## OverviewThis record bundles three certificate-facing artifacts from the *Collatz Final-Gate* program: 1) **Main paper (v8. 0a) **: an **EB-only closure packet** (Level-2A/2B), structured for auditable, pipeline-style verification. 2) **Supplement / Gate-B bottleneck note (v1. 3b) **: a deterministic interval baseline and an explicit “upgrade interface” that isolates the *exact remaining obstruction* for certificate-grade Gate-B uniformity. 3) **Auditable Level-2B demo packet (L0=16, r2) **: a reproducible packet that emits a **machine-checkable verdict certificate** and a standardized **FINAL VERDICT** console box. The unifying goal is to turn “assumptions” into **certificate inputs** that are enforced by audit scripts, so that a single verdict object can mechanically trigger the All-\ (L\) promotion interface when the required witness is present. --- ## Closed results (what this record already certifies) - **EB-only closure (Level-2A/2B) ** is packaged in a self-auditing format (manifest + hashes + audit scripts). - The demo packet provides a **Level-2B-style, certificate-grade** workflow at \ (L₀=16\), including: - schema-complete JSON reports, - manifest enforcement, - deterministic replayability via `auditₐll. py`. --- ## What is new in r2 (turning-point strengthening) The demo packet is tightened so that **Corr (\ (\) ) /TwGap is not merely stated but enforced as a required checked inequality**: - `gatebcertificate. json` now contains: - `verdict. checkedᵢnequalities` as an **explicit list of inequality IDs** (not a mere count), - mandatory inclusion of the Corr/TwGap witness entry: - **`I4corrdeltaₜwgapwitness`** (required; missing ⇒ FAIL). - The audit logic is strengthened so that: - an empty or missing inequalities/checklist field ⇒ **FAIL**, - missing `I4corrdeltaₜwgapwitness` in `checkedᵢnequalities` ⇒ **FAIL**, - the console output **must** include the standardized line: - `corrdeltaₜwgapwitness: OK (id=I4corrdeltaₜwgapwitness) ` - The wording is harmonized so that: - **paper (Chapter 9 listing) ** ↔ **README template** ↔ **JSON field** ↔ **audit output** are aligned 1: 1 at the phrase level for “one-second referee comprehension. ” --- ## Mathematical content in the Gate-B bottleneck note (v1. 3b) The note isolates two components: 1) **Deterministic baseline (unconditional). **On contiguous intervals \ (I=a, a+K-1\), a parity-cylinder decomposition gives an explicit AP–dispersion bound for \ (TL (n) q\), with a deterministic \ (2L\) superposition cost. 2) **Upgrade interface (Corr (\ (\) ) /TwGap removes \ (2L\) ). **Assuming a correlation/spectral witness (Corr (\ (\) ) / TwGap-type input), a van der Corput step plus a carry-buffered normal form removes the \ (2L\) union-bound loss and yields an interval AP–dispersion estimate of the form\ (2^-₁ L + q/K + 2^-c₀ L\). A theorem-level condition (GB1c) is stated for promotion to an admissible window class. This separates: - (i) a deterministic, literature-comparable baseline output, from- (ii) the unique remaining certificate obstruction for window-uniform Gate-B. --- ## Reproducibility (demo packet) - Unzip the demo packet. - Run: - `python -B scripts/auditₐll. py --packetdir. `- The audit prints a **FINAL VERDICT** box and verifies: - manifest integrity, - report schema completeness, - Gate-B certificate validity, including the required Corr/TwGap witness line. --- ## Files in this record- Main paper PDF (v8. 0a): EB-only closure packet. - Gate-B bottleneck note PDF (v1. 3b): deterministic baseline + Corr/TwGap upgrade interface + Final Verdict Protocol listing. - Demo packet ZIP (L0=16, r2): auditable Level-2B workflow, including `results/gatebcertificate. json`, manifests, and audit scripts. --- ## Scope and program targetThis record is a *certificate and protocol* milestone: it enforces the Corr (\ (\) ) /TwGap witness as an auditable input format and ties it to an automatic verdict trigger. Full Gate-B uniform closure depends on producing such a witness object (as a certificate-grade input), after which the All-\ (L\) promotion interface is designed to activate mechanically. ========================= Author: Lee Byoungwoo leeclinic@protonmail. com
Byoungwoo Lee (Wed,) studied this question.