The Weinberg Angle as Orbifold Survival Fraction: sin² (thetaW) = Nₒrbi / Nₜotal = 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 sin² (thetaW) = Nₒrbi / Nₜotal = 3/13 = 0. 23077 as the orbifold survival fraction of integrable SU (2) ₖ representations on the Seifert-fibered trefoil complement M = S³ minus N (T (2, 3) ) at the minimal Chern-Simons level kₘin = 2pq = 12, within Discrete Topological Torsion Theory (DTTT). The result matches the PDG 2024 effective-leptonic Z-pole value sin² (thetaW) 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 (MT (p, q), Ad) = 1/ (2pq) = 1/12 at the trefoil. Step 2 (Witten 1989, Comm. Math. Phys. 121: 351): phase quantisation kₘin = 2pq = 12 as the minimal level admitting a trivial adjoint flat connection. Step 3 (Kac-Peterson 1984, Adv. Math. 53: 125): Nₜotal = kₘin + 1 = 13 integrable SU (2) ₁2 representations. Step 4 (Chinese Remainder Theorem): Nₒrbi = |2j in {0, pq, 2pq}| = 3 spin-j representations satisfying 2j ≡ 0 mod p AND mod q. Step 5 (structural identification): sin² (thetaW) = Nₒrbi / Nₜotal = 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) sin² (thetaW) ᵖred within 1% of the PDG effective-leptonic value AND (ii) the Diophantine condition Nₒrbi = q' via CRT count at k = 2p'q'. Verified by direct enumeration in verifyweinbergₐngle. py. Route B (QCD beta-function b₀ 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 mₚ). 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 (sin² (thetaW) ᵒn-shell = 1 - MW² / MZ² = 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. sin² (thetaW) is the second leg of the DTTT gauge triptych: alpha^-1 = 4 pi³ + pi² + pi via Peter-Weyl decomposition (BATCH-2. 1 ALPHA at DOI DOI-ALPHA) ; sin² (thetaW) = 3/13 via CS rep-counting (this paper) ; b₀ (nf=3) = 9/ (4 pi) via Chen-Ruan inertia decomposition (BATCH-3. 2 ALPHA-S). The fourth gauge-sector parameter thetaQCD = 0 is established by the rationality of CS (M, Ad) = 1/12. Triple-gauge-ratio corollaries F-363, F-368, F-369 (sec: triplegauge) jointly close the O2 triple-gauge-ratio open problem. The integer Nₜotal = 13 appears also as (a) the lambdafiber = Phi₆ (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 DeltaCos 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.