# Summary (v9. 1) ## OverviewThis record releases a **certificate-oriented closure interface** for the Collatz conjecture, together with a **sealed Level-2B/Level-3 packet** that produces and verifies a *verdict certificate* under explicit thresholds and mechanically enforced audit rules. The program replaces an informal “last arithmetic bottleneck” with a **finite, hash-locked, mechanically auditable attack surface**: the Corr\ ( () \) /TwGap obligation is exposed as a **mandatory checked entry (I4) ** in the Gate-B certificate, and the All-\ (L\) promotion is triggered by a **single PASS event** rather than additional ad-hoc arithmetic arguments. Scope note: this is not a claim of a purely pen-and-paper proof. The statement is **certificate-graded** and conditional on the published verifier/auditor under its declared numeric model and thresholds (see “Trust base (TCB) ”). --- ## Contents of this record ### Papers (PDF + TeX) - Collatz Final Gate v9. 1 (main paper) - Gate B Bottleneck v2. 1 (companion) - L0=16 Complete Certificate Checklist v9. 1 (reviewer-facing checklist) ### Sealed packet (ZIP) - CollatzL2Bᵥ9. 1. zip Unzips to a single root directory: - pktᵥ9₁/ Key directories/files inside pktᵥ9₁/: - scripts/ (auditors/verifiers) - results/ (certificates, reports, witnesses) - thresholds. json, numericₘodel. json- packetᵢdentity. json, paperᵥersion. json- manifest. json (hash-locked canonical file list) - provenance/ (seal + environment/toolchain fingerprints) - replication/ (runA/runB semantic replication + comparator) - historical/ (archived legacy material; excluded from the manifest) Important: - historical/ is non-canonical archive storage; it is explicitly excluded from manifest. json and never participates in PASS. --- ## Quickstart (fresh unzip) Run exactly the following from a clean unzip: 1) unzip CollatzL2Bᵥ9. 1. zip2) cd pktᵥ9₁3) python -B scripts/auditₐll. py --mode sealedfullₙonsynthₕardenedₗevel3 Expected result: - PASS If any mandatory field/inequality/file/hash is missing or inconsistent, the audit must FAIL (no partial acceptance). --- ## Closed results (interface-level, v9. 1) ### Gate-A: budget trigger (certificate-grade) The program requires an explicit budget domination trigger inequality at the packet-declared base scale: \ₓ₎ₓ₀₋ < ₀, as a checked entry in the verdict certificate. ### Gate-B: Corr (delta) /TwGap bottleneck as a mandatory checked entry (I4) - The Corr\ ( () \) /TwGap bottleneck is exposed as a mandatory Gate-B certificate obligation (I4). - The verifier enforces I4 as a checked inequality (not a narrative assumption). - The audit fails if the I4 entry is missing. Consumption rule (localization): - The deterministic map / finite validation grid enters the logic only via a certified grid-envelope scalar. - Corr\ ( () \) /TwGap is consumed solely through this finite-grid envelope plus enforced margins (no hidden global Corr\ ( () \) interpretation elsewhere). --- ## Frozen constants (prevents “definition shift”) The packet freezes a conservative max-extension rule for the q-sweep proxy via an audited coverage slack constant. Definitions: - Let \ (\) be the audited q-sweep proxy maximum over the observed modulus keys. - Let \ (s₂₎ₕ 0\) be the audited coverage slack. - Define the frozen worst-case proxy constant: \_: = + s₂₎ₕ. \ Semantic contract (soundness role of coverage slack): \ₐ ₐ_₃₄₂₋MS (q) _. \ Certified margin (on PASS): - Let \ (_\) be the enforced threshold. - Define the proxy margin: ₑ₎ₗₘ: = _ - _. PASS, \ (mₑ₎ₗₘ 0\). Interpretation: - \ (s₂₎ₕ\) is fixed as the conservative slack needed to extend the observed maximum to the declared modulus set. - This makes the grid-envelope consumption explicit and auditable; no other Corr\ ( () \) semantics are assumed. --- ## What is new in v9. 1 (reader-facing changes) - Dependency map (1 page): the main paper includes a compact table stating exactly where the certified grid-envelope is used (and nowhere else). - Frozen proxy/coverage constants: the companion paper fixes the semantic contract for coverage slack and proxy margin as certificate-grade constants. - Minimal packet hygiene: the sealed ZIP has a clean top-level surface (pktᵥ9₁/) ; legacy artifacts are moved under historical/ and excluded from the manifest. - Audit-first interface: mandatory certificate entries are enforced mechanically (missing entry ⇒ FAIL). --- ## Trust base (TCB) and numeric soundnessThe main claim is certificate-graded and conditional on: - correctness of the published auditor/verifier implementation, - correctness of the declared numeric model for the checked inequalities, - correctness of the manifest/seal binding the verified objects. In the standard computational-proof sense, the implication“Level-3 PASS ⇒ program conclusion”is meaningful only under the above trust base. A key target going forward is reducing the TCB further: - interval/rational enforcement for critical scalars/inequalities, and/or- an independent verifier. --- ## Scope it is excluded from the canonical manifest and does not affect PASS. (End of description. ) ========================= Author: Lee Byoungwoo leeclinic@protonmail. com
Byoungwoo Lee (Sat,) studied this question.