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):** - `Collatz_Final_Gate_v9.2.pdf` - `Collatz_Final_Gate_v9.2.tex` - **Companion / interface paper (PDF + TeX):** - `GateB_Bottleneck_v2.2.pdf` - `GateB_Bottleneck_v2.2.tex` - **Reviewer checklist (base-scale trigger instance, PDF + TeX):** - `L0_16_complete_certificate_checklist_v9.2.pdf` - `L0_16_complete_certificate_checklist_v9.2.tex` - **Sealed packet (ZIP):** - `Collatz_L2B_v9.2.zip` ## How to reproduce (packet) Unzip the packet and run the hardened audit mode: - `python3 -B scripts/audit_all.py --mode sealed_full_nonsynth_hardened_level3` 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:** \( η < κ_0 \)- **GateB bottleneck item:** a checked Corr\((δ)\)/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 & non-toy status This record should be read as a **computational-proof interface**:- It does **not** ask the reader to trust an opaque computation.- It supplies a **sealed, auditable artifact** whose outputs are verified by **independent checkers** and enforced by a deterministic audit script. The intended evaluation axis is therefore: “Does the sealed artifact, under a transparent audit policy,produce the required finite witness and checked inequalities?”—rather than “Does the narrative alone persuade?” ## Program closure and targets The long-term closure target can be stated at the interface level: - **Soundness target (program-level):** a theorem of the form \[ (Level-3 PASS for the sealed packet) \;\; Collatz conjecture. \]- **Interface target (artifact-level):** the packet must output at least one base-scale trigger instance (e.g., \(L_0=16\)) 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 enforcingcoverage_slack = 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: Collatz_L2B_v9.2r1_coverage_enforced.zipReproduce: python3 -B scripts/audit_all.py --mode sealed_full_nonsynth_hardened_level3 ================================================ # 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/.../pkt_v9_1_r8/...`) 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 `qsweep_report.json` files referenced by a `best_by_M[*].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/gateb_tau_sweep_report_level3.json`: - `runs[]`: per-\(M\), per-variant metrics table (including \(K\), \(Kref\), \(τraw\), \(τeff\)) - `best_by_M`: selected best variant per \(M\) and the corresponding tau metrics Verifier2 checks: 1) **Existence / linkage**: For each \(M\) in `best_by_M`, the chosen variant must exist in `runs[]`.2) **Metric consistency**: The `best_by_M[M].tau_raw` and `tau_eff` must match the corresponding `runs[]` entry (within a fixed numeric tolerance policy).3) **Penalty rule correctness**: The conservative normalization \[ τeff = τraw · max\!(1, {K}{Kref}) \] must hold for every run entry, and the `best_by_M` value must be consistent with this rule.4) **Best-variant correctness**: For each \(M\), the selected variant is required to achieve the minimum \(τeff\) among all variants in `runs[]` for that \(M\). This eliminates the requirement to open per-run `qsweep_report.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., `packet_root`, `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/audit_all.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 `best_by_M` 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 \(τeff\) among candidates for that \(M\).- Any report digest mismatch arises fro