Collatz Final Gate v9.2.1: Certificate-Triggered Closure Program (Sealed Level-2B Packet + GateB Bottleneck Interface)

# Overview This record releases **Collatz Final Gate v9. 2** together with a **sealed, mechanically auditable Level-2B packet**implementing a **certificate-triggered closure program** for the Collatz conjecture. The core idea is to expose the remaining “bottleneck” as a **finite, hash-locked proof interface**: a single sealed packet deterministically produces a **verdict certificate** whose checked inequalities are designedto be sufficient to trigger an **All-\ (L\) ** promotion mechanism (analytic upgrade), rather than relying on informalclaims or non-reproducible computations. ## Contents - **Main paper (PDF + TeX): ** - `CollatzFinalGateᵥ9. 2. pdf` - `CollatzFinalGateᵥ9. 2. tex` - **Companion / interface paper (PDF + TeX): ** - `GateBBottleneckᵥ2. 2. pdf` - `GateBBottleneckᵥ2. 2. tex` - **Reviewer checklist (base-scale trigger instance, PDF + TeX): ** - `L0₁6completecertificatechecklistᵥ9. 2. pdf` - `L0₁6completecertificatechecklistᵥ9. 2. tex` - **Sealed packet (ZIP): ** - `CollatzL2Bᵥ9. 2. zip` ## How to reproduce (packet) Unzip the packet and run the hardened audit mode: - `python3 -B scripts/auditₐll. py --mode sealedfullₙonsynthₕardenedₗevel3` A PASS indicates that: 1) all schema-critical files exist, 2) manifests/hashes match, 3) the certificate outputs match the independently re-computed verifier outputs, and4) the packet satisfies bytecode hygiene and provenance constraints enforced by the audit rules. ## Closed results (interface-level) The v9. 2 packet is organized around **mandatory, checked entries** in the verdict certificate. In particular, the auditexpects the certificate to bind to nontrivial numeric inequalities such as: - **Budget domination: ** \ (< ₀ \) - **GateB bottleneck item: ** a checked Corr\ ( () \) /TwGap entry (as a mandatory interface field, not as prose) The program is “certificate-triggered” in the sense that once the above checked entries pass, the analytic upgradepath to All-\ (L\) becomes a fixed, non-interactive argument in the paper (i. e. , not a moving target of additionalcomputational claims). ## What is new in v9. 2 (reader-facing changes) Version v9. 2 is a **reviewer-facing consistency and soundness-hygiene upgrade** on top of v9. 1. The analytic skeleton (Gate A / Gate B separation and the All-\ (L\) promotion mechanism) is unchanged. What is strengthened is the **public proof interface** and **TCB visibility**. ### (1) TCB reduction is made visible on the critical path Certificate-grade scalars and inequalities are additionally verified by: - an **interval-based checker** for key bounds, and- a **rational re-computation** for derived scalars where the witness admits an exact reduction. The audit fails if these independent checks disagree with the primary pipeline outputs. ### (2) Proxy \ (\) Corr/TwGap coverage and slack are frozen as explicit constants The proxy-to-target bridge is accompanied by a single, explicit set of numeric constants (coverage domain, slack margins, and guard thresholds) recorded in machine-readable reports. The audit enforces that the certificate binds to **those constants**, not to informal prose. ### (3) “Option 1” verification hardening: self-contained Level-3 tau-sweep verification In v9. 2, the verifier path is hardened to avoid fragile dependencies on per-run file paths. Verifier2 is upgraded to validate Level-3 tau-sweep consistency using only theself-contained summary report (runs + best-by-\ (M\) table), so that audit digests are invariantunder extraction paths and reviewer environments. (See the included one-page verification design note for details. ) ## Scope \; Collatz conjecture. \]- **Interface target (artifact-level): ** the packet must output at least one base-scale trigger instance (e. g. , \ (L₀=16\) ) together with GateB mandatory checks and a strict audit PASS. v9. 2 focuses on strengthening the **artifact-side guarantees** (TCB visibility, frozen constants, and audit invariance) so that the remaining debate concentrates on the soundness theorem rather than on tooling ambiguity. ## Keywords Collatz conjecture; computational proof; certified computation; reproducibility; auditable artifact; proof interface; hash-locked certificate; interval arithmetic; rational witness. ## License / notes Choose a permissive license appropriate for reproducible research artifacts (code + data + PDFs). If you mirror the record, preserve the sealed ZIP unchanged; any modification requires re-sealing and re-auditing. ================================================ ## Addendum (coverage enforcement) This update strengthens the Proxy→Corr (δ) /TwGap interface by enforcingcoverageₛlack = 0 via an explicit full-enumeration coverage report. The sealed packet now fails audit unless the reported q-domain enumerationcount and q-list hash match an independently recomputed enumeration. Download: CollatzL2Bᵥ9. 2r1coverageₑnforced. zipReproduce: python3 -B scripts/auditₐll. py --mode sealedfullₙonsynthₕardenedₗevel3 ================================================ # Verification Design Note (v9. 2) — “Option 1” Self-Contained Level-3 Tau-Sweep Verification ## Context / problem Earlier packet iterations exposed a reviewer-facing fragility: 1) **Path-dependent reports**: Some verifier reports embedded absolute extraction paths (e. g. , `/mnt/. . . /pktᵥ9₁ᵣ8/. . . `) inside the report payload. If an auditor re-computed the same checks in a different directory, the semantic content matched but the report digest differed, causing audit failure. 2) **Non-essential external-file dependencies**: Verifier2’s Level-3 tau-sweep consistency checks sometimes relied on per-run `qsweepᵣeport. json` files referenced by a `bestbyM*. path` field. In practice, a sealed packet may store only a self-contained summary table (runs + best-by-\ (M\) ), while omitting some per-run files to keep the packet minimal. This made the verification brittle even when the summary table fully determined the consistency checks. The result was an avoidable failure mode: the audit could fail for “packaging reasons” rather than mathematical reasons. ## Design goals (v9. 2) - **G1. Audit invariance under extraction paths**: Digests should not depend on absolute filesystem locations. - **G2. Self-contained verification**: If a summary report contains enough information to verify a claim, verifiers should not require additional per-run files. - **G3. Preserve strictness**: Self-contained verification must remain as strong as the original check (i. e. , no weakening into a mere smoke test). - **G4. Keep the TCB small and visible**: The critical-path checks should be independently re-computable with small, readable code. ## Implementation summary (Option 1) ### (A) Verifier2: self-contained Level-3 tau-sweep verification In v9. 2, Verifier2 validates Level-3 tau-sweep consistency using only: - `results/gatebₜauₛweepᵣeportₗevel3. json`: - `runs`: per-\ (M\), per-variant metrics table (including \ (K\), \ (Kₑ₄₅\), \ (ₑ₀ₖ\), \ (₄₅₅\) ) - `bestbyM`: selected best variant per \ (M\) and the corresponding tau metrics Verifier2 checks: 1) **Existence / linkage**: For each \ (M\) in `bestbyM`, the chosen variant must exist in `runs`. 2) **Metric consistency**: The `bestbyMM. tauᵣaw` and `tauₑff` must match the corresponding `runs` entry (within a fixed numeric tolerance policy). 3) **Penalty rule correctness**: The conservative normalization \ ₄₅₅ = ₑ₀ₖ \! (1, KKₑ₄₅) \ must hold for every run entry, and the `bestbyM` value must be consistent with this rule. 4) **Best-variant correctness**: For each \ (M\), the selected variant is required to achieve the minimum \ (₄₅₅\) among all variants in `runs` for that \ (M\). This eliminates the requirement to open per-run `qsweepᵣeport. json` paths while preserving the same logical strength: the best-variant claim and tau-sweep penalty rule are fully determined by the summary table. ### (B) Report path hygiene (digest invariance) Verifier reports (and any audit-digested artifacts) must not include absolute extraction paths. Where a path-like field is needed, v9. 2 uses one of: - relative paths from packet root, or- canonical labels (e. g. , `packetᵣoot`, `results/. . . `) that remain invariant across extraction locations. This ensures audit digests depend on semantic check outputs, not on the auditor’s filesystem. ## Audit policy impact - `scripts/auditₐll. py` continues to re-compute verifier outputs and compare digests / key fields. - The policy is strict: if the recomputed self-contained checks disagree with on-disk reports, the audit FAILs. - The policy is now robust: an auditor can extract the ZIP anywhere and obtain identical PASS/FAIL outcomes. ## Failure conditions (explicit) The audit must FAIL if any of the following occur: - Verifier2 cannot match a `bestbyM` selection to a corresponding run entry. - Any tau-sweep penalty rule identity fails for any run entry. - The selected best variant is not the minimum \ (₄₅₅\) among candidates for that \ (M\). - Any report digest mismatch arises fro