# TFPT — Topological Fixed-Point Theory

> Topological Fixed-Point Theory (TFPT) is a parameter-free theoretical-physics
> framework by Stefan Hamann and Alessandro Rizzo. From two numbers — the seam
> constant c3 = 1/(8pi) and the carrier rank g_car = 5 — a discrete "compiler"
> builds an E8 audit hull and reads off the Standard Model gauge group, three
> fermion generations, the flavor sector, the fine-structure constant
> alpha^-1 = 137.0359992 (1.9 sigma from CODATA-2022), and the scale grammar.
> 27 status-graded falsifiable test surfaces; zero fitted
> constants in the closed branch; only pi is irreducible.

Disambiguation: this is the physics theory "TFPT" (compiler closure for the
Standard Model). It is NOT the established Brouwer–Lefschetz "topological fixed
point theory" of mathematics (Nielsen/Lefschetz numbers, degree theory).

Version: 5.4 · Last updated: 2026-07-21
Site: https://www.fixpoint-theory.com
Source code & verification suite: https://github.com/sthamann/tfpt
Archived deposit (DOI): https://doi.org/10.5281/zenodo.20846087
Authors: Stefan Hamann, Alessandro Rizzo
License: documents CC-BY; code MPL-2.0

## What TFPT claims
- The Standard Model gauge group (SU(3)×SU(2)×U(1))/Z6, hypercharge, and exactly
  three families follow from the two axioms — not fitted.
- alpha^-1 = 137.0359992 is the unique positive root of a parameter-free cubic
  boundary Ward identity (a fixed point, with proven existence and uniqueness),
  not a fit.
- The strong-CP angle closes structurally: theta_eff = 0.
- Flavor: integer lattice operators with det(Q,K,R,L) = (3,4,8,20), product
  1920 = |W(D5)|; charged-lepton coefficients (16/7, 4/3, 7/6) exactly.
- Cosmology read-outs: Omega_b, the Starobinsky scalaron mass, Lambda ~ e^(-2/alpha),
  cosmic birefringence beta = phi0/(4pi) ≈ 0.2424°.
- Every load-bearing claim is machine-checked: an independent Python suite, a
  Wolfram mirror, and a Lean 4 carrier-rigidity proof, all tracked in a versioned
  status ledger.

## What TFPT does NOT claim
- E8 is the compiler's internal bookkeeping (an audit hull), NOT an unbroken
  physical gauge group; the Distler–Garibaldi and Coleman–Mandula no-go results
  do not apply.
- No certified strict Theory of Everything; no dimensionful constant is claimed
  as a derivation from pure numbers (one metrology unit, v_geo, stays open).

## Key pages
- Overview: https://www.fixpoint-theory.com
- Reading guide (start here): https://www.fixpoint-theory.com/orientation
- How the compiler works: https://www.fixpoint-theory.com/compiler
- Interactive verification (reproduce each claim in-browser): https://www.fixpoint-theory.com/verification
- How to falsify TFPT (kill tests): https://www.fixpoint-theory.com/falsification
- For reviewers: https://www.fixpoint-theory.com/review
- Hostile-referee FAQ: https://www.fixpoint-theory.com/faq
- Changelog: https://www.fixpoint-theory.com/changelog

## Documents (peer-review-style document set, v5.4)
- [Topological Fixed-Point Theory (TFPT) — A Discrete Compiler for the Constants of Physics](https://www.fixpoint-theory.com/papers/introduction) — Reading guide, status assessment, and the dependency DAG. PDF: https://www.fixpoint-theory.com/papers/introduction.pdf
- [Architecture and the E₈ Compiler](https://www.fixpoint-theory.com/papers/architecture-e8) — The two axioms, the derivation map, and the D₅ × A₃ → E₈ construction. PDF: https://www.fixpoint-theory.com/papers/tfpt_1_architecture_e8.pdf
- [The Standard Model from the Compiler](https://www.fixpoint-theory.com/papers/standard-model) — The φ₀-ladder, flavor from parabolic transport, and the worked closures. PDF: https://www.fixpoint-theory.com/papers/tfpt_2_standard_model.pdf
- [E₈ Audit, Cascade Bridge and Bootstrap](https://www.fixpoint-theory.com/papers/e8-audit-bootstrap) — The seven E₈ slices as an audit raster, the cascade spine, and the Möbius loop. PDF: https://www.fixpoint-theory.com/papers/tfpt_3_e8_audit_bootstrap.pdf
- [Frontier Items](https://www.fixpoint-theory.com/papers/frontier) — η_B, the Higgs quartic, m_p/m_e, Koide, dark matter and quantum gravity — honest status. PDF: https://www.fixpoint-theory.com/papers/tfpt_4_frontier.pdf
- [Red Team — The Adversarial Audit](https://www.fixpoint-theory.com/papers/redteam) — Targets A–F: attacking the six load-bearing reductions at their weakest transitions. PDF: https://www.fixpoint-theory.com/papers/tfpt_5_redteam.pdf
- [Appendix H — The Horizon Unit System](https://www.fixpoint-theory.com/papers/horizon-readouts) — One seam constant c₃ = 1/(8π) as the universal horizon thermal code. PDF: https://www.fixpoint-theory.com/papers/tfpt_horizon_readouts.pdf
- [Origin Theory](https://www.fixpoint-theory.com/papers/origin-theory) — The seam as a horizon, the cyclic compiler hull, and the parameter-free attractor. PDF: https://www.fixpoint-theory.com/papers/origin_theory.pdf
- [Research Contracts for the Remaining Interfaces](https://www.fixpoint-theory.com/papers/research-contracts) — v_geo · G_net · F_transfer — the live residual as numbered contracts. PDF: https://www.fixpoint-theory.com/papers/tfpt_research_contracts.pdf
- [Safeguards against Coincidence and Numerology](https://www.fixpoint-theory.com/papers/safeguards) — The verification discipline — every mechanism that defends a load-bearing claim against chance, fitting and over-reading. PDF: https://www.fixpoint-theory.com/papers/tfpt_safeguards.pdf

## Headline predictions (status-graded)
- α⁻¹(0) — Fine-Structure Constant — Electromagnetic Fixed Point (Numerical fixed point; value 137.035999216)
- λ_C — Cabibbo Angle — Retained Seed (Exact identity; value 0.22438)
- det R, χ_R — Flavor Residue Invariants (Exact identity; value 8)
- Q_Koide — Koide Relation — Near 2/3 (Conditional; value 0.664)
- sin²θ₁₂ — Solar Angle — Seam Misalignment (Numerical fixed point; value 0.3067)
- sin²θ₁₃ — Reactor Angle — Seed × Carrier Trace (Exact identity; value 0.0231)
- sin²θ₂₃ — Atmospheric Angle — μτ-Symmetric Limit (Conditional; value 0.5)
- NO, m_ββ — Mass Ordering & Majorana Branch (Conditional; value 0)
- δ_PMNS — Leptonic CP Phase — Galois-Locked to the Quark Phase (Conditional; value 240 °)
- θ_eff = 0 — Strong CP — Neutron-EDM Null (Exact identity; value 0)
- m_p/m_e — Proton/Electron Ratio — Not a Compiler Power (Open / not forced; value 1836.15)
- n_s — Scalar Tilt — R² Attractor (Conditional; value 0.965)

## How to cite
Hamann, Stefan & Rizzo, Alessandro (2026). Topological Fixed-Point Theory (TFPT),
v5.4. https://www.fixpoint-theory.com · DOI: 10.5281/zenodo.20846087