TFPT — Topological Fixed-Point Theory: Two Axioms, One Compiler, the Standard-Model Skeleton Derived

Hamann, Stefan; Rizzo, Alessandro

Verification & workflow

Every number is machine-checked.

TFPT is built like a compiler, and verified like one. Nothing is marked closed that is not re-derived from the two axioms by an independent script, and no dimensionful quantity is claimed as a derivation from pure numbers. The graph below shows exactly how each result is computed, on which inputs it depends, and which scripts check it — including the self-consistency loop that fixes the inputs themselves. Browser-compatible scripts are runnable live in your browser via Pyodide / WebAssembly. Native interval certificates instead show the pinned local reproduction command.

Open the repository Jump to the dependency graph

The discipline

Four independent paths, one ledger

A claim is only as strong as the way it is checked. The dimensionless skeleton is verified three independent ways and typed in a single versioned ledger, so the papers, the suite and the ledger stay in lock-step.

Python suite

1028 checks

One file per claim cluster; run_all.py ends ALL CHECKS PASSED. Exact sympy where possible, mpmath/numpy otherwise.

Wolfram mirror

116/116 + 603/603

An independent second path for the exact algebraic / identity / lattice results, on the Wolfram Engine: the verified base file plus the v84+ extension file.

Lean carrier

0 sorry

The P2 carrier algebra — hypercharge, anomaly-freedom, integer rigidity — formalised in Lean 4 with only kernel axioms.

Status ledger

1 source of truth

status_ledger.csv types every claim (id, status, location, dependency, script) and is versioned. If the text and the ledger disagree, the ledger wins.

Status markers[E]exact / machine-proven (fine types: identity, lattice/Lie, Lean-formalised, numerical fixed point)[C]conditional — holds under named hypotheses (fine type: physical bridge / readout)[O]open / axiom — declared input or genuine gap[X]falsifiable kill test (committed in advance)

For the external reviewer

The attack surface, at a glance

What is exact, what is conditional, what is an open premise, and how to kill it — so a reviewer can decide where to push without reading 1028 scripts first. Nothing here is hidden: the exact kernel stands on its own; the conditional and open layers are explicitly typed.

[E]Exact kernel

Anchor a = (1,1,2); the E₈ glue D₅⊕A₃+μ₄ ⇒ E₈
Gauge group, 3 families, hypercharge (Lean-formalised)
α⁻¹ = 137.0359992 — unique root of the explicit cubic F_U(1)=0
Flavor operator ladder (Q,K,R,L), quark/lepton ratios
θ_QCD = 0, Higgs uniqueness, the order-30 Coxeter cycle

[C]Conditional physics

Reflection positivity + gap (Decoupling Theorem, Δ_eff = 1.648)
G_net algebra + AQFT machinery [E]; seam coupling [C] closed modulo cited theorems (v367/v368 + v376–v379, ground-state witnesses v489/v490)
F_transfer: Koide, η_B, axion relic, m_p/m_e — runnable typed solvers (v371–v375)
Scheme layer / QCD + EW matching for absolute masses

[O]Open premises

P1, P2 — declared inputs, reduced to the anchor a=(1,1,2)+π (not free dials)
v_geo — the one dimensionful scale (metrology primitive, No-Unit Theorem)
SEAM.EQUIV.01 — [C] closed modulo cited theorems (lattice v367/v368 + S3 stack v376–v379, ground-state witnesses v489/v490, Lean FORM.SEAM.MMST.01); the only [O] residual is the cited continuum scaling-limit existence (v336; extension leg on the peer-reviewed crossed-product package, realisation at invariant level, v469; stays [O])

[X]Kill tests

JUNO: sin²θ₁₂ ≈ 0.3067 (prediction of record, frozen)
CMB tensor-to-scalar r ≈ 0.004 (Starobinsky); n_s vs DESI
Normal neutrino ordering with small m_ββ; θ_eff = 0 (nEDM)
A fourth chiral generation (N_fam ≠ 3) or w ≠ −1 breaks the core

The honest one-line claim is conditional, not messianic: a two-input discrete compiler whose algebraic kernel derives the Standard-Model skeleton and several dimensionless readouts, with every physical transfer layer explicitly typed — and the keystone SEAM.EQUIV.01 (the raw RP seam = the holomorphic (E₈)₁ net at τ=i) is closed modulo cited theorems: the target net is pinned at every computable level (lattice v367/v368 + S3 stack v376–v379, ground-state witnesses v489/v490) and Lean-pinned (FORM.SEAM.MMST.01) to the published MMST/Adamo theorems, so the AQFT closure to (E₈)₁ follows modulo the cited continuum-existence residual (v336; extension leg on the peer-reviewed crossed-product package, realisation at invariant level, v469; stays [O]); its conformal-deck face QGEO.SYM.01 is a corollary.

Origin Theory · one clock

The translation clock: discrete ↔ dynamic is 5 × 6

The bridge between the static (lattice/spectrum) data and the dynamic (recovery/relaxation) data is not a map but a clock — the order-30 Coxeter element — and, because gcd(5,6)=1, it factorizes into two coprime hands: a static carrier ring ℤ/5 = g_car (golden √5, no rate) and a dynamic family ring ℤ/6 = 2·N_fam (the recovery rate (2/3)⁶, exponent 6). The dynamic hand runs 0,1,2,3,4,5 — position 0 is the conserved law (the attractor, rate 0), 1..5 the live phases; the static hand runs 1,2,3,4,5. So '0,1,2,3,4,5' is the law-inclusive reading and '1,2,3,4,5' the live-only reading of the same clock (v319).

Two concentric clock rings. The outer ring has six ticks labelled 0,1,2,3,4,5 — the dynamic family hand ℤ/6 = 2·N_fam carrying the recovery rate (2/3)^6 (exponent 6); the tick at position 0 is highlighted as the conserved law (rate 0, the attractor). The inner ring has five ticks labelled 1,2,3,4,5 — the static carrier hand ℤ/5 = g_car carrying the golden √5 and no rate. The centre reads ℤ/30 = 5 × 6, the order-30 Coxeter clock. — The order-30 Coxeter clock as two coprime hands: dynamic ℤ/6 = 2·N_fam (0..5, rate (2/3)⁶, position 0 = the law) and static ℤ/5 = g_car (1..5, golden √5, no rate). One full turn is lcm(5,6) = 30 = h(E₈)— the arithmetic is [E], the “the bridge is one clock” reading is [C] (v319).

One principle · animated

The universal spectral gap — one operator, many sectors

Every TFPT sector is the same object: a gapped operator with a unique attractor (the physics) and a spectral gap (the reason there is no free parameter). The same gap also sizes each sector's first correction — a typed budget, not a uniform band — and the gap rate is the prime-3 (2/3)⁶ or prime-5 golden facet of the order-30 Coxeter clock. Watch the sectors race to their attractor; the gapless r=1 track never forgets its start (the Lean theorem). Beside it: the entire-form-factor graviton (finite, ghost-free, UV-soft) and the residual matrix that is certification, not construction (v383–v391).

A · every sector is the same gapped operator

iterₙ = x⋆ + rⁿ(x₀ − x⋆) → the unique attractor

step n = 0.0the rate r is the first-correction size (v388); r = 1 never forgets x₀ (Lean FORM.SPECTRALGAP.01)

B · the entire-form-factor graviton (v386 / v389)

dressed / GR ratio e^−u, u = p²/M²

u = 0.00, e^−u = 1.00e+0ghost-free, UV-soft, finite at every loop order by power counting (v389)

C · historical compiler-residual audit (v384)

compiler residual classified; physical TOE gates T1–T8 remain open

AExternal math proof

SEAM.EQUIV.01cited continuum theorem (v336); extension leg certified via the crossed-product package, realisation at invariant level (v469); stays [O]
ALPHA.QUILLEN.EXACT.01the Quillen ζ-functional, narrowed stepwise (v382 → v472): α³ level = the computed inflow Chern |C| = 1, F-normalisation = k_Y = 5/3, det-line curvature over the U(1)-twist moduli = 1; only the continuum ζ-det identification stays [O]
QG.AMB.01constructive measure, a [C] redundancy (v369)

BTheorem-forbidden

v_geothe one metrology unit — No-Unit theorem (v153/v364)

CExternal physics

F_transferQCD / Boltzmann / relic (v371–v374, firewall v187)

Open physical TOE gatesT1–T8 · all [O]

External cross-check

The F_transfer running — confirmed by an independent tool

The renormalization-group running that carries the boundary data to the observables, computed at two loops. The gauge β-functions are reproduced verbatim by the third-party generator PyR@TE 3 (v159); the strong coupling reaches confinement at Λ_QCD ≈ 0.4 GeV (v164); and the Higgs quartic falls to ≈0 with β_λ ≈ 0 at the Planck scale — the free-seam near-criticality that ties the Higgs mass to the boundary fixed point (v166).

Two-panel plot. Left: the inverse gauge couplings versus log10(energy) from the QCD confinement scale to the Planck mass — the U(1) line has slope b1 = 41/10, the strong coupling inverse goes to zero at about 0.4 GeV, and the three couplings do not unify. Right: the Higgs quartic coupling falls from 0.13 at the electroweak scale to about 0.002 near the reduced Planck mass with vanishing beta function — the Standard-Model near-criticality. — Two-loop Standard-Model running. Left: gauge couplings — the U(1)_Y slope is the abelian index b₁ = 41/10 (v159), confinement α₃⁻¹ → 0 at Λ_QCD ≈ 0.4 GeV (v164), no SM unification. Right: the Higgs quartic reaches λ(M̄_Pl) ≈ 0.002, β_λ ≈ 0 — the free-seam near-criticality (v166).

Structural compression

The E₈ audit atlas is one projection of two alphabets

The seven maximal E₈ slices (each a way of cutting 248) are not seven separate hits: every block is built from just two typed alphabets — the anchor power sums P = (3,4,6,10) and the Sheet-Diamond operator determinants (3,4,8,14,20,32), which sum to 81 = N_fam⁴. The atlas is therefore a closed grammar over admissible invariants, exactly what the No-Free-Pattern discipline demands (v170).

Seven horizontal stacked bars, one per maximal E8 subalgebra (D8, D5xA3, E6xA2, E7xA1, F4xG2, A4xA4, A8), each summing to 248. Blocks are coloured by their alphabet source: blue for anchor power-sum blocks, gold for Sheet-Diamond determinant blocks (which sum to 81 = N_fam^4), green for symmetric or glue blocks. — Each of the seven E₈ slices is one projection of two alphabets — blue power-sum blocks P=(3,4,6,10) and gold determinant blocks (3,4,8,14,20,32) with Σdet = 81 = N_fam⁴; green = symmetric/glue. The full lemma (K₄ edge graph, row-budget cross, dim E₆ = 78 = p₂·Δ) is in the E₈-audit paper (v170).

The journey & what remains

From two axioms to one geometric premise

The verification suite grew in phases — foundations, the Standard-Model readouts, the seam=horizon geometry, the reductions and external cross-checks, the AQFT closure, the icosahedral capstone, the seam-equivalence arc and the certification round (1028 scripts today). The historical closure arc narrows the finite seam/compiler lane; it does not close the physical TOE gates. Round 4 (v1026–v1030) closes the fixed-M=1, Ny=8 relaxed TEL-B norm below 2.995906 and adds narrow T3/T4/T6–T8 identities and counterbounds. FE-GEN/ALG-EXH, every T1–T8 gate and the shared 3+1D parent remain open. Both views below are live HTML, not images: browser-compatible script chips run in Pyodide, and native-only certificates provide their local command.

The suite timeline — 17 phases, liveclick a script to run it in your browser

v1–v23
Foundations
Carrier D₅+A₃+μ₄, the E₈ glue, the α⁻¹ fixed point; the anchor a = (1,1,2) to which {c₃, g_car} reduce — the two axioms become {a, π}.
v24–v53
Standard-Model readouts
The φ₀ mass ladder, the flavor operators (Q, K, R, L), lepton/quark ratios, hypercharge, the compiler core (5,3).
v54–v100
Seam = horizon
One-sided Gauss–Bonnet c₃ = 1/(8π), the Coxeter-30 cycle, the gapped attractor (2/3)⁶, the frozen registry + look-elsewhere null-MC.
v101–v140
Horizon + flavor geometry
Nariai = anchor, monodromy = W(A₃) = S₄, cusp weights {0, 1/3, 2/3}, H¹ cohomology, the canonical map.
v141–v158
R1–R5 + premise (A)
Deck selection, the EH mechanism, the No-Unit Theorem, the simple-current (E₈)₁, the free c = 8 fixed point isolated and stable.
v159–v169
PyR@TE cross-checks
SM gauge/Yukawa/Higgs RGEs confirmed by an independent third-party generator (b₁ = 41/10), Λ_QCD, m_H near-criticality, η_B leptogenesis.
v170–v174
AQFT bridges
E₈ slice compression, OS moment/Sugawara, the trace-anomaly seed 4/3, the strong-CP Pfaffian, the Fock readings.
v175–v181
AQFT closure → geometric bedrock
Net existence + full-cone reflection positivity [E]; the residual collapses to one premise: the carrier μ₄ clock is the seam's conformal deck.
v182–v213
F_transfer functor + frontier
The reviewer residual map; F_transfer as one typed functor (Koide, η_B, axion, m_p/m_e) + the machine wording guard.
v214–v218
Pillowcase + Sheet Diamond
The QGEO pillowcase (cross-ratio 2 ⇒ j = 1728), the four marks from Gauss–Bonnet, the Sheet-Diamond axis geometry.
v219–v237
Icosahedral capstone
McKay: why {2,3,5}; the CM norms 41/7, CP triality, the Kleinian seam, det K = 1 = the Kitaev E₈ phase — the (2,3,5) Brieskorn singularity generates ALL.
v238–v261
NCG / Modular Spectral Closure
The 96-dim spectral triple, Dirac = covariance induction, cutoff = KMS weight; seam, carrier and E₈ on one K3: one relative object.
v262–v275
Frontier closure + QFT4D fork
F_QCD m_p/m_e, the M_ν seesaw, S_pert (Epstein–Glaser, 1-loop + gauge betas), scale over-determination, the QG.AMB roadmap.
v276–v299
The Gral: SEAM.EQUIV.01 + Flat-Away
Seam = (E₈)₁ at τ=i named as ONE theorem; both routes reduced to one shared input (Flat-Away), the heat a₂ proved + Lean (v295/v296).
v300–v302
Closing arc: residual pinned
Flat-Away hard-pinned from the (E₈)₁ Steklov data (v300); Route A invertible via free fermions (v301); the gap = derived 6 ln(3/2) > 0 (v302).
v303–v407
Solvers + gravity equation (no free Newton dial [C]) + closure
Typed F_transfer solvers (Koide/η_B/m_p-m_e/axion, v371–v375/v402); the classical covariant Einstein equation under the named QFT/entanglement premises, full nonlinear (v359) [C]; QG.AMB a [C] redundancy (v369).
v408–v481
Certification + external bridges
The α-Quillen residual narrowed to the continuum ζ-det face (computed inflow level |C| = 1, det-line moduli; v470/v472); the sheet/deck complement + degree-ladder audits (v430/v431); the entropic-action bridge typed and bounded (v473–v478); the Kronheimer quiver bridge (v479), the four-interval μ₄ geometry (v480), the seesaw carrier-ladder candidate (v481).

The two axioms c₃ = 1/(8π) and g_car = 5 are not free knobs: they reduce to the single parabolic anchor a = (1,1,2) plus π, and g_car = 5 is a bootstrap fixed point (forced three ways, Lean-formalised). The full script→check map follows in the index below.

The finite seam/compiler reduction chain — live

This historical seam/certification lane narrows, one checked step at a time, to whether the raw seam is (E₈)₁ at τ=i. It is not the full quantum-gravity or strict-TOE problem: T1–T8 remain [O].

startNaive residual
“build a quantum-gravity measure”
v175Net existence + full-cone RP
discharged to [E] (the CAR functor)
v176–v181One geometric premise
QGEO.SYM.01: the carrier μ₄ clock = the seam's conformal deck
v194–v201Non-circular form
state-invariance ω∘ρ = ω; the DtN map mark-local (ℤ₄)
v234–v235ONE condition: holomorphy
no abelian sector ⇔ det K = 1 (the Kitaev E₈ tower)
v276Flat all-orders closure (Lean)
flat τ=i ⇒ [ρ,H] = 0 to ALL orders (FORM.QGEO.04)
v282Two faces, ONE object
χ_E8(i) = 12: the flat τ=i geometry = (E₈)₁ holomorphy
v284–v285Two routes, one open lemma
RP-uniqueness 5/6 + condensation 3/4; the open lemmas coincide
v286–v288SEAM.EQUIV.01 named + attacked
import firewall (v286); Route A 4/5 (v287); Route B proves the full-L² ℤ₄ lift (v288)
v289–v297Flat-Away: 3 routes reduced
heat a₂ closed + Lean (v292/v295/v296), spectral Hessian PD (v293), Troyanov (v294); Route A citable stack (v297)
v300–v302Closing arc: shared fact pinned
Flat-Away hard-pinned from the (E₈)₁ Steklov data (v300); Route A invertible via free fermions (v301); the gap = derived 6 ln(3/2) > 0 (v302)
v335–v379Closed modulo cited theorems
QGEO.SYM.01 = a corollary of SEAM.EQUIV.01 (v335); the gapped lattice model (v367/v368) + the S3 stack (v376–v379, ground-state witnesses v489/v490), Lean-pinned MMST
v458–v480Certification round
the 128-spinor extension leg certified by the peer-reviewed crossed-product package, realisation at invariant level (v469); the R3 graph→geometry bridge discharged to Kronheimer 1989 (v479); the four-interval μ₄ mechanism exhibited, ω∘ρ = ω manifest (v480)
BEDROCKFinite MMST/compiler lane reduced
no further finite assumption in this lane; SEAM.EQUIV.01 still has the cited continuum scaling-limit existence (v336) and stays [O]. This does not close T1–T8 or supply a shared 3+1D parent

Everything above the bedrock is a theorem, a Lean proof or an established citation; the bedrock, once a definition (QGEO.SYM.01), is now one falsifiable physical question — and the emergent-QFT layer (v258–v261, the Modular Spectral Closure) collapses onto this same bedrock, so the boundary QFT adds no new open item. The full step-by-step reduction lives on the /changelog page.

The OS twistor bridge — WOIT.OS.TWISTOR.01

research contract · [O]

The actual bridge from compiler to physics: a global real structure Θ plus interacting reflection positivity plus OS reconstruction on the twistorial algebra. Three of its preregistered stages are executed; the contract itself stays open.

αv519

Θ exists at the free level; free RP selects the same family

β₁v522

the μ₄ clock is the euclidean rotation; gaugeable datum = the GSO ℤ₂; gauge-fixed RP holds

β₂v524

OS quotient explicit: H_phys positive definite, the clock a positive transfer operator with spectral calculus

β₃open

next — under the straddle-law constraint

Toy-level kill, honestly banked (v529). On the first genuinely interacting seam toy, Kill-Test 2 fires: reflection positivity breaks for every g > 0 following the straddle law (RP fails exactly on quartet-straddled cuts, 24/24). Fenced — one toy, one interaction class — this is a named threat and the first hard selection principle: every candidate A_hol must pass the straddle filter. Executed as a selector (v534), the filter keeps exactly one member alive: reflection positivity dynamically selects the alignment bit δ = π/2 with positive coupling — the symmetric straddling of the self-mirror member protects RP where every asymmetric member dies. No marker moves; the contract stays [O].

In its historical finite seam/compiler scope, the bedrock is narrowed modulo cited theorems via the Seam Equivalence Theorem (SEAM.EQUIV.01) — the raw RP seam IS the holomorphic (E₈)₁ net at τ=i (its conformal-deck face QGEO.SYM.01 is a corollary): the target is pinned at every computable level by an explicit lattice model (v367/v368) and the S3 closure stack (v376–v379, ground-state witnesses v489/v490), Lean-pinned (FORM.SEAM.MMST.01) to the published MMST/Adamo theorems, with the cited continuum scaling-limit existence still [O] (v336; extension leg on the peer-reviewed crossed-product package, realisation at invariant level, v469; stays [O]). The emergent-QFT layer (v258–v261, the Modular Spectral Closure) collapses onto this same bedrock — the boundary QFT is one relative object in that scoped lane — and v_geostays the one no-unit primitive. The strict TOE accounting is separate: native v1026, using v1022/v1025, discharges the former CF/DG estimates needed for the fixed-M=1, Ny=8 relaxed TEL-B norm ||R_N||HS < 2.995906 < 3 for every even N≥16. Round 4's other modules add narrow T3/T4/T6–T8 results only. FE-GEN/ALG-EXH, every T1–T8 gate and the shared 3+1D parent remain open; v1029 requires global zero-mode removal and is not a TFPT embedding. Round 7 (v1031–v1035; 233 typed checks including the quantum prerequisite) adds a free positive covariant helicity-±2 curvature field, auxiliary charged corners, a factorized mirror bound and prescribed-source Ward propagation. No common-parent or nonlinear-coupling theorem follows. Source-auditing v1033/v1034 require the full local repository. The 2026-09-06 v1033 amendment removes an eigensolver-dependent zero-mode vacuum choice: it declares a ground-space mixture and reports energy bounds across all pure zero-mode fillings. This finite-grid diagnostic is not a conformal-weight or all-size scaling theorem.

The classical covariant field equation carries no free dimensionless Newton coupling under the named QFT/entanglement premises [C] — three origins of c₃ converge

v358 / v359 · [C]

The entanglement first law $\delta S = \delta\langle K\rangle$ (Jacobson; Faulkner et al.), run with TFPT’s atoms, gives the full covariant Einstein equation (fixed-volume stationarity → the Einstein tensor, v359) with both coefficients fixed under the named QFT/entanglement premises [C] — no free dimensionless Newton coupling. The seam constant c₃ = 1/(8π) arrives by three independent routes that all agree.

Anchorv23

a = (1,1,2)

c_3 = \frac{1}{2\,e_1(a)\,\pi}

e₁(a) = 4

Geometryv58

one-sided Gauss–Bonnet

c_3 = \frac{1}{|\mathbb{Z}_2|\,2\pi\,\chi(S^2)}

|Z₂|·χ = 2·2

Thermodynamicsv358

entanglement first law

c_3 = \frac{\eta}{2\pi},\ \eta = \frac{1}{|\mu_4|}

δS = δ⟨K⟩

triply over-determined

c_3 = \tfrac{1}{8\pi}\quad\Longrightarrow\quad c_3^{-1} = 8\pi

thermo origin

2\pi/\eta

= geo origin

|\mathbb{Z}_2|\,2\pi\,\chi

iff |μ₄| = |Z₂|·χ = 4

classical covariant field equation [C]

G_{ab} + \Lambda g_{ab} = c_3^{-1}\, T_{ab} = 8\pi\, T_{ab}

no free dimensionless Newton coupling under the named QFT/entanglement premises [C]; G is the one unit v_geo

Honest residual. This closes the full covariant equation with no free dimensionless Newton coupling under the named QFT/entanglement premises [C] (v359: fixed-volume → the Einstein tensor, with Lovelock making matter conservation an output); the matter flux is assembled (the Casini–Huerta–Myers ball modular Hamiltonian, boost via Bisognano–Wichmann, v323) and the entropy density is atom-fixed (1/4 = 1/|μ₄|, central charge c = g_car + N_fam = 8). What remains is the Jacobson equation-of-state status (an external candidate action — Bianconi's entropic action, PRD 111, 066001 (2025) — is quantified in v473–v478: β′_B = c₃/6 pinned, the R² kill test executed and resolved as a scale-measure datum, the compression conjecture made well-posed — all without changing the typing), the global ambient measure (QG.AMB.01), and the absolute scale v_geo.

The seam is thermal — three legs of c₃

v526 · [E]/[C]

The Hawking normalisation is no longer only postulated into the horizon sector: the reconstructed free seam OS quotient admits exactly one detailed-balance thermal representation, and its temperature comes out T_seam = 4c₃ — the same Bisognano–Wichmann/Hawking normalisation the geometry and anomaly legs carry. Temperature is the third leg of c₃ = 1/(8π).

Geometryv58

one-sided Gauss–Bonnet

c_3 = \frac{1}{|\mathbb{Z}_2|\cdot 2\pi\,\chi(S^2)} = \frac{1}{8\pi}

seam winding 8 = 2|μ₄|

Anomaly / modularv208

Bisognano–Wichmann KMS

T_H = \frac{\kappa}{2\pi},\quad \frac{1}{2\pi} = 4c_3

Δ^{it} = e^{−2πtK_H}

Temperature — measuredv526

detailed balance on the free OS quotient

\beta_{\mathrm{angle}} = 2\pi\ \text{exact} \;\Rightarrow\; T_{\mathrm{seam}} = 4c_3

β = N clock steps (N = 8 and 16)

geometry + anomaly + temperature

c_3 = \tfrac{1}{8\pi}, \qquad T_H = \tfrac{c_3}{M} = \tfrac{1}{8\pi M}

the axiom is the Hawking coefficient — now derived from the seam KMS structure; SdS

\tfrac{1}{4\pi} = 2c_3

, Nariai

T_N = 4c_3\sqrt{\Lambda}

Honest fences.The [C]-typed reading is “seam euclidean circle = thermal circle of the reconstructed horizon dynamics”; “the seam isa horizon” stays [C]. The temperature bridge closes, the entropy-fraction bridge (the v129 fractions {1/3, 2/3}, the v190 floor) honestly does not; the non-compact control has T = 0 — the temperature hangs exactly on compactness of the euclidean circle (v526, no marker moves).

Honest boundaries

What is still open?

After the compiler closure the live compiler residual is Rest = v_geo ⊕ G_net ⊕ F_transfer. Beside it sits the strict-physical-TOE accounting Rest_TOE: every T1–T8 gate and TFPT.TOE.COMPLETE.01 remains [O], and the shared 3+1D parent is absent. The one-condition/E₈ reduction applies only to the historical finite seam/compiler and boundary-net/MMST lane. Round 4 (v1026–v1030) closes the fixed-M=1, Ny=8 relaxed TEL-B norm below 2.995906 and adds narrow T3/T4/T6–T8 results; FE-GEN/ALG-EXH and all full gates stay open. Round 7 (v1031–v1035) quantizes the zero-mean target and constructs a free covariant helicity-±2 curvature field, auxiliary charged corners, a factorized mirror bound and prescribed-source Ward propagation. This is not a microscopic TFPT embedding or universal nonlinear coupling.

The historical finite seam/compiler residual is one condition — the seam carries no nontrivial abelian sector

The metric inclusion (G_net), the carrier P2 and red-team Target A are not three gates but three faces of a single condition — and all three force E₈:

AQFT

boundary net holomorphic (μ-index 1)

Geometry

seam link a homology 3-sphere (Γ perfect ⟺ 2I)

Rep theory

exactly one 1-dim irrep (no abelian charge)

\#(\text{1-dim irreps}) = |\Gamma^{\mathrm{ab}}| = |H_1(S^3/\Gamma)| = 1 \iff E_8 \iff |\det K| = 1

In words: The seam has exactly one one-dimensional representation (a trivial abelianisation, a homology-sphere link) if and only if it is E₈ — equivalently, the K-matrix has determinant one.

In short. Within the historical finite boundary-net/MMST lane, the Modular Spectral Closure (v258–v261) reduces the listed Dirac, cutoff, gauging and glue data to this one premise, so SEAM.EQUIV.01 is closed modulo cited theorems: the explicit lattice model (v367/v368) and the S3 stack (v376–v379, ground-state witnesses v489/v490) pin the target at every computable level, Lean-pinned (FORM.SEAM.MMST.01) to the published MMST/Adamo theorems — the only residual [O] is their abstract continuum-existence (v336; its 128-spinor extension leg is certified at net level by the peer-reviewed crossed-product package v469, LR/Böckenhauer/KLM, with AGT/AMT an independent second witness; stays [O]). The ambient measure QG.AMB.01 is a [C] redundancy (v369/v379) — a certification object, not missing dynamics — and perturbative spin-2 graviton unitarity is established (the Stelle ghost is a truncation artefact, v304/v370/v380).

＋ The full argument (Modular Spectral Closure · the QG-decoupling theorem · the 4D-QFT fork)

Modular Spectral Closure (v258–v261). The finite Dirac is a covariance induction of the seam KMS state, the spectral-action cutoff is that KMS weight (f₂/f₀ = 1), and seam, carrier-16 and E₈ live on one Kummer/K3 surface — so Dirac, cutoff, gauging, glue and orientability are readouts of one seam state. QGEO.SYM.01 is then a corollary (a conformal net's vacuum is rotation-invariant by axiom, v335).

Seam keystone.The keystone's residual is a composition of standard cited theorems (OS/clustering, the Kitaev free-fermion classification, the AQFT stack) over established TFPT facts — Lean-pinned to those named steps plus the derived gap Δ = 6·ln(3/2) ≈ 2.43 > 0, with no further finite MMST/compiler assumption in that lane. The strict physical TOE obligations remain separate. Its two heavy legs are literature-anchored (v336): the MMST continuum scaling limit and the Adamo–Moriwaki–Tanimoto OS reconstruction, with (E₈)₁ inside their range (c=8, rank 8, 16 Majoranas) — and the 128-spinor extension leg is now certified at net level by the peer-reviewed crossed-product package (v469, LR/Böckenhauer/KLM), with the AGT/AMT route demoted to an independent second witness; stays [O].

QG decoupling is a theorem (v337).Every readout factors through the gapped admissible spectrum (χ = 729/665, margin 1.648 > 0), so TFPT provably does not need the ambient measure.

4D-QFT fork (v265). Finite boundary-net/MMST lane — closed modulo cited theorems; ambient QG — discharged as redundancy [C]; 4D GUT — not claimed (E₈ is the audit hull, not a 4D gauge group); optional UV branch — carrier-native Pati–Salam, falsifiable by thresholds and proton decay. The sprint-by-sprint reduction (v234 → v302) is on the changelog.

Strict TOE scope.Round 4 (v1026–v1030) closes the relaxed TEL-B norm at fixed M=1, Ny=8: ||R_N||HS < 2.995906 < 3 for every even N≥16. Native v1026 uses v1022/v1025 to discharge the former CF/DG estimates for this bound; the other modules add only narrow T3/T4/T6–T8 identities and counterbounds. FE-GEN/ALG-EXH, every T1–T8 gate and TFPT.TOE.COMPLETE.01 remain [O], and a shared 3+1D parent is still absent. The v1029 tensor target requires global zero-mode removal and is not a TFPT embedding. Round 7 (v1031–v1035) quantizes this free target and constructs a positive local covariant curvature field with helicities ±2. It also gives auxiliary flux-register charged corners, the factorized onsite mirror bound 1+√3/2, and linear Ward propagation for prescribed sources. The quarter twist has h=1/4, not hλ=1; the mirror bound is not an interacting hopping theorem; universal quantum coupling remains open.

Rest = v_geo ⊕ G_net ⊕ F_transfercompiler rest · beside it Rest_TOE (ten named [O] contracts / eight TFPT.TOE.COMPLETE.01 gates; nothing claimed closed)

v_geo[O]

Closed metrology structure (R₊ unit torsor; one dimensionful calibration necessary by dimensional analysis — No-Unit v153/v725), over-determined (gravity = dark energy = M̄_Pl to 0.11%, v274); the same unit as gravity's 1/G. Display stays [O], no physical [E].

G_net[E] target · [C] seam

Metric inclusion: the (E₈)₁ target is closed and pinned at every computable level (lattice v367/v368 + S3 stack v376–v379, ground-state witnesses v489/v490); the seam coupling SEAM.EQUIV.01 now carries two named routes (route split 2026-07-22, no status change in substance): the MMST route SEAM.EQUIV.MMST.01 is closed modulo cited theorems (Lean FORM.SEAM.MMST.01), residual [O] = the cited MMST continuum existence only (v336; extension leg crossed-product certified, v469); the twistor route SEAM.EQUIV.TWISTOR.01 (Costello–Li construction, prepared by CELEST.SEAM.01) is [O]; the parent closes if either route closes and stays [O] as an unconditional claim.

F_transfer[C]

One typed runnable solver suite (v371–v374) + a prediction-observatory CI (v375), each with a kill test — bridges, never primitive outputs.

Interface 1[O]

v_geo — the one scale anchor

Quark ratios are closed (Readout Rigidity, c_u/c_d = 55/117); only the absolute scale v_geo remains. CLOSED METROLOGY STRUCTURE (R₊ unit torsor; one dimensionful calibration necessary by dimensional analysis — the No-Unit theorem v153/v725): the same unit as gravity's 1/G, over-determined to 0.11% by gravity vs dark energy (v274). A unit, not a gap; display stays [O], no physical [E]. The interface is structurally closed as an R₊ scale torsor in calibration form (v725: complete export table, dimension-matrix rank 1, all conditions λ-homogeneous; one external reference selects the point, everything else is prediction) — no scale derivation; the one remaining dimensionless ratio is named: H_EW = ln(M̄_Pl/v_EW) = 37.1776. Unit choice closed; dimensionless hierarchies (M_Pl/v_EW, M_R/M_scal, Λ_QCD/v_EW, T_reh/M_scal) remain predictive obligations.

\det R = 8,\ \operatorname{Spec}(Q_+)=\{1,2,3\};\ U_{\mathrm{point}} \to v_{\mathrm{geo}}

Interface 2[E] target · [C] seam

G_net — the metric-sector inclusion

The (E₈)₁ target is closed [E] (v154/v175). The seam coupling SEAM.EQUIV.01 is closed modulo cited theorems: the lattice model (v367/v368) + the S3 stack pin it at every computable level (c = 8, the (E₈)₁ character, torus GSD = 1, RP; v376–v379, ground-state witnesses v489/v490), Lean-pinned (FORM.SEAM.MMST.01). Residual [O] = the cited MMST continuum-existence only (v336; extension leg on the peer-reviewed crossed-product package, realisation at invariant level, v469; stays [O]). First exact index/arithmetic witnesses (2026-08-03): the NS/R grading is the parity character of E8(Z[i])/(1+i) = F₂⁴ — the Ramond projection is (1+i)-adic (v722) — and the Pimsner–Popa/Watatani index is exactly 4 on the CAR ladder with μ₄ derived from the clock (v726); the Q-system identification stays the registered open half (GNET.RAMIFIED.01).

[\,(E_8)_1 : (D_5)_1\times(A_3)_1\,] = 4 = |\mu_4| \Rightarrow \text{holomorphic } c = 8

Interface 3[C]

F_transfer — one typed runnable solver suite

One typed, runnable solver suite — Koide (v371), η_B (v372), axion relic (v373), m_p/m_e (v374) — each with a kill test, folded into a prediction-observatory CI (v375). A guard (v187) keeps them [C] bridges, never compiler outputs. The external-clock typing is now confirmed from the inside: both thermal-time routes for internalizing the four clocks are machine-killed (v723 STT-KILLED, v724 T3B-DEAD) — the anchor census sharpens to {one unit (v_geo-class), one angle θᵢ, one lattice O(1) C_p} — and the frozen external-clock contract is now EXECUTED on its expected K1 kill (v777, NO-COMMON-CONNECTION: no common continuous clock exists; the fibered functor with the constant seam cocycle {0, ±Δ²/2} survives). The coset assignment of the fibered functor's clock fibers is now a theorem from KMS/modular typing (v792, KMS-COSET-THEOREM: S = −(ln λ)²/2 from the time-1 holonomy multiplier; blind protocol, exact unblinding). Honest residual (wave 3, FTRANSFER.SK.RHO0.01 [O]; mechanism executed wave 4, v1000): cosmological transfers need the pair (S, ρ₀), not W[J] alone — FTRANSFER.GENERATING.01 covers equilibrium correlators; KMS/FDT ~10⁻¹⁷, ρ₀-contrast 3.550×10⁻³ vs static 10⁻¹⁶; the typed candidate is θ_i = 3π/5 unique in the μ₄ lift [C] (no cosmological solve).

F_transfer: one shape, four rates

Every transfer instance is a gapped relaxation to a unique attractor — the same shape as the main-branch update to the E₈ marks.

Only F_pole (Koide) runs at the seam rate — its Möbius multiplier is (2/3)⁶ = λ₂, the seam-clock eigenvalue. The other three share the shape with external rates (thermal washout, cosmological freeze, RG flow). So F_transfer is the readout end of the one discrete→dynamic principle, not a bolt-on (v303).

The gates are written up as numbered research contracts.Read the contracts

The dependency graph

How each result is computed

Two axioms at the source, the E₈ compiler in the middle, the observables as sinks. Click any node to see what it is, its status marker, its inputs and outputs, how it can fail, and the scripts that machine-check it — then click a script for its reproduction path. Browser-compatible scripts run live; native-only certificates show the pinned local command. The dashed rose edges are the bootstrap: the E₈ closure feeds back and fixes the inputs.

Interactive dependency graph — click any node

axiomlatticeidentitynumericalconditionalself-consistency

E₈ — the μ₄ glue (audit hull)

[E] lattice theorem

D₅ and A₃ share the discriminant group ℤ₄; their glue norms add to the root norm, q(D₅) + q(A₃) = 5/4 + 3/4 = 2. So E₈ = (D₅ ⊕ A₃) + μ₄ closes as a lattice theorem — the unimodular audit hull, not a gauge group; 240 = 16·5·3 roots, 248 = 240 + 8. It is the forced choice, not a guess: no nontrivial abelian sector ⇔ holomorphy ⇔ det K = 1 selects (E₈)₁ uniquely (v234/v235/v237), and the whole (2,3,5) skeleton drops out of the Brieskorn capstone (v236). The golden ratio φ is not a fitted number but the icosahedral (2,3,5) signature — an output of the bootstrap-forced geometry, not an input (v349/v354) — and a reverse numerology audit of the unmapped E₈ region finds no missed primary readout under a strict discriminator (v355). The sheet/deck complement (v430) closes the adversarial follow-up: the seam's “other side” (the double-cover deck |ℤ₂| = 2 and the conjugate half-spinor S⁻ = part of the 128-spinor, ∑deg) is matched structure, forced-disjoint from the five unmapped Casimir degrees {12,14,18,20,24}. Those five are not diffuse overhead either: they are the forced two-family decomposition deg(E₈) = 6·spine{2,3,4,5} ⊔ ({2}∪det-ladder{8,14,20}), the residue classes {0,2} mod 6 forced by h = 30 = 2·3·5 (v431) — exact arithmetic, with the functorial flavour identification honestly kept [P].

Inputs

D₅
A₃

Outputs

240 roots, 248 = dim E₈

How it can fail

Not even-unimodular; glue norms do not sum to 2.

Verified by (241 scripts) — click for its reproduction path

Reproduce it yourself

Five commands, from the repository root

Dependencies: a LaTeX distribution, Python 3 with sympy / mpmath / numpy / matplotlib; optionally the Wolfram Engine and Lean 4. Every script cited in run_all.py is also cited inline in the documents, and the status heatmap is generated directly from the ledger.

reproduce / verify

# 1. Compile the active document set      ->  "10 ok, 0 failed"
bash build.sh notes

# 2. Run the Python verification suite    ->  "ALL CHECKS PASSED"
cd verification && python run_all.py

# 3. Independent Wolfram path (optional)  ->  "ALL WOLFRAM CHECKS PASSED"
wolframscript -file verification/wolfram/tfpt_readouts.wl

# 4. Lean carrier-rigidity proof (optional) ->  "AUDIT: PASS"
cd experiments/lean4-carrier-rigidity && lake exe cache get && bash scripts/audit.sh

# 5. The sync audit (papers <-> suite <-> ledger <-> website)  ->  "AUDIT OK"
bash build.sh audit

# 6. Regenerate the reproducibility manifests (run last)
python verification/make_manifest.py

The script index

What each script checks

The full verification suite, grouped by what it proves. Every script links to its source in the public repository.

The suite is organised by what it proves. Each script is one claim cluster, cited inline in the documents via \veri{vN} and registered in run_all.py, which ends ALL CHECKS PASSED. Click any script to inspect it. Browser-compatible scripts run live; native-only certificates show their pinned local command.

1028 scripts

Compiler core & the E₈ glue

494

Why the two axioms build E₈, why the carrier rank is forced, and the integer skeleton that follows.

Electromagnetic fixed point

The fine-structure constant as the unique root of the boundary U(1) Ward identity.

Flavor matrix & operators

The integer operator ladder (R, K, Q, L) and its spectral invariants — the flavor signature.

Masses, leptons & quark ratios

The φ₀-ladder mass formula, the exact lepton coefficients, and the integer-Plücker quark ratios.

Neutrinos & the solar angle

The Majorana texture, the dual anchor, and the previously open solar angle from the seam.

Gravity, inflation & cosmology

The R + R² spectral-action shadow, the seam-fixed scalaron, and the cosmological readouts.

Horizon code & Origin Theory (self-consistency)

The seam as the universal horizon code, the order-30 Coxeter cycle, and the gapped unique attractor that makes parameter-freeness a theorem.

Open gate (U_wall) — the flavor wall

The parabolic wall-selection contract: the quark ratios are closed; only the absolute amplitude scale stays open.

Open gate (G_metric) & the frontier

251

The quantum-gravity measure contract, the audit ledger, the data scorecard, and the honestly-typed frontier items.

Blind registry & red-team follow-ups (v84–v175)

184

The frozen prediction registry and the follow-up rounds: Target A merged to one residual, the CP residual quantified, N★ from reheating, the F_transfer gauge inputs cross-checked with an external RGE tool, and the AQFT closure round — net existence and full-cone reflection positivity discharged to [E], leaving the seam realisation as the single open premise. Every freeze machine-enforced.

Independent scrutiny is the point

These checks are reproducible by anyone — that is what “independent” means here: the same result falls out on your machine, on an independent Wolfram path, and (for the carrier) in Lean. We do not claim external endorsements; we invite review. Open questions and known limitations are tracked openly in the research contracts and the status ledger — disagreements go in the issue tracker.

Source, scripts & ledger Open an issue / review

Every number is machine-checked.

Four independent paths, one ledger

The attack surface, at a glance

The translation clock: discrete ↔ dynamic is 5 × 6

The universal spectral gap — one operator, many sectors

The F_transfer running — confirmed by an independent tool

The E₈ audit atlas is one projection of two alphabets

From two axioms to one geometric premise

Foundations

Standard-Model readouts

Seam = horizon

Horizon + flavor geometry

R1–R5 + premise (A)

PyR@TE cross-checks

AQFT bridges

AQFT closure → geometric bedrock

F_transfer functor + frontier

Pillowcase + Sheet Diamond

Icosahedral capstone

NCG / Modular Spectral Closure

Frontier closure + QFT4D fork

The Gral: SEAM.EQUIV.01 + Flat-Away

Closing arc: residual pinned

Solvers + gravity equation (no free Newton dial [C]) + closure

Certification + external bridges