Proof Capacity Algebra and Automatic Threshold Ladder Both Ruled Closed: the Global Capacity Theorem and Free Promotion Theorem Are Established, but the Global 0.835 Heavy Certificate Remains COMPUTE-DEFERRED

Round 15 (AMRAL-LUC-FC-R15, 2026-09-19) upgrades Round 14's per-claim PASS/FAIL verdict into a quantitative proof capacity: without recomputing every shard, how far can the existing distributed evidence graph push the lower target, and once that point is exceeded, exactly which proof regions actually need recomputing. The round first flags that the root itself has a capacity — the legitimacy of the outside-root region rests on the a-priori bound ≥0.835, so even if internal shards later carry enormous slack, the theorem target cannot simply be pushed to 0.8353 unless the outside-root proof also supports 0.8353 or the root was built with a higher master target to begin with; it therefore defines Root-Domain Capacity C_root (conservatively set to C_root = T_M, the phase target), and defines Direct Claim Capacity D(p) as the maximum capacity among all currently compatible claims at a node, then recursively defines Recursive Evidence Capacity C(p) = max(D(p), min(C(p0), C(p1))), with C_evidence = C(ε) at the root, proving the Evidence Capacity Theorem (Theorem 6.1) by structural induction on the proof tree: for any T ≤ C(p), the registered evidence graph proves Φ(x) ≥ T for all x in B(p). The round's central quantity is the Global Proof Capacity C_global = min(C_root, C_evidence), and it proves the Free Promotion Theorem — as long as a new target T1 ≤ C_global, promoting T0→T1 needs no new geometry certificate at all, only three steps: recomputing eligibility with the semantic validator, a global merge, and the publication arithmetic gate — from which the Capacity Staircase structure follows: eligibility only changes at each claim's own capacity breakpoint as the target rises, so evidence reuse is stepwise. The round also defines the Target Feasibility Boolean F_T(p) and the Deficit Frontier Δ_T (the first layer of obligations not yet closed by existing evidence), and proposes a Minimum-Cost Repair DP, K_T(p) = min(J_T(p), K_T(p0)+K_T(p1)), proved optimal by standard dynamic-programming induction (the Repair DP Theorem), which promotes Round 13's Certified Ancestor Contraction from a purely race-safe recovery mechanism to an active cost-optimization option. To avoid re-expanding the global root every time the target rises by 10⁻⁴ — which would churn shard reuse and scheduling — the round proposes a Phase Master Root strategy, demonstrated with the design example T_M = 0.8365 (explicitly declared not a new lower bound, only a phase root ceiling): the corresponding a-priori half-widths are t3≈0.201023529007, t5≈0.202618286230, t7≈0.201473445786 (versus t3≈0.194856180909, t5≈0.197820670401, t7≈0.196935046771 at T=0.835, giving half-width ratios of about 1.03165 / 1.02425 / 1.02305), with the enclosing-box volume over the four base translation coordinates growing by about 11.66% and the B7 translation square by about 4.66% — a domain-amortization tradeoff; it further notes that the numerical search ceiling for the official four-body D+B3+B5+B7 is about 0.836494901, so 0.8365 is simply a natural B7-phase engineering ceiling (a PHASE-DESIGN-CHOICE, not a theorem). The round runs this entire mechanism end to end on Round 14's local reference example (e=10⁻⁸): four leaf capacities (seed 00/11 via shard-A at 0.8350003869877956, seed 01 via the corrected shard-B at 0.8350009574949350, seed 10 via shard-C at 0.8350003545377508) yield C_ref = C(ε) = 0.8350003545377508, explicitly stated to NOT be the global Lebesgue lower bound; it demonstrates that the original target 0.835 can be freely promoted, with no new certificate, to 0.83500035 (the machine planner returns READY), then walks through the bottleneck progression: at target 0.83500036 only shard-C/seed-10 falls short (Δ_T={10}); at 0.83500039 shard-A (covering both seed 00 and seed 11) also falls short, so the nominal deficit is the three points {00,10,11}, but the job planner should recognize that 00 and 11 share the same shard-A claim, so the actual work is just "upgrade shard-A + upgrade shard-C," two jobs rather than three independent ones; the deficit stays at {00,10,11} up to 0.83500095, and only at 0.83500096 does shard-B also fall short, so all four seeds need new evidence. The round further defines four Proof Capacity Status Labels (FREE-PROMOTABLE / SHARD-BOTTLENECK / ROOT-BOTTLENECK / MIXED-BOTTLENECK) and an automatic threshold-ladder production loop, lists four COMPUTE-DEFERRED items (C15-1 through C15-4: switching the production root to a phase master, storing per-shard capacity metadata, building a real global capacity tree, and auto-emitting bottleneck/repair-plan data at each checkpoint), and assigns Round 16 the topic of a Capacity-Aware Global Pilot and Bottleneck Profiler. In the reproducibility checklist, Root capacity is FORMALIZED; Recursive evidence capacity, the Global capacity theorem, Free promotion, and the Minimum-cost repair DP are all PROVED; Capacity staircase is FORMALIZED + REFERENCE TESTED; Deficit frontier is IMPLEMENTED; Phase-master root is FORMALIZED; the Reference capacity planner is PASS — but the Global 0.8350 heavy certificate remains COMPUTE-DEFERRED: the global bound a_Leb≥0.835 gains no new certification this round, whose actual output is a reusable proof-capacity algebra and scheduling architecture rather than a new numerical lower bound. 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.