← AMRAL · PROGRAM-COLLATZ
Collatz Conjecture (the 3x+1 problem) · Operation Translation Series
Nine core papers by Neo.K, co-organized with Aletheia, starting from the exact finite affine structure of parity words and building a Local Affine Atlas — proving that every finite word corresponds to an exact affine operator under the Collatz map — then generalizing that structure from Collatz's specific coefficients (3, 1) to the broader Residue-Class Operation Translation (RCOT) class, and delineating the algebraic domain of validity for each theorem. The series reaches its capstone at Paper 09, which explicitly states that the global-quantifier gap remains uncrossed. The follow-on Hard-Zeta research program, developed independently afterward, proposes a proof research program aimed at exactly that gap — it is not the series' tenth paper.
Does not claim to prove or disprove the Collatz conjecture. Every paper in the series has its own "what this paper does not prove" section; the source documents' own non-claim language is carried directly into each page's claim-box. Independent secondary verification and formal reproduction run on a separate line — see the link below.
Ordered by the source documents' own "next paper" pointers; recommended reading order. SSSP Repaired v1.0, full provenance (every paper has a before/after SHA-256 comparison).
From Bidirectional Trees and Decimal Reduction to Local Affine Atlases — an accounting of the author's own 2025-2026 Collatz research.
Paper 02 · v0.1.1Finite-Word Affine Closure, Count/Order Decomposition, and a Word-Order Correction.
Paper 03 · v0.1.1The exact domain of validity, 2-adic splitting, and local trivialization of the Collatz Local Affine Atlas.
Paper 04 · v0.1From the Collatz Local Affine Atlas to exact inverse fibers, the odd skeleton, and bidirectional-tree reconstruction.
Paper 05 · v0.1From exact affine drift and a word-order correction to a purely combinatorial account of the 89.4943% figure.
Paper 06 · v0.1From run-length encoding of parity words and exact v₂ drift to a valuation-order correction.
Paper 07 · v0.1.1From the Collatz special case to commutative scalar affine dynamics, a phase boundary, and a generalized local atlas.
Paper 08 · v0.1.1The domain of applicability of Residue-Class Operation Translation, from commutative integral domains to non-commutative and nonlinear dynamics.
Paper 09 · v0.1.1From the Local Affine Atlas and a descent sieve to an integer-anchored hard branch — the capstone of the series.
Developed independently on top of the nine-paper core series, co-organized with Aletheia. The document states plainly that this is a research program, not the series' tenth paper.
The source material uses the SSSP (SSSP-assisted scholarly repair) protocol to preserve provenance — the before/after SHA-256 for every paper, per-file diffs, and an aggregate hash are all preserved in full. The early experiments (bidirectional inverse-tree theory, etc., predating the nine-paper series) are not counted in this series; they are background material, with the original package preserved in the amral-research-trees source tree — no detail pages are built here to duplicate them.
Independent secondary verification and machine-readable cross-checking: see the collatz-verification-zhuiheng line (a separate sub-site, in preparation).