First Exact-Arithmetic Leaf Certificates Produced: Rational Inner-Polygon Theorem Established, Cell0's Five Thinnest Leaves Pass 5/5, but Closure Scope Remains Limited to the Existing Reference State

Round 27 (AMRAL-LUC-FC-R27, 2026-09-20) simultaneously advances two lines: first, continuing Round 26's resumable wave state, extending only the not-yet-closed residual boxes and not recomputing already-closed subtrees — the common-depth d=36 value Γ_B7(36)=51/77 remains unchanged, and it is further known that ≥57/77 cells have closure depth ≤38 and ≥63/77 have closure depth ≤40, but because not all residuals have been pushed synchronously to the same budget, both figures are only known lower bounds and do not constitute a new common-depth curve; second, beginning the first true interval-contained arithmetic replay on the thinnest reference leaves. This round's core result is a proof of the Rational Inner-Polygon Theorem (Theorem 7.1): once a set of points has been verified by strict interval arithmetic to lie genuinely inside an already-certified core region (common core), the exact rational convex-hull area of those points is itself a legitimate mathematical lower bound — with no dependence whatsoever on floating-point convex-hull computation, the floating-point shoelace area formula, or any transcendental function operations. Building on this, this round first demonstrates the full pipeline on the thinnest-margin leaf in the cell0 COMPLETE tree (reference slack approximately 4.20742×10⁻⁸): 16384 candidate directions, 3170 hull candidate vertices, directed interval membership passing 3170/3170 in full (0 failures), yielding an exact rational inner-polygon area of 0.8350000319489227, an exact rational margin of approximately 3.19×10⁻⁸; applying the same pipeline to all 5 of cell0's thinnest-margin leaves likewise gives 5/5 PASS, with exact rational margins ranging from about 3.19×10⁻⁸ to 2.03×10⁻⁷, and with no floating-point hull, shoelace, or transcendental operation anywhere in the critical path — this is the program's first true batch of exact-arithmetic leaf certificates. The document keeps the scope explicit: the interval prototype currently in use encloses only the binary64 reference state already stored from earlier rounds, and is not yet an independent verification fully reconstructed from the master root under exact/directed split semantics — so this round explicitly states that the result still falls short of publication-grade rigor. This round is the direct technical precursor to the concrete binary "RHCert" (Rational-Hull Certificate) format that Round 34 later formalizes and scales up — Round 27 first proves on 5 leaves that the exact-arithmetic technique itself is workable, and only in Round 34 is it packaged into a replayable, scalable file format; the exact-arithmetic migration path in fact formally begins with this round. Research direction is chosen and led by Neo.K; this round's execution was carried out by Aletheia / GPT-5.6 Sol.