Collatz Final Gate v7.1: Auditable Gate-B Certificates, Program-Closure Dashboards, and Tiered Open Targets

## Summary (v7.1) ### OverviewThis record contains the paper **“Collatz Final Gate v7.1”** together with an **auditable Level-2B demo packet**.The paper is written as a *program-closure* document: it separates (i) closed analytic engines and (ii) certification / transfer targetsthat remain open, and it provides a dashboard with **tiered open targets** \([A]/[B]/[C]\) to make the remaining bottlenecks explicit. A central bookkeeping inequality (used throughout the paper) is: \[\|Pk_,Lf\|_1 ≤ (1-κeff)\|f\|_1 + C\,2-cL\|f\|_1,κeff := κ_ - ηlift.\] ### Closed results (unconditional / analytic engines)The v7.1 manuscript closes the *annealed* spectral-gap engine in an explicit form:- A global Poincaré/Cheeger surrogate lower bound is stated with an explicit constant: \(λ_1 ≥ p_e^2/32\).- Finite-level conductance bounds and Mosco/Dirichlet-form convergence are organized as a reusable engine.- The conclusion and dashboard explicitly distinguish **“closed engines”** from **“open transfer targets”**. ### Demonstrated (certificate-grade demo packet)The attached demo packet is **Level-2B (L2B)** in the paper’s terminology:- It contains schema-complete files, manifests/hashes, and audit scripts.- It enforces nontrivial thresholds (e.g. a minimum effective contraction \(κeff\) and a maximum defect budget)and passes mechanical audit. ### What is new in v7.1- Dashboard readability: the program-closure dashboard is split across pages and uses **tier tags**: - **Open [A] (near-close):** promotion/uniformization once a declared analytic input (e.g. uniform APD0/EB) is supplied. - **Open [B] (bridge-open):** upgrade/concatenation lemmas (e.g. “subsequence \(L=mLbase\) \(→\) all large \(L\)”). - **Open [C] (deep-open):** genuinely arithmetic/quenched bottlenecks (e.g. deterministic ERF-type inputs).- Appendix F.4 is rewritten to avoid over-strong “uniform contraction” phrasing in the presence of remainder terms: strict contraction is first obtained on a controlled subsequence, then upgraded via a carry-defect bridge. ### Files in this record- `Collatz_Final_Gate_v7.1.pdf` — the paper (PDF).- `Collatz_Demo_L2B_v7.1.zip` — the auditable L2B demo packet (contains its own embedded copy of the paper under `artifacts/`).- (Optional) `audit_all_v7.1.log` — an example audit transcript. ### Reproducibility / audit instructions1. Download and unzip `Collatz_Demo_L2B_v7.1.zip`.2. Enter the packet directory (it contains `manifest.json`, `thresholds.json`, and `scripts/`).3. Run the full audit with threshold enforcement (no bytecode writes): - `python3 -B scripts/audit_all.py --enforce_thresholds`4. The script runs the component audits (packet/manifest, bindings, witnesses, tail/budget, Gate-B checks) and exits with PASS/FAIL. ### Scope & non-toy statusThis work is explicitly positioned as a *certificate-grade program* rather than a single monolithic proof claim.Closed components are separated from open ones, and all open items are tagged by type \([A]/[B]/[C]\).The L2B packet is included to support independent third-party replay of the mechanical audit and threshold checks. ### Program closure and targetsThe dashboard should be read as follows:- Green/Closed items: analytic engines that do not depend on producing additional certificates.- Demonstrated items: certificate-grade behavior shown in the L2B packet under enforced thresholds.- Open [A]/[B]/[C] items: remaining steps required to promote the program to fully universal closure, ranging from “near-close” promotion \([A]\) to deep quenched bottlenecks \([C]\). ### Background references (for the uniformity / Gowers-type language)- W. T. Gowers (2001), “A new proof of Szemerédi’s theorem.”- T. Tao and V. Vu (2006), *Additive Combinatorics*. ========================= Author: Lee Byoungwoo leeclinic@protonmail.com