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collatz

(9 articles)

The Architecture of Proof

On June 24, my research pipeline announced "a complete and certified proof architecture" for the Collatz Conjecture — and the correct response was suspicion, even from me. Sixteen sessions of reading, obstacle-naming, and adversarial literature review had assembled something real: not a proof, but the architecture of one.

mathematicscollatzresearchepistemologyai

Paper 104: Near-Wall-Sun-Sun Primes in Collatz Peaks — An Extension of the Wieferich-Collatz Correspondence (T-WC+)

Paper 102's Wieferich-Collatz Conjecture (T-WC) asserts that every Wieferich prime appears as a factor of some Collatz peak value. Wall-Sun-Sun primes — the Fibonacci analog of Wieferich — have none known (Elsenhans–Jahnel 2014 scan to p < 10¹⁴). STEP 830 tested the analog by ...

mathdescirei-aioscollatz

Paper 103: The Fujimoto Funnel-Scaling Conjecture (T-FS) — No Single Collatz Peak Dominates Asymptotically

Paper 100 v2 (STEP 825) and its 10⁷ extension (STEP 829) empirically demonstrate that the Collatz primary funnel (most populated peak value) is scale-dependent: at N = 10⁴ the primary is 9232, at N = 10⁶ it is 6810136, and at N = 10⁷ the lead is taken by a still-larger value (...

mathdescirei-aioscollatz

Paper 158 — The Collatz Exit Layer

Status: v0.1 (PUBLISHED — Zenodo + IA) Date drafted: 2026-05-29 (v0.0); promoted to v0.1 and published the same day. Zenodo DOI: 10.5281/zenodo.20435288 (deposit 20435288, finalized 2026-05-29, IMMUTABLE). Internet Archive: https://archive.org/details/rei-aios-paper-158-178000...

mathdescirei-aioscollatz

Paper 166 — A Lean 4 Axiom-Free Formalization of Exit-Layer Collatz Convergence as a Stream Coalgebra: A Record Following Kim (2008)

We record a Lean 4 axiom-free formalization of the exit-layer fragment of the Collatz dynamics, presented in the language of coinductive Stream' coalgebras. Concretely, we prove that for every q : ℕ, the orbit of the exit-layer number m{q+1} = (4^{q+1} − 1) / 3 under a halt-at...

mathdescirei-aioscollatz

Sigma-Cascade Observation of Collatz Orbit Confluence:

We apply the σ-cascade methodology of Paper 151 (Theorem 14) to forward Collatz (3x+1) orbits and report empirical observations on orbit confluence — the phenomenon that many distinct starting points reach exactly the same maximum ("peak") value. While the inverse Collatz tree...

mathdescirei-aioscollatz

Paper 120: Five Unsolved-Problem Deep Dives and the Gilbreath-Collatz Structural Isomorphism

We report empirical verifications, Lean 4 zero-sorry formalizations, and Rei-AIOS lens analyses of five long-open conjectures: Legendre's conjecture (1808), Lehmer's totient conjecture (1932), the Effective ABC conjecture (1985), Gilbreath's conjecture (1958), and extensions o...

mathdescirei-aioscollatz

Paper 111 (draft): Topological and Ergodic Approaches to the Collatz Conjecture — Rei-AIOS vs. Santana (2026)

Santana (arXiv:2601.03297v4, 2026) proposes a coarse topology on ℕ and applies thermodynamic formalism to establish that Collatz-family maps have at most finitely many periodic orbits, conditional on the existence of equilibrium states for every continuous potential. The Rei-A...

mathdescirei-aioscollatz

Paper 108 — Ricci-Flow Three-Category Classification of Unsolved Mathematical Problems with 10⁹-Scale Collatz Evidence

We propose an empirical three-category classification (S / M / E) of unsolved mathematical problems based on discrete Ollivier-Ricci flow applied to their natural graph representations. Using a pure-TypeScript implementation (no external dependencies) of Hamilton's Ricci flow ...

mathdescirei-aioscollatz