The gauge group, three families, hypercharge and the flavor matrix follow from two numbers. α⁻¹ = 137.0359992, 1.9σ from CODATA-2022. 20 status-graded test surfaces; zero fitted constants in the closed branch. Every claim is ledger-typed, reproducible, or explicitly open.
Topological Fixed-Point Theory reads the Standard Model, the constants and the scale grammar off a single built from the and the . The exceptional lattice E₈ is the compiler’s internal bookkeeping, not a gauge group. The even re-derives its own inputs, so the core is overdetermined, not fitted.
Two numbers enter: the seam constant c₃ = 1/(8π) on the boundary, and a five-slot carrier g_car = 5. Nothing else — no gauge group, no families, no α — is inserted by hand.
240 roots, det 1 — the unimodular audit hull; the SM is a readout after projection
Read this as a checkable research artifact, not a landing page. Each result is typed, machine-verified, or explicitly open — and a single versioned ledger is the source of truth.
It takes roughly twenty numbers as input and fits the flavor sector by hand. It does not say why there are three families, why the hypercharges are what they are, or why the strong-CP phase is essentially zero. TFPT asks whether these share one boundary origin — and answers with a status-graded derivation, not a fit.
The seven original papers reproduced dozens of numbers but could not say why the same small integers (2, 3, 5, 16, 240, 248) kept reappearing. The compiler closure answers exactly that: one tiny machine, fed by two inputs, builds E₈ and reads off the Standard Model, the constants, and the scale grammar.
Everything starts from the seam constant c₃ = 1/(8π) (P1) and the five-slot carrier g_car = 5 (P2). No SM gauge group, no families, no α is inserted by hand — they are consequences.
The carrier gives the D₅ half-spinor, the family geometry ℙ¹∖μ₄ gives A₃, and the μ₄ glue closes E₈ = (D₅ ⊕ A₃) + μ₄ as a lattice theorem. E₈ is the audit hull; the SM is a readout after projection.
The E₈ closure feeds back and fixes the inputs: g_car = 5 is forced three ways and the 8 in c₃ equals rank E₈ = h(D₅) = φ(30). The discrete core is overdetermined — only π stays irreducible.
Every claim carries a grade — [I] identity, [L] lattice theorem, [F] formalised, [N] numerical fixed point, [P] conditional, [A] open — and resolves to a single machine-checked ledger. The ledger wins on any disagreement.
D₅ = so(10) (spinor 16) and A₃ = su(4) (the four-puncture family geometry) share the same discriminant group ℤ₄. Their glue norms add to the E₈ root norm:
A closed lattice-theoretic construction — not a blind positing of 248. The E₈ numbers are carrier traces, not inputs.
The fine-structure constant is the unique positive root of a parameter-free cubic built only from c₃ and the abelian coefficient 41 = 10 b₁ — existence and uniqueness are proved:
CODATA-2022 recommends 137.035 999 177(21); the deviation is 2.9 × 10⁻¹⁰, about 1.9σ of its uncertainty — a fixed point, not a fit.
Reading rule. The two axioms at the top are the foundation; each lower band depends on the bands above. Exact identities and lattice theorems fail with a single counterexample; numerical fixed points fail against a declared comparison; conditional and open items fail only as named hypotheses. The single source of truth for which claim sits where is the machine-checked status ledger — if the text and the ledger ever disagree, the ledger wins.
After the compiler closure the entire residual is Rest = (U_wall) ⊕ (G_metric) ⊕ (F_frontier): one flavor wall-selection, one quantum-gravity measure, and a set of deliberately typed frontier interfaces. None of these is hidden, and none is overclaimed.
The quark mass ratios are closed (Readout Rigidity, c_u/c_d = 55/117 on the discrete selector stratum). Only the absolute amplitude normalisation U_point stays open — finite, algebraic and falsifiable.
R + R² is heat-kernel grounded (G2) and the admissible IR sector is gap-decoupled (G5, Δ_eff = 1.648 > 0). The ambient projective limit G6 remains open — so a strict physical TOE is not certified.
Koide (near-miss), η_B (downstream readout), the axion relic scale, and m_p/m_e are honestly typed interfaces. They are deliberately not claimed as forced compiler outputs.
TFPT is a directed acyclic graph: two axioms at the source, the E₈ compiler in the middle, the observables as sinks. The compiler factorises into two engines — a discrete closure (from g_car) and a boundary dressing (from c₃) — that branch into three readouts, and the bootstrap loop feeds the output back to fix the inputs.
Doc 0 is the reading guide (introduction). Docs 1–4 are the core papers — architecture & E₈, the Standard Model, the E₈ audit & bootstrap, the honest frontier. Docs 5–7 are the companions — Appendix H (horizon unit system), the Origin Theory synthesis, and the research contracts. Each opens with its inputs, contribution, what it does not claim, and its falsification surface.
The two axioms, the derivation map, and the D₅ × A₃ → E₈ construction
The architecture layer: how the two axioms c₃ = 1/(8π) and g_car = 5 build the Coxeter–cyclotomic compiler — the carrier C⁺ = D₅, the family geometry ℙ¹∖μ₄ = A₃, the μ₄ glue D₅ ⊕ A₃ + μ₄ ⇒ E₈, the electromagnetic fixed point α⁻¹ (with its ablation), and the whole number alphabet 16, 40, 41, 48, 240, 248 as carrier traces.
The even-Hamming code on five slots is the D₅ half-spinor: its dimension is the Pascal sum 1 + 5 + 10 = 16, which forces g_car = 5 uniquely. The E₈ root count is then a pure carrier trace.
D₅ = so(10) (spinor 16) and A₃ = su(4) (the four-puncture family geometry ℙ¹∖μ₄) have the same discriminant group ℤ₄. Their discriminant-form norms are two TFPT constants that add to the E₈ root norm, so the glue closes as a lattice theorem — not a posited 248.
The Coxeter number of E₈ is h = 30 = 2·3·5 — exactly the three discrete atoms (sheet ℤ₂, families ℤ₃, carrier g_car = 5). The rank is the count of live phases of the order-30 cycle.
The fine-structure constant is the unique positive root of a parameter-free cubic built only from c₃, the abelian coefficient (Σ L + N_Φ = 41 = 10 b₁) and the exact seam generating function. Existence and uniqueness are proved; the value lands 1.9σ from CODATA-2022.
The same α⁻¹ ≈ 137 generates the electroweak scale (divided by the carrier 5), the cosmological constant (times 2) and the Hubble scale (via the square root) — the action ladder 1 : 5 : 10 is the Pascal row of the carrier.
With , , and from the carrier packet, the seam opening
enters the closure function
and the prediction is the unique positive root.
Hand-shaped sketch — the residual is shown enlarged so it is visible. The actual numerical separation is 39.8 parts per billion in α⁻¹.
| Element | Comes from | Must not |
|---|---|---|
| Axiom P1 — the seam constant | CODATA fitting | |
| Doc 1 — abelian index coefficient | Empirical post-tuning | |
| Doc 2 — word-lengths + Higgs index | Free parameter | |
| Doc 1 — exact seam opening | Freezing at inside the root equation | |
| CODATA 2022 | External comparison row | Being used as input |
Every quantity on the left is fixed by an upstream paper before the closure equation is touched. Anything in the right column would silently turn the α prediction into a fit.
The seam opening depends on α itself, so α is fixed as the unique positive root of a closure equation that contains α inside its own opening — not as a freely adjustable parameter.
The φ₀-ladder, flavor from parabolic transport, and the worked closures
The fermion spectrum — masses, Yukawa structure, CKM, the PMNS skeleton and neutrinos — follows from one master formula with one seed φ₀, the carrier base λ_Y = √(φ₀(1−φ₀)), and the residue matrix of the compiler. Plus the flavor block from parabolic transport on ℙ¹∖μ₄, the five worked closures (θ₁₂, quark c, the explicit mass gap, Starobinsky M, the H2 splitting), and gravity/QG as the seam response.
Every fermion mass is the same ladder: the geometric VEV times the carrier base raised to a compiler word-length, times an O(1) residue. The word-lengths are the fixed residue matrix of the compiler — not free parameters.
The word-length matrix L = R + 6·(winding) carries only compiler numbers: its trace is N_fam², its determinant is h(D₅) = 8, its principal 2-minors are (2,3,5) with product 30 = h(E₈), and its Frobenius norm is dim E₆ = 78.
The lepton amplitudes are the rationals (16/7, 4/3, 7/6) with product 2⁵/N_fam² = 32/9, and the masses are exact φ₀-powers. Applied to the down sector the lepton law provably fails — the quark c's live on the parabolic wall.
The quark mass ratios are pure integer Plücker readouts on the derived selector stratum — no transcendental solve. The absolute amplitude reduces to one overall scale v_geo (ratios + Grand Mass Volume), the same dimensionful anchor as gravity's 1/G.
Tri-bimaximal gives 1/3; the charged-lepton 1–2 misalignment is the seam ε = q(A₃)φ₀ = (3/4)φ₀, and TBM geometry gives the only previously open SM angle as a conditional derivation — 0.1% from NuFIT 6.0.
The carrier polynomial is not an entry assumption. It is the algebraic shadow of the 3+2 split forced by the five-slot carrier — the same split that gives the D₅ half-spinor and the hypercharge. The four steps below show the order in which the compiler reads this off.
The second axiom
The carrier is five slots — 3 colour + 2 weak. Its even-Hamming code is the D₅ half-spinor. This is the only structural input on the matter side.
Even exterior code
The half-spinor dimension is the Pascal sum on the carrier slots. The identity 2^(g−1) = C(g,0)+C(g,1)+C(g,2) holds only at g = 5, which fixes the carrier rank uniquely.
Unique integer split
The colour/weak split of the five slots is the unique integer solution of b + s = 5 with b² + s² = 13 = |R(A₃)| + 1. No SM data is imported — the split is forced arithmetically.
Polynomial as corollary
With (b, s) = (3, 2) the determinant-normalized generator is unique: Y = −1/3 on the colour block, +1/2 on the weak block. The carrier polynomial appears only as the minimal polynomial of these two derived roots.
The polynomial is not assumed. It is the algebraic shadow of the forced 3+2 split.
| Multiplet | SU(3)×SU(2)×U(1) | Y | Dim |
|---|---|---|---|
| Q_L | (3, 2, 1/6) | 1/6 | 6 |
| u_R | (3, 1, 2/3) | 2/3 | 3 |
| d_R | (3, 1, −1/3) | −1/3 | 3 |
| L_L | (1, 2, −1/2) | −1/2 | 2 |
| e_R | (1, 1, −1) | −1 | 1 |
| ν_R | (1, 1, 0) | 0 | 1 |
| One chiral family (incl. ν_R) | 16 | ||
For any essential split E_b ⊕ E_s with determinant-normalized roots −1/b and 1/s, the two-point generator satisfies the family
The carrier and family arguments fix . The coefficient 6 = 3 · 2 is the determinant-periodized block normalization of the rigid 3+2 carrier — not phenomenologically tuned, not guessed.
The seven E₈ slices as an audit raster, the cascade spine, and the Möbius loop
E₈ as an audit container, not a mystery: the seven maximal slices of 248 as a falsification raster (every load-bearing number must appear in at least one projection), the bridge showing the old E₈ orbit cascade D = 60 − 2n is the same even-integer spine as the compiler, and the Möbius bootstrap in which g_car = 5 and the '8' in c₃ are overdetermined E₈-closure fixed points — only π irreducible.
Each maximal subalgebra of E₈ projects a TFPT module. The strongest new hit: E₆ × A₂ reads the flavor residue matrix, with E₆ reading its Frobenius norm and A₂ the three-family symmetry.
The old E₈ orbit cascade is the same even-integer spine: it starts at 60 = 2·3·10, ends at 8 = h(D₅) (the flavor selector), passes the Coxeter rung 30 = h(E₈), and the product of endpoints recovers the root count.
The carrier rank is an overdetermined E₈-closure fixed point: rank-fill (g + 3 = 8), Coxeter-match (h(D_g) = 2g − 2 = 8), and the integer-glue/norm closure whose reverse-glue quadratic has nontrivial root μ = 4.
The seam denominator is fixed five concordant ways. The two axioms collapse to one continuous primitive (π, from Möbius/Gauss–Bonnet) plus one discrete fixed point (the E₈ closure).
η_B, m_p/m_e, Koide, dark matter and quantum gravity — honest status
The honest frontier: which physics has a genuine TFPT handle and which does not. For each of η_B, m_p/m_e, the Koide relation, dark matter and full quantum gravity, this note states the genuine structural handle, the precision it currently lands at, and — crucially — what is not a clean compiler power and is deliberately not forced onto the ladder. This document is the status authority for the frontier items.
From the closed baryon fraction Ω_b = (4π − 1)β_rad, the asymmetry follows as a cosmological readout. As a fundamental compiler power it is not closed — the leptogenesis Boltzmann solve is not carried out.
The source-level Koide quotient from the lepton φ₀-ladder is 0.664, 0.33% below the democratic compiler target 2/3 = |ℤ₂|/N_fam. A source→pole transfer conjecture brings it onto 2/3, but is not a derivation.
The candidate is the determinant-line axion of the strong-CP sector; WIMPs are ruled out (no spare E₈ singlet). The misalignment angle is closed; the decay constant is a conjecture.
c₃ = 1/(8π) is the gravitational seam constant; the spectral action gives R + R² structurally (G2), and the closed admissible sector is gap-decoupled from the un-built ambient (G5, Decoupling Theorem). The ambient measure (G6) is holographically reduced to a finite seam-boundary measure — the strict-TOE completion target, no longer a bulk problem.
One seam constant c₃ = 1/(8π) as the universal horizon thermal code
A change of bookkeeping, not new gravitational physics: if gravity is the geometry-channel readout of the seam, then all horizons read the same boundary constant c₃ = 1/(8π). This note collects the readouts — Hawking, de Sitter and Unruh temperature, black-hole thermodynamics, the Page time, scrambling, the Nariai bound, v_GW = c and cosmic birefringence — in seam units, with two genuine compiler fingerprints (1920 = |W(D₅)|, |μ₄| = 4).
The factor that appears in every horizon temperature is the seam constant itself. Black holes, de Sitter and Unruh therefore share one thermal grammar.
Temperature, entropy, power and lifetime all read off c₃, with the Hawking power denominator carrying the compiler fingerprint 1920 = |W(D₅)| (the Weyl group order of D₅).
The Page time is a fixed fraction of the evaporation time, and the scrambling time carries the second fingerprint |μ₄| = 4. The Page-recovery kernel decays at the same λ₂ = (2/3)⁶ that sets the SM flavor gap.
The de Sitter entropy and the cosmic-birefringence angle are the same seam readouts; v_GW = c follows with no measurable dispersion.
The seam as a horizon, the cyclic compiler hull, and the parameter-free attractor
Why the two TFPT inputs leave no free fundamental number. Two layers, kept strictly apart: a structural [I]/[L] core (exact, machine-checked identities) — the (g_car, N_fam) = (5,3) skeleton, the triply-forced 8 (geometry = lattice = gravity), the order-30 Coxeter cycle, one boundary transport for both flavor and horizon, and a gapped unique attractor — plus one honestly-typed [P] interpretation: the cyclic self-reproduction reading.
The integer alphabet of the theory falls out of (g_car, N_fam) = (5,3): the E₈ rank, the sheet and glue counts, and the Pythagorean mass volume as a difference of squares.
The seam denominator is fixed three independent ways. If the seam is a horizon, the gravitational 8π forces c₃; it must then coincide with the geometric 2|μ₄| (Gauss–Bonnet) and the lattice rank E₈ — all three give 8.
The boundary transport spectrum {1, (2/3)⁶, (1/3)⁶} has a sub-leading eigenvalue that appears in both sectors: the SM flavor gap and the horizon Page recovery are the same number.
The transport gap is positive, so by Perron–Frobenius the operator has a unique dominant eigenvector and iterating from any start converges to the same fixed direction. Parameter-freeness is an attractor, not a tuning.
(U_wall) the parabolic flavor wall-selection · (G_metric) the full QG measure
After the compiler closure the entire residual is Rest = (U_wall) ⊕ (G_metric) ⊕ (F_frontier). This note turns the two genuine research gates into contracts: a numbered chain of lemmas, the single theorem that closes each gate, and — for every step — whether it is machine-certifiable today. F_frontier is not a gate. Priority: (U_wall) first (finite, algebraic, falsifiable), then (G_metric) (deep analytic programme).
The goal is to select the one D₄-symmetric realisation on the family curve. The selectors det R = 8 and Spec(Q₊) = {1,2,3} are read off the bundle; the quark ratio is closed by Readout Rigidity, leaving only the absolute amplitude scale.
The remaining flavor bridge splits into four pieces: unitarity (polystable ⇒ unitary, finite linear algebra), the H2 readoff, the Λ² readout rigidity, and U_point (the full amplitude normalisation) — which now reduces to the single overall scale v_geo (the same anchor as 1/G). Gate 1 is complete.
The goal is the reflection-positive projective-limit measure over the diffeomorphism-quotiented metric sector. G2 (Seeley–DeWitt R + R²) and G5 (gap dominance, Decoupling Theorem) are certified; the projective limit G6 is now holographically reduced — because the seam is a finite causal boundary, the bulk measure is reconstructed from a finite seam-boundary (Calderón) measure, so G6 is a boundary projective limit (conditional on RP + tightness), not a diffuse bulk problem.
(U_wall) is finite, algebraic and falsifiable today; (G_metric) is a deep analytic programme. The recommended order freezes the frontier status in between.
Every row is a single readout with an explicit status marker, dependency class, and a stated kill or pressure criterion. Each links to the source document that derives it — and the freeze file commits the decisive kill criteria in advance. Some rows are closed numerical tests, some are structural kill tests, some are conditional, and a few are honestly-typed non-claims.
The fine-structure constant is the unique positive root of the parameter-free cubic built from c₃ and the abelian coefficient 41 = 10 b₁. Existence and uniqueness are proved; the value lands 2.9×10⁻¹⁰ (1.9σ) from CODATA-2022.
The Cabibbo angle is the carrier base of the φ₀-ladder — the same seed that fixes the reactor angle and the birefringence amplitude.
The flavor matrix carries only compiler numbers: determinant h(D₅) = 8, principal 2-minors (2,3,5) with product h(E₈) = 30, trace N_fam². Any future global fit must satisfy these.
The source-level Koide quotient is 0.664, 0.33% below the democratic compiler target 2/3. Near-miss, not an exact derivation — the source→pole transfer is a conjecture.
Previously the only open SM angle. Tri-bimaximal 1/3 plus the seam misalignment ε = (3/4)φ₀ gives 0.3067 — 0.1% from NuFIT 6.0. JUNO (data since Aug 2025) is the sharpest live test.
The reactor angle is the seed times the carrier-trace factor e⁻⁵ᐟ⁶ (γ = 5/6). Daya Bay / RENO / JUNO compare; PDG 0.0220.
The atmospheric angle is near-maximal in the μτ-symmetric limit; the octant is not selected by the present transport. DUNE addresses the ambiguity.
The Majorana neutrino sector prefers normal ordering with a small effective mass. LEGEND / nEXO / DUNE / KATRIN are the comparison surface.
θ_eff = 0 follows from three structural facts (γ₅-Hermiticity, polar structure, sheet involution) plus reflection positivity — without a mass gap. PSI nEDM / SNS test it.
Listed for honesty: m_p/m_e is a cross-sector ratio, computable at scheme level but genuinely not a compiler power. It is deliberately not forced onto the ladder.
The scalar tilt comes from the same R² (Starobinsky) attractor that fixes the scalaron mass — Planck measures 0.965; CMB-S4 sharpens.
The tensor ratio of the R² scalaron is already below the current bound and within reach of CMB-S4 (σ_r ≤ 5×10⁻⁴, ~2033).
Generic Starobinsky fits the scalaron mass to A_s; TFPT fixes it by the seam, (M/M̄)² = c₃⁷, so A_s becomes a prediction (and the measured A_s predicts N★ ≈ 56).
The scalaron mass comes out exactly at the canonical Starobinsky value, with the exponent 7 = 48 − 41 = Ω_adm − 10 b₁ fixed by the seam. A former input is now an output.
The baryon fraction reads off the determinant-line angle β_rad. Planck comparison row 0.04930.
A downstream cosmological readout from the closed Ω_b h² (observed 6.1×10⁻¹⁰). As a fundamental compiler power it is not closed — the leptogenesis Boltzmann solve is an interface.
The electroweak scale, the cosmological constant and the Hubble scale are all powers of one exponential engine on the carrier — the same α⁻¹ ≈ 137. SH0ES / DESI / Planck (Hubble tension) test it.
The carrier index fixes exactly one weak doublet (N_Φ = g_car − |μ₄| = 1). A structural prohibition, not a fit.
The determinant-line / Chern–Simons response of the seam. ACT DR6 measures 0.215° ± 0.074° — within 0.4σ of the TFPT value.
The candidate is the determinant-line axion of the strong-CP sector (WIMPs ruled out — no spare E₈ singlet). The misalignment angle is closed; f_a = M_scal/128 is a conjecture.
The introduction states the compiler closure, the dependency DAG, the before/after against the seven original papers, the predictions, and the single proof ledger.
Open the reading guideThe full TFPT 5.0 document set — the introduction reading guide, the four core documents, and the three companions (Appendix H, the Origin Theory synthesis, and the research contracts). All distributed for academic use.
The two axioms, the derivation map, and the D₅ × A₃ → E₈ construction
The φ₀-ladder, flavor from parabolic transport, and the worked closures
The seven E₈ slices as an audit raster, the cascade spine, and the Möbius loop
One seam constant c₃ = 1/(8π) as the universal horizon thermal code
The seam as a horizon, the cyclic compiler hull, and the parameter-free attractor
(U_wall) the parabolic flavor wall-selection · (G_metric) the full QG measure
Every claim marked exact identity, lattice theorem or numerical fixed point is re-derived from the two axioms by a self-contained Python verification suite, mirrored in an independent Wolfram path, and recorded in a single machine-checked status ledger. The carrier algebra (P2) is Lean-formalised (0 sorry, only kernel axioms). The full source — theory documents, scripts and ledger — lives in one public repository. If the text and the ledger ever disagree, the ledger wins.