[kim-em]

Algebraic source preparation retained checked coefficient identities, but consumers lacked a proof connecting actual producer/checker acceptance to the original real statement and its original divisor conditions. This adds AlgebraicCells.accepts, its completeness/soundness equivalence, and transport through the checked source tuple and shared frontend equivalence.

Inputs.acceptanceIff target registry constructs that ordinary-kernel equivalence from retained algebraic expressions. It checks the exact expected target, reconstructs and compares the complete ordered original divisor inventory, and uses the checked source formula. Removing or replacing both public guard arrays is rejected. Runtime proposals and the separate runtime formula/quantifier do not enter the proof. Proof construction continues the aggregate preparation budget and returns usage for later stages. Both preparation and acceptance preserve lowered recursion limits, and failures restore caller state. The result is an equivalence; acceptance evidence is still needed to prove the goal.

Fresh consumers cover true/false sources, original division/inverse guards, cancelled zero divisors, reversed domains, local aliases, exact source order, runtime-only mutations, corrupted identities, forged guard inventories and inner recursion limits. Proof-node and source-node tests check continued accounting and exact/insufficient cumulative limits; distinct alias views catch charging the original source twice. Unchanged-record controls verify recovery after forgeries without diagnostic-string classification. Fresh importing audits check nine actual theorem bodies, their complete standard-three-axiom closure and required owner-law dependencies. The manual includes the runnable helper and its limitations.

Validation:

• At 6f0ae423ab, all 263 RCF conformance roots, 16 base-test roots, both libraries, actual caller π/e proof probes and full manual pass: 14,610 jobs, 25m29.596s, 10.7 GiB/no swap.
• At cdc9b747aa, the additional budget/alias/control tests, fresh body audit, public-import consumer, libraries, caller probes and full manual pass: 14,322 jobs, 7m53.580s, 7.8 GiB/no swap. Production definitions differ from 6f only by a public API docstring. The final head adds report/evidence only.
• Fresh full-manual render and joint audit/manual import pass; six desktop/narrow captures were opened with zero local changed-content overflow. DAG/umbrella, copyright, line, admission, source/artifact binding, relative-link and diff checks pass. All own local builds remain serial under a verified hard aggregate 32 GiB/no-swap cap with two Lean threads.
• Retained evidence distinguishes current and historical scopes and binds exact sources, commands, logs, captures and live resource limits. Final hosted CI remains pending; no merge is claimed.

Frozen proposal/provider-history authentication, general source quotation, strict compiled missing-record coverage, resource-scheduled completeness, realization, the native stress runtime and remaining evidence stay on the open implementation issue. This tranche installs no new tactic handler. Publication and owner arithmetic are outside this change.

Refs #10358
Refs #10331

[kim-em]

Updated head 6f0ae42 addresses the fresh post-opening review's guard-inventory and recursion concerns. acceptanceIff reconstructs the ordered divisor inventory from the exact target using the existing frontend collector in the continued reflection budget. It rejects removed/replaced public guard arrays, including a same-count valid replacement for a semantically zero divisor. Preparation now preserves a lowered caller recursion limit too; per-entry cancellation checks and explicit budget-threading documentation are included.

The strengthened cases pass (11,032 Lake jobs, 3m16.654s, peak6.9GiB/no swap), including inner recursion tests entered at depth zero, corrupted divisor identity, complete rollback, exact budget boundaries and forged guard inventories. All four metadata checks pass. Full263+16/library/probe/manual compatibility and a fresh follow-up Opus review are running. Old-head CI38898250524 was intentionally cancelled; its final state is cancelled, not a code failure. New exact-head CI38901991810/job116976428165 has one120-second monitor. Auto-merge remains held until review, tests and CI are green. No final green or full issue completion is claimed.

All own local builds remain serial under a hard aggregate32GiB cap, MemorySwapMax0 and LEAN_NUM_THREADS2.

[kim-em]

Fresh follow-up Opus review confirmed the original-divisor inventory and recursion fixes with no blocking soundness finding. The remaining test follow-up adds source-node growth, exact two-view accounting, measured source-node budget boundaries and successful unchanged-record controls after guard forgeries; it documents the public guard scanner's structured budget errors. It changes no production definition. It is prepared but not yet pushed or claimed passing.

The full local compatibility run remains on the unchanged 6f0ae42 source snapshot, under the verified hard aggregate32GiB/no-swap cap. Its fresh actual proof-body/axiom audit has passed; the overall run remains active. After it finishes, the test follow-up gets focused bounded validation and its source-bound evidence. The final test/evidence head will be pushed together to avoid extra CI runs for evidence-only commits.

The superseded6f hosted run 38901991810 is intentionally cancelled (final job/run state verified); it is not a code failure. Its one watcher has exited. No replacement CI or auto-merge is claimed yet. The existing PR and #10358 remain open; no duplicate PR, new issue, owner worker or CI fan-out was created.

[kim-em]

Final pushed head: db4041d.

The fresh post-opening and follow-up Opus findings are addressed. Original guards are reconstructed from the exact target; removing or replacing both stored arrays fails. Lowered recursion limits are preserved in preparation and acceptance. Per-entry cancellation checks, divisor-identity mutations, exact proof/source-budget boundaries, distinct original/closed alias-view accounting and unchanged-record controls pass. Controls preserve caller metavariables; no failure is classified by its diagnostic string. The source-budget review's distinct-alias gap is covered by the final regression and the actual end-of-run marker.

The retained report and validation record distinguish the full 263+16/library/probe/manual run at 6f0ae423ab (14,610 jobs, 25m29.596s, 10.7 GiB/no swap) and the stronger focused follow-up at cdc9b747aa (14,322 jobs, 7m53.580s, 7.8 GiB/no swap). The follow-up changes only tests and the guard scanner's docstring; the final head adds report/evidence only. Fresh actual-body/axiom audits, the joint audit/manual import, full render, all six opened captures, metadata, relative links and source/artifact hashes pass. The live hard aggregate 32 GiB/no-swap limit is bound to the full run's actual Lake process; all own builds remain serial with two Lean threads.

Exact-head CI 38906847559 / job116992442321 is in progress, with one 120-second monitor. No green CI or merge is claimed; auto-merge remains held until the final gate passes. The two superseded runs were intentionally cancelled, not diagnosed as code failures.

This is a Refs tranche. #10358 remains open for generic frozen source/proposal and provider-history authentication, strict compiled missing-record rejection, general accepted-source quotation/completeness, realization, the native stress runtime and remaining evidence. Existing owner requests remain on #10378; #10967 is still open and has not been assumed merged. No new issue, downstream worker, arithmetic framework or CI fan-out was introduced.