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Lebesgue Universal Covering Problem II · Computation and Reproduction BCODR · Bidirectional Circular Overlap Decomposition 懸衡 (Xuán Héng) NOT AS STATED PASS

BCODR: the method and LUC 01–05 and 07

BCODR's numbers nearly all reproduce — on the exact grid, in exact fractions, at 40 to 50 digits. What falls is the reading: 64 contact pairs, not 35; 127's provenance "flip" is a tie; 101, 103 and 104 are four-way ties; a germ's second-order stiffness needs the Lagrangian weight (127's is 0.2366, not 0.4543), so the two germs "a tenth of a percent apart" differ by a factor of 1.92; the candidate vector LUC 07 calls unrecorded is in Cycle 17's diagnostic, and 131's corner holds the barrier itself, lower still. LUC 06 is not in the archive.

NOT AS STATED · PASS This page is the second layer: the finite claims of the layer-1 document "BCODR · Bidirectional Circular Overlap Decomposition" recomputed, its constructions reproduced, and its theorem-level steps read against the source. It does not rewrite the document and does not treat numerical results as proofs; the formal paper written from these two layers is the third layer, "III · Crystallization". Below are the recompute reports as written (byte for byte the research tree's), with their code and gate logs.

Connections

I · Precursor (the document)BCODR · Bidirectional Circular Overlap Decomposition · BCODR Bidirectional Circular Overlap Decomposition and Recomposition v0.1: A General Methodology That Simultaneously Overlaps Outward-Expanding and Inward-Contracting Circular-Kernel Fields, Cuts Them Apart, and Traceably Recomposes the Candidate Space — Not Specific to the Lebesgue Problem
PreviousLESR · Skew-Field Variational Reconstruction · II · Computation and Reproduction

Recompute reports · Runs

RUN-062BCODR: a closure architecture, as it says of itself; its two differences are metrics on discs, its arrangement's cells are counted right (O(N²), exactly N(N−1) + 2 when every pair crosses), its certificate weights and its UESFCM link are consistent; its root-6217 map would start from the 195-root ledger that leaves out 14.6 % of the box

Verdict: PASS. BCODR is a methodology: it computes nothing, and it says so (§78). Its one status phrase in a judgment position is a label. Its mathematical statements hold where they can be instantiated, and its links to UESFCM and to the certificate grades are consistent. One thing is noted: the root-6217 map it proposes as the next step starts from an incomplete ledger.

The report, the code and the gate log: report RUN-062.md · run062_bcodr.py · gate log run062-bcodr.json

Loads when opened…

RUN-063BCODR LUC 01: the arrival-time field over root 6217 rebuilds from the 200 subroots, cell for cell, and the most balanced eight are the text's; but the boxes touch the prune seeds in 64 pairs, not 35, and for 178 of the 195 cells the "distance to the critical ray" is the distance to its end ρ = 0.003

Verdict: PASS for the domain, every arrival time, balance, entropy and score, the nexus, the finite-radius cells, the most balanced eight, and the records. Not as stated: the active/prune boundary contacts are 64 pairs, not 35. Noted: the frontier is set mostly by where the ray is cut.

The report, the code and the gate log: report RUN-063.md · run063_bcodr_luc01.py · gate log run063-bcodr-luc01.json

Loads when opened…

RUN-064BCODR LUC 02: the 288 children, their signatures and the 17 classes rebuild exactly; but 127's provenance "flip" is a tie — it is 1/8 from 144 and from 155 alike — and a BOTH child is one that holds the point where the later field first touches its parent, which is all 127's "thick corridor" is

Verdict: PASS for the cut, all 288 signatures, the 17 classes and the seven BOTH classes, the second cut's counts, and the records. Not as stated: the provenance flip at 127, and 127 as a thick, provenance-switching corridor. Noted: at its parent's radius a BOTH child is one that holds a point where the later field first touches the parent; the 17 classes need corner contacts and the parent's ρ class.

The report, the code and the gate log: report RUN-064.md · run064_bcodr_luc02.py · gate log run064-bcodr-luc02.json

Loads when opened…

RUN-065BCODR LUC 03: the true overlap times, the modes and 127's one-dimensional reduction rebuild to 40 digits; but 101, 103 and 104 have no single provenance — four prune boxes tie exactly there — and the "reverse numerical error" is LUC 01's ray grid, not the optimizers

Verdict: PASS for the theorem, every true overlap time and gap, the modes, 131's midpoint, and 127's value, provenance and reduction, and for the records. Not as stated: the provenance of 101, 103 and 104, where four boxes tie exactly. Noted: the seven outward-limited gaps are exactly zero, and the ledger's negative ones come from LUC 01's ray grid; the printed r* and λ* are right only to about 10⁻¹¹; at 127 the provenance 155 is right, but no flip led to it.

The report, the code and the gate log: report RUN-065.md · run065_bcodr_luc03.py · gate log run065-bcodr-luc03.json

Loads when opened…

RUN-066BCODR LUC 04: 127's germ, its KKT relation, its critical tangent space and both reduced Hessians rebuild at 50 digits, and the germ is isolated; but the overlap's second-order law is the Lagrangian α H₊ + (1 − α) H₋, softest 0.2366, not the straight-line 0.4543 — the lens grows to 1.036 √ε, not 0.748 √ε

Verdict: PASS for the meeting point, the KKT relation, the rank and the critical tangent space, D₊, H± and Q±, the stiffness as §14 defines it, and the isolation of the birth germ. Not as stated: that stiffness as the overlap's second-order law, and the growth law built on it (§16–19, §22). Noted: the printed z, g±, α and slope carry the printed r's error; the printed T is LUC 03's 5-D value.

The report, the code and the gate log: report RUN-066.md · run066_bcodr_luc04.py · gate log run066-bcodr-luc04.json

Loads when opened…

RUN-067BCODR LUC 05: root 131's interior-midpoint germ rebuilds at 50 digits, and there the straight-line stiffness is the right one; but the "same second-order mechanism" — κ₁₃₁ / κ₁₂₇ = 1.0008 — is a ratio of 1.92 for the overlaps' own constants, and the taxonomy's rule can neither reach the unstable class it defines nor certify isolation

Verdict: PASS for 131's nearest pair, T*, the midpoint and its place inside 131, the rank-1 first order and the four-dimensional critical space, H±, κ₁₃₁, its C_R and C_H, and the isolation of both germs. Not as stated: the comparison of the two germs (§13–14), and §22's rule as a test of isolation. Noted: the midpoint theorem needs the nearest pair of all, and its proof the meeting point's own nearest points; §3 "reproduces" LUC 03 to 8×10⁻¹³; LUC 06 is not in the archive.

The report, the code and the gate log: report RUN-067.md · run067_bcodr_luc05.py · gate log run067-bcodr-luc05.json

Loads when opened…

RUN-068BCODR LUC 07: the midpoint-separation and mirror theorems hold, and Paper 07's barrier is exactly up/down symmetric, hence achiral; but the candidate vector that §6 says is not recorded is in Cycle 17's diagnostic — 7.7×10⁻⁴ from the ray, 0.109 from the IMBG point — and it is not 131's lowest point: 131's corner holds the barrier, 1.38×10⁻⁷ lower

Verdict: PASS for the midpoint separation theorem, T* and κ₁₃₁, the mirror algebra and the support-function covariance, and the up/down symmetry of Paper 07's barrier (CERT-07.4, replayed body by body and at 50 digits). Not as stated: that Cycle 17's candidate is recorded by value only (§6, §20), and the positions of the B5 slivers (§22). Noted: the printed p, q and z* are LUC 03's optimizer values, 5×10⁻⁷ from the exact pair; LUC 06, and the schema and verifier that §23 describes, are not in the archive.

The report, the code and the gate log: report RUN-068.md · run068_bcodr_luc07.py · gate log run068-bcodr-luc07.json

Loads when opened…