Kingdom Come: A Quaternionic Geometric Approach to Number Theory and Physics

Spine: this repo — manuscript + pedagogical Python
Shared math: flux_hopf_lib — SoT for Hopf / quaternion / Hurwitz 24
Engine: qga_engine (scenes, Rust math) · qga_gpu (frame)
This repo: the book. Not a GPU runtime and not an empirical attack.

Start here: START_HERE.md

Title: Kingdom Come: A Quaternionic Geometric Approach to Number Theory and Physics
Short name: QGA

Mission. Lift Hatcher’s visual, diagrammatic number theory from the Farey plane and binary quadratic forms to unit quaternions, the Hopf fibration, and gauged flux lattices—keeping pure geometry and arithmetic defensible, and isolating physical models and observational hypotheses so they never masquerade as theorems.

How to read claims

Label	Meaning	Here
Theorem	Geometry or arithmetic already on this spine	Quaternion orders, Hopf facts, Hatcher lift
Model	Dynamics or construction that uses the spine	Flux flywheels, gauged-lattice adjacency, flux topographs, (Z\mapsto) map, arena mix (Q = Q_O + \sigma(Z) Q_C)
Hypothesis	Observational claim that can fail	(350/\pi) (H1a–H1e); not a theorem
Software fact	True of current code or figures	What a helper or portal function returns today

Short note: notes/spine/ (QGA_Spine_Note.pdf). The book remains Release draft-D.

Hatcher map. Parallel reading: HATCHER_MAP.md · Hatcher’s Topology of Numbers (PDF).

Spine vs not. This repo is the manuscript spine. Quaternion orders, Hopf, and the Hatcher lift stay theorems. Flux flywheels, flux topographs, the (Z\mapsto) map, (350/\pi), and the arena mix are Model or Hypothesis. arena is a labeled Model companion; it is not a second spine.

What this is

An original book and research project that maps Allen Hatcher’s geometric number theory (Topology of Numbers) onto unit quaternions (S^3), the Hopf fibration (S^3 \to S^2), and gauged flux lattices.

It is not a modified reprint of Hatcher’s PDF. Hatcher is free to read and AMS-published; this is a companion extension in the same visual, diagrammatic spirit.

Source	Role
Hatcher, Topology of Numbers	Geometric backbone: Farey diagram, continued fractions, topographs, class groups
flux_hopf_lib	Shared Hopf / quaternion / conduit primitives
kingdom_come	Portal and figures for models and observations — not a theorem source

Core thesis (one paragraph)

Hatcher shows that elementary number theory is spatial: mediants, zigzag paths, topographs, and (SL(2,\mathbb{Z})) symmetries make integers visible. This book develops the lift: quaternionic norms and Hurwitz integers as the four-square upgrade of sums of two squares; Hopf fibers as the higher-dimensional analogue of Farey edges; flux topographs as Conway topographs on a gauged lattice. Flywheels, the (Z\mapsto) map, and Magic Islands are labeled Models. Speculative observational claims (e.g. (350/\pi)) are Hypotheses with explicit validation criteria.

Project layout

qga/
├── README.md
├── TOC.md
├── HATCHER_MAP.md
├── SYNOPSIS.md
├── book/                     # manuscript (Markdown)
│   ├── *.md                  # chapters 0–10, preface, HOW_TO_USE
│   ├── figures/              # static figures
│   ├── latex/                # PDF production (main.tex + generated chapters)
│   └── Kingdom_Come_QGA.pdf  # latest build (also book/latex/main.pdf)
├── START_HERE.md             # 20-minute door
├── lib/                      # pedagogical-only; not a second SoT; not on PyPI
├── scripts/                  # figure generators + md_to_latex + build_latex
│                             # (+ lab harness op1_adjacency / op2_topograph)
├── refs/
└── notes/                    # open_problems; PIPELINES.md is the lab harness, not the book

Build the PDF

python3 -m pip install -e .
python3 -m pip install -e ".[portal]"   # flux-hopf-lib (PyPI) + kingdom-come (git pin)
./scripts/build_latex.sh
# → book/latex/main.pdf  and  book/Kingdom_Come_QGA.pdf

See book/latex/README.md for details.

Two pipelines share vertices and hopf_map, not a graph. This README is the manuscript. The OP1/OP2 harness is the lab notebook notes/PIPELINES.md. OP1 stays Open. OAM 2×2 tiles are not a step in either pipe.

Status

• Scaffold + TOC + mapping: done
• Manuscript body: Complete draft Parts I–V (Ch. 0–10)
• Code/figures: pedagogical lib/ (not a competing package) + figure generators under book/figures/
• GitHub: https://github.com/qga-lab/qga

Quick links

• Table of Contents
• How to Use the Figures and Code
• Hatcher mapping
• Synopsis & assessment
• Hatcher TN: https://pi.math.cornell.edu/~hatcher/TN/TNbook.pdf
• Shared primitives: https://github.com/qga-lab/flux_hopf_lib
• Portal: https://github.com/kinaar8340/kingdom_come
• This book: https://github.com/qga-lab/qga

X: @kinaar8340

Geometry libraries are MIT. Several VQC repos are PolyForm Noncommercial plus patent notice US 63/913,110. This repo is MIT.