Paper CXXVI: (v2) · The Seifert Surface of the Fano Valve: Chirality, Orientation, and the Sign of All CP Violation in the Standard Model

The quark–lepton boundary B(1,4) of the One-Octonion Brane-Bulk Framework is a knot in S³. We identify the 7×7 Gram matrix G = −(1+γ)I₇ + γJ₇ (Paper CXXIV) as the Seifert form of this knot, and compute its signature σ(B(1,4)) = −7. The five physical consequences are: (1) δ_CP = +67.79°; (2) SM neutrinos are left-handed; (3) δM > 0 (matter dominates); (4) q_e < 0; (5) all CKM CP asymmetries share a definite sign. In this version (v2) we add the Clauwens (1987) confirmation. Clauwens' Theorem 1 states that for a group G of odd order d, the Arf invariant of the associated quadratic form is Ω(d), where Ω(d) = 1 iff d ≡ ±3 mod 8. Applied to the odd-order subgroup of PSL(2,7) = Aut(Klein quartic): |PSL(2,7)| = 168 = 2³ × 21, odd part = 21 = 3×7, 21 mod 8 = 5 ≡ -3 mod 8 Arf(B(1,4)) = Ω(21) = 1 (Clauwens 1987, Theorem 1) ✓ This is the third independent confirmation of Arf = 1 (after Paper CXXI's spin structure analysis and the Gram matrix determinant |det(2G)| ≈ 21). The number 21 = 3×7 appears simultaneously as (i) the odd part of |PSL(2,7)|, (ii) the Fano aliasing factor F = 3×7 of Paper CXXV, and (iii) |det(2G)| at γ_eff = 0.155. The master unification: QCD confined ⇔ σ(G) = -7 ⇔ Arf(B(1,4)) = 1 ⇔ universe is right-chiral 1–4. Seifert Surface, Gram Matrix, Signature, and Five Chirality Theorems (Contents of §1–4 are unchanged from v1. Summary below.) The Gram Matrix as Seifert Form The Fano valve Σ (Paper CXVIII: inclination θ = 4.054°, twist τ = 67.79°) is the Seifert surface of the B(1,4) knot. The 7×7 Gram matrix G = −(1+γ)I₇ + γJ₇ is its Seifert form. Theorem 1. For γ (0, 1/6): both eigenvalues μ₀ = −1+6γ < 0 and μ₁ = −(1+γ) < 0. Therefore σ(B(1,4)) ∈ = 0 − 7 = −7. G₂ SO(7) Fixes the Chirality ⊂ G₂ is a subgroup of SO(7) (not just O(7)): every G₂ automorphism preserves the volume form on ⁷. ℝ The Seifert surface Σ inherits a definite right-chiral orientation. There is no orientation-reversing G₂ automorphism. Theorem 2. The Fano valve Σ carries a definite right-chiral orientation. σ = −7 < 0. This orientation is an absolute geometric fact. The Five Chirality Theorems Coroll ary Observable Status before CXXVI Status after CXXVI 1 δ_CP = +67.79° (positive sign) Magnitude only Theorem from σ < 0 2 SM neutrinos are left- handed Gauge group assumption Theorem from σ = −7 3 Matter > antimatter (δM > 0) Magnitude free Theorem from σ < 0 4 Electron charge is negative Convention Theorem from Seifert orientation 5 All CKM CP phases share sign Parametric Theorem from... Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.