Collatz Final Gate v9.2: 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. ================================================ # 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 from non-semantic fields (e. g. , absolute paths), indicating the packet is not invariant. ## Rationale Option 1 is intentionally “the cleanest” approach: it makes the packet’s verification logic depend only onfinite, explicitly declared summary objects whose semantics are stable under packaging, extraction paths, andreviewer environments. This tightens the reproducibility contract without weakening the mathematical claims checked by the artifact. ================================= Author: Lee By