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Document 1 of the TFPT 5.0 setCompiler core
TFPT 5.02026-06-08928 KBSHA-256 2a5bc1bc…142e
Paper 1Compiler core

Architecture and the E₈ Compiler

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.

Inputs
  • ›P1: the boundary kernel c₃ = 1/(8π) (Gauss–Bonnet hardenable).
  • ›P2: the five-slot carrier g_car = 5 (3 colour + 2 weak); P2 algebra is Lean-formalised.
Contribution
  • ›The glue theorem E₈ = (D₅ ⊕ A₃) + μ₄: common discriminant ℤ₄, glue index |μ₄| = 4, and q(D₅) + q(A₃) = 5/4 + 3/4 = 2 (the E₈ root norm).
  • ›240 = 16·5·3 and 248 = 240 + 8 derived as carrier traces; b₁ = 41/10 and the hypercharge polynomial from the 3+2 split.
  • ›The electromagnetic fixed point α⁻¹ = 137.0359992168… as the unique root of F_U(1)(α) = 0.
Not claimed here
  • ›E₈ is the unimodular audit/compiler hull, not an unbroken physical gauge group; the SM is a readout after projection.
  • ›No dimensionful mass ladder, no full quantum-gravity measure, no cosmology fit.
Falsification surface
  • ›Fails if D₅ and A₃ do not share the ℤ₄ discriminant, if the glue norms do not sum to 2, or if F_U(1)(α) = 0 has no/second admissible root.
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Highlights
E₈ glueD₅ ⊕ A₃ + μ₄Closed lattice construction, not a posited 248
α⁻¹137.0359992Unique root of F_U(1)(α) = 0; 1.9σ from CODATA-2022
q(D₅)+q(A₃)5/4 + 3/4 = 2The even glue condition — the E₈ root norm
rank E₈8 = φ(30)Live phases of the order-30 Coxeter cycle

Key formulas

  • Glue theorem
    E8=(D5⊕A3)+μ4E_8 = (D_5 \oplus A_3) + \mu_4
    disc = ℤ₄, glue index 4, q(D₅)+q(A₃) = 2. [L]
  • Carrier traces
    240=16⋅5⋅3,248=240+8240 = 16\cdot 5\cdot 3, \qquad 248 = 240 + 8
    E₈ numbers as traces over the 3+2 carrier, not inputs. [I]
  • EM fixed point
    FU(1)(α⋆)=0⇒α−1=137.0359992168…F_{U(1)}(\alpha_\star) = 0 \Rightarrow \alpha^{-1} = 137.0359992168\ldots
    Unique root; CODATA-2022 137.035999177(21), dev 2.9×10⁻¹⁰ (1.9σ). [I/N]
  • Abelian coefficient
    10 b1=41=∑f,jLf,j+NΦ10\,b_1 = 41 = \textstyle\sum_{f,j} L_{f,j} + N_\Phi
    b₁ = 41/10 as a carrier trace.

The Pascal compiler on five carrier slots

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.

dim⁡S+=2gcar−1=(gcar0)+(gcar1)+(gcar2)  ⟺  gcar=5\dim S^+ = 2^{g_{\mathrm{car}}-1} = \binom{g_{\mathrm{car}}}{0}+\binom{g_{\mathrm{car}}}{1}+\binom{g_{\mathrm{car}}}{2} \iff g_{\mathrm{car}} = 5
∣R(E8)∣=dim⁡S+(dim⁡S+−1)=16⋅15=240|R(E_8)| = \dim S^+(\dim S^+ - 1) = 16 \cdot 15 = 240

The μ₄ glue: how E₈ is really built

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.

disc⁡(D5)=disc⁡(A3)=Z4,[E8:D5⊕A3]=∣μ4∣=4\operatorname{disc}(D_5) = \operatorname{disc}(A_3) = \mathbb{Z}_4, \qquad [E_8 : D_5 \oplus A_3] = |\mu_4| = 4
q(D5)+q(A3)=54+34=2=∣E8 root∣2q(D_5) + q(A_3) = \tfrac{5}{4} + \tfrac{3}{4} = 2 = |\text{$E_8$ root}|^2

The Z₃₀ = 2·3·5 cyclotomic Coxeter compiler

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.

h=∣Z2∣⋅Nfam⋅gcar=2⋅3⋅5=30h = |\mathbb{Z}_2|\cdot N_{\mathrm{fam}}\cdot g_{\mathrm{car}} = 2\cdot 3\cdot 5 = 30
∣R(E8)∣=rh=240,dim⁡E8=r(h+1)=8⋅31=248,r=φ(30)=8|R(E_8)| = r h = 240, \qquad \dim E_8 = r(h+1) = 8\cdot 31 = 248, \qquad r = \varphi(30) = 8

The electromagnetic fixed point

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.

FU(1)(α)=α3−2c33 α2−45c36(∑f,jLf,j+NΦ)log⁡1φseam(α)=0F_{U(1)}(\alpha) = \alpha^3 - 2c_3^3\,\alpha^2 - \tfrac{4}{5}c_3^6\Big(\textstyle\sum_{f,j}L_{f,j} + N_\Phi\Big)\log\tfrac{1}{\varphi_{\mathrm{seam}}(\alpha)} = 0
α−1=137.035 999 216 8…\alpha^{-1} = 137.035\,999\,216\,8\ldots

The scale grammar: one exponential engine

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.

AEW:AH:AΛ=1:5:10=(50):(51):(52)A_{\mathrm{EW}} : A_H : A_\Lambda = 1 : 5 : 10 = \tbinom{5}{0} : \tbinom{5}{1} : \tbinom{5}{2}
vEW∼e−α−1/5,Λ∼e−2α−1,H0∼Λv_{\mathrm{EW}} \sim e^{-\alpha^{-1}/5}, \qquad \Lambda \sim e^{-2\alpha^{-1}}, \qquad H_0 \sim \sqrt{\Lambda}

Key formulas at a glance

  • Glue theorem
    E8=(D5⊕A3)+μ4E_8 = (D_5 \oplus A_3) + \mu_4

    disc = ℤ₄, glue index 4, q(D₅)+q(A₃) = 2. [L]

  • Carrier traces
    240=16⋅5⋅3,248=240+8240 = 16\cdot 5\cdot 3, \qquad 248 = 240 + 8

    E₈ numbers as traces over the 3+2 carrier, not inputs. [I]

  • EM fixed point
    FU(1)(α⋆)=0⇒α−1=137.0359992168…F_{U(1)}(\alpha_\star) = 0 \Rightarrow \alpha^{-1} = 137.0359992168\ldots

    Unique root; CODATA-2022 137.035999177(21), dev 2.9×10⁻¹⁰ (1.9σ). [I/N]

  • Abelian coefficient
    10 b1=41=∑f,jLf,j+NΦ10\,b_1 = 41 = \textstyle\sum_{f,j} L_{f,j} + N_\Phi

    b₁ = 41/10 as a carrier trace.