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.
TFPT reproduces dozens of real numbers of physics — and the compiler closure says why the same small integers (2, 3, 5, 16, 240, 248) keep reappearing in every sector: 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 — [E] identity, [E] lattice theorem, [E] formalised, [E] numerical fixed point, [C] conditional, [O] 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 live residual is Rest = v_geo ⊕ G_net ⊕ F_transfer: one dimensionful scale anchor, one metric-sector inclusion theorem, and one downstream transfer functor. None of these is hidden, and none is overclaimed. (The historical labels U_wall / G_metric / F_frontier are kept only for ledger continuity.)
The quark mass ratios are closed (Readout Rigidity, c_u/c_d = 55/117 on the discrete selector stratum) and the selector triangle pins R columnwise (the dual pair (d,n), v136/v139). Only the absolute amplitude scale v_geo remains — the same dimensionful anchor as gravity's 1/G; finite, algebraic and falsifiable. (Historical label: U_wall.)
R + R² is heat-kernel grounded (G2) and the admissible IR sector is gap-decoupled (G5, Δ_eff = 1.648 > 0). The closing statement is exactly ONE theorem: the seam–Calderón inclusion has Jones index 4 = |μ₄| (the μ₄ simple-current extension of the carrier net), from which holomorphy and the unique (E₈)₁ bulk follow (LR/KLM/BKLR). Full QG closure is a certification layer, not a prerequisite for testing the readouts. (Historical label: G_metric.)
Koide, η_B, the axion relic scale and m_p/m_e are four instances of one missing functor F_transfer — the continuous transport from compiler source data to measured observables (Koide source→pole; η_B source→Boltzmann relic; axion scale→abundance; m_p/m_e→QCD/EW matching). Each has a genuine handle but is a transfer target, deliberately not claimed as a primitive compiler output.
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.
A structure test with zero free parameters: write classical black-hole mechanics inside the de Sitter bulk in seam units, and every coefficient lands on a compiler atom that is already load-bearing elsewhere. Machine-checked in v101/v102; the carrier-in-the-bulk reading is typed [C].
At the maximal mass the horizon equation becomes with roots — exactly the traceless form of the anchor that generates the whole compiler grammar.
The maximal black hole carries exactly of the de Sitter entropy — the same constant that sits at the Koide branch point of the flavor sector. Zero adjustable parameters on either side.
Flavor relaxation and black-hole evaporation both flow away from the anchor configuration — a stationary repeller with grammar-constant curvature ( in flavor, in gravity). Reading it as one variational principle stays [C].
Doc 0 is the reading guide (introduction). Papers 1–5 are the numbered documents — architecture & E₈, the Standard Model, the E₈ audit & bootstrap, the honest frontier, and the adversarial Red Team audit. Three companions complete the set — 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.
The seam owns exactly one measuring device — a single scalar two-point kernel — and four theorems pin what it can do. A scalar kernel exists iff it pairs the two sheets (exactly 2 = |ℤ₂| kernels = the glue ambiguity, v110); the certified channel counts the code by itself — one neutral kernel per code state, graded (1,5,10), so the Pascal closure is two countings of one set, not a condition (v112); pair transport is minimally complete — degree ≤ 1 generates nothing, degree 2 generates every code operation (v111); and the carrier net is 16 free Majorana fermions whose tower carrier → SO(16)₁ → E8₁ never changes the field content — only the certificate grows, and the central charge is the rank of the one kernel: 5 on the carrier block, 8 on the seam hull (v113). The interior is free; the structure is the certificate. Honest residue: the premise 'the seam is the free c=8 net' is the G_net gate itself — one theorem now closes both the metric story and the carrier choice.
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. The remaining ℤ₃ deck choice is since derived: the integer deck pairs the Q₊=1 line with the self-conjugate character 2, so the geometric boundary deck is the sheet-twisted class and the cusp exponential is excluded (v141) — GATE.QGEO keeps only its realisation premise, with no discrete freedom left — and that premise sits at its floor: the full Möbius D₄ of the seam curve matches the integer model parity by parity (ι = T_A exactly, δι = Σ; v146).
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.
At the two branch points of the anchor-block double cover the block is rank 1, with integer kernels — at the carrier point the kernel is the democratic vector itself. Rank 1 forces the kernel image onto the antisymmetric direction (−1,1,0): up and down are the deck-odd pair, and the lepton pairing vanishes — the leptons sit on the ramification (Koide is leptonic). The anchor-forced cusp conjugation T_A (with a = e₂+e₃, the conjugation-symmetric vector) realises the same deck action, and the dictionary 'Q₊ grading = A₃ discriminant grading' is now derived (G = T_A·Σ acts integrally as the B₁⊕E decomposition on the cusp basis): the sheet question carries no separate [C] — its residual coincides with the one existing Q-geometry gate.
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.
An explicit, fully declared null model: the entire complexity-matched formula grammar (provably containing every scored TFPT formula) is enumerated exactly against conservative data windows over the 13 scored frozen-registry observables; Monte-Carlo pseudo-theories and negative controls (seed perturbation, data shuffle) demonstrate the test's power; all 94 500 variants of the F_U(1) equation are root-solved with exactly one CODATA hit. The result is a null-model rejection conditional on the declared grammar — never 'certainty'.
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 relaxation now has a canonical generator — dq/dt = (Δ/N_fam)·det B(q), the gap times the anchor-block quadric, whose time-1 map is the forced Möbius attractor — and the discrete-vs-continuous question is experimental: n = 3 = N_fam transfer steps corresponds to m_τ = 1776.9427 MeV (+0.14σ; n = 2 excluded at −2.9σ), decidable at σ(m_τ) ~ 0.01 MeV.
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.
Targets A–E: attacking the five load-bearing reductions at their weakest transitions
The deliberately adversarial layer: instead of confirming TFPT, this document attacks the five load-bearing reductions (Targets A–E) at their weakest logical transitions. Each target runs through one fixed protocol — minimal statement, assumptions, logical chain, counterexample search, limiting cases, alternative structures, verdict. A red-team check asserts an adversarial fact (a counterexample really exists, a hidden assumption is really needed, a firewall really holds); the honest outcome lives in the status of each target, never in a green pass. Verdicts: A reduced (one residual), B/D/E survive narrowed, C survives; none broken.
Each reduction is treated as a hostile witness under one fixed protocol. Allowed outcomes: survives (stands as worded), survives narrowed (stands only after a silent assumption is made explicit), reduced not closed (the conservative wording is correct). A confirmatory script that always passes is worthless here.
Level-1 primary counting (det Cartan: D₈ has 4, E₈ has 1) makes holomorphy necessary AND sufficient — a holomorphic c = 8 chiral CFT is the lattice theory of the unique even unimodular rank-8 lattice. Bulk uniqueness is not independent: for a holomorphic net Rep(A) = Vect, so the bulk pairing is unique (machine contrast: SO(16)₁ admits six modular invariants). Target A therefore collapses to one residual statement.
B: the Pascal ladder 2^{g−1} = Σ_{k≤2} C(g,k) is exactly equivalent to the degree-2 truncation; the residual is the QBL premise, since merged with the boundary-net gate. C: the replica chain is derived; the absolute 1/G stays the one anchor. D: the frozen CP phase survives at +0.98σ with a decision threshold σ_γ ≤ 0.96°. E: v_geo carries the dimensionless theory; EW/reheating scales are typed interfaces.
Two machine-checked follow-up rounds (v83–v100, v141–v144) moved Target A from three residuals to one and hardened the firewalls (numerology null test: P ≤ 10⁻³⁰·⁷ conditional on the declared grammar). The front summary table states the final reduction; the historical development is kept below it, honestly dated.
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.
Put a black hole into the de Sitter bulk: at the maximal (Nariai) mass the horizon cubic has roots (1,1,−2) — exactly the traceless projection of the anchor a = (1,1,2) — and the total entropy bound is exactly the Koide branch value 2/3 = |ℤ₂|/N_fam (each horizon carries S_dS/3). The interpolation is (x²+1)/Φ₃(x) with the N_fam cyclotomic; the three-root entropy total |ℤ₂|·S_dS is conserved for every mass; the mass line is itself a split double cover whose deck involution is the horizon swap; and evaporation always flows away from the anchor point — the same repeller/attractor orientation as the flavor relaxation. Six independent landings on already-load-bearing atoms, zero free parameters; the carrier-in-the-bulk reading stays [C].
The flavor relaxation is the gradient flow of a cubic potential whose critical points are exactly the two branch points, with stationary curvatures ±Δ (the transfer gap) and a constant Lyapunov rate Δ. The SdS entropy functional has the Nariai/anchor point as its unique stationary point with curvature 2/9 = |ℤ₂|/N_fam², and evaporation ascends the entropy away from it. Both sectors flow away from an anchor-stationary configuration with grammar-constant curvatures; reading this as one variational principle of the seam stays [C], with the disanalogies recorded honestly.
The SdS horizon cubic is uniformized by angle trisection (r = 2cos θ turns it into cos 3θ = −3m; the ℤ₃ trisection deck is the triality of coker Q = ℤ/N_fam). In the centered angle the mass is a pure cosine, m = cos(ψ)/N_fam, and the entropy collapses to ONE cosine of glue atoms with canonical curvature (2/3)³ at the anchor — the Koide constant to the family power. The invariant slope dσ/dm at Nariai is −8/9 = −rank E₈/N_fam². The flavor invariant is a rate, (2/3)^{2N_fam} per transport step; the gravity invariant is a curvature, (2/3)^{N_fam}: same base, exponent ratio |ℤ₂|. The gravity-side clock asked for here has since been constructed (v124–v133, sections below): one clock, two known geometries — the identification reading stays [C].
The classical half of the clock question is pure GR: linearizing around the Nariai geometry dS₂×S², the static mode φ(ρ) = ρ solves the static-patch equation exactly with m² = −2Λ = −|ℤ₂|Λ — the exact SdS family itself pins the modulus mass (Ginsparg–Perry tower: exactly one negative mode). In Hubble units the clock's characteristic polynomial is (λ−1)(λ+2) — the anchor quadratic: its eigenvalues are the distinct anchor roots, and the Nariai cubic factors as (t−1)·χ_clock. The anchor appears a third time: configuration roots, curvature base, clock spectrum. The honest (2/3)-test is negative for the classical clock (integer eigenvalues); the quantum clock — the one-loop conversion of curvature into rate — was the remaining [C] and is resolved by the resummed-clock chain below.
The quantum clock now has a closed form: rate(n) = −p₂ ln(1 − n/N_fam) — the three-level spectrum is forced by the pole at N_fam, and the bend log₃∕₂3 is its n = 2 value. Its weights are the Mehta–Seshadri parabolic weights of the exact anchor residue A₀* (v126); the geometric tail is the standard log-determinant/RPA ring resummation, one tower per hexagon site (v127); the rate is an entropy power law Γ ∝ (S/S_dS)^{p₂} in Gibbons–Hawking form (v129); the exponent p₂ = 2h follows from mode counting plus the Born rule, h = N_fam = half the zero-mode count (v130); the per-mode S^{1/2} is the zero-mode area norm ‖Y₁ₘ‖² = A/(4π) exactly (v131); and the scaling anomaly of the non-zero-mode S² determinant is exactly −2/3 = −|ℤ₂|/N_fam — the Koide constant as a spectral anomaly (v132). The ζ(0) budget computed both ways selects the reduced seam reading: per sector −2/3, total −4/3 = minus the seed gain, while the naive 4d route gives −109/45, no atom (v133). The residue of the clock question is one finite budget — the graviton/ghost heat coefficients on S²×S² [C]. The det-ratio step is since derived within the SdS family: e₂-rigidity gives r_b·r_c = 1 − Δ²/3 exactly, so the non-zero-mode determinant ratio is (1 − Δ²/3)^{4/3} with no first-order term in the horizon split (v144); the finite-weight absorption stays [C] with its obstruction stated sharply. The ring sum is now identified as the Born-squared Gaussian zero-mode integral (variance = area ratio, forced by v131), and the quantum bend log_{3/2}3 is determinant-clean — both Nariai weights share one geometry (v147); the residue is the measure identification plus one reference normalisation.
The Nariai pattern is stored inside the flavor compiler as a dual invariant: d := aᵀR⁻¹ = aᵀL⁻¹ = (−1/2, −1/2, 1), with d·1 = 0, d·a = 1 and (1,1,−2) = −2d. The invariance is structural (Sherman–Morrison): a covector is winding-invariant iff it annihilates R⁻¹1 = (1,1,−1)/4 — the anchor does, while 1, e₁ and the torsion normal n do not (the membership is special). Together (d, n) form the dual normal pair of the flavor boundary: d reads the traceless horizon structure, n reads first-generation torsion. A third, purely algebraic leg of the flavor↔horizon bridge, beside the shared clock spectrum (v126) and the entropy power law (v129) [E]; the bridge reading stays [C].
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 [E] 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 [C] 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.
v_geo · G_net · F_transfer — the live residual as numbered contracts
After the compiler closure the live residual is Rest = v_geo ⊕ G_net ⊕ F_transfer (the historical labels U_wall / G_metric / F_frontier are kept only for ledger continuity). This note turns the open interfaces into contracts: a numbered chain of lemmas, the single theorem that closes each, and — for every step — whether it is machine-certifiable today. Priority: the selector-triangle pairings and the v_geo scale anchor first (finite, algebraic, falsifiable), then the G_net inclusion theorem (deep analytic programme); F_transfer is the downstream interface.
The flavor interface (historically U_wall) is reduced to the selector triangle: with the dual anchor d = a·R⁻¹ and the torsion normal n, each column of R is the unique lattice point of the address box (v136/v139). The selectors det R = 8 and Spec(Q₊) = {1,2,3} are read off the bundle; the quark ratios are closed by Readout Rigidity, leaving only the absolute amplitude scale = v_geo.
The dual normal pair (d, n) pins R columnwise; d = (3/2)a − 2·1 is pure anchor data (the first selector is derived), and n is the unique covector with atom pairings (2, 8, 121) on the frame (1, a, σ) of determinant 11. Frame integrality cuts this further: integer covectors form an index-11 sublattice, so the σ-pairing is forced mod 11 — and the pairing values themselves are atom identities given the exponent-duality lift — including the new closure |μ₄| + g_car = N_fam² (4+5=9) — so the residue is exactly one lift map (v145); the historical U_wall machinery is over-engineering for the ratios.
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 ambient measure is holographically reduced to a finite seam-boundary (Calderón) measure. The single closing statement is: the seam-Calderón inclusion has Jones index 4 = |μ₄| (the μ₄ simple-current extension of the carrier net), from which holomorphy and the unique (E₈)₁ bulk follow. The identification now holds at the finite Lie level: the ℤ₄-graded hull carries exact coset duality, the glue average is the carrier projector (index 4), and each sector is a single Weyl orbit — the ℂ[ℤ₄] Q-system realised in 248 dimensions (v143); exactly one conformal-net statement remains — and it is scoped: the odd glue sectors are twisted (Ramond-type) modules with zero support on the untwisted Fock model, so the statement is irreducibly about the twisted-sector structure (v148).
The selector-triangle pairings and the v_geo scale anchor are finite, algebraic and falsifiable today; G_net is a deep analytic programme; F_transfer is the downstream interface. 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 cubic closure 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. Typed [E]: the coefficients are exact identities, the Ward-origin reading of the equation is conditional [C] (ledger EM.FP.01).
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 = c₃ + 36 c₃⁴ (leading term c₃, fourth-order puncture correction exact) gives the prediction of record sin²θ₁₂_seed = 0.306747. This single seed value is the frozen prediction (blind registry predictions_frozen.json, 2026-06-09, machine-enforced by v84); the seam-corrected 0.306808 and non-linear 0.307020 are scheme diagnostics of the same texture along the F_transfer orbit, not rescue branches. Live status: JUNO's first 59.1-day run reports sin²θ₁₂ = 0.3092 ± 0.0087 — compatible, precision phase pending; NuFIT 6.0 (0.307) is the pre-JUNO global-fit baseline.
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. The frozen registry keeps n_s as a band over N★ ∈ [50,60]; the scalaron-reheating chain (v86, Higgs channel) sharpens it conditionally to N★ = 51.4 ⇒ n_s = 0.9611 — recorded with its tensions (+0.9σ below Planck; the same chain underpredicts A_s, so the point is conditional on the decay channel).
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). The frozen registry keeps r as a band; the scalaron-reheating chain (v86) sharpens it conditionally to r = 0.0045.
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 — the measured A_s prefers N★ = 56.2. Honest record (v86): the slow Higgs-channel reheating point N★ = 51.4 underpredicts A_s by 11σ, so the measured A_s requires near-instantaneous reheating; A_s arbitrates the reheating speed.
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 predictions, and the single proof ledger.
Open the reading guideThe full TFPT 5.1 document set — the introduction reading guide, the five numbered documents (architecture, Standard Model, audit & bootstrap, frontier, Red Team), and the three companions (Appendix H, the Origin Theory synthesis, and the research contracts). All distributed for academic use.
Reading guide, status assessment, and the dependency DAG
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
Targets A–E: attacking the five load-bearing reductions at their weakest transitions
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
v_geo · G_net · F_transfer — the live residual as numbered contracts
The canonical dated changelog of every change to the theory, the suite, the papers and the website.
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.