What question did this study set out to answer?

This work aims to reduce the Collatz conjecture's arithmetic complexity into a finite, auditable format, making progress measurable and reproducible.

February 9, 2026Open Access

Collatz Final Gate v8.5 — Main Paper + Auditable Level-2B Demo Packet + Replication Toolkit (GateB companion v1.7r2)

Key Points

This work aims to reduce the Collatz conjecture's arithmetic complexity into a finite, auditable format, making progress measurable and reproducible.
Isolation of arithmetic bottlenecks into Gate B.
Implementation of a sealed Level-2B proof-input packet for mechanical auditing.
Creation of a semantic replication toolkit for validated results.
The methodology exposes checked inequalities with explicit margins in a hash-locked format.
A successful mechanical audit indicated that all tests passed for the demo packet.
The replication toolkit was provided for checking results across different environments.

Abstract

## Overview This record is a versioned “reduction + certification” package toward the Collatz conjecture. The main paper (v8. 5) isolates the remaining arithmetic bottleneck into a finite, auditable attack surface (“Gate B”) and ships a sealed Level-2B proof-input packet that passes mechanical audit with`thresholds. json` enforcement enabled. **Core methodological claim (carefully stated): **the remaining hard input (Corr (δ) /TwGap/AP-dispersion, etc. ) is not hidden inside prose;it is exposed as *checked inequalities with explicit margins* inside a hash-locked certificate object, so that progress is measurable, reproducible, and version-stable (Four-Color-style workflow). ## What is new in v8. 5 - **Manifest-SHA sealing (cycle-avoidance + binder binding): ** the verdict binder records `binder. manifestₛha256 = SHA256 (manifest. json) ` while excluding the binder itself from the manifest to avoid a hash cycle. - **Final Verdict Protocol (trigger instance at fixed base scale L0): ** a single audited base-scale packet (e. g. L0 = 16) is treated as a *trigger instance*; once its one-line inequality is certified with positive margin, the analytic “All-L promotion” mechanism applies. - **Two-environment semantic replication toolkit: ** contract-driven snapshot collection + comparison script + replication audit script. ## Included files (this Zenodo record) 1. `CollatzFinalGateᵥ8. 5. pdf` — main paper. 2. `GateBBottleneckᵥ1. 7r2. pdf` — companion note (Gate-B bottleneck + certificate interface). 3. `CollatzDemoL2Bᵥ8. 5ₛealedfullPASSₘanifestsha. zip` — sealed Level-2B demo packet (mechanical audit PASS). 4. `CollatzReplicationToolkitᵥ8. 5. zip` — two-environment semantic replication toolkit. 5. `L0₁6completecertificatechecklist. pdf` — one-page checklist for a complete trigger instance at L0=16. **Minor naming note (cosmetic only): **inside the demo packet, `packetᵢd` may still read `. . . v8. 4ᵣeplicationᵣeady` for backward compatibility, while `paperᵥersion. json` and the shipped paper hash bind to v8. 5. The audits do not depend on this label. ## How to reproduce (mechanical audit) Unzip the demo packet and run: ```bashunzip CollatzDemoL2Bᵥ8. 5ₛealedfullPASSₘanifestsha. zipcd python -B scripts/auditₐll. py --mode sealedfull Expected outcome: PASS: all audits completed. How to run semantic replication Unzip CollatzReplicationToolkitᵥ8. 5. zip and follow READMEREPLICATION. txt. The toolkit includes: specs/reprocontract. json scripts/collectₘinimalᵣun. py scripts/compareᵣuns. py scripts/auditᵣeplication. py Scope certification; reproducibility; audited computation; Markov operators; contraction; spectral gap; hash-locked evidence; Gate B; TwGap; Corr (δ) ========================= Author: Lee Byoungwoo leeclinic@protonmail. com

Collatz Final Gate v8.5 — Main Paper + Auditable Level-2B Demo Packet + Replication Toolkit (GateB companion v1.7r2)

Key Points

Abstract

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