← Lebesgue Universal Covering Problem / II · Computation and Reproduction / LESR · Skew-Field Variational Reconstruction

Lebesgue Universal Covering Problem II · Computation and Reproduction LESR · Skew-Field Variational Reconstruction 懸衡 (Xuán Héng) NOT ESTABLISHED NOT AS STATED PASS

LESR: Papers 00–09B and the midpoint handoff

Every formula LESR derives holds, every constant of the barrier reproduces to the last digit, and Paper 09B's critical velocity is exact at 70–90 digits. But its first-order model is not the first-order behaviour on the Lebesgue kernel: Papers 03–05's LP, flow and dual certify placements that are not stationary. Paper 07's translation KKT is not established as argued, Paper 08's reason for criticality is not the right one (the fact holds numerically), and the second-order slivers open at the antipodal walls, not where the papers look.

NOT ESTABLISHED · NOT AS STATED · PASS This page is the second layer: the finite claims of the layer-1 document "LESR · Skew-Field Variational Reconstruction" 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)LESR · Skew-Field Variational Reconstruction · LESR Paper 00: From Skew Fields to Variational Shape Closure — Reusing the Existing Support-Skew-Field and Shape-Update Formula to Establish a Shape-Closure Research Line for the Lebesgue Universal Covering Problem, Independent of Area-Value Closure
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Recompute reports · Runs

RUN-050LESR Paper 00: every formula the bridge uses holds, symbolically and on bodies; its one programmatic principle needs a curvature hypothesis at corners, where the first variation of an inward move is zero and the area drops only at second order; the predecessors it builds on are not in the archive

Verdict: PASS. The paper is a bridge, not a result: it states definitions and classical facts, and marks its principles as programmatic. Every formula it uses holds. Two things are noted (see the report).

The report, the code and the gate log: report RUN-050.md · run050_lesr_paper00.py · gate log run050-lesr-paper00.json

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RUN-051LESR Paper 01: the topology layer's two theorems hold on the Lebesgue kernel — at most two crossings per piece pair, and an area that is analytic inside a contact stratum and kinks where the stratum changes; the compiler scaffold implements the monotone envelope but not the stability test the text states

Verdict: PASS. Paper 01 is a framework: definitions, two small theorems, and lemmas on monotonicity and stability. The theorems hold where the series uses them, on the Lebesgue kernel. One thing is noted (see the report).

The report, the code and the gate log: report RUN-051.md · run051_lesr_paper01.py · gate log run051-lesr-paper01.json

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RUN-052LESR Paper 02: the group layer holds — its projectors, its gcd law, and its first-variation theorem, on symmetric bodies and on the kernel's own hull; but on the kernel's seed no pure Fourier mode survives the convexity test, and breaking the symmetry costs area at first order — the recorded seed's asymmetry is search noise

Verdict: PASS. Paper 02 is a framework built from classical facts, and every fact it states holds. Three things are noted. The first two concern the kernel itself (see the report).

The report, the code and the gate log: report RUN-052.md · run052_lesr_paper02.py · gate log run052-lesr-paper02.json

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RUN-053LESR Paper 03: the calculus holds along affine support paths — every formula, the step caps, the contact variation; but on the kernel, where the container is the hull of its witnesses, a witness move turns the container's edges and the first-order change is the edge average, not the affine maximum — so the finite LP finds no descent even where the area's gradient has length 0.39

Verdict: PASS for everything Paper 03 derives. Its first-order model of a witness move is the affine one, and on the kernel that model is not the first-order behaviour (see the report).

The report, the code and the gate log: report RUN-053.md · run053_lesr_paper03.py · gate log run053-lesr-paper03.json

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RUN-054LESR Paper 04: the metric layer and the projection theorem hold (for a cone the descent inequality is an equality); but on the kernel the projected velocity is zero at every placement tried, so the flow never starts — at a generic placement the area falls at 0.390 per unit along its steepest descent

Verdict: PASS for everything Paper 04 derives. Its flow runs on Paper 03's affine cone, and on the kernel that cone holds no descent, so the flow never starts (see the report).

The report, the code and the gate log: report RUN-054.md · run054_lesr_paper04.py · gate log run054-lesr-paper04.json

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RUN-055LESR Paper 05: the finite duality is Farkas and holds; the smooth No-Slack theorem holds and carries the curvature hypothesis Paper 00 lacked; but on the kernel the finite dual certificate exists wherever it is solved — at a generic placement whose area gradient has length 0.39 too

Verdict: PASS for everything Paper 05 proves. §68 says the finite dual can be solved now on the Lebesgue line's own witnesses; solved, it certifies a placement that is not stationary (see the report).

The report, the code and the gate log: report RUN-055.md · run055_lesr_paper05.py · gate log run055-lesr-paper05.json

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RUN-056LESR Paper 06: the barrier and its four-direction orbit reproduce, and target-specific closure is the right accounting; but Task B has a one-line answer — no measure on the four directions can match the barrier's curvature once cos 2θ is in the library — and the ledger still marks the ρ₅ = 0 slice closed, which RUN-048 found is not established

Verdict: PASS for what Paper 06 proves and computes. Its accounting (value proof, barrier shape and global shape kept apart; proof closure not inherited by a shape target) is right. Three things are noted (see the report).

The report, the code and the gate log: report RUN-056.md · run056_lesr_paper06.py · gate log run056-lesr-paper06.json

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RUN-057LESR Paper 07: every constant of the root-6217 barrier reproduces from the contact equations to the last printed digit, and so does the 13-interval signature; but the translation KKT is not established as argued — the claimed subgradient (g₃, 0) fails in a direction where B3 and B5 move left together; the barrier is still the constrained minimum, with a multiplier of 0.047, not 0.123

Verdict: PASS for the reconstruction; NOT ESTABLISHED for the translation KKT as argued (§15–16); the conclusion it draws holds numerically.

The report, the code and the gate log: report RUN-057.md · run057_lesr_paper07.py · gate log run057-lesr-paper07.json

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RUN-058LESR Paper 08: the two-gate reduction and its arithmetic hold, the 10 / 141 / 44 split reproduces, and the rotation is first-order critical on the true hull; but not for the reason given — in the finite active-contact model the paper names, turning B5 costs +0.029 at first order — and the "honest" 195-root shape ledger leaves out the 14.6 % of root 6217's box that Cycle 11 closed for A₃⁺ only

Verdict: PASS for the reduction, its arithmetic and its partition. Two things are not as the paper states them (see the report).

The report, the code and the gate log: report RUN-058.md · run058_lesr_paper08.py · gate log run058-lesr-paper08.json

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RUN-059LESR Paper 09A v0.2: the 18-piece signature and both primary O(ρ) arcs reproduce, and the branch's velocity tends to (5.157, −0.1892, 0); but at properly minimized placements the three "secondary slivers" have zero width — the diagnostic's come from a stored sample 1.5×10⁻⁹ above the minimum — and the first epi-derivative has a single zero, with a slope in every direction

Verdict: PASS for the withdrawals, the blow-up chart, the 18-piece count, both primary arcs, the numerical centre and ratio, and the box. One thing is not as the paper suggests: at the minimum, the three secondary slivers have zero width.

The report, the code and the gate log: report RUN-059.md · run059_lesr_paper09a.py · gate log run059-lesr-paper09a.json

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RUN-060LESR Paper 09B: every wall constant, the critical velocity and the transition speeds reproduce from the barrier's geometry, and on the primary chart the first area variation is exactly γ_w·w — the u and v terms cancel identically; the full Φ₁ stays non-negative along −w only by 0.035 per unit, and along the critical velocity the second-order slivers open at the antipodal walls, not where §18 looks

Verdict: PASS for the wall constants, the collapse equations and their solution, w* = 0, the transition speeds, the primary 15-piece structure and Φ₁^primary = γ_w·w. Two things are noted: how thin the full Φ₁'s margin is along −w, and where the second-order slivers actually open.

The report, the code and the gate log: report RUN-060.md · run060_lesr_paper09b.py · gate log run060-lesr-paper09b.json

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RUN-061the LESR midpoint handoff: every number it records is this tree's to the printed digit, and its three ledgers and target-specific closure are right; of what it lists as done in Paper 07, B3's subgradient is not; run here, the counter-certificate line finds no lower branch, finds the new topology, and finds the finite dual uninformative

Verdict: PASS for the barrier state, b* and g₃, the three ledgers and target-specific closure, cell 131, the artifact index and the package. Four things are not as the document states them, and the counter-certificate line it assigns has, in this tree, a first answer.

The report, the code and the gate log: report RUN-061.md · run061_lesr_midpoint_handoff.py · gate log run061-lesr-midpoint-handoff.json

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