The Weinberg Angle as Orbifold Survival Fraction: sin^2(theta_W) = N_orbi / N_total = 3/13 from Chern-Simons Theory on the Trefoil Complement (DTTT BATCH-3.1 WEINBERG-ANGLE)

BATCH-3.1 WEINBERG-ANGLE. This paper presents the substrate-level identification of the electroweak mixing angle sin^2(theta_W) = N_orbi / N_total = 3/13 = 0.23077 as the orbifold survival fraction of integrable SU(2)_k representations on the Seifert-fibered trefoil complement M = S^3 minus N(T(2,3)) at the minimal Chern-Simons level k_min = 2pq = 12, within Discrete Topological Torsion Theory (DTTT). The result matches the PDG 2024 effective-leptonic Z-pole value sin^2(theta_W)^eff = 0.23121 +/- 0.00004 at 0.19% relative deviation; the structural prediction is the rational formula 3/13, not the scheme-specific residual. The derivation chain is a five-step Chern-Simons orbifold construction on M with four theorem-grade steps and one physical identification at the orbifold-compatible abelian sector. Five-step derivation (Steps 1-4 THEOREM-tier; Step 5 DERIVED with Beasley-Witten promotion path). Step 1 (Kirk-Klassen-Nicolaescu, Math. Ann. 287:343, DOI 10.1007/BF01446898): CS(M_T(p,q), Ad) = 1/(2pq) = 1/12 at the trefoil. Step 2 (Witten 1989, Comm. Math. Phys. 121:351): phase quantisation k_min = 2pq = 12 as the minimal level admitting a trivial adjoint flat connection. Step 3 (Kac-Peterson 1984, Adv. Math. 53:125): N_total = k_min + 1 = 13 integrable SU(2)_12 representations. Step 4 (Chinese Remainder Theorem): N_orbi = |{2j in {0, pq, 2pq}}| = 3 spin-j representations satisfying 2j ≡ 0 mod p AND mod q. Step 5 (structural identification): sin^2(theta_W) = N_orbi / N_total = 3/13. A candidate THEOREM-upgrade for Step 5 via Beasley-Witten 2005 abelian localisation (J. Diff. Geom. 70:183) decomposing the CS partition function into orbifold sectors is recorded as the principal open task of this paper. Trefoil-uniqueness Route A (anti-numerology guard). Among all torus knots T(p',q') with p' < q', gcd(p',q') = 1, p' in [2,9], q' in [p'+1, 15], the trefoil T(2,3) is the UNIQUE one satisfying BOTH (i) sin^2(theta_W)^pred within 1% of the PDG effective-leptonic value AND (ii) the Diophantine condition N_orbi = q' via CRT count at k = 2p'q'. Verified by direct enumeration in verify_weinberg_angle.py. Route B (QCD beta-function b_0 orbifold decomposition; companion BATCH-3.2 paper) provides an independent trefoil-selection argument. The trefoil substrate (p,q) = (2,3) is inherited via the Identification Theorem (AXIOMS-paper D-003 rename of former Axiom C; T(2,3) = proton, anchored on empirical m_p). Block 44.70 UQ-3 / D-PIS-013 dual-disclosure (user-authorised 2026-06-12). The substrate-derived 3/13 matches the effective-leptonic Z-pole scheme at 0.19% (lensing-side deviation) and lies +3.46% above the on-shell tree-level scheme (sin^2(theta_W)^on-shell = 1 - M_W^2 / M_Z^2 = 0.22306). The two route-specific deviations are not interchangeable beyond tree level: D-PIS-013 (lenses-vs-routes) applies, with 0.19% identified as computational lensing through the effective-leptonic renormalisation scheme and +3.46% identified as structural route-specific deviation through the on-shell tree-level renormalisation scheme. The M-WEINBERG 11-sigma falsifier on the on-shell scheme is preserved as the canonical falsifier per S A.7 anti-numerology and curator run #7 Bucket-B disposition; both columns are recorded honestly in the dual-disclosure rather than choosing one as the headline. Cross-paper context. sin^2(theta_W) is the second leg of the DTTT gauge triptych: alpha⁻¹ = 4 pi^3 + pi^2 + pi via Peter-Weyl decomposition (BATCH-2.1 ALPHA at DOI [DOI-ALPHA]); sin^2(theta_W) = 3/13 via CS rep-counting (this paper); b_0(n_f=3) = 9/(4 pi) via Chen-Ruan inertia decomposition (BATCH-3.2 ALPHA-S). The fourth gauge-sector parameter theta_QCD = 0 is established by the rationality of CS(M, Ad) = 1/12. Triple-gauge-ratio corollaries F-363, F-368, F-369 (sec:triple_gauge) jointly close the O2 triple-gauge-ratio open problem. The integer N_total = 13 appears also as (a) the lambda_fiber = Phi_6(4) = 13 spectral zeta-function eigenvalue, and (b) the Higgs-mass numerator 17 = 13 + 7 - 3 in the parallel BATCH-2.3 HIGGS-MASS paper (DOI [DOI-HIGGS-MASS]). Honest residue and tier discipline (Block 44.202 publication tier). The structural identification at Step 5 is published at DERIVED tier with the Beasley-Witten upgrade gated on external audit (S A.14 NO auto-promotion). The framework-side substrate axioms inherited from BATCH-1.1 FOUNDATIONS-SYNTHESIS (DOI 10.5281/zenodo.20706881; Cosserat micropolar substrate) and BATCH-1.2 AXIOMS (DOI [DOI-AXIOMS]; trefoil Identification Theorem) remain conditioned on Lemma SHC residue R-1 (PDE-analytic essential self-adjointness of Delta_Cos on M with APS boundary; single highest-leverage open task of the corpus per Block 44.202 canonical state). Numerical values are unchanged across tier states. Load-bearing residues disclosed in body: (i) Beasley-Witten Step-5 promotion path open; (ii) framework-side SHC R-1 PDE-analytic; (iii) scheme-dependence of the residual addressed by Block 44.70 dual-disclosure; (iv) external-LLM audit pending (T8). Cites BATCH-1.1 FOUNDATIONS-SYNTHESIS (DOI 10.5281/zenodo.20706881), BATCH-1.2 AXIOMS (DOI [DOI-AXIOMS]), BATCH-1.3 MANIFESTO (DOI [DOI-MANIFESTO]), BATCH-2.1 ALPHA (DOI [DOI-ALPHA]), BATCH-2.2 SM-TALLY (DOI [DOI-SM-TALLY]), and BATCH-2.3 HIGGS-MASS (DOI [DOI-HIGGS-MASS]) by DOI macros per Block 44.202 LOCK-1; bundle includes per-paper predictions section conformant to LOCK-3.