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...
We present OctaTheoria (オクタテオリア / 八軸観測装置), a multi-domain observation framework that projects heterogeneous time-series data (theory-chart category bins, arXiv submission hourly counts, cryptocurrency 1-minute candles, FX currency pair rates, and v0.2 Earth+Cosmos additions: L...
We present a synthesis-friendly Verilog implementation of the D-FUMT₈ Arithmetic Logic Unit, targeting the Sipeed Tang Console NEO development board (GW5AST-138B FPGA, FPG676 package). The ALU realizes eight discrete logic values — FALSE, TRUE, NEITHER, BOTH, ZERO, FLOWING, SE...
We report a single-day 12-file Lean 4 formalization sweep across seven long-open conjectures plus infrastructure extensions, producing 158 new zero-sorry theorems under Mathlib v4.27.0. The conjectures attacked include: 1. Lehmer's totient (1932, extended with k-Lehmer Q23 clo...
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...
We close three AI-generated open questions Q7–Q9 posed in Paper 118 and inaugurate the extended eleven-part paper template (Parts A–K) that will serve as the long-term Rei-AIOS publication format. Key results: - Q7 FALSIFIED empirically. The Paper-118 strong-uniqueness claim —...
We report five structurally independent zero-sorry Lean 4 formalizations from the Rei-AIOS unsolved-problems corpus, together with observed findings, AI-generated new open questions, and a unified D-FUMT₈ solution-status summary. This is the first Rei-AIOS paper to adopt the f...
The Erdős-Straus conjecture (1948) states that for every integer n ≥ 2 there exist positive integers a, b, c with 4/n = 1/a + 1/b + 1/c. It is empirically verified for n < 10^17 (Salez and others) and closed for many residue classes, but remains open in full generality. We pro...
Andrica's conjecture (1985) asserts An = √p{n+1} − √pn < 1 for every n ≥ 1, where pn denotes the n-th prime. The conjecture is empirically verified for n ≤ 1.3 × 10^16 but remains open. We provide a Mathlib-native Lean 4 formalization with 33 zero-sorry theorems: 24 concrete c...
The Twin Primes conjecture — that infinitely many pairs (p, p+2) are both prime — remains open. We reinterpret the modern bounded-gap landscape (Zhang 2013, Maynard-Tao 2015, Polymath8b) under the Rei-AIOS 8-valued logic (D-FUMT₈) and the Ricci-flow category classification (Pa...
We reproduce the classical random 3-SAT phase transition (Mitchell-Selman-Levesque 1992; Mertens-Mézard-Zecchina 2006; Ding-Sly-Sun 2014) using a minimal DPLL solver and find that at n = 30 variables with 20 samples per clause-to-variable ratio α, the DPLL backtrack count peak...
QR codes (Denso Wave 1994) encode up to 23,624 data bits in a 177×177 binary grid, with four fixed error-correction levels and a rigid byte-stream payload model. We propose a complementary 2D symbol system — the Nested Colored-Dot (NestedDot) — whose payload is a recursively s...
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...
We apply Rei-AIOS's discrete Ollivier-Ricci flow three-category taxonomy (Paper 108) to the Erdős-Straus conjecture 4/n = 1/a + 1/b + 1/c and find that among classifiable small n in [2, 1000] (i.e. those for which the partition graph has at least 3 edges with a ≤ 60), 84.3% be...
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 ...
Paper 33 (Fujimoto 2026, DOI 10.5281/zenodo.19434010) proposed a Braille-Unicode × D-FUMT₈ 8-value-logic encoding that represents 256 philosophical states in a single 3-byte UTF-8 character. The present paper contrasts this encoding with three widely deployed multi-modal embed...