Reuleaux Erosion Generalizes to a Generic Convex Common-Core Theorem, Nested Base/Lift Certificate Soundness Closed: B7's Root Domain at the 0.8350 Threshold Is Fixed, the Global Lower Bound Remains COMPUTE-DEFERRED

Round 08 (AMRAL-LUC-FC-R08, 2026-09-18) picks up the task that follows Round 07's fixing of the first improvement milestone at T1=0.835 and its rewriting of the five-dimensional sublevel region Q<T1 of the base family F3={D,B3,B5} as a threshold-conditioned minimizer atlas: this round does not claim that a large-scale atlas has already been completed, but instead builds the full proof grammar for "base atlas + cell-local witness lift + independent verifier." Conclusion A abstracts Mishra's erosion lemma into a general convex common-core theorem (Theorem 2.1): for a compact convex witness K⊆B_RK(0) and a placement box with half-widths (h_φ,h_x,h_y), the motion bound δ=√(h_x²+h_y²)+2R_K sin(h_φ/2) is such that, whenever a candidate core C satisfies C+δB⊆K0, C⊆Kg holds for every configuration in the entire placement box. Conclusion B proves that the official Reuleaux erosion (radius 1↦1-δ) is exactly the closed-form specialization of this general theorem when K=∩_j B(V_j,1). Conclusion C shows that Round 02's general Fourier/asymmetric witnesses plug directly into the same lift machinery: the verifier only needs K0, δ, a candidate core C, and the inclusion certificate C+δB⊆K0, with no witness-specific proof logic required. Conclusion D establishes the nested base/lift certificate theorem (Theorem 12.1, the Nested Certificate Soundness Theorem): every base-atlas leaf must be classified as either BASE-PRUNED (L_F3(Cj)≥T1) or LIFT-CLOSED (a witness Kj together with a finite local lift tree that covers its placement root, with every lift leaf's hull area ≥T1); once all base leaves are legally classified, Λ(F3∪{Kj})≥T1. Conclusion E defines an independent certificate grammar that forbids any unclassified or unresolved leaf, and lists the ten items the verifier must independently reconstruct (the base root, the split, each base leaf, each witness lift root, each lift subtree, each leaf's common core, each lower hull area, the floating-point/interval error margin, tree structural identity, and all referenced hashes). Conclusion F fixes B7's local lift root at T1=0.835: R7=1/(2sin(3π/7))≈0.512858431636, with Round 07's threshold-dependent a-priori domain giving |t7|≤0.196935046771 and rotation symmetry φ7∈[0,2π/7). Conclusion G audits the geometric headroom: the official rigorous three-body outer ceiling is 0.834781190917, only 0.000218809083 short of the milestone; the official four-body D+B3+B5+B7 search-only optimum is 0.836494901, which is 0.001494901 above the milestone (about 6.83 times the required increase) — this is numerical headroom, not a proof. The round's overall verdict is three boxed conclusions: CONDITIONAL-LIFT CERTIFICATE COMPILER: CLOSED, GENERIC CONVEX COMMON-CORE INTERFACE: CLOSED, and a_Leb≥0.8350: COMPUTE-DEFERRED — what this round completes is the certificate architecture itself and the witness root-domain constants; the actual construction of the base atlas, the actual B7 conditional-lift run, and the independent replay are all listed in Section 34 of the source document as COMPUTE-DEFERRED (C08-1 through C08-5) and have not yet been executed. Research direction and methodology are due to Neo.K; this round's AI collaborating researcher and primary executor were Aletheia / ChatGPT, GPT-5.6 Sol.