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A Math→Physics Representation Library

Six mathematical faculties, each formalized as a physical-representation module. The modules compose hierarchically into one modular synthesis, and every rung of the ladder adds an emergent representational property that the level below cannot express.

Live site: https://math-phy-library.vercel.app  ·  Repository: https://github.com/YonedaAI/math-phy-library

Overview

This project decomposes a single representation-stack manuscript into six focused papers plus a synthesis, seven in all. Each paper takes one mathematical domain and formalizes it as a module in a shared representation stack with a common realization pipeline

Obs_α(M) = Obs( Real_α( Φ(M) ) )

The modules stay separate and compose instead of merging into a monolith. The synthesis proves a closure theorem: Part I introduces the representation stack abstractly, and the Part VI capstone discharges it via descent-theoretic stackification. The isotropy keystone Aut(x) ≃ Stab_G(x) carries the argument, and its physical face is gauge redundancy ≅ quantum high-availability.

Every paper has been peer reviewed, code reviewed, and formally verified (see below).

Papers

Part Title Pages Read PDF
I Foundations: The Representation Stack and the Realization Pipeline 25 read pdf
II Motives, Periods, and Amplitudes: The Coalgebraic Anatomy of Physical Representation 26 read pdf
III Algebraic Geometry: Spaces of Physical Possibility 27 read pdf
IV Algebraic Topology: Conserved Global Information 24 read pdf
V Category Theory and Homotopy Type Theory: Composition and Identity 24 read pdf
VI Sheaves, Stacks, Gauge Redundancy, and Quantum High-Availability 23 read pdf
— A Modular Representation Synthesis: Composing Six Mathematical Faculties into Physical Representation 24 read pdf

173 pages total.

The Modular Composition Ladder

Part I    Foundations ──────────── the representation stack + realization pipeline
   │  ↳ coalgebraic anatomy
Part II   Motives / Periods / Amplitudes
   │  ↳ geometric variation
Part III  Algebraic Geometry
   │  ↳ functorial conservation
Part IV   Algebraic Topology
   │  ↳ universal compositional grammar
Part V    Category Theory & HoTT
   │  ↳ high availability / fault tolerance
Part VI   Sheaves, Stacks, Gauge Redundancy, Quantum High-Availability
   ↓
Synthesis  closure theorem — the ladder closes on itself

Each arrow names the emergent property that appears only after composition.

Formal Verification

Each domain paper comes with machine-checked companions:

  • Haskell (src/<topic>/): every major theorem has a QuickCheck property and an equational Proofs.hs derivation; all modules compile under ghc -Wall -Wextra -Werror, and Main exits 0. (15 to 34 QuickCheck properties plus 8 to 10 equational proofs per topic, all passing.)
  • Lean 4 (lean/<topic>/): best-effort formalization sketches of the core structures and theorem signatures; they elaborate standalone.

Review. Every paper went through iterative peer review by agy (Antigravity CLI, Gemini 3.1 Pro), which caught genuine mathematical errors across the series: a Z₂^{4g} surface-code logical-group correction, a Conditional-Amplitude-Decomposition category error, a translation-strength inconsistency, and more. Each paper also had a Codex (gpt-5.5) formatting and code review, and both passes ran in fix loops until the reviewers were satisfied. Round-by-round reviews are in reviews/.

Repository Layout

papers/latex/   LaTeX sources (7 papers)          website/     Next.js static site (deployed to Vercel)
papers/pdf/     compiled PDFs                       docs/papers/ pandoc HTML conversions
src/<topic>/    Haskell modules + Properties/Proofs posts/       social posts (4 platforms × 7 papers)
lean/<topic>/   Lean 4 sketches                     reviews/     agy + Codex review transcripts
images/         cover images (300 DPI)              context/     original source material

Build Locally

# Papers
cd papers/latex && pdflatex -interaction=nonstopmode <topic>.tex   # ×2

# Haskell verification (per topic)
cd src/<topic> && ghc -Wall -Wextra -Werror -o test Main.hs *.hs -package QuickCheck && ./test

# Website
cd website && npm install && npm run build      # static export to website/out/

Author

Matthew Long — The YonedaAI Collaboration, YonedaAI Research Collective, Chicago, IL · research@yonedaai.com · https://yonedaai.com

License

Research artifacts released for academic use.