Paper P-05c: Collatz Finite Duration and Cap-Free Lock-B Closure - Internal FRC Computational Theorem-Chain Proof Object

Paper P-05c: Collatz Finite Duration and Cap-Free Lock-B Closure - Internal FRC Computational Theorem-Chain Proof Object Abstract This record publishes Paper P-05c of the FRC programme: Collatz Finite Duration and Cap-Free Lock-B Closure. The paper records an internal FRC computational theorem-chain for the Collatz problem, supported by a replayable proof chain, hostile audit notes, appendices and a corrected v7.209 final theorem assembly artefact. The central claim is deliberately scoped. Inside the internal FRC computational theorem chain, the v7.206-v7.209 sequence closes the residual A6 obstruction, lifts the closure into the global A6 grammar, assembles A5/G1 and A6 under cap-free Lock-B and marks universal Collatz proved inside that generated chain. Outside that chain, external mathematical peer review remains open. The submission includes the main P-05c proof paper, Appendix A on Real Collatz and Symbolic Collatz, Appendix B documenting the development journey, a support pack containing v7.206-v7.212 clean proof-chain artefacts, an independent v7.209 replay verifier, hostile audit notes and the FRC programme table. The publication is intended as a complete, citable record for external audit, criticism, replay and peer review. Description This Zenodo record contains the complete P-05c Collatz submission package prepared for external audit and peer review. The submission is part of the Finite Reality Closure (FRC) programme and is positioned after P-05a finite prime-support closure work and P-05b prime structural masonry work. P-05c addresses the Collatz problem through the FRC finite-duration and cap-free Lock-B framework. The proof-bearing claim is not that Zenodo publication itself constitutes peer review. The claim is that the uploaded internal FRC computational theorem-chain package reports closure inside its own replayable chain. The external status remains: not externally peer reviewed. Included files: 1. Main proof paper: P-05c Collatz Finite Duration and Cap-Free Lock-B Closure, Rev03, with closure model figure. 2. Appendix A: Real Collatz and Symbolic Collatz, a conceptual companion explaining the memory-profile and symbolic/reality distinction. 3. Appendix B: The Journey, a narrative audit appendix describing how the proof route changed through reverse-corridor analysis, owner locks, overshoot, Lock A/B, A5/A6 splitting, positive-growth SCC obstruction, source-row reconstruction and final assembly. 4. Supporting proof pack: v7.206-v7.212 artefacts, including the corrected v7.209 final theorem assembly freeze, hostile audit notes and an independent replay script. 5. FRC programme table: Framework paper numbering and relationship table, with DOI/URL fields to be updated after final publication. Key internal chain:- v7.206: exact affine exclusion of source-composable residual cycles.- v7.207: residual SCC affine exclusion proof review.- v7.208: A6 global lift integration.- v7.209: cap-free Lock-B final theorem assembly.- v7.210-v7.212: replay certificate, external review pack, and final survival/claim-scope pack. Corrected v7.209 SHA-256:092bb64c6c818aa4c887c008b57b4dfbc566b52d2289c59e5ed9a5fe6a81b183 Corrected full submission package SHA-256:0a2929b96abc52cd2517741654a280d78887b7ba11ad8e3881705c417b7044ef This record publishes a complete internal FRC proof-chain submission for the Collatz conjecture. The work claims closure inside the uploaded computational theorem chain, not external mathematical acceptance. The purpose of this Zenodo record is to make the paper, appendices, proof artefacts, replay script and audit materials citable and available for independent review. This upload is not externally peer reviewed at the time of publication. It is published to create a permanent, citable record for external mathematical audit, independent replay, criticism and peer review. Status: internal FRC computational theorem-chain closure claimed; external peer review open. FRC Programme at time of Publishing this paper.