Independent Second Arithmetic Backend libMPFR/GMP Clears Full Replay, marker70/B7 Shard Becomes the Research Line's First Publication Candidate: Root, Upper Bound, and All 9,278 Lower Leaves Pass, Global Bound Still Unproven

Round 31 (AMRAL-LUC-FC-R31, 2026-09-20) addresses the last outstanding worry left over from Rounds 28–30: the two existing verifiers — the emitter and Round 30's independent A1 implementation — actually share the same underlying arithmetic, mpmath.iv/libmp, so a single rounding defect in that stack could in principle contaminate both results at once. This round finds, sitting unused in the system's dynamic libraries, libmpfr.so.6 and libgmp.so.10 (the environment has no gmpy2, python-flint, or Arb Python binding), and builds a deliberately minimal directed-rounding wrapper directly on the C ABI/ctypes — exposing only exact integer/rational load, RNDD/RNDU, the four arithmetic operations, sqrt, sin, cos, acos, atan2, pi, comparison, and exact dyadic endpoint extraction, rather than treating an entire CAS as a proof dependency — and fixes the exact-rational→binary64→interval pathway that Round 30 had already found to destroy ultra-narrow intervals, replacing it with an exact-Fraction→MPFR-directed→exact-dyadic-Fraction round trip. Using this MPFR/GMP backend, the root is rebuilt from scratch by monotone directed bisection starting at T_M=1673/2000 (the resulting d* interval width is about 2.97×10⁻⁶⁷), and all three of t3, t5, t7 overlap-PASS against the Round 28 mpmath/libmp intervals; the marker70 upper bound swaps out only the Reuleaux support sector's transcendental arithmetic for MPFR, leaving the exact rational marker point, finite integer normals, rational half-plane outer polygon, and exact rational polar hull/line-intersection/shoelace machinery unchanged, and recomputes to 0.8349075014501105<0.835 (margin≈9.24985499×10⁻⁵), matching the value already displayed in Round 29; and the full lower tree — rebuilt leaf by leaf from the MPFR-reconstructed root, explicit topology, explicit witness split axes, defining-disk common-core geometry, an independent support-candidate generator, fixed-grid integer rationalization, an exact integer monotone hull, and exact rational area comparison against the same A_rational>167/200 threshold — passes all 9,278 leaves with 0 inconclusive, reproducing the identical density histogram from Round 30's A1 implementation (512:8174, 1024:869, 2048:163, 4096:56, 8192:12, 16384:4 — the same finite candidate policy, just a different arithmetic backend underneath), with a thinnest leaf margin of about 3.5423707501×10⁻⁹, strictly positive. This round also logs a process correction, R31-MPFR-MARKER-001: the first version of the MPFR marker verifier additionally included Reuleaux pieces adjacent to the active sector as extra upper candidates — still a safe one-sided upper bound, but loose enough to push U to about 0.9201 and fail to pass, which is not a false proof but a safe, conservative false negative/inconclusive result, fixed by restricting the candidate set to pieces that can genuinely be active within the directed sector interval, after which the exact outer certificate passes; a second, purely mechanical issue, R31-BACKEND-LOCK-001 (an early ldconfig-parsing helper generated an invalid "/" path and failed before manifest generation), is also fixed with no mathematical effect. Because the root, the marker upper bound, and every one of the 9,278 lower leaves now each carry two independent implementations — Backend A (mpmath.iv/libmp: the Round 28 emitter plus Round 30's A1) and Backend B (libMPFR 4.2.2/GMP: Round 31's full replay) — any backend-specific silent rounding defect would have to produce compatible errors in two different arithmetic stacks simultaneously to slip through, which substantially raises confidence, though the document is explicit that this is still not a formal proof of either library: the real mathematical certificate still rests on a software trust base, and the round's approach — shrinking the trust surface, pinning source and binary, using different libraries and different implementations, and finishing with strict rational comparisons — is practical computer-assisted-proof engineering, not foundational formal verification. Round 31 uses this to define the research line's own local six-condition PUBLICATION-CANDIDATE-SHARD gate (two-backend root/path replay, two-backend marker-upper replay, an independent lower-tree A1 implementation, an independent lower-tree arithmetic-backend replay, every claimed inequality strict, and dependency hashes pinned), and marker70/B7 satisfies all six, becoming the research line's first shard ever to reach this tier — but the document is equally explicit that a publication-candidate shard is not the global theorem: global geometry remains incomplete (the latest common budget is Γ_B7(36)=51/77, with higher known budgets Γ_B7(38)≥57 and Γ_B7(40)≥63 not yet adopted as the new common budget), so promoting a single shard cannot change the global bound's status. The round also routes geometry-complete cells into a newly defined ARITHMETIC-MIGRATION QUEUE (each cell proceeding through explicit topology/axis streams, directed root/path, marker upper, bulk lower migration, implementation A1, MPFR cross-backend, and publication-candidate promotion, in that order), while geometry-residual cells continue the B7 closure wave; an early observation that arithmetic migration itself has caused zero leaves to require resplitting across Rounds 28/30/31 (0 for marker70) supports — as an engineering hypothesis still to be verified shard by shard, not yet a proven fact — the conjecture that most geometry-complete shards with a comfortable margin can be bulk-migrated directly. marker70/B7's local status climbs the lattice from REFERENCE-COMPLETE through ARITHMETICALLY-CLOSED-PROTOTYPE and A1-VERIFIED-PINNED-PROTOTYPE to PUBLICATION-CANDIDATE-SHARD, while the global THEOREM-READY flag stays false; the preview of Round 32 splits the work into an arithmetic team that will bulk-migrate the next, cheapest batch of cells from the geometry-complete queue and tally a publication-candidate coverage fraction, and a geometry team that will independently push a new common budget at d=40, 42, and beyond. The global bound a_Leb≥0.835 remains NOT CERTIFIED. 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.