Iris Number System Deduction & Inference Engine

Author: Frédéric Blondel Custer

MOVED TO: https://github.com/chemoelectric/the-iris-number-system

An interactive deduction framework, analytical workbench, and comprehensive textbook environment for the Counting-Iris Number System (INS) in number theory, Clifford algebra Cl(4,1,1), nonstandard analysis, Jaynesian Maximum Entropy probability, and constructive multivector analysis.

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🏛️ Project Architecture & Overview

The Iris Number System Deduction & Inference Engine provides a constructive first-principles mathematical framework and software environment for formulating, evaluating, and proving mathematical propositions without relying on non-constructive set theory or ungrounded abstractions.

Core Modules & Features

1. Iris Textbook (AsciiDoc & LatexMath Renderer)

   • Complete interactive rendered textbook sourced from public/Iris_Number_System.adoc.
   • Dynamic Table of Contents, section navigation, full-text search, and inline KaTeX equation rendering.
   • Automatically updated Index of Formal Statements indexing all Postulates, Theorems, Definitions, Axioms, and Lemmas.
   • Direct dynamic AsciiDoc file generation and download option (/public/Iris_Number_System.adoc).

2. Search & Inference Engine Prover

   • Server-side inference integration powered by Gemini 3.6 Flash.
   • Generates multi-step, rigorous Iris deductions from natural language or mathematical conjectures.
   • Restricts logical inferences to the Counting-Iris framework, Master Field Equation, and Cl(4,1,1) metric preservation.
   • Instant step-by-step verification and direct import into the active deduction workspace.

3. Deduction Framework & Proof Builder

   • Step-by-step formal proof builder with status flags (valid, invalid, hypothetical, pending).
   • Automated step integrity check and rule justification inspector.
   • High-contrast formatted display with one-click LaTeX Export and JSON workspace state export/import.

4. Iris Calculator & Multivector Workbench

   • Full implementation of the Cl(4,1,1) 6-generator Clifford algebra signature (+,+,+,+,-,0).
   • Embedded bivector quaternion algebra (i = e_23, j = e_31, k = e_12).
   • Discrete Spectrum Arithmetic (x + ε · residual + ω · scale) and Nonstandard Analysis Main Scale Projection operator ↓(x).

5. Zeta & Prime Spectrum Visualizer

   • Interactive Recharts visualization of the Iris Zeta function ζ_I(s) along the critical line.
   • Numerical zero candidate search and phase spectrum density analysis.
   • Iris Prime Distribution spiral and residue distribution modeling.

6. Theorem Library & Axiom Workbench

   • Complete repository of formal axioms, postulates, definitions, and proven theorems.
   • Filterable by domain (Tautological Discrete Arithmetic, Clifford Algebra Cl(4,1,1), Jaynesian MaxEnt Probability, Spectral Topology, etc.).

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🚀 Getting Started

Prerequisites

• Node.js (v18+ or v20+)
• npm

Environment Setup

Create a .env file or export your Gemini API key:

GEMINI_API_KEY=your_gemini_api_key_here

Installation & Development

# Install dependencies
npm install

# Start the development server (runs express backend + Vite frontend on port 3000)
npm run dev

Production Build

# Build Vite client and bundle server.ts with esbuild
npm run build

# Launch the production server
npm start

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📜 Mathematical Foundations

The inference engine operates strictly within the Counting-Iris number system:

• Basis Elements: {1, ι, ϖ, ϑ} with ι² = τ - 1 where τ = (1+√5)/2.
• Multiscale Resolution Analysis (MSRA): Multiscale resolution numbers (short name m-res numbers) x* ∈ ℝ* constructed as physical vernier aperture measurement quantities on ultra-refined grid G_ω.
• MSRA Calculus:
  • Differential Calculus: Constructive derivatives dΨ/dx = ↓((Ψ(x+δ_ω) - Ψ(x))/δ_ω) with exact proofs of product, quotient, and chain rules, plus step-by-step worked polynomial examples.
  • Integral Calculus: Indefinite and definite integrals defined as discrete vernier aperture sums with exact proof of the Fundamental Theorem of MSRA Calculus and explicit step-by-step sum calculations.
  • Contour Integration: Multivector path line integrals in Cl(4,1,1) with the vernier Cauchy-Goursat theorem, residue-free loop circulations, and closed circular path examples.
• Clifford Cl(4,1,1): 6 basis generators {e1, e2, e3, e4, e+, e-} satisfying e1²=e2²=e3²=1, e4²=0, e+²=1, e-²=-1.
• Conformal Null Vectors: e_∞ = e_+ + e_-, e_0 = 1/2(e_- - e_+).
• Master Field Equation: D F_total = J_total where D = ∇ + e4 (1/c) D_t.

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🛠️ Project Structure

├── public/
│   └── Iris_Number_System.adoc   # Canonical AsciiDoc source textbook
├── src/
│   ├── components/               # React UI modules (Navbar, Textbook, Prover, Calculator, etc.)
│   ├── data/
│   │   └── textbookData.ts       # Structured JSON textbook parser & AsciiDoc generator
│   ├── lib/
│   │   └── irisEngine.ts         # Clifford Cl(4,1,1) multivector math & Iris arithmetic engine
│   ├── types.ts                  # Shared TypeScript interfaces and domain models
│   ├── App.tsx                   # Main React application shell
│   └── main.tsx                  # Application entry point
├── AGENTS.md                     # System directives and project rules
├── server.ts                     # Express server & Gemini inference endpoint proxy
├── package.json                  # Dependencies & execution scripts
└── README.md                     # Documentation (kept in sync)