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Document 2 of the TFPT 5.0 setCompiler core
TFPT 5.02026-06-08928 KBSHA-256 de0522d8…d9f6
Paper 2Compiler core

The Standard Model from the Compiler

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.

Inputs
  • ›The two axioms and the E₈ compiler of Document 1.
  • ›The seed φ₀ = 1/(6π) + 3/(256π⁴) and the carrier base λ_Y = √(φ₀(1−φ₀)).
Contribution
  • ›One master mass formula for all nine masses, Yukawa, CKM, PMNS and neutrinos, with the word-lengths read off the compiler residue matrix.
  • ›The residue matrix R with det R = 8 = h(D₅), principal 2-minors (2,3,5), and χ_R = t³ − 9t² + 10t − 8.
  • ›The solar angle sin²θ₁₂ = 1/3 − φ₀/2 = 0.3067 from the seam misalignment ε = q(A₃)φ₀.
Not claimed here
  • ›Charged-lepton masses and quark mass ratios are closed; the absolute quark amplitude scale reduces to one overall scale v_geo (Grand Mass Volume + ratios) — the same dimensionful anchor as gravity's 1/G.
  • ›Dimensionful m_W, m_Z, m_H, sin²θ_W, α_s are RG scheme-layer projections, not compiler outputs.
Falsification surface
  • ›Fails if the residue invariants (det 8, minors 2,3,5, χ_R) are not respected by a future global CKM/PMNS fit, or if the lepton φ₀-ladder mismatches the observed hierarchy.
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Highlights
Masses1 formulaAll nine masses + mixings from one φ₀-ladder
det R8 = h(D₅)Flavor matrix determinant is a compiler number
sin²θ₁₂0.30671/3 − φ₀/2; 0.1% from NuFIT 6.0
Quark ratios55/117, 34/47, 3/26Integer Plücker readouts on the selector stratum
Anchor plane(3x+2)(3x+5)det B(K+xQ): 2/3 (Koide/gap) & 5/3 (D₅/A₃) are its singular points [I/L]
Double coverKoide = branch pty² = det B(K+xQ): Koide −2/3 & carrier −5/3 are the two branch points (deck 2 = |ℤ₂|, disc = N_fam⁴) [I/L]

Key formulas

  • Master mass formula
    m^f,j=vgeo2 λY Lf,j Λf,j\hat m_{f,j} = \frac{v_{\mathrm{geo}}}{\sqrt2}\,\lambda_Y^{\,L_{f,j}}\,\Lambda_{f,j}
    One seed φ₀, one carrier base, the compiler residue matrix.
  • Flavor invariants
    det⁡R=8,minors=(2,3,5),χR=t3−9t2+10t−8\det R = 8,\quad \mathrm{minors}=(2,3,5),\quad \chi_R = t^3 - 9t^2 + 10t - 8
    Exact compiler signature any future fit must satisfy. [I]
  • Solar angle
    sin⁡2θ12=13−φ02=0.3067\sin^2\theta_{12} = \tfrac{1}{3} - \tfrac{\varphi_0}{2} = 0.3067
    Previously open; now conditionally derived (seam ε = (3/4)φ₀). [N/P]
  • Lepton product
    cecμcτ=2gcarNfam2=329c_e c_\mu c_\tau = \frac{2^{g_{\mathrm{car}}}}{N_{\mathrm{fam}}^2} = \frac{32}{9}
    Charged-lepton amplitudes closed in φ₀.

One master formula instead of many Yukawas

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.

m^f,j=vgeo2 λY Lf,j Λf,j,λY=φ0(1−φ0)\hat m_{f,j} = \frac{v_{\mathrm{geo}}}{\sqrt2}\,\lambda_Y^{\,L_{f,j}}\,\Lambda_{f,j}, \qquad \lambda_Y = \sqrt{\varphi_0(1-\varphi_0)}
φ0=16π+3256π4=0.05317…\varphi_0 = \frac{1}{6\pi} + \frac{3}{256\pi^4} = 0.05317\ldots

The flavor residue matrix is the compiler signature

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.

R=(130152253),det⁡R=8,∥R∥F2=78R = \begin{pmatrix} 1 & 3 & 0 \\ 1 & 5 & 2 \\ 2 & 5 & 3 \end{pmatrix}, \qquad \det R = 8, \quad \|R\|_F^2 = 78
χR(t)=t3−9⏟Nfam2t2+10⏟(52)t−8⏟h(D5)\chi_R(t) = t^3 - \underbrace{9}_{N_{\mathrm{fam}}^2}t^2 + \underbrace{10}_{\binom{5}{2}}t - \underbrace{8}_{h(D_5)}

Charged leptons: completely closed in φ₀

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.

(m^e,m^μ,m^τ)=vgeoπ2(167(φ0)5, 43(φ0)3, 76(φ0)2)(\hat m_e, \hat m_\mu, \hat m_\tau) = \frac{v_{\mathrm{geo}}\pi}{\sqrt2}\Big(\tfrac{16}{7}(\varphi_0)^5,\ \tfrac{4}{3}(\varphi_0)^3,\ \tfrac{7}{6}(\varphi_0)^2\Big)
ce cμ cτ=2gcarNfam2=329c_e\,c_\mu\,c_\tau = \frac{2^{g_{\mathrm{car}}}}{N_{\mathrm{fam}}^2} = \frac{32}{9}

Quark ratios from the same word-lengths

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.

cucd=gcar⋅11Nfam2 ΔQ=55117,cccs=3447,ctcb=326\frac{c_u}{c_d} = \frac{g_{\mathrm{car}}\cdot 11}{N_{\mathrm{fam}}^2\,\Delta_Q} = \frac{55}{117}, \quad \frac{c_c}{c_s} = \frac{34}{47}, \quad \frac{c_t}{c_b} = \frac{3}{26}
m^t/m^b=326(φ0)−2=40.81\hat m_t/\hat m_b = \tfrac{3}{26}(\varphi_0)^{-2} = 40.81

The solar angle θ₁₂ from the seam

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.

ε=q(A3) φ0=34φ0,q(A3)=Nfam∣μ4∣\varepsilon = q(A_3)\,\varphi_0 = \tfrac{3}{4}\varphi_0, \qquad q(A_3) = \frac{N_{\mathrm{fam}}}{|\mu_4|}
sin⁡2θ12=13−23ε=13−φ02=0.3067\sin^2\theta_{12} = \tfrac{1}{3} - \tfrac{2}{3}\varepsilon = \tfrac{1}{3} - \tfrac{\varphi_0}{2} = 0.3067

Key formulas at a glance

  • Master mass formula
    m^f,j=vgeo2 λY Lf,j Λf,j\hat m_{f,j} = \frac{v_{\mathrm{geo}}}{\sqrt2}\,\lambda_Y^{\,L_{f,j}}\,\Lambda_{f,j}

    One seed φ₀, one carrier base, the compiler residue matrix.

  • Flavor invariants
    det⁡R=8,minors=(2,3,5),χR=t3−9t2+10t−8\det R = 8,\quad \mathrm{minors}=(2,3,5),\quad \chi_R = t^3 - 9t^2 + 10t - 8

    Exact compiler signature any future fit must satisfy. [I]

  • Solar angle
    sin⁡2θ12=13−φ02=0.3067\sin^2\theta_{12} = \tfrac{1}{3} - \tfrac{\varphi_0}{2} = 0.3067

    Previously open; now conditionally derived (seam ε = (3/4)φ₀). [N/P]

  • Lepton product
    cecμcτ=2gcarNfam2=329c_e c_\mu c_\tau = \frac{2^{g_{\mathrm{car}}}}{N_{\mathrm{fam}}^2} = \frac{32}{9}

    Charged-lepton amplitudes closed in φ₀.