Proof Progress Becomes a Staged Coverage Dashboard for the First Time: Cell69 Reaches PUBLICATION-CANDIDATE-SHARD Status as Geometry Completion Climbs to 66/77

Round 32 (AMRAL-LUC-FC-R32, 2026-09-20) is the first round to split proof progress out of a single completion percentage and into a staged coverage dashboard — tracking five separate stages in sequence, Geometry, Arithmetic, A1, Cross-Backend, and Publication Candidate, rather than asking only how many cells have closed. On the geometry side, three new strict closures land this round: cell31 (depth 42), cell27 (depth 44), and cell7 (depth 46, whose unresolved volume shrinks in sequence from 9.43×10⁻¹⁰ to 1.34×10⁻¹⁰ to 1.93×10⁻¹², and finally to zero), bringing the known strict geometry-complete count to 66/77 (about 85.71%) — but this is not a uniform common-depth statement: the only fully synchronized common-budget result remains the depth-36 figure Γ_B7(36)=51/77, and higher budgets, since not all residuals are synchronized yet, can only be reported as lower bounds: Γ(38)≥57, Γ(40)≥63, Γ(42)≥64, Γ(44)≥65, Γ(46)≥66. On the arithmetic side, cell69 (15711 nodes, 7856 terminal leaves, closure depth 30) completes migration on two independent backends: the first, an mpmath/libmp exact-rational path, reaches 7856/7856 PASS (minimum rational margin 1.6525×10⁻⁹); the second, a libMPFR/GMP cross-backend path, hits one remaining leaf that at 32768-direction density first returns A-167/200≈-3.07×10⁻⁹ and is marked INCONCLUSIVE under the fail-closed rule — raising the density on that single leaf alone to 65536 directions flips the value positive, to 2.5329×10⁻⁹>0, with no whole-tree geometry resplit required, and the backend likewise finishes at 7856/7856 PASS. Marker69's exact-rational outer bound comes to 0.8349723974082344<0.835 (margin 2.76026×10⁻⁵), with both the independent exact polar replay and the MPFR support replay passing, so cell69 satisfies the local publication-candidate gate and is promoted to PUBLICATION-CANDIDATE-SHARD — joining the existing marker70 to bring Publication-Candidate coverage to 2/77 (about 2.60%; 2/66≈3.03% relative to the 66 geometry-complete cells). Cell47 (18455 nodes, 9228 leaves, depth 30) completes its mpmath/libmp rational lower bound this round (9228/9228 PASS, minimum margin 6.7502×10⁻⁹), adding to the rational upper PASS and independent exact polar replay PASS it already had from Round 29 — but because a full MPFR lower cross-backend replay is not yet done, it strictly remains at ARITHMETICALLY-CLOSED-PROTOTYPE rather than advancing to publication-candidate, bringing Arithmetic-Closed-or-Better to 3/77. The round also establishes a methodological lesson: cell69's MPFR leaf is the research line's first genuine trigger of selective arithmetic refinement — raising the density on a single leaf — rather than a whole-tree geometry resplit, empirically confirming the correct fallback order: raise finite-certificate density first, then raise arithmetic precision, and only then consider a geometry resplit; production now maintains two independent, mutually non-blocking queues — 11 geometry residuals dedicated to B7 closure continuation, and 63 geometry-complete-but-arithmetic-pending cells scheduled separately. The stage coverage dashboard currently reads: Geometry Complete 66, Arithmetic Closed-or-Better 3, Publication Candidate 2, Geometry Complete-but-Arithmetic-Pending 63, Geometry Residual 11. The global bound a_Leb≥0.835 remains NOT CERTIFIED, unchanged by this round; Round 33 is previewed as prioritizing arithmetic migration for the depth-30 batch (cells 47, 48, 68, 75, 76) and continuing work on the remaining 11 geometry residuals.