← Lebesgue Universal Covering Problem / Round 06 · Witness Exchange Compiler

Lebesgue Universal Covering Problem Round 06 · Witness Exchange Compiler Neo.K

The Witness-Exchange Eventual-Success Theorem Is Established: A Non-Saturated Family Is Guaranteed to Find a Separating Witness Batch in Finitely Many Steps, but This Round Adds No New Numerical Lower Bound

Round 06 (AMRAL-LUC-FC-R06, 2026-09-18) compiles Round 05's existence claim — that whenever the current witness family is not yet saturated, some larger finite family is guaranteed to raise the lower bound — into a replayable, finitely verifiable Witness Exchange Compiler. The round takes Mishra 2026's public certified result Λ(D,B_3,B_5) ≥ 0.8344 (a five-dimensional placement space, exhaustively subdivided into 486,799,600 certificate nodes, with a reported floating-point error bound of 1.72×10⁻⁹) as its formal starting point, and first lays out the classical seed ladder: λ_1 = π/4 ≈ 0.7853981634, Λ(D,B_3) ≥ π/8 + √3/4 ≈ 0.8257117836, and Λ(D,B_3,B_5) ≥ 0.8344, a strictly increasing chain (sanity check S3: PASS). The core proofs are: (1) the low-area configuration domain of a finite family is a compact set, and for m = 3 the continuous placement dimension is d_m = 3m − 4 = 5, exactly matching Mishra's certified five-dimensional search (sanity check S1: PASS); (2) the Uniform Minimizer Violation Theorem (Theorem 11.1), which shows that if Λ(F) < a_Leb, then on the compact set 𝔐(F) of all area-minimizing covers, the infimum of the worst-target margin, δ_F = min_{U∈𝔐(F)} W(U), is strictly positive; (3) Finite Dictionary Separation (Theorem 14.1) and the Exchange Theorem (Theorem 18.1), which show that a positive separation margin S_F(B) > 0 for a finite batch B is exactly the condition needed to guarantee Λ(F∪B) > Λ(F); (4) the Witness-Exchange Eventual-Success Theorem (Theorem 24.1 — this round's main termination theorem), which proves that as the minimizer-cell radius τ_n, target-dictionary error ε_n, and placement-certificate error ζ_n all shrink to 0, a non-saturated family is certified, at some finite level n*, to yield a separating witness batch; and (5) the Witness Dominance Theorem (Theorem 26.1), which formally classifies Mishra's replacement of the regular polygons P_3, P_5 with the Reuleaux bodies B_3, B_5 as a DOMINANCE-UPGRADE rather than merely adding more test shapes. The round also supplies eight witness-ancestry labels (REUSED / DOMINANCE-UPGRADE / NEW-SEPARATOR / BATCH-ONLY / REDUNDANT / SEARCH-ONLY / FALSE-POSITIVE / COMPUTE-DEFERRED) and a nine-step Full Witness Exchange Compiler pseudocode for Round 07 and the local computation layer to execute directly. The document states plainly that this round does not find any new certified witness family beyond 0.8344, does not claim that the Mishra family's exact minimum equals 0.8344, does not claim that the regular Reuleaux shape B_7 or any other specific shape must be the next hard witness, and does not prove the Finite Witness Attainment Conjecture; the actual numerical work — Mishra certificate ingestion, near-minimizer reconstruction, the adversarial target dictionary, the robust incidence matrix, solving for the separating batch, and a rigorous lower certificate for the new family (C06-1 through C06-6) — is entirely marked COMPUTE-DEFERRED and handed to Round 07. Research direction and methodology are due to Neo.K; this round's AI collaborating researcher and primary executor was Aletheia / ChatGPT, GPT-5.6 Sol.

Round 06 compiles Round 05's existence claim — that a non-saturated witness family is guaranteed to have some larger family that raises the lower bound — into a replayable, finitely verifiable Witness Exchange Compiler. It proves that a finite family's low-area configuration domain is compact, with continuous placement dimension d_3 = 5 at m = 3, exactly matching Mishra 2026's external certified five-dimensional search, and it proves the Uniform Minimizer Violation Theorem: whenever Λ(F) < a_Leb, the infimum δ_F of the worst-target margin over the minimizer set 𝔐(F) is strictly positive. The core Witness-Exchange Eventual-Success Theorem goes further, proving that once the minimizer-cell, target-dictionary, and placement-certificate errors are small enough, a non-saturated family is certified to yield, at some finite level, a separating witness batch B with Λ(F∪B) > Λ(F). This page's point is not a new numerical lower bound, but compiling the intuition that a non-saturated family always has a separating witness batch into a terminating, certifiable algorithm specification. Round 06 states plainly that it does not find any certified witness family beyond Mishra 2026's external 0.8344 bound, does not claim that Λ(D,B_3,B_5)'s exact minimum equals 0.8344, and does not identify the regular Reuleaux shape B_7 or any other specific shape as necessarily the next witness; the actual numerical work (certificate ingestion, minimizer-cell reconstruction, adversarial target search, a new family's lower certificate) is entirely marked COMPUTE-DEFERRED, left to Round 07 and the local computation layer. The exact value of the global bound a_Leb remains undetermined — known only to lie between the proven lower bound 0.8344 (external, not new to this round) and the proven upper bound 0.8440935944 — and this round neither narrows nor claims to narrow that interval.

Connections · Connections

PreviousRound 05 · Saturation Gate · Finite-Witness Completeness Proved, the Saturation Gate Formalized: Finite Witness Attainment Remains an Open Conjecture
NextRound 07 · Minimizer Atlas · Independently Reproducing Mishra's Official 0.8344 Certificate, B7 Remains Only a Search Prior: Threshold-Conditioned Minimizer Atlas and Witness-Switching Lower Bounds

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