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Document 0 of the TFPT 5.1 setReading guide
TFPT 5.12026-06-131.13 MBSHA-256 b9cb38e3…55f4
Paper 0Reading guide

Topological Fixed-Point Theory (TFPT) — A Discrete Compiler for the Constants of Physics

Reading guide, status assessment, and the dependency DAG

The entry document. From two axioms — the seam constant c₃ = 1/(8π) (P1) and the carrier rank g_car = 5 (P2) — TFPT constructs a discrete compiler for the Standard-Model skeleton (gauge group, three families, hypercharges, the flavor matrix), with E₈ (D₅ ⊕ A₃ + μ₄ ⇒ E₈) as the consistency checksum. The algebraic core is machine-checkable and the dimensionless constants follow as fixed points; physical readouts (scales, masses, inflation, gravity, cosmological transfers) run through explicitly named, status-typed bridges (v_geo, G_net, F_transfer), not as free outputs. This note is the reading guide: the architecture, the predictions, the dependency DAG, and the single proof ledger.

Inputs
  • ›The two axioms {c₃, g_car}; everything else is a consequence.
Contribution
  • ›States the compiler closure, the two-engine picture, the dependency DAG, the proof ledger, and the live experimental tests in one place.
Not claimed here
  • ›No new physics is introduced here — the introduction is a map. Load-bearing derivations live in the companion documents.
Falsification surface
  • ›Fails as a guide if the dependency order is misstated, if a status marker disagrees with the ledger, or if a claim is promoted past the grade the companion document carries.
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Highlights
Axioms2c₃ = 1/(8π) and g_car = 5 — the rest is a consequence
CompilerZ₃₀ = 2·3·5Coxeter–cyclotomic generator behind every sector
Free primitiveπThe one genuinely irreducible continuous number
Documents8Introduction + 4 core docs + Appendix H + Origin Theory + contracts

Key formulas

  • Compiler closure
    {c3,gcar}⇒D5⊕A3+μ4⇒E8\{c_3, g_{\mathrm{car}}\} \Rightarrow D_5 \oplus A_3 + \mu_4 \Rightarrow E_8
    Two axioms build the E₈ audit hull; the SM is read off by projection.
  • Bootstrap loop
    E8 closure⇒gcar=5,  8=rank⁡E8E_8\text{ closure} \Rightarrow g_{\mathrm{car}}{=}5,\ \ 8 = \operatorname{rank}E_8
    Inputs and output are mutually locked — only π stays irreducible.
  • Reduction in one line
    247 pages⇝2 inputs+1 machine247\,\text{pages} \rightsquigarrow \text{2 inputs} + \text{1 machine}
    The number of independent structural assumptions drops to two.

Two inputs, one machine

Two numbers go in — a boundary number 1/(8π) and a five-slot carrier — and the discrete Standard-Model core, the dimensionless constants and several scale readouts come out. The dashed loop is the point: the machine reproduces the very two numbers it started from, so the discrete core is overdetermined rather than fitted.

{c3,gcar}  ⇒  D5⊕A3→  μ4  E8  ⇒  (SM, constants, scale grammar)\{c_3, g_{\mathrm{car}}\} \;\Rightarrow\; D_5 \oplus A_3 \xrightarrow{\;\mu_4\;} E_8 \;\Rightarrow\; (\text{SM},\ \text{constants},\ \text{scale grammar})

The master story is two engines

Read from the two axioms, the theory factorises into exactly two engines: a discrete closure (from g_car = 5) that builds E₈ and the SM packet, and a boundary dressing (from c₃) that produces the seed, α⁻¹ and the scale grammar. Gravity is not a third block — it is the geometry channel of Engine 2.

Engine 1: gcar=5→E8→(Nfam,Ωadm,b1,R)\text{Engine 1: } g_{\mathrm{car}}{=}5 \to E_8 \to (N_{\mathrm{fam}}, \Omega_{\mathrm{adm}}, b_1, R)
Engine 2: c3=18π→(u=φ0,α−1,ξ,Λ,H0)\text{Engine 2: } c_3{=}\tfrac{1}{8\pi} \to (u{=}\varphi_0, \alpha^{-1}, \xi, \Lambda, H_0)

The compact status formula

TFPT 5.1 closes the discrete compiler, the algebraic SM readout, the EM fixed point, the admissible gapped IR sector and the R + R² spectral-action shadow. It does not yet certify a strict physical TOE. The residual is one flavor wall-selection, one quantum-gravity measure, and a set of deliberately typed interfaces.

compiler⏟closed ∣ admissible IR physics⏟conditional (RP, gap) ∣ strict physical TOE⏟open\underbrace{\text{compiler}}_{\text{closed}}\ \Big|\ \underbrace{\text{admissible IR physics}}_{\text{conditional (RP, gap)}}\ \Big|\ \underbrace{\text{strict physical TOE}}_{\text{open}}
Rest=(Uwall)⊕(Gmetric)⊕(Ffrontier)\text{Rest} = (U_{\mathrm{wall}}) \oplus (G_{\mathrm{metric}}) \oplus (F_{\mathrm{frontier}})

The anchor: one number a = (1,1,2)

The two axioms are not even independent: they are the elementary symmetric polynomials of the single parabolic anchor a = (1,1,2), and its power sums generate the big Lie data directly. The inputs collapse to the anchor plus the lone continuous primitive π.

e1(a)=4=∣μ4∣,e2(a)=5=gcar,e3(a)=2=∣Z2∣e_1(a)=4=|\mu_4|,\quad e_2(a)=5=g_{\mathrm{car}},\quad e_3(a)=2=|\mathbb{Z}_2|
c3=12 e1(a) π=18π,∣R(E8)∣=p1p2p3=240c_3 = \frac{1}{2\,e_1(a)\,\pi} = \frac{1}{8\pi}, \qquad |R(E_8)| = p_1 p_2 p_3 = 240

Key formulas at a glance

  • Compiler closure
    {c3,gcar}⇒D5⊕A3+μ4⇒E8\{c_3, g_{\mathrm{car}}\} \Rightarrow D_5 \oplus A_3 + \mu_4 \Rightarrow E_8

    Two axioms build the E₈ audit hull; the SM is read off by projection.

  • Bootstrap loop
    E8 closure⇒gcar=5,  8=rank⁡E8E_8\text{ closure} \Rightarrow g_{\mathrm{car}}{=}5,\ \ 8 = \operatorname{rank}E_8

    Inputs and output are mutually locked — only π stays irreducible.

  • Reduction in one line
    247 pages⇝2 inputs+1 machine247\,\text{pages} \rightsquigarrow \text{2 inputs} + \text{1 machine}

    The number of independent structural assumptions drops to two.