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Architecture

How the pieces lock

The dependency chain from two axioms to the Standard-Model skeleton, the seam=horizon identification, and the parameter-free Einstein equation — archived here so the homepage stays a narrative, not a museum.

The compiler pipeline

From two axioms to the observables

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. On top, the boundary QFT assembles into one relative object — the Modular Spectral Closure — reduced to a single seam premise.

1

Two axioms

Axiom — declared input
c3=18π,gcar=5c_3 = \tfrac{1}{8\pi}, \qquad g_{\mathrm{car}} = 5
Input
Declared (P1 Gauss–Bonnet-hardenable, P2 Lean-formalised)
What is fixed
The seam constant and the five-slot carrier — the only structural inputs
Not claimed here
No SM gauge group, no families, no α inserted by hand
How it can fail
No reflection-positive seam, or the wrong family/charge lattice
2

The two atoms

Lattice / identity
C+=D5 (16),P1∖μ4=A3C^+ = D_5\ (16), \qquad \mathbb{P}^1\setminus\mu_4 = A_3
Input
The carrier g_car = 5 and the four-puncture family geometry
What is fixed
The D₅ half-spinor (dim 16, hypercharge Y) and the A₃ family geometry (N_fam = 3)
Not claimed here
Not yet E₈ — the atoms are intermediate products of the inputs
How it can fail
Group/matter mismatch, or the wrong family multiplicity
3

The μ₄ glue ⇒ E₈

Lattice / identity
E8=(D5⊕A3)+μ4E_8 = (D_5 \oplus A_3) + \mu_4
Input
D₅, A₃ with the common discriminant ℤ₄
What is fixed
240 = 16·5·3 roots, 248 = 240 + 8; q(D₅)+q(A₃) = 5/4+3/4 = 2
Not claimed here
E₈ is the unimodular audit hull, not a physical gauge group
How it can fail
Not even-unimodular, or the glue norms do not sum to the root norm 2
4a

Standard Model — Engine 1

Readout
Nfam=3, Ωadm=48, b1=4110N_{\mathrm{fam}}=3,\ \Omega_{\mathrm{adm}}=48,\ b_1=\tfrac{41}{10}
Input
E₈ projection + carrier traces
What is fixed
The gauge group, 3 families, hypercharge, b₁ = 41/10, and the residue matrix R (det 8)
Not claimed here
No absolute quark amplitude scale (the v_geo anchor)
How it can fail
Hierarchy mismatch, or a robust fourth generation
4b

Constants — Engine 2

Readout
α−1=137.0359992,θeff=0\alpha^{-1} = 137.0359992,\quad \theta_{\mathrm{eff}} = 0
Input
The seed u = φ₀ and the abelian coefficient 41 = 10 b₁
What is fixed
α⁻¹, the Cabibbo angle λ_C, sin²θ₁₃, β_rad, and the strong-CP null
Not claimed here
Not the dimensionful EW/QCD masses (those sit on the RG scheme layer)
How it can fail
No root or a second root of F_U(1)(α), or a robust nonzero neutron EDM
4c

Gravity & cosmos — Engine 2

Geometry channel
Gab+Λgab=c3−1TabG_{ab} + \Lambda g_{ab} = c_3^{-1} T_{ab}
Input
The seam constant via the geometry channel (entanglement first law + R + R²)
What is fixed
Parameter-free Einstein equation Gₐᵦ+Λgₐᵦ=c₃⁻¹Tₐᵦ, both coefficients fixed (v358/v359); scalaron mass 3.06×10¹³ GeV, n_s = 0.965, r ≈ 0.004, Λ ∼ e⁻²ᵅ⁻¹, H₀ ∼ √Λ
Not claimed here
The absolute dark-energy density is a typed cosmology interface
How it can fail
A robust r ≳ 0.01, or a robust w ≠ −1
5

The φ₀-ladder & flavor matrix

Bridge readout
m^f,j=vgeo2 λY Lf,j Λf,j\hat m_{f,j} = \tfrac{v_{\mathrm{geo}}}{\sqrt2}\,\lambda_Y^{\,L_{f,j}}\,\Lambda_{f,j}
Input
The residue matrix R and the seed φ₀
What is fixed
All nine masses, CKM, the PMNS skeleton; the solar angle sin²θ₁₂ = 1/3 − φ₀/2
Not claimed here
Quark ratios closed; the absolute quark scale is the U_point anchor
How it can fail
The lepton φ₀-ladder mismatches the observed charged-lepton hierarchy
6

The bootstrap loop

Lattice / identity
E8 closure⇒gcar=5,  8=rank⁡E8E_8\text{ closure} \Rightarrow g_{\mathrm{car}}{=}5,\ \ 8 = \operatorname{rank}E_8
Input
The E₈ closure fed back to the inputs
What is fixed
The two inputs are re-derived — the discrete core is overdetermined, not fitted; only π stays free
Not claimed here
Not creation from nothing — two inputs remain
How it can fail
g_car = 5 not forced three ways, or the reverse glue μ² − 5μ + 4 = 0 does not pick μ = 4
7

Boundary QFT — the Modular Spectral Closure

Bridge readout
TFPTQFT=(AΣ,ωΣ,ΔΣ,ρ,AF,HF,DF,J,γ,Srel)\mathsf{TFPT}_{\mathrm{QFT}} = (\mathcal{A}_\Sigma, \omega_\Sigma, \Delta_\Sigma, \rho, A_F, H_F, D_F, J, \gamma, S_{\mathrm{rel}})
Input
The seam KMS state + the 96-dim carrier finite spectral triple
What is fixed
One relative object: D_F = covariance induction of the seam state (v258), cutoff = the KMS weight ⇒ f₂/f₀ = 1 (v259), seam + carrier-16 + E₈ on one Kummer/K3 (v260); cross-checked 4 = [B:A] = |μ₄| = 2χ = |(ℤ/2)²| (v261)
Not claimed here
Closed modulo cited theorems via the keystone SEAM.EQUIV.01 (QGEO.SYM.01 is its corollary, v335; extension leg on the peer-reviewed crossed-product package, realisation at invariant level, v469; stays [O]); the ambient QG measure is discharged as a redundancy [C] (v369+v379)
How it can fail
An invariant disagrees across the round (index ≠ marks, two carrier-16s, or two different gaps)
8

The residual: two gates + interfaces

Open / frontier
Rest=(Uwall)⊕(Gmetric)⊕(Ffrontier)\text{Rest} = (U_{\mathrm{wall}}) \oplus (G_{\mathrm{metric}}) \oplus (F_{\mathrm{frontier}})
Input
The compiler closure
What is fixed
One flavor wall-selection (v_geo, [O]); the ambient quantum-gravity measure now discharged as a redundancy [C] (v369+v379; the local field equation is itself parameter-free, v358/v359); and a set of typed runnable frontier solvers (F_transfer, v371–v375) — the whole boundary-QFT layer collapses onto the G_metric keystone (step 7), closed modulo cited theorems, adding no new open item
Not claimed here
No strict physical TOE certified end-to-end (the keystone SEAM.EQUIV.01 is closed only on its MMST route SEAM.EQUIV.MMST.01, modulo a cited published theorem, v336 — extension leg crossed-product certified, v469; stays [O] — and v_geo stays the one unit). The ambient measure G6 is not an open hole — it is discharged as a redundancy [C]
How it can fail
A gate's closing theorem asserted before its lemma chain completes
Status discipline.The diagram is a status map of the dependency DAG, not a claim that all displayed outputs share the same grade. Axioms, lattice/identity steps, readouts, the bridge ladder and the open residual are all visibly distinct — and each carries its own “not claimed here” and “how it can fail” rows. The machine-checked ledger is the single source of truth.
Explore the interactive dependency graph
Seam = horizon

A black hole in the bulk: every coefficient lands on a compiler atom

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].

The maximal black hole in the de Sitter bulk

cosmological horizon (de Sitter bulk)black hole growsmaximal case (Nariai): both horizons meet — roots (1, 1, −2) = the anchor

One orientation: away from the anchor, toward the democratic endpoint

anchor configuration (stationary repeller)democratic endpoint (attractor)

Nariai = the anchor

[E]

At the maximal mass the horizon equation becomes t3−3t+2=(t−1)2(t+2)t^3-3t+2=(t-1)^2(t+2) with roots (1,1,−2)(1,1,-2) — exactly the traceless form of the anchor a=(1,1,2)a=(1,1,2) that generates the whole compiler grammar.

The Koide 2/3 is the entropy bound

[E]

The maximal black hole carries exactly 23=∣Z2∣/Nfam\tfrac{2}{3}=|\mathbb{Z}_2|/N_{\mathrm{fam}} 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.

One orientation in both sectors

[E] + [C]

Flavor relaxation and black-hole evaporation both flow away from the anchor configuration — a stationary repeller with grammar-constant curvature (±Δ\pm\Delta in flavor, 2/9=∣Z2∣/Nfam22/9=|\mathbb{Z}_2|/N_{\mathrm{fam}}^2 in gravity). Reading it as one variational principle stays [C].

Gravity, parameter-free

The Einstein equation falls out — with no free dial

The entanglement first law, run with TFPT's atoms, gives the full covariant Einstein equation Gₐᵦ + Λgₐᵦ = c₃⁻¹Tₐᵦ; the seam constant c₃ arrives by three independent routes that all agree — and its temperature leg is now measured on the seam itself (T_seam = 4c₃, v526).

Gravity is parameter-free — three origins of c₃ converge

v358 / v359 · [E]

The entanglement first law δS=δ⟨K⟩\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 — no free dimensionless dial. The seam constant c₃ = 1/(8π) arrives by three independent routes that all agree.

Anchorv23
a = (1,1,2)
c3=12 e1(a) πc_3 = \frac{1}{2\,e_1(a)\,\pi}
e₁(a) = 4
Geometryv58
one-sided Gauss–Bonnet
c3=1∣Z2∣ 2π χ(S2)c_3 = \frac{1}{|\mathbb{Z}_2|\,2\pi\,\chi(S^2)}
|Z₂|·χ = 2·2
Thermodynamicsv358
entanglement first law
c3=η2π, η=1∣μ4∣c_3 = \frac{\eta}{2\pi},\ \eta = \frac{1}{|\mu_4|}
δS = δ⟨K⟩
triply over-determined
c3=18π⟹c3−1=8πc_3 = \tfrac{1}{8\pi}\quad\Longrightarrow\quad c_3^{-1} = 8\pi
thermo origin 2π/η2\pi/\eta = geo origin ∣Z2∣ 2π χ|\mathbb{Z}_2|\,2\pi\,\chi  iff  |μ₄| = |Z₂|·χ = 4
parameter-free Einstein equation (full covariant)
Gab+Λgab=c3−1 Tab=8π TabG_{ab} + \Lambda g_{ab} = c_3^{-1}\, T_{ab} = 8\pi\, T_{ab}
no free dimensionless Newton dial — G is the one unit v_geo
Honest residual. This closes the full covariant equation parameter-free (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
c3=1∣Z2∣⋅2π χ(S2)=18πc_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
TH=κ2π,12π=4c3T_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
βangle=2π exact  ⇒  Tseam=4c3\beta_{\mathrm{angle}} = 2\pi\ \text{exact} \;\Rightarrow\; T_{\mathrm{seam}} = 4c_3
β = N clock steps (N = 8 and 16)
geometry + anomaly + temperature
c3=18π,TH=c3M=18πMc_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 14π=2c3\tfrac{1}{4\pi} = 2c_3, Nariai TN=4c3ΛT_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).