What question did this study set out to answer?

This research aims to establish a finite arithmetic theorem related to the weak mixing angle in electroweak theory.

May 14, 2026Open Access

The Broken Symmetry is an Integer Fraction: A 3:5:5 Trace Partition, Hypercharge Lexicon, and the 3/13 Electroweak Mixing Skeleton in TWIST-J

Key Points

This research aims to establish a finite arithmetic theorem related to the weak mixing angle in electroweak theory.
Utilized the cyclotomic ring ℤ[ζ₅] from TWIST-J to extract structural information.
Analyzed orbits and partitions using the Klein four-group V₄ and coefficient trace components.
Modeled the mixing ratio and hypercharge parameters through a finite carrier approach in the context of Standard Model dynamics.
Establishes a tree-level weak mixing angle of sin²θ_W(tree) = 3/13.
Shows the adjusted value sin²θ_W(M_Z) = 0.231231186… which is slightly above the standard accepted value.
Confirms the coupling rule gᵢ² ∝ Nᵢ while highlighting areas for further investigation in electroweak completion.

Abstract

Standard electroweak theory leaves the weak mixing angle sin²θW as a continuous parameter fixed by experiment. This paper proves a finite arithmetic theorem in the coefficient shadow 𝔽₅⁴ of the cyclotomic ring ℤζ₅ from TWIST-J and uses it to extract a structural skeleton for hypercharge and the weak mixing ratio. The Klein four-group V₄ = ⟨b, d⟩, generated by piston swap‑and‑negate and by the affine reflection d (ψ) = 2p₀ − ψ with p₀ = (1, 3, 4, 2), has 169 = 13² orbits. These orbits split under the coefficient trace into 39: 65: 65 = 3: 5: 5. Under the decoder dictionary YF = (B − L) + 2T₃ᴿ, with quark baryon number Bquark = 1 / dim ker (Tr) = 1/3, the finite carrier ker (Tr) ⊂ 𝔽₅⁴ gives the color dimension and the dictionary reproduces the full first‑generation Standard Model hypercharge table, including the optional sterile νR and the Higgs doublet. Assigning the trace‑zero sector S₀ to the U (1) Y channel and the paired nonzero‑trace sectors S₊ ∪ S₋ to the SU (2) L channel, and assuming the explicit orbit‑channel coupling rule gᵢ² ∝ Nᵢ, the partition yields g′² / gw² = 3/10 and therefore the tree‑level value sin²θW (tree) = 3 / (3 + 10) = 3/13. Adding the canonical TWIST‑J Ω‑projection X = 1/ (32π²φ⁴) built from the spectral data of the multiplication matrix MJ gives sin²θW (MZ) = 3/13 + 1/ (32π²φ⁴) = 0. 231231186… This lies +1. 12 × 10⁻⁵ above the PDG 2025 MS value ŝ²Z (MZ) = 0. 23122 ± 0. 00006, corresponding to +0. 19 σ or +48 ppm relative. The comparison is made to the MS Z‑pole parameter, not to the effective leptonic weak mixing angle. The paper does not claim a first‑principles derivation of the electroweak action, Higgs‑sector dynamics, electroweak mass generation, or renormalization‑group running. The action‑density origin of the coupling rule gᵢ² ∝ Nᵢ and the scheme map from the TWIST‑J Ω‑projection to the MS renormalization prescription remain explicit open obligations. The result is a finite arithmetic electroweak skeleton: the integer structure is closed; the action‑level completion is the next test.

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