Certified Ancestor Contraction Theorem Proved: A Late Strong Certificate Can Absorb an Already-Split Subtree, Closing the Autonomous Scheduler/Merger ABI and Passing a Live Concurrency-Race Test

Round 13 (AMRAL-LUC-FC-R13, 2026-09-19) takes up the question that follows directly from Round 12's persistent distributed proof checkpoint: once a checkpoint still has a pending frontier, how should an AI or local worker autonomously pull work and return results without a race condition or a stale worker corrupting the canonical proof state? The round starts by sharply narrowing worker authority: a worker may never directly modify the frontier, active claims, theorem-ready status, pending seeds, or any other canonical state; it may only produce an immutable Worker Proposal P=(proposal_id, type, job_id, CP, H_dep, p, payload) in response to an immutable Job Contract J=(id, type, CP, H_dep, p, scope, caps, lease), restricted to six proposal types (CERTIFY, EXPAND, AUDIT_PASS, AUDIT_FAIL, COUNTEREXAMPLE, INCONCLUSIVE), with "Proposal → Validated State Transition" as the only legal write path, gated by the Canonical Merger's five-point check (payload hash, dependency fingerprint, seed identity, artifact hashes, proposal type schema) that can reject a STALE-PROPOSAL outright. The round's central new result is the Certified Ancestor Contraction Theorem (Theorem 10.1): if the descendant set D_p below an ancestor seed p in the current frontier completely covers the box B(p), and there exists a still-current-dependency valid certificate c_p that directly proves Φ(x)≥T for every x∈B(p), then contracting D_p back down to the single node p — with c_p as its active claim — still yields a legitimate complete frontier. Paired with the OBSOLETE-EXPANSION rule for stale EXPAND proposals, the OBSOLETE-COVERED-BY-CERTIFIED-ANCESTOR rule for a late EXPAND arriving after an ancestor is already closed, a deterministic min-canonical-claim-hash rule for selecting the active claim when multiple valid CERTIFY proposals exist, and an append-only retention policy for descendant evidence, the round goes on to prove the Atomic Merge Theorem (Theorem 20.1): as long as canonical state transitions use only six defined primitives (pending seed → verified claim; pending seed → complete descendant frontier; complete descendant frontier → verified ancestor claim; active-claim rejection → pending seed; appending an audit/auxiliary claim; dependency-stale filtering), then after any finite sequence of accepted transitions the frontier remains prefix-free and complete, and whenever theorem_ready=true every frontier seed carries a current, valid active claim. This is not proved on paper alone: starting from Round 12's checkpoint_v4_rejected_B (in which seed 01 is the sole pending seed), the round deliberately dispatches two parallel jobs to simulate a race — Job A, EXPAND_SEED(01) → {010,011}, and Job B, CERTIFY_SEED(01), using the real reference evidence of Round 12's V5 corrected shard-B claim. Merging A first takes the frontier to {00,010,011,10,11} with theorem_ready=false; when B arrives late, the Canonical Merger confirms that {010,011} completely covers B(01) and that c_p is still current, so it performs the ancestor contraction {010,011}→{01}, returning the frontier to {00,01,10,11} with theorem_ready=true; a further, late EXPAND(01) submitted afterward is then correctly marked OBSOLETE-COVERED-BY-CERTIFICATE and leaves the frontier unchanged — the expand, late-certify-plus-contraction, and late-expand steps are each verified PASS, so the round's central asynchronous-race scenario has actually been run, not just argued. The round also fixes the authority split explicitly: the Scheduler may only decide dispatch order and caps, the Worker may only produce proposals, the Verifier may only produce audit evidence, and only the Canonical Merger may change canonical proof state. 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. The round's own verdicts are AUTONOMOUS SCHEDULER / MERGER ABI: CLOSED and CONCURRENT RACE SAFETY: TESTED, but it states just as plainly that GLOBAL 0.8350 HEAVY COMPUTE: STILL DEFERRED — wiring the real global T=0.835 checkpoint into the scheduler (C13-1), building the local worker adapter (C13-2) and lagged AI audit adapter (C13-3), auto-generating a new checkpoint and State Crystal on every merge (C13-4), and testing multi-worker races such as duplicate certify, expand/expand, certify/audit-fail, and mid-run stale dependencies (C13-5) are all listed as COMPUTE-DEFERRED work still to be done before Round 14, and the global bound a_Leb≥0.835 gains no new evidence in this round.