Second Independent Implementation Replays the marker70/B7 Shard (9278/9278 Lower Leaves Pass), Backend Now Pinned: Shared Arithmetic Trust Still Blocks Publication-Candidate Status

Round 30 (AMRAL-LUC-FC-R30, 2026-09-20) closes a gap Round 29 left open: Round 29 had established the first arithmetic whole-shard prototype — marker70's upper bound → the directed root/path → the 9278-leaf B7 lower tree — but still lacked a genuinely independent second full lower replay. This round supplies that missing implementation (A1) across all three pieces. For the root/path, Round 30 redoes the monotone directed bisection from the exact value T_M=1673/2000 and independently recomputes t₃, t₅, and t₇; after fixing a verifier script bug — an exact rational d* interval about 10⁻⁶⁴ wide had been cast to binary64 before use, perturbing its position by about 10⁻¹⁷, now logged formally as R30-A1-ROOT-001 — all three values overlap-PASS against Round 28's directed root. For marker70's upper bound, instead of the emitter's SciPy candidate-constraint reduction, Round 30 builds the exact cyclic polar hull directly from the full set of rational support constraints, which PASSes with outer bound 0.8349075014501105 and margin 9.24985499×10⁻⁵. For the 9278 leaves of the B7 lower tree, the second implementation avoids every piece of Round 28's machinery — the Body.support/contact candidate generator, the SciPy convex-hull candidate reduction, Round 28's density decisions, and Round 28's Fraction-based hull code — and instead builds support candidates directly from the defining-disk representation G=⋂ⱼD(Vⱼ,ρ) for each direction, takes the union support winner across bodies, quantizes onto a fixed 10⁻¹⁷ℤ² rational grid, computes an exact integer monotone hull from an integer orientation predicate, checks directed point membership against both the disks and the Reuleaux common cores, and finally compares the exact shoelace numerator (at denominator Q=10¹⁷) against 167/200 by integer cross-multiplication — with no floating-point acceptance anywhere in the chain. The result is 9278/9278 A1-PASS with zero inconclusive; the second verifier's thinnest leaf margin is 3.54237075×10⁻⁹, thinner than — and on a different path from — Round 28 emitter's thinnest margin of about 3.1949×10⁻⁸, though the document is explicit that this does not mean Round 30 is closer to the true hull minimum, only that the two different inner-polygon approximations carry different losses; the two implementations' density distributions also differ sharply (Round 30: 8174 leaves at density 512, 869 at 1024, 163 at 2048, 56 at 4096, 12 at 8192, 4 at 16384, none needing 32768), further confirming that these are genuinely two different finite-certificate generators, whose shared conclusion is min A_A1 > 167/200. Round 30 therefore upgrades the marker70/B7 shard's status to A1-VERIFIED-PINNED-PROTOTYPE-SHARD, and fully pins the arithmetic backend — the Python interpreter version, the mpmath version, the hash of the entire mpmath Python source package, the hash of every libmp/*.py file, the fractions.py hash, and the A1 verifier script hashes — under the identity PINNED-MPMATH-LIBMP-PROTOTYPE-v0.1; but the document is explicit that reproducibility ≠ backend trust proof — pinning only guarantees that a future replay knows exactly which source bytes were used, not that the backend has passed an independent correctly-rounded audit — so the backend trust state is PINNED-REPRODUCIBLE-PROTOTYPE yet still TRUST-AUDIT-OPEN, and Round 29's originally planned PUBLICATION-CANDIDATE-SHARD is still not granted. The round also explicitly does not overreach: Round 29 had already achieved 77/77 marker emitter arithmetic PASS, and this round's whole-shard gate needs only marker70, which already PASSes; extending A1 to all 77 markers was attempted but not completed, because the high-density markers make interactive batch computation too costly, so this is deferred to sharded asynchronous proof jobs and does not affect marker70's local A1 status. The global geometry closure ledger is completely unchanged this round — the latest complete common-budget remains 51/77 at d=36, with known higher-budget closures of at least Γ_B7(38)≥57 and Γ_B7(40)≥63 — and arithmetic A1 success must not be mistaken for global geometry completion. The global bound a_Leb≥0.835 remains unproven. Research direction and methodology are due to Neo.K; this round's AI collaborating researchers and primary executors were Aletheia / ChatGPT, GPT-5.6 Sol.