# AMRAL × Lebesgue Universal Covering — Round 12 ## Global Checkpoint Crystal and Multi-AI Shard Audit Protocol **Document ID:** AMRAL-LUC-FC-R12 **Version:** v0.1 **Date:** 2026-09-19 **Research status:** Round 12 / Persistent distributed proof-state **Research mode:** Human-Directed + Semi-Autonomous AI Mathematical Research **Research initiation and methodology source:** Neo.K **AI collaborating researcher and primary executor:** Aletheia / ChatGPT, GPT-5.6 Sol **Predecessor documents:** AMRAL-LUC-FC-R00 v0.2; R01–R11 v0.1 --- # 0. This Round's Summary and Verdict Round 11 had already rewritten the global exhaustive proof as: $$ \boxed{ \text{complete prefix-free frontier} + \text{independently verifiable shards.} } $$ Round 12 goes a step further: > Not just storing each round's results as a research summary, but storing the entire **distributed proof state** itself as a Checkpoint Crystal that is recoverable, auditable, invalidatable, and correctable. This round completes: 1. a machine-readable checkpoint schema; 2. a proof-critical dependency fingerprint; 3. separation of performance-only dependencies; 4. an append-only claim ledger; 5. an audit ledger; 6. stale propagation; 7. local rejection / global aggregate invalidation; 8. append-only correction; 9. a checkpoint sufficiency theorem; 10. a multi-AI lagged audit protocol; 11. an actual Round 11 reference proof-state transition dry run. This round's verdict: $$ \boxed{ \text{DISTRIBUTED PROOF CHECKPOINT ABI: CLOSED} } $$ $$ \boxed{ \text{STALE / CORRECTION SEMANTICS: CLOSED} } $$ $$ \boxed{ \text{GLOBAL }0.8350\text{ COMPUTE: STILL DEFERRED} } $$ --- # 1. The Checkpoint Crystal Is No Longer Just a Summary The previous purpose of the State Crystal: > Use the minimum information to let the next conversation know how far the research has progressed. Round 12 adds a second layer: > Use a minimal but sufficient **machine-readable proof state**, so that any worker / AI / verifier can recover the global computation without depending on past scheduler history. So a checkpoint contains: $$ \boxed{ \mathsf{CP} = ( R, T, D, \mathcal P, \mathcal C, \mathcal A, G ) } $$ where: - $R$: root identity; - $T$: target; - $D$: dependency bundle; - $\mathcal P$: current frontier; - $\mathcal C$: claim ledger; - $\mathcal A$: audit ledger; - $G$: aggregate state. --- # 2. Proof-critical dependencies Definition: $$ D_{\rm proof} $$ includes at least: 1. global root-domain hash; 2. the target: $$ T; $$ 3. split-semantics hash; 4. geometry / common-core hash; 5. arithmetic-policy hash; 6. witness-catalog hash; 7. verifier-semantics hash; 8. certificate-format-semantics hash. Define: $$ \boxed{ H_{\rm dep} = H(D_{\rm proof}). } $$ Every shard claim retains: $$ H_{\rm dep}^{\rm claim}. $$ --- # 3. Performance-only dependencies The following information affects cost, but does not affect theorem meaning: - scheduler; - worker count; - queue order; - machine identity; - wall-time limit; - node cap; - cache; - priority score. Denote as: $$ D_{\rm perf}. $$ It may enter the checkpoint, but: $$ \boxed{ D_{\rm perf} \text{ does not enter }H_{\rm dep}. } $$ So switching machines / switching schedulers will not needlessly stale a valid shard. --- # 4. Dependency Eligibility Rule A claim: $$ c $$ is eligible to enter the final aggregate only when: $$ \boxed{ H_{\rm dep}^{c} = H_{\rm dep}^{\rm current} } $$ If they are not equal: $$ \boxed{ \operatorname{status}_{\rm effective}(c) = \texttt{STALE}. } $$ This does not require modifying the old claim file. Staleness can be computed dynamically by the validator. --- # 5. Stale-Safety Theorem ## Theorem 5.1 If the finalizer accepts only claims with: $$ H_{\rm dep}^{c} = H_{\rm dep}^{\rm current} $$ then after any proof-critical dependency is updated, no claim that depends on the old semantics can silently enter the new final proof. ### Proof Old claim: $$ H_{\rm dep}^{c} = H_{\rm old}. $$ New checkpoint: $$ H_{\rm dep}^{\rm current} = H_{\rm new}. $$ If the proof-critical bundle changes, the cryptographic canonical payload also changes: $$ H_{\rm old}\ne H_{\rm new} $$ barring a hash collision. The validator classifies it as: `STALE` and excludes it. Q.E.D. --- # 6. Selective invalidation Not every dependency change necessarily staleifies all shards. The dependency can be split into: $$ D_{\rm global} $$ and: $$ D_{\rm local}(c). $$ For example: ## Global - target; - root; - split semantics; - common arithmetic policy. A global change usually affects all claims. ## Witness-Local If a certain shard only uses: $$ B_7 $$ while others use the Fourier witness, then modifying the Fourier witness geometry only needs to stale claims that use that geometry hash. So production can further store: ```text claim_dependency_hashes: global witness_ids_used geometry_modules_used arithmetic_policy verifier_modes_used ``` This round's reference schema, for now, uses a single aggregate fingerprint. --- # 7. Claim ledger Claims are always append-only. Basic states: ## `EMITTED` Certificate bytes already exist. ## `VERIFIED-INDEPENDENT-REPLAY` The canonical independent verifier PASSes. ## `REJECTED` A valid failure has been found. ## `STALE` The dependency is no longer current. ## `SUPERSEDED` Replaced by a new claim. ## `CORRECTED` A historical marker; the formally usable evidence should still be a new verified claim. --- # 8. Why append-only Reasons for not overwriting an erroneous shard: 1. correction ancestry can be traced; 2. what was wrong at the time can be reproduced; 3. the lagged verifier can reference the original claim; 4. a future audit can compare; 5. it prevents "fixing history so that it looks like it was never wrong." So: $$ \boxed{ \text{evidence history monotone} } $$ but: $$ \boxed{ \text{belief / active status can roll back}. } $$ --- # 9. Claim supersession If: $$ c_2 $$ corrects: $$ c_1, $$ then: ```text c2.supersedes = c1.claim_id ``` Active claim selection excludes the superseded ancestor. The old claim remains in the ledger. --- # 10. Audit ledger An audit record contains at least: ```text audit_id claim_id audit_level verifier_hash result failure_class counterexample metrics auditor timestamp ``` Audits themselves are also append-only. --- # 11. Audit levels This round formalizes: ## A0 — Emitter Self-Check Cannot alone enter the final theorem. ## A1 — Canonical Independent Replay Every final terminal shard must have at least one: $$ \boxed{ \text{A1 PASS}. } $$ The checkpoint's: `theorem_ready` treats this as a necessary condition. ## A2 — Independent Implementation / Lagged AI A second implementation, a second AI, a lagged branch. It does not replace A1. ## A3 — Adversarial Mathematical Audit Actively looks for: - branch omission; - false core; - stale dependency; - sign error; - arithmetic mismatch; - hidden symmetry assumption. --- # 12. Multi-AI Is Not Majority Vote If: - AI-A: PASS; - AI-B: PASS; - AI-C: FAIL, and gives a valid counterexample; then the result: $$ \boxed{ \text{FAIL}. } $$ not: $$ 2:1\text{ PASS}. $$ Likewise: 100 PASSes will not cancel out one valid counterexample. --- # 13. Lagged verifier Following the Round 00 design: the secondary verifier, by default, lags the canonical by: $$ 2\text{–}4 $$ rounds. For a shard: 1. first read the designated checkpoint; 2. do not read future corrections; 3. independently rebuild; 4. return the audit. This reduces information contamination. --- # 14. Checkpoint validity The checkpoint itself must satisfy: 1. payload hash; 2. frontier prefix-free; 3. frontier complete; 4. dependency fingerprint internal consistency; 5. claim ids unique; 6. active claim selection deterministic; 7. every theorem-ready seed exactly one eligible active claim; 8. the eligible claim has an A1 PASS; 9. no duplicate active coverage. --- # 15. Theorem-Ready State For the current frontier: $$ \mathcal P, $$ if for every: $$ p\in\mathcal P $$ there is exactly one active claim: $$ c_p $$ such that: 1. the dependency is current; 2. status: `VERIFIED-INDEPENDENT-REPLAY`; 3. at least one A1 PASS; 4. no newer rejection / supersession; 5. the frontier is complete; then: $$ \boxed{ \texttt{theorem\_ready}=true. } $$ This only means: > the scope declared by the checkpoint has been completely certified. The scope may still be the Round 10 local reference, not the global Lebesgue theorem. --- # 16. Scope is proof-critical metadata Must always be preserved: ```text scope global_lebesgue_claim ``` Round 12 reference: ```text scope = ROUND10-LOCAL-REFERENCE-SHARDS global_lebesgue_claim = false ``` So even if: `theorem_ready=true` it must not be misread as: $$ a_{\mathrm{Leb}}\ge0.835. $$ --- # 17. Checkpoint Sufficiency Theorem ## Theorem 17.1 If the checkpoint preserves: 1. root identity; 2. current complete frontier; 3. current proof dependencies; 4. complete shard claims + certificate hashes; 5. pending frontier; 6. deterministic split semantics; then, in order to **continue or verify the proof computation**, scheduler history is not needed. ### Reasoning Complete seeds: - evidence can be retrieved again via the certificate hash; - can be independently replayed. Pending seeds: - seed path + root + split semantics is enough to rebuild the box; - can be reassigned to any worker. History: - which machine ran it; - which seed ran first; - how many times it was queued; none of this affects domain coverage. So: $$ \boxed{ \text{checkpoint crystal is the sufficient state for global proof continuation}. } $$ --- # 18. Local rejection propagation If: $$ c_B $$ covers only shard-B seeds, and an audit proves it wrong: $$ c_B\to\texttt{REJECTED}, $$ then: - the shard-A claim can still be retained; - the shard-C claim can still be retained; - the B seeds become pending; - the global aggregate is no longer theorem-ready; - the global merge hash / Merkle summary is stale. So invalidation is: $$ \boxed{ \text{local evidence invalidation} + \text{global aggregate invalidation}. } $$ Not a full-system recomputation. --- # 19. Correction propagation If a corrected claim: $$ c_B' $$ is appended, and: $$ c_B'.supersedes=c_B, $$ and it passes the current A1 audit, then the B seeds again have an eligible active claim. If the remaining seeds have stayed valid: $$ \boxed{ \text{global theorem-ready can be restored}. } $$ There is no need to re-verify sibling certificate bytes that have no dependency relationship with the correction. The final merger should still recompute the global aggregate summary / Merkle root. --- # 20. Proof-critical change propagation If the arithmetic policy: $$ A_1 \to A_2, $$ and all shards originally depended on: $$ A_1, $$ then: $$ \boxed{ \text{all old claims' effective status = STALE.} } $$ The checkpoint becomes: `NOT-THEOREM-READY` until compatible claims are re-verified / re-issued. --- # 21. Performance-only change If only: $$ \text{scheduler v1} \to \text{scheduler v2}, $$ changes, and the proof-critical bundle does not change, then: $$ H_{\rm dep} $$ does not change. All valid shard claims remain eligible. So: $$ \boxed{ \text{switching schedulers does not require re-proving.} } $$ --- # 22. Reference checkpoint dry run Round 12 uses Round 11's reference sharded proof as its source. Frontier: $$ \boxed{ \{00,01,10,11\} } $$ Shards: - A: $\{00,11\}$; - B: $\{01\}$; - C: $\{10\}$. All of them already have a Round 11 independent replay PASS. --- # 23. Reference Checkpoint V1 Establish: `checkpoint_v1.json` State: $$ \boxed{ \texttt{theorem\_ready}=true } $$ but scope: $$ \boxed{ \texttt{ROUND10-LOCAL-REFERENCE-SHARDS} } $$ and: $$ \boxed{ \texttt{global\_lebesgue\_claim}=false. } $$ --- # 24. Transition V2: Performance-Only Only changes: $$ \text{scheduler hash}. $$ The proof-critical fingerprint does not change. Validator: $$ \boxed{ \texttt{theorem\_ready}=true. } $$ PASS. --- # 25. Transition V3: Proof-Critical Arithmetic Change Changes: $$ e_{\rm fp}=10^{-8} $$ the reference policy to: $$ 2\times10^{-8} $$ without re-issuing shards. The new dependency fingerprint changes. All three old shard claims become: $$ \boxed{ \texttt{STALE}. } $$ All four seeds in the frontier become pending. $$ \boxed{ \texttt{theorem\_ready}=false. } $$ PASS. --- # 26. Transition V4: Local Shard Rejection Under the original proof dependencies, mark the shard-B claim: `REJECTED` Siblings A/C are retained. Result: - seeds: $$ 00,10,11 $$ still have valid claims; - seed: $$ 01 $$ is pending; - aggregate: $$ \boxed{ \texttt{theorem\_ready}=false. } $$ PASS. --- # 27. Transition V5: Append-Only Correction Add: `claim-shard-B-v2-corrected` and: ```text supersedes = claim-shard-B-v1 ``` The new claim has an A1 PASS. The old rejected claim remains in the ledger. The validator again obtains: $$ \boxed{ \texttt{theorem\_ready}=true. } $$ PASS. --- # 28. Reference transition summary Round 12's actual machine test yields: | State | Change | theorem_ready | |---|---|---:| | V1 | baseline | `true` | | V2 | scheduler only | `true` | | V3 | arithmetic proof dependency | `false` | | V4 | shard-B rejected | `false` | | V5 | append corrected B claim | `true` | Therefore: $$ \boxed{ \text{stale / rejection / correction semantics actually run through successfully.} } $$ --- # 29. Persistent checkpoint naming Suggested for production: ```text AMRAL_LUC_FC_CP___.json ``` For example: ```text AMRAL_LUC_FC_CP_08350_000127_ab12cd34.json ``` Each checkpoint is appended. Do not overwrite the single: `latest.json` as the sole history. There may separately be: `LATEST_POINTER.json` purely for navigation convenience. --- # 30. Checkpoint lineage Each: $$ \mathsf{CP}_n $$ preserves: ```text checkpoint_id parent_checkpoint_id ``` forming: $$ \boxed{ \text{append-only checkpoint DAG / chain}. } $$ If two workers fork simultaneously from the same checkpoint, there can exist: $$ CP_{n+1}^{(A)}, \qquad CP_{n+1}^{(B)}. $$ later merged into: $$ CP_{n+2}. $$ The merge must resolve: - seed ownership; - duplicate claims; - supersession; - dependency compatibility. --- # 31. Multi-AI audit output destination Canonical proof state: `99_State_Crystals` Secondary AI audit artifacts: `01_Verification_Expansion` Audit file naming: ```text AUDIT____.json ``` After the canonical conversation ingests an audit: - it does not directly overwrite the shard; - appends the audit ledger; - appends a correction claim when necessary; - produces a new checkpoint crystal. --- # 32. Mandatory claim ancestry Any correction must record at least: ```text claim_id supersedes reason source_certificate_hash new_certificate_hash audit_ids ``` If it is only a dependency re-verification that does not change the bytes: `new_certificate_hash` may stay the same, but the claim id and audit evidence must be new. --- # 33. Stale aggregate Whenever any terminal claim is stale / rejected: the global aggregate must be marked: $$ \boxed{ \texttt{NOT-THEOREM-READY} } $$ even if: - 99.999% of seeds are still verified; - only one tiny seed remains. A coverage proof allows no percentage leniency. --- # 34. Partial progress metrics Although 99% cannot be treated as a theorem, the checkpoint can report: ```text verified_seed_count pending_seed_count verified_domain_weight_estimate verified_nodes pending_nodes_estimate hard_shards ``` These are progress metadata. They do not enter the final proof's premises. --- # 35. Global $0.8350$ production checkpoint A real heavy run should establish: ```text scope = GLOBAL-F3-BASE-ATLAS-T08350 global_lebesgue_claim = conditional ``` Before the complete base + witness certificates are all finished: ```text global_lebesgue_claim = false aggregate_status = IN-PROGRESS ``` Only after the final independent merge + publication arithmetic gate passes can it be raised to: ```text global_lebesgue_claim = true ``` --- # 36. Publication gate This round recommends that a final declaration of: $$ a_{\mathrm{Leb}}\ge0.8350 $$ requires at least: 1. complete global frontier; 2. every terminal seed exactly one active claim; 3. all active claim dependencies current; 4. every active claim A1 PASS; 5. no active reject/stale; 6. global merge PASS; 7. forest identities; 8. publication-grade arithmetic theorem; 9. source / certificate hashes; 10. correction ancestry complete. A2/A3 additionally increase confidence, but do not replace the hard gates with voting. --- # 37. Files produced in Round 12 This round's package contains: - canonical Round 12 MD; - State Crystal; - checkpoint JSON schema; - Multi-AI audit protocol; - checkpoint validator; - state-transition demo; - reference checkpoint V1–V5; - transition summary; - source Round 11 shard metadata needed by demo; - validation report. --- # 38. COMPUTE-DEFERRED ## C12-1 Turn the global: $$ T=0.835 $$ heavy run's current frontier into an actual checkpoint crystal. ## C12-2 Let the lagged verifier chat begin accepting shard audit jobs. ## C12-3 Build automatic stale detection against: - geometry hash; - arithmetic hash; - witness hash; - verifier hash. ## C12-4 Ingest local-side worker output directly into the claim ledger. --- # 39. Round 13 Assigned Topic ## AMRAL-LUC-FC-R13 ### Proof-State Scheduler and Autonomous Work Allocation If heavy global compute has not yet returned: the next round can turn the checkpoint directly into an autonomous scheduler interface: 1. estimate cost from the pending frontier; 2. dispatch base-atlas jobs / B7 lift jobs; 3. automatically choose local / cloud / secondary AI; 4. reclaim the descendant frontier according to the node cap; 5. automatically update the checkpoint; 6. never let any worker directly modify canonical truth; 7. the canonical merger is the sole state-transition authority. If global compute has already returned: directly ingest the real checkpoint. --- # 40. Reproducibility checklist ## Checkpoint schema `IMPLEMENTED` ## Dependency fingerprint `IMPLEMENTED` ## Stale-safety semantics `PROVED + TESTED` ## Performance-only non-invalidation `TESTED` ## Local rejection propagation `TESTED` ## Append-only correction `TESTED` ## Multi-AI audit protocol `SPECIFIED` ## Reference checkpoint transition V1–V5 `PASS` ## Global $0.8350$ checkpoint `COMPUTE-DEFERRED` --- # 41. Shortest Handoff Conclusion Round 12 elevates the "cloud-sync crystallization method" from research memory to: $$ \boxed{ \text{persistent distributed proof state}. } $$ Now a checkpoint crystal can already answer: - what the global root is; - where the frontier is; - which shards are already proven; - which are pending; - which proof version each claim depends on; - which are stale; - which have been rejected; - which are corrections; - which audits have passed; - whether the current scope is theorem-ready. So going forward, even if: - the conversation changes; - the AI changes; - a worker dies; - a shard is recomputed; - the proof formula is upgraded; none of it needs to rely on "some AI remembering what happened before." All that's needed is: $$ \boxed{ \text{the latest checkpoint crystal} + \text{append-only evidence ledger}. } $$