Exact Split Scheduler Proved, Core-Dominance Reuse Established: The Adaptive Certificate-Cost Layer Closes This Round While the Heavy 0.8350 Certificate Remains Compute-Deferred

Round 09 (AMRAL-LUC-FC-R09, 2026-09-18) builds on the nested certificate grammar Round 08 completed (base atlas → cell-local witness lift → independent verifier); this round adds no new proof domain, and instead builds, for certificate cost, a set of adaptive strategies that leave soundness completely untouched. Its core is the family-cell Hausdorff-radius theorem: τ(C) = max{δ3, δ5}, where δ3=√(hx3²+hy3²) and δ5=√(hx5²+hy5²)+2R5 sin(hφ5/2), which converts the size of the five-dimensional placement box directly into the hull's geometric uncertainty; the inradius-perimeter inequality (every relevant hull contains B_{1/2}(0)) then tightens Round 07's perimeter bound from 4πT to 4T, correspondingly tightening the area-transfer error term from 4πTτ+πτ² to Ω_T(τ)=4Tτ+πτ². From this follows the explicit margin-to-resolution formula τ⋆(m,T)=(√(4T²+πm)−2T)/π: on the D+B3+B5 root cell at T=0.8350, this round computes τ_root≈0.6046803712, and one-step splitting shows x3 and y3 leave the current maximum radius unchanged, φ5 gives τ≈0.4442456056, and x5=y5 give τ≈0.5460899294 — so the first cut should be φ5, the exact radius-bound-optimal split under this round's certified radius bound, and the round proves that the official Mishra-code split weight R h_φ is exactly the first-order Taylor approximation of the exact term 2R sin(h_φ/2) (with a next-order error of size O(h_φ³)). The round also lays out the prune cascade explicitly in tiers (§16: P0 a-priori single-point/translation bound, P1 representative-area transfer, P2 common-core lower hull, P3 an optional stronger witness-specific lower bound, P4 split by the exact radius-gain scheduler), together with five proof-mode tags (APR/REP/CORE/SUPPORT-CORE/OTHER-VERIFIED); and it proves the Core-Dominance Reuse Theorem (Theorem 19.1: when G(C_a)⊇G(C_b), the same witness lift tree — under the same witness root/split tree and the same witness-core construction — can be safely reused for C_a) and the Single-Lift Witness Dominance Theorem (Theorem 22.1: K⪯L implies J_L(H)≥J_K(H), letting the witness pool be pruned by geometric dominance first — when L dominates K at no higher cost, K can be dropped from the priority pool entirely). It further proves that, given a positive margin m>0 and a full bisection every round, τ_r≤2^{-r}S_0, so a finite depth r≥⌈log2(S0/τ⋆(m,T))⌉ is guaranteed to achieve representative closure — an extremely conservative worst-case bound that practice should easily beat. §29 runs one more sanity check, explicitly marked SEARCH-ONLY and not a rigorous lower bound: substituting Round 07's official four-body search headroom m_search=0.001494901 gives the new bound τ⋆≈4.47387×10⁻⁴, about 3.14 times looser than the old Round 07 conservative bound's required ≈1.42462×10⁻⁴. The document itself closes with three boxed verdicts: ADAPTIVE CERTIFICATE-COST LAYER: CLOSED, SOUNDNESS-PRESERVING REUSE RULES: CLOSED, and HEAVY 0.8350 CERTIFICATE: STILL COMPUTE-DEFERRED; its reproducibility checklist lists 8 items as PROVED (family-cell Hausdorff radius, sharpened area transfer, the margin-to-resolution formula, the exact one-step split scheduler "proved for the certified radius bound," representative-area prune, common-core dominance reuse, single-lift witness dominance, and the margin-to-depth finite bound), lists the schedulers' A/B performance comparison (C09-5) as COMPUTE-DEFERRED, and lists the 0.8350 theorem itself as NOT YET CERTIFIED. None of the five engineering tasks C09-1 through C09-5 listed in §30 were implemented this round; all are left for later. Round 10's assigned topic is a scaled-down reference atlas emitter and B7 conditional-lift dry run. 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.