a segment box | 1 | W5 (also P1) | | a prepared tree's axis digest | 1 | W6 | | the dashboard's count | 1 | W7 | | the scripts' facts | 1 | W8 | | the merge spec's segment count; the crystal's 10,539 | 2 | P1 | | a VALIDATION digest; a VALIDATION check | 2 | P2 | | a review dropped | 1 | R1 | Four controls: - the unmodified inputs; - the document with CRLF line endings; - the chain record with its keys sorted; - the batch file with its keys sorted. **17 / 17 plants, 4 / 4 controls, 231 / 231 must-stay-green assertions; no check without a plant.** | check blinded | cases reported blind | plants naming it | | --- | ---: | ---: | | A0 | 1 | 1 | | W1 | 2 | 2 | | W2 | 1 | 1 | | W3 | 1 | 1 | | W4 | 2 | 2 | | W5 | 3 | 3 | | W6 | 1 | 1 | | W7 | 1 | 1 | | W8 | 1 | 1 | | P1 | 5 | 5 | | P2 | 2 | 2 | | R1 | 1 | 1 | ## What this round does not establish - **The MPFR marker replays and dominance audits.** Their records are not shipped, and there is no MPFR here. - **The merge in general.** For these files, containment is checked here box by box. The shipped merge alone would accept a segment that strayed outside its slice. - **The other ten residual frontiers.** Round 37 recovers none, and Round 38 is to begin. - **The global theorem**, which Round 37 does not claim.