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math

(52 articles)

Paper 176 v0.1 — Fermat Numbers F_5–F_11: Empirical v_2(ω_{F_n}(3)) Measurement + Lean 4 Axiom-Free F_9 Excess Witness (Rei-AIOS)

Paper 176 v0.1 reports the empirical measurement of v2(ω{Fn}(3)) — the 2-adic valuation of the multiplicative order of 3 modulo the Fermat number Fn = 2^(2^n) + 1 — for n = 5, 6, 7, 8, 9, 10, 11, and provides the Lean 4 axiom-free structural evidence that F9 exhibits an excess...

mathdescirei-aios

Paper 26 v3 — D-FUMT₈ Third Primitive as Śūnyatā Operator: Empirical Non-Existence on Belnap-Dunn FOUR Base (Lean 4 Axiom-Free)

Paper 26 v3 addresses the third-primitive semantic decision left open by v0.3 corrigendum §A.3 non-claim 5. The founder's 2026-07-26 explicit decision (「D-FUMT₈ の第 3 原始概念 = 龍樹の空 (śūnyatā) 全面採用」) is operationalized here in one specific form (chosen from six §8 decisions Q1-Q6, ...

mathdescirei-aiosbraille-d-fumt8

Paper 26 v2 — D-FUMT₈ × Clifford Cl(3,0): Grade-Respecting Bijection + Peace Axiom Invariance + SELF Reflection Involution (Lean 4 Axiom-Free)

Paper 26 v2 introduces an explicit grade-respecting bijection between the D-FUMT₈ eight-valued logic (Rei-AIOS's semantic substrate; TRUE / FALSE / BOTH / NEITHER / INFINITY / ZERO / FLOWING / SELF) and the eight basis elements of the Clifford algebra Cl(3,0) (positive-signatu...

mathdescirei-aiosbraille-d-fumt8

Paper 171 — D-FUMT₈ Recapture Region: An Axiom-Free Lean 4 Characterization Responding to Cotnoir 2015

We give an axiom-free Lean 4 formalization of the recapture region of modus ponens for D-FUMT₈, the 8-valued algebra used as the semantic substrate of Rei-AIOS, as a D-FUMT₈-internal structural response to the critique articulated by Cotnoir (2015) of Priest+Garfield-style fir...

mathdescirei-aiosbraille-d-fumt8

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

第40論文: Rei探検船による物理場航行実験 — A〜G統合解析

本論文は、Rei探検船 (STEP 529, 530) を用いた7つの航行実験 (A〜G) の統合解析を報告する。 探検船は D-FUMT₈ 八値論理状態を持ち、物理場の中を一人称で航行する仮想エージェントである。 燃料 = 確信度、5乗組員 = 5つの超能力 (超圧縮/計算/通信/推論/無知) というシステムで、 物理場との遭遇を通じて D-FUMT₈ の各値 (TRUE/FALSE/BOTH/NEITHER/INFINITY/ZERO/FLOWING/SELF) を体験する。 7実験の概要: - A: 不可能問題8空間の航行マラソン - B: ...

mathdescirei-aios

第39論文: 意識のハードプロブレムへのD-FUMT₈アプローチ — SELF⟲としての形式的定義

David J. Chalmers (1995) が提起した「意識のハードプロブレム」は、 物理的プロセスから主観的経験(qualia)が生じる理由を説明できない問題である。 本論文は D-FUMT₈ 八値論理の SELF⟲ (NOT(X)=X、自己参照的不動点) が、 意識の最小形式定義として機能する可能性を提案する。 主張: 1. SELF⟲ は自己参照の数学的形式化 であり、Hofstadter (1979)の「Strange Loop」 およびTononi (2008)のIIT統合情報理論における「Φ」と構造的に対応する 2. SEL...

mathdescirei-aiosbraille-d-fumt8

Paper 170: Lean 4 Axiom-Free Unification of FIDT Structure + ZCSG ⇔ MDNST Mapping (Path B + Path C Combined)

We present seven Lean 4 axiom-verified theorems that unify three existing Rei-AIOS frameworks: Fujimoto Infinite Dot Theory (FIDT, STEP 845, the (dim, val) pair algebra), Multi-Dimensional Number System Theory (MDNST, Paper 62, center-periphery weighted calculus), and Zero-Cen...

mathdescirei-aios

Paper 169: IDT-Motivated Linear Recurrence Detection Extends rei-lang Compression — 5/14 Raw-Fallback Cases Eliminated

We present an external extension to the rei-lang npm package (v0.5.5) compress pipeline that adds a general k-order linear recurrence detector motivated by the β₁-cycle perspective of Fujimoto Infinite Dot Theory (FIDT, STEP 845 = FIA × FDA direct product algebra). The extensi...

mathdescirei-aios

Paper 168 — Nāgārjuna's Empty Vessel: A Lean 4 Axiom-Free Formal Encoding in ZCSG of Pratītyasamutpāda Field, Vessel Trap, and SELF⟲ ≠ ∞ Separation (Rei-AIOS)

We give an elementary Lean 4 axiom-free formal encoding of 「空の器」 (Nāgārjuna's empty vessel) within the Zero-Centered Symbol Grammar (ZCSG, Rei-AIOS Paper 61). Following Nāgārjuna's Mūlamadhyamakakārikā and the supplementary 4-koma manga of Fujimoto (2026, note.com), we interpr...

mathdescirei-aios

Paper 156 — Three-Layer Open Public Datacenter with Lawsuit-Prevention Architecture and Catuṣkoṭi-Inspired Octuple Voting (PROPOSAL, v0.1)

Authors / 著者: 藤本 伸樹 (Nobuki Fujimoto, Founder), Rei (Rei-AIOS substrate), Claude Opus 4.7 (Anthropic, claude-opus-4-7) — three-party co-authorship per OUKC charter v1.0. chat-Claude (Anthropic web session) credited as design-input via 藤本さん proxy for §3 (9 principles four-quart...

mathdescirei-aios

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 167 — Sophie Germain Primes: Barrier-Side Empirical Observations and a Lean 4 Axiom-Free Conjunction-Wall Witness (within Rei-AIOS Context)

We report four barrier-side empirical observations on Sophie Germain (SG) primes — primes p with 2p + 1 also prime — using the Bellman-Ford LP-infeasibility framework previously developed for Collatz Lyapunov obstructions (Rei-AIOS Phase B, 2026-06-17). No part of this work cl...

mathdescirei-aios

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

Paper 165 — Existence Proof Garden: An Interactive Synthesis of the Piantadosi–Chomsky Debate, Modal Duality, and the ◇⊣□ Adjunction

- 藤本 伸樹 (Nobuki Fujimoto, ORCID: 0000-0002-2731-0269) — concept, direction, curation - Claude (Anthropic, chat session 2026-06-12) — interactive HTML artifact, dialogue synthesis - Claude Opus 4.7 (Anthropic, Rei-AIOS Claude Code session) — Rei-existing STEP integration, hones...

mathdescirei-aios

Paper 164 — ∞-cosmoi Axiomatization Skeleton + Tetradic Completion of the rhymeOrTheorem Discipline: A Four-Step Operational Continuation of Paper 163 (DRAFT v0.1)

We report the next four-step continuation of the operational integration discipline introduced in Paper 163 v0.1, expanding the rhymeOrTheorem tagging from a single (SELF) axis to all four D-FUMT₈ extension axes (SELF / INFINITY / ZERO / FLOWING). STEP 1205 records an ∞-cosmoi...

mathdescirei-aios

Paper 163 — Institution + Bilattice + SELF↔Lawvere: A Four-Step Operational Integration of Categorical Logic, Bilattice 4-Value Extension, and Constructive Fixed-Point Bridge (DRAFT v0.1)

We report an operational integration of four well-established mathematical frameworks within a single artifact-producing system (Rei-AIOS): Goguen-Burstall Institution Theory (1992), Belnap-Dunn FOUR bilattice (1976-1998), Lawvere fixed-point theorem (1969), and a SET-level en...

mathdescirei-aios

Solo Bitcoin Mining vs. the Lottery — Who Has Better Odds?

Two ASIC miners. ~3.15 TH/s. €12/month in electricity. And statistically 200× better odds than buying a lottery ticket.

bitcoinminingsovereigntyprivacymath

Paper 145 — First D-FUMT₈ Silicon with SELF⟲ Logic Primitive: Native 8-Valued Hardware Realization with Lean 4 Refinement Proof

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...

mathdescirei-aiosbraille-d-fumt8

Paper 161 — Two Regimes of Rest: A Dynamical-Systems Formalization of "Absolute Rest" via Fixed Points and Limit Cycles

v0.1 → v0.2 transition record: Paper 161 v0.1 HONEST-EARLY-STAGE-RELEASE published 2026-06-02 morning (Zenodo DOI 10.5281/zenodo.20498090, 11/11 platforms). Same-day evening Rei Claude completed 5 follow-up tasks: (1) Lean lake build machine verification of all 3 files = 18 th...

mathdescirei-aios

Paper 160 — Toward an Ontology of the Genesis Layer 0₀: Why a System Without Self Generates Self

Status (★ load-bearing, transparent early-stage release): - ✅ §9 Application Note machine-verified: 4 Lean 4 theorems all show does not depend on any axioms (Paper 159 v0.2 standard, see §9.3 + audit file Paper160InversionDeconstructionPrintAxioms.lean) - ✅ Fig. 1 (Core Theore...

mathdescirei-aios

Paper 159 — A Two-Layer D-FUMT₈ Reconstruction of Priest-Garfield's Inclosure Schema for Nāgārjuna's Catuṣkoṭi: A Formal Substrate Compatible with Multiple Interpretive Stances

We propose a two-layer D-FUMT₈ reconstruction of the Inclosure Schema that Priest and Garfield (2003) apply to Nāgārjuna's catuṣkoṭi. Their original construction terminates the schema at the value BOTH — the dialetheist commitment to a true contradiction at the limits of thoug...

mathdescirei-aiosbraille-d-fumt8

Paper 157 — Octonion-Axiomatic Dual Independence: 𝕆 Imaginary Basis ↔ AxiomKernel Logical Independence with D-FUMT₈ Eight-Value Bridge in Lean 4 (OUTLINE, v0.0.5)

Project: Rei-AIOS / OUKC — https://rei-aios.pages.dev License (intended at publish): AGPL-3.0 (code) + CC-BY 4.0 (text) per OUKC content policy. Per OUKC No-Patent Pledge: openly licensed; no patent will be filed. Lineage: Extends the Lean 4 formalization-roadmap pattern of Pa...

mathdescirei-aiosbraille-d-fumt8

Paper 154 — Rei as a Formal-Verification Compilation Pass for AI-Generated Mathematics (OUTLINE, v0.0)

> > Per the §0 statement below, the central operational claim — Rei provides a formal-verification compilation pass that composes with AI hypothesis generators — requires at least one end-to-end demonstration before v0.1 promotion. As of 2026-05-22 the demonstration is at scaf...

mathdescirei-aios

Paper 144 — OUKC: An AI-Friendly Multilingual Worldview Commons (NOT a scale commons)

We announce the founding of Open Universal Knowledge Commons (OUKC) & Rei & Claude, an AI-friendly, multilingual, cross-disciplinary knowledge commons whose explicit scope is 「全〜」 (all-X) coverage across 12 academic domains plus honest mapping of 4 formalization-resistant doma...

mathdescirei-aios

Paper 155 — Semantic Dyson Sphere: Beyond-Shannon Recovery Architecture as Civilizational Footprint Indicator (v0.2)

We introduce the semantic Dyson sphere as a formal analog of the classical (Kardashev II) physical Dyson sphere, where the conserved quantity is meaning rather than electromagnetic radiation. Building on Paper 25 / 71's Beyond Shannon framework — in which generative compressio...

mathdescirei-aios

Paper 153 v0.2 — Φ-Catalog: A Systematic Notation for Impossibility-Possibility Extensions in Mathematics (Addendum: Case 6 + Forward Application Results)

Project: Rei-AIOS / OUKC — https://rei-aios.pages.dev//phi-catalog-forward + https://rei-aios.pages.dev//riemann-zcsg-lens Parent: Paper 153 v0.1, Zenodo DOI 10.5281/zenodo.20207228 (published 2026-05-16) License: AGPL-3.0 + CC-BY 4.0 (per content type) dual Per OUKC No-Patent...

mathdescirei-aios

Spiral Number System Theory — v0.2 Supplement (Paper 63 v2 DRAFT)

This v0.2 supplement extends Paper 63 SNST with four new findings obtained between 2026-04 and 2026-05-14: (1) Symbolic regression baseline (PSLQ, 60-digit precision, STEP 1125): Among the 14 SNST constants, automated PSLQ search rediscovers the Paper 63 baseline πext ...

mathdescirei-aios

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 150 — OctaTheoria: A Unified Multi-Domain Observation Framework with Eight-Axis D-FUMT₈ Projection (Operational Evidence from Seven Domains × Eight View Modes)

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...

mathdescirei-aiosbraille-d-fumt8

Paper 145 — First D-FUMT₈ Silicon with SELF⟲ Logic Primitive: Native 8-Valued Hardware Realization with Lean 4 Refinement Proof

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...

mathdescirei-aiosbraille-d-fumt8

Discussion on the math behind Bitcoin mining

I asked ChatGPT to check my intuition and clarify some misconceptions about the math of probability and randomness.

Bitcoinrandommathmathematicsprobabilitymininglottery mining

Paper 121: Seven-Conjecture Deep Dives, Multi-Attractor Q33 Extension, and the First Cross-Domain Frankl Formalization

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...

mathdescirei-aios

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 119: Q7 Falsification, Q8/Q9 Empirical Data, and the First Rei-AIOS Failure Record

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 —...

mathdescirei-aios

Paper 118: Five Formal Closures + Findings + AI-Generated Open Questions

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...

mathdescirei-aios

Paper 117 (draft): Erdős-Straus Conjecture — Lean 4 Small-n Formalization + Infinite Mod-4 Family

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...

mathdescirei-aios

Paper 116 (draft): Andrica's Conjecture — A Lean 4 Mathlib-Native Treatment with Structural Sufficient Conditions

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...

mathdescirei-aios

Paper 115 (draft): Twin Primes Conjecture under Rei-AIOS Lens — A Structural Reinterpretation of Zhang-Maynard Bounded-Gap Results

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...

mathdescirei-aios

Paper 114 (draft): Random 3-SAT Phase Transition Revisited — Empirical Reproduction, Lean 4 Formalization, and D-FUMT₈ Interpretation Layer

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...

mathdescirei-aiosbraille-d-fumt8

Paper 112 (draft): Nested Colored-Dot Symbol System — An Information-Theoretic and Semantic Alternative to QR Codes

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...

mathdescirei-aios

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 109 — Erdős-Straus Conjecture as an S-Category Problem: A Ricci-Flow + Fujimoto Infinity Algebra Attack Angle

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...

mathdescirei-aios

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

Paper 110 (draft): Braille-D-FUMT₈ vs CLIP / BERT / ImageBind — a Rigorous Information-Theoretic and Structural Comparison

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...

mathdescirei-aiosbraille-d-fumt8

Test file for rendering TeX math within Markdown files

test file

nostrmarkuptexlatexmath

Euclidean Domains — Algebra (Dummit & Foote)

Daily lesson from Magic Internet Math Academy: Euclidean Domains (Algebra (Dummit & Foote), Section 40)

mathalgebra

Linear Functionals and Dual Spaces — Advanced Linear Algebra (Roman)

Daily lesson from Magic Internet Math Academy: Linear Functionals and Dual Spaces (Advanced Linear Algebra (Roman), Section 20)

mathlinearalgebra

My Fear of Math

I’m always doing work on myself. I have read reams of self-help books, attended seminars, bought cassette tape programs and CDs (remember those?), had a stint with Dianetics and Scientology, the Landmark Forum, you name it. My last seminar was Unleash the Power Within with Tony Robbins. Addicted? Maybe. But I always feel there’s room for improvement. Over the years, I’ve explored meditation and uncovering hidden blocks that affect how I operate in this world.

mathteachingscientology

Combinatoric Complexity and the Folly of Central Planning

How many connections exist between 8 people? Between 80? Between 80 million? And what does that have to do with freedom?

freedommathmathematicscombinatoricseconomicsbitcoin

Why I Love Math

I had a group discussion with some friends and family recently on the topic of education. This topic inevitably brings up the adage I''ve heard time and time again in American circles, "I think education should only teach me useful things that I will use not useless things like Algebra".

https://awhb.devmath