← Lebesgue Universal Covering Problem · PROGRAM-UNIVERSAL-COVERING · MCC-OE
Measured-Construct Closure and Observable Exhaustion · Lebesgue case, layer one (continued): a local B01 certification line for root 6217 — 28 papers and one evidence package
MCC-OE (Measured-Construct Closure / Observable Exhaustion) is a methodology-plus-certification line Neo.K wrote for the 'final closure stage' of the Lebesgue case; its header calls it an 'AMRAL / NIOT Bridge Methodology'. The idea: instead of exhausting all geometry, compile every construction into observable intervals (O+), compile theorem failure into threat regions (O−), look for their intersection (the hit set), and require a proof that the probe family used is 'sufficient' for the task; 'untested is not safe', and closure at the level of observables must never be written as global closure. From paper 06 on it becomes a conventional certificate pipeline for one concrete object (interval arithmetic, adaptive covers, Taylor/Dini bridging, convexity lift): near the symmetric triple contact of root 6217, the convex-hull area A of a finite witness family, with the target A ≥ A0 + 0.1ρ² (A0 ≈ 0.83498611). These documents do not define the members of that witness family and ship no compiler code.
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: none of these 28 papers claims to improve or complete the global Lebesgue bound — aLeb ≥ 0.835 remains NOT CERTIFIED, and every closing section lists 'global Lebesgue solved' as not claimed. The constants that appear (A0 ≈ 0.83498611, Aref ≈ 0.8349865, A3⁺ ≈ 0.834781191) are local thresholds, all below 0.835, and never tied to aLeb. What is established holds only at the authors' own 'research-interval' authority (mpmath.iv interval arithmetic; inputs from LESR 09E–09I, which are not in the MCC-OE package but on LESR · Root 6217 local analysis; per-cell areas in double precision under an assumed 1e-10 error contract; no independent directed-rounding backend): (1) the B01 regular charts, boundary and parent (24, 25, 26): A > A0 + 0.1ρ², R2fin > 0.105758 and a uniform margin > 5.758e-11 on the declared box — paper 27 itself calls this 'redundant research provenance', since LESR 09K had already closed B01; (2) the S100 continuum tube (23): C_total ≤ 718.573 < 728.571. Still open: the tail [0.003, π/120] (173 active parents → 29 hard → 10 closed → 19 remaining), the positive rotation and full shape of root 6217, other roots, an independent publication audit, global 'task sufficiency', and global Lebesgue.
Not yet covered by layer two: 懸衡's layer two · Computation and Reproduction recomputed the 42 documents of layer one and the material up to LESR Paper 09B; it has not recomputed this batch. How layer two's existing findings on root 6217 (RUN-048, 049, 058: the ρ5 = 0 'Gate 6217-A' is numerically true but not established; 44% of the root-6217 box is closed only by a Cycle-11 search; Paper 08's 195-root ledger omits the 14.6% closed only for A₃⁺) relate to this series' paper 27 (which lists 'ZERO ROTATION: RESEARCH-CLOSED' and a corrected ledger 200 → 192 → 187 → 182) is not addressed in these documents — they may concern the same gate, and this site draws no conclusion here.
Authorship and provenance: the surviving original headers are signed only 'Author: Neo.K; Institution: EveMissLab / 一言諾科技有限公司', with no AI-collaborator line (the LUC-FC rounds of this case carry 'AI research collaborator and principal executor: Aletheia / ChatGPT, GPT-5.6 Sol'). This site does not guess and will add the line once it is confirmed. The two shipped verifiers (26, 27) only check flags and set arithmetic.
Corrections inside the series, and missing files: 04 corrects the winner-margin definition of 02 and 03; 16 corrects 15 (the compression is an invertible affine map, not a continuous dimension compression); 17 supersedes the 'five slivers' and the vertical term, and finds that the float64 'C↓ ≈ 600 threats' vanish at 80-digit precision; 19 withdraws 18's C² bridge and records the raw Hausdorff bridge as REJECTED (0 of 10,485,760 pass); 22 turns 21's 'quadratic' into 'affine'. Missing, and recovered: the paper MCC-OE-24 itself (on which paper 25 depends) was recovered by GPT from the ChatGPT Library on 2026-10-07 (the recovery bundle's SHA-256 is 5532652b…, and all 27 checksums inside it match), see the page of paper 24; paper 28 is announced, but GPT's search did not find it either (the bundle lists it as MISSING and ships only the paper-27 integrated-state package as the nearest context), the original remains unavailable; a clearly labelled reconstruction from paper 27 is now provided. The load-bearing inputs LESR 09C–09U, BCODR-LUC-06, the 09V snapshot and the Open Questions are on this case's LESR · Root 6217 local analysis (the original Paper 09V text is unavailable; a reconstructed text dated 2026-10-08 is now provided).
Reading path: start with 00 (charter and non-claims), 01 (the ledger) and 27 (the state-of-play ledger); to see how the method evolved read 02, 05, 06; for the failures and corrections the authors wrote down themselves read 10, 16, 17, 19; for the original wording of the 'research-interval closed' claims read 23, 24, 25, 26, together with their authority qualifiers. The other 13 papers have no pages of their own: they are listed in the table below with a one-line claim and a flag, and are available for download.
15 of the 28 papers have pages of their own; the rest are in the table below. This batch adds complete Chinese-English full texts to 15 registered reader pages (15 body parts); authoritative originals and full-text translations can be downloaded separately. Translation is not independent mathematical verification and does not upgrade the original evidence level. The existing reconstructed paper 28 remains unchanged; the other 21 download-only originals below are outside this batch of full-text bilingual coverage and retain their original bytes.
The original bytes of the 15 papers with pages and of the remaining files, unmodified; the files recovered on 2026-10-07 are in the recovered/ subfolder. The other 13 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). The full archive is kept in AMRAL's research archive and verified per file by SHA-256. Original bytes are retained; reconstructed material is distinguished by its directory and page labels.