← Lebesgue Universal Covering Problem · PROGRAM-UNIVERSAL-COVERING · BCODR-LUC

BCODR-LUC: the bidirectional circular-overlap method applied to root 6217

Lebesgue case, layer one (continued): seven applications of the BCODR methodology to the root-6217 local minimum, LUC-01 to LUC-07; layer two has recomputed six of them

BCODR (Bidirectional Circular Overlap Decomposition and Recomposition) is a general candidate-space search and bookkeeping method designed by Neo.K (see the methodology paper and its layer-two recompute); the LUC series is seven applications of it to the local minimum at root 6217 of the Lebesgue problem, all seven dated 2026-09-21. In the five-dimensional parameter space (x₃, y₃, ρ₅, x₅, y₅) of root 6217 it grows two arrival-time fields — outward from the "candidate structure" (the analytic critical ray of Paper 09B) and inward from the "certificate boundary" (the boxes already pruned by A_ref certificates) — records where and when they meet, and then cuts, separates and recomposes the meeting regions step by step, leaving finite certificates and local calculus to the surviving responsibility spine. It proves no lower bound itself. LUC-01 builds the arrival-time fields; LUC-02 cuts the selected frontier into 288 children and recomposes them into 17 classes; LUC-03 corrects "box contact" into true common-point overlap; LUC-04 and 05 split the earliest overlap points of roots 127 and 131 into normal and tangent parts and classify them by second-order stiffness; LUC-06 corrects the parent ledger and shrinks the selected frontier from 9 to 1 (its page is on the LESR · Root 6217 case, recovered only on 2026-10-07); LUC-07 separates the candidate shape, the certificate germ and the mirror/chirality obligation.

Honest status: this site has read the conclusions of all seven papers and the non-claims and open items each lists, and has seen none that claims to improve or complete the global Lebesgue bound (a_Leb ≥ 0.835 remains NOT CERTIFIED) or that the shape at root 6217 is proved or excluded; for example LUC-01 §38 lists as not claimed that root 131 contains the exact barrier minimizer, and LUC-07 §20 lists the alignment of the candidate with the germ, the global extremal shape, attainment and uniqueness as still open. They are bookkeeping and geometric analysis of the candidate space of one local minimum, with numbers that are research-level outputs of the authors' AI runtime; the formal certification LUC-04 lists (CERT-04A to 04C) and the eight local certificate obligations of LUC-07 have not been done. 'PASS', 'closed' and 'isolated' are the authors' own status words and hold only under the definitions given in each paper.

Layer two has recomputed it (懸衡): layer two · Computation and Reproduction has recomputed the methodology paper and LUC-01 to 05 and 07 (RUN-062 to 068), with the verdict NOT AS STATED · PASS: the numbers nearly all reproduce — on the exact grid, in exact fractions, at 40 to 50 digits; what falls is some of the reading. The contacts are 64 pairs, not the paper's 35, and for 178 of the 195 cells the "distance to the critical ray" is the distance to the ray's end ρ = 0.003 (LUC-01); 127's provenance "flip" is a tie, 1/8 from 144 and from 155 alike (LUC-02); 101, 103 and 104 have four pruned boxes tied exactly and no single provenance (LUC-03); 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, while the isolation of the germs themselves stands (LUC-04, 05); the candidate vector LUC-07 calls unrecorded is in Cycle 17's diagnostic, and 131's corner holds the barrier itself, 1.38×10⁻⁷ lower than the candidate (LUC-07). The "note" on each page gives the result for that paper. LUC-06 was not in the archive then, and layer two has not recomputed it.

Authorship and provenance: all seven headers read 'Author: Neo.K; AI collaborator: Aletheia / ChatGPT, GPT-5.6 Sol; Institution: EveMissLab / 一言諾科技有限公司', dated 2026-09-21. LUC-01 to 05 each ship a package: the paper, JSON/CSV ledgers, a relation graph, a verifier script, STATE and VALIDATION. The hash records inside the packages all match their files (checked here); the five verifier scripts were re-run here and all pass, but they only check the ledgers' own counts, flags and inequalities (LUC-01's script, for instance, merely prints the 35) and recompute no geometry; STATE and VALIDATION are records of the authors' own runtime, not an independent replay. LUC-07 is a single md with no package.

Corrections inside the series, and missing files: LUC-02 renames LUC-01's "zero-radius contact nexus at 131" a "coarse bicontact container" (it has no BOTH child after the cut); LUC-03 proves that the box distance is only a lower bound (T∩true ≥ T_box), turns 131 into a "false zero-radius nexus" and separates "BOTH by arrival" from "balanced true overlap"; the next node LUC-05 proposes is exactly LUC-06, which then shrinks the selected frontier from 9 to 1. The schema and verifier that LUC-07 §23 describes are not among this series' files. LUC-01 to 07 are now complete: LUC-06 was recovered on 2026-10-07, and Neo handed over the other six on 2026-10-08 to be published.

Reading path: to see how the method meets the data read LUC-01 → 02; for the turning point of the whole line read LUC-03 (box contact corrected into true overlap, and 131 corrected); for the germs read LUC-04 → 05; for the ledger read LUC-06 (on the LESR · Root 6217 case); for the candidate shape and chirality read LUC-07. Read each together with the layer-two result on its page; the methodology paper is the BCODR page of the Lebesgue home.