← Lebesgue Universal Covering Problem · PROGRAM-UNIVERSAL-COVERING · LESR-6217
Lebesgue case, layer one (continued): from the second-order coefficient to a local 'research-interval' closure of [0, 0.003] — 19 papers, one handoff, one open-questions list, and the recovered BCODR-LUC-06 and 09V snapshot
LESR (Lebesgue Extremal Shape Reconstruction) is the line on which Neo.K moved the variational technique of this site's skew-field case onto the Lebesgue problem; layer one of this case carries its Paper 00 and layer two recomputes up to Paper 09B. This batch is the continuation: Papers 09C to 09U, the 2026-09-24 Root-6217 handoff, and a 20-question 'open questions' list; on 2026-10-07 BCODR-LUC-06 and the 09V tail snapshot were recovered as well. They study the root-6217 local minimum — positive rotation near the convex hull of a disc, a Reuleaux-3 and a Reuleaux-5 (area A0 ≈ 0.8349861, below 0.835). The authors split the rotation radius ρ into a micro range [0, 1e-4], eleven bridge cells B01–B11 (up to 0.003) and a tail [0.003, π/120], and aim to prove A(ρ) ≥ A0 + 0.1ρ². The local range [0, 0.003] is considered closed at the authors' own 'research-interval' authority; the tail only in part; positive rotation at root 6217 itself and global Lebesgue are open.
Reconstruction update · 2026-10-08: reconstructed from surviving records, with sources and limits stated; originals and new implementations are distinguished. Overview and download bundle.
Honest status: no paper claims to improve or complete the global Lebesgue bound; every paper ends with non-claims, the handoff (§26) says 'GLOBAL LEBESGUE: OPEN' and 'ROOT6217 POSITIVE ROTATION: OPEN', and 09K says aLeb ≥ 0.835 is 'NOT CERTIFIED'. A0 ≈ 0.8349861 is below 0.835, and the papers study a constrained local minimum, so completing them would not by itself give a ≥ 0.835 certificate for that cell (this site's reading, not the papers' statement). 'RESEARCH-CLOSED' is an author-defined status word: it holds over the declared velocity box, with mpmath.iv interval jets and an assumed 1e-10 cell-centre evaluation error contract; every numerical result is research-level and unrefereed. The handoff keeps RESEARCH-CLOSED, LOCAL-REPLAY-PASSED and PUBLICATION-CERTIFIED as three separate levels, and only the first has been reached. A spot check with independent convex-hull code, made while preparing these pages, supports the local anchor (A0 ≈ 0.83498611055; the critical path's (A−A0)/ρ² → ≈ 0.34802; the 09C secondary path → ≈ 0.22559); B01–B11, the micro range, the tail and 09T's 64-QP step were not independently checked.
Not yet covered by layer two: layer two · Computation and Reproduction recomputed the 42 documents of layer one and the material up to Paper 09B; it has not recomputed this batch. Layer two's RUN-049 (a boundary-locking lemma) and RUN-057 (barrier constants) support two inputs of this batch; how its other findings on root 6217 (RUN-048, 058) relate to this batch's handoff ledger is discussed on the MCC-OE hub. The 64-QP step of 09T and 09U (pairing the universal lower bound of a first-order sign region with any finite chart's higher-order data) is the logical hinge of the micro-range seal, and layer two should test it first.
Authorship and provenance: the headers of the 19 papers, the handoff and the PPS series read 'Author: Neo.K; AI research collaborator: Aletheia / ChatGPT, GPT-5.6 Sol; Institution: EveMissLab / 一言諾科技有限公司'. The open-questions list carries no signature, date or status header. Some scripts write to /mnt/data (a ChatGPT sandbox path); 09L and 09N mention a private research-drive folder (informational only). The recovered BCODR-LUC-06 carries 'Author: Neo.K' and 'AI collaborator: Aletheia / ChatGPT, GPT-5.6 Sol'.
Corrections inside the series, and missing files: 09E corrects 09C and 09D on the vertical term and the 'five slivers' count, and withdraws 09D's flat-minimizer-face reading; 09J's remainder is replaced by 09K; 09L's B04 split is dropped by 09M; 09N's Route A is replaced by 09O's explicit coefficients; 09F's micro plan is replaced by 09U. Missing, and recovered: the original Paper 09V text is unavailable; a reconstructed text dated 2026-10-08 is now provided (the recovery bundle leaves only an inferred working title, which this site does not adopt). The 125-file LESR_09V_Tail_Campaign_Snapshot_2026-09-24.zip and BCODR-LUC-06 were recovered on 2026-10-07 — see the next note and the BCODR-LUC-06 page; the cloud checkpoint files named in 09L and 09N have still not been found. Inconsistencies of presentation: the ladder order (the open-questions list puts Finite-Branch below Counterexample-Resistant; PPS-06 reverses them) and 'B' meaning witness bodies, bridge cells and branch classes at once.
The recovered 09V snapshot and BCODR-LUC-06: recovered by GPT from the ChatGPT Library / Drive on 2026-10-07; all 27 checksums inside the recovery bundle (SHA-256 5532652b…) match, and this site additionally replayed the hash and byte count of each of the 125 files in the snapshot's embedded manifest, with zero mismatches. The snapshot is a mid-run state saved against timeouts, not a conclusion: its manifest records 9 of the 29 hard parents closed (1, 13, 45, 47, 95, 143, 169, 178, 186) and states tail_gate: OPEN, root6217_positive_rotation: OPEN and global_lebesgue: OPEN. It is the earlier 9/29 state: the handoff's cloud-freeze list has one more parent, parent 0, and MCC-OE-27 states explicitly that the final handoff of 9/24 added parent 0 to the earlier nine (9/29 → 10/29), so the snapshot's state file for parent 0 is still OPEN (10 boxes in the queue). So '9 closed' has file support in this snapshot, while the tenth (parent 0) and the '19 remaining' rest on the statements of the handoff and MCC-OE-27, with no replayable file on this site. The snapshot zip is 33.6 MB (about 359 MB unpacked), above the hosting platform's limit for a single static file (25 MiB), so it is not on the site and is kept in full in AMRAL's research archive; the site carries only its embedded manifest, the eight checkpoint texts, the original BCODR-LUC-06 package zip, its two JSON files and the verifier script. The BCODR-LUC-06 verifier depends on the UESFCM Cycle 17 kernel, which is in no recovered package, so that step was not re-run here; only the JSON's structure and the paper's table were checked, and the VALIDATION.json inside the original package records the authors' own runtime replay (PASS), not an independent one.
Reading path: start with the open-questions list (the authors' own list of what cannot yet be excluded) and the handoff (the state-of-play ledger); for one complete certificate chain read 09C → 09E (with its self-correction) → 09H (research interval versus audited kernel) → 09I → 09K (the recipe for closing the first parent) → 09M → 09O → 09T → 09U. Papers 09G, 09L, 09N, 09P, 09Q, 09R and 09S have no pages of their own: they are listed in the table below and available for download. ANWIS_08 arrived with this batch but has nothing to do with Lebesgue (it is a draft of a different series) and is not included here; for the PPS series (Papers 02–06) see Projection Possibility Space.
The open-questions list, 12 LESR papers, the handoff and BCODR-LUC-06 have pages of their own; the other 7 papers are in the table below.
The original bytes of the 15 documents with pages and of the remaining files, unmodified; the 20 original zips are attached too (paper, verifier script and JSON ledger). The files recovered on 2026-10-07 are in the recovered/ subfolder; the 09V snapshot zip (33.6 MB) exceeds the hosting platform's limit for a single static file (25 MiB) and is not on the site, only its embedded manifest and the eight checkpoint texts are. The other 7 papers carry a one-line claim and a flag ('none' = no overclaim found; 'mild' = wording a reader could over-read, or corrected later in the series). Original bytes are retained; reconstructed material is distinguished by its directory and page labels.