Semantic Implication Joins Exact-Hash Matching as a Legal Path, Hot/Cold Compaction Proved to Leave Theorem-Readiness Unchanged: The Semantic Rebase Layer and the Hot-Core Compaction Theorem Both Close

Round 14 (AMRAL-LUC-FC-R14, 2026-09-19) addresses five problems left over from running the AMRAL distributed proof system long-term: whether every old claim must be recomputed once a proof dependency's hash changes; when two claims can both prove the same region, which one should stay in hot state; whether the checkpoint balloons without bound once rejected / stale / obsolete / superseded history accumulates into the millions of records; whether moving history out of hot state breaks theorem soundness; and how to safely rehydrate old evidence if it is needed again later. The first problem is resolved by the semantic compatibility relation (c⪰_sem O) and a scalar error-budget margin credit: for the reference leaf's worst verified slack s_c=min[L-e_c-T_c], as long as a new policy satisfies T'+e'≤T_c+e_c+s_c, the old raw witnesses still suffice to prove the new obligation, from which claim capacity is defined as Cap_e'(c)=T_c+s_c+e_c-e'. The document verifies this on three active shards from the Round 11 local reference: worst slacks s_A≈3.8698779558643537×10⁻⁷, s_B≈9.574949350055206×10⁻⁷, s_C≈3.545377508240577×10⁻⁷, giving capacities of about 0.8350003769877956, 0.8350009474949350, and 0.8350003445377508 under e'=2×10⁻⁸, whose minimum is the aggregate capacity T_cap^ref≈0.8350003445377508 (the document explicitly flags this as Round 10/11 local reference scope only, not the global Lebesgue lower bound); an actual migration test to T'=0.8350001, e'=2×10⁻⁸ leaves all three claims with positive margin and theorem_ready=true, while pushing further to T''=0.8350004 leaves at least A and C under-capacity, giving theorem_ready=false — proving that the semantic validator is genuinely bounded by the margin rather than simply passing everything through whenever a hash fails to match. The second problem is resolved by Claim Dominance (c1⪰_Π c2 requires all four of: domain coverage, capacity at least as high, compatible local dependencies, and the required A1 evidence) and by the definition of a Minimal Active Proof Core, with the Hot-Core Compaction Theorem (Theorem 22.1) proving that whenever scope/root, semantic policy, frontier, active coverage claims, and theorem-critical A1 evidence agree, the full state S and the compacted state H yield identical theorem_ready verdicts — because rejected, superseded, and obsolete history never enters the current proof implication. Round 14 actually compacts the Round 13 reference checkpoint (active A, rejected B-v1, active corrected B-v2, active C, 5 audit records, an orchestration race ledger) down to a hot core holding only the three active evidence summaries A, B-v2, C, moving the complete checkpoint into a cold archive bound by content hash plus a Merkle root; a machine test verifies the archive object's SHA-256, reads claim-shard-B-v1 back from the cold snapshot, confirms its status remains REJECTED, and confirms the certificate hash stays consistent — all PASS. The document explicitly defines GC as hot-state eviction, not physical deletion, and registers four migration rules (TARGET-MONOTONICITY-V0.1, SCALAR-ERROR-MARGIN-CREDIT-V0.1, WITNESS-CATALOG-SUPERSET-V0.1, ANCESTOR-DOMAIN-COVERAGE-V0.1) that the canonical merger may use automatically, while listing eight kinds of change — an arbitrary geometry rewrite, changed normalization, a witness replaced by a non-equivalent shape, and the like — that are never automatically compatible and still require new proof. Items C14-1 through C14-5 — converting the future global 0.835 checkpoint to hot-core format, building a migration registry validator, building an archive rehydration CLI, and more — are listed as COMPUTE-DEFERRED and have not yet been run against the actual global proof state. This round presents no new numerical lower bound; 0.835 here serves only as the Round 10/11 local reference's illustrative threshold. The global Lebesgue universal-covering constant a_Leb≥0.835 is not itself the object of this round's work and remains unproven. Research direction and methodology are due to Neo.K; this round's AI collaborating researchers and primary executors were Aletheia / ChatGPT, GPT-5.6 Sol.