What question did this study set out to answer?

This research aims to identify a theoretical origin for the 25 free parameters of the Standard Model using a novel framework.

June 18, 2026Open Access

A Deterministic Number-Theoretic Framework for the Standard Model Parameter Spectrum

Key Points

This research aims to identify a theoretical origin for the 25 free parameters of the Standard Model using a novel framework.
Developed a deterministic arithmetic cascade based on the Universal Generative Principle (UGP).
Utilized Lean 4 for machine-checking the derivation with no errors.
Classified all sectors using a five-status claim taxonomy.
Bare gauge couplings are calculated exact rationals and closely match PDG values.
Nine charged-fermion masses predicted with zero active parameters yield a low error rate.
A neutrino mass-squared ratio is derived parameter-free, matching NuFIT 6.0 results.

Abstract

The Standard Model's 25 free parameters-gauge couplings, fermion masses, mixing angles, and the cosmological constant-have no accepted theoretical origin. We present the Universal Generative Principle (UGP): the Perfect Self-Containment (PSC) foundational principle uniquely selects a single integer seed (1, 73, 823) at ridge level n=10, and a deterministic arithmetic cascade generates the Category-A structural backbone of the SM parameter spectrum. The framework is complementary to QFT: it fixes the numerical constants that a QFT Lagrangian takes as input. The Category-A derivation chain is machine-checked in Lean 4 with zero sorry and the standard Mathlib axiom signature; all sectors are classified by a five-status claim taxonomy (machine-certified, MDL-optimal, partially-derived, calibrated, and exploratory;: status). Headline results: (1) Bare gauge couplings g₁², g₂², g₃² are Lean-certified exact rationals; the blind evaluation ₛ (MZ) = 0. 11822 sits at +0. 24 of PDG 2024, pre-committed 43 days before the first PDG comparison (Appendix app: alphaₛₚrecommit). (2) Nine charged-fermion masses at 0. 293\% RMS with zero active parameters at prediction time; a 1000-permutation null confirms arithmetic structure. (3) Neutrino mass-squared ratio m²₂1/ m²₃1 = 0. 0294 at 0. 16 from NuFIT 6. 0, parameter-free, from Braid Atlas b-values and the structural exponent 29/9. (4) Cosmological constant from the derived model bit-length Lₘodel = ₂ (2000/3) at 0. 31 from Planck 2018 (H₀ is the sole external input). (5) Higgs quartic = phi/ (4) \! (1+IPT-12cH+1) is machine-certified (Category ALean; SpivackGTEUnification, SpivackSRRG) ; with vPSC=246. 16 GeV, this gives mH = 125. 2499 GeV (-0. 0003 PDG 2022), closing the prior -2. 08 tension. The lepton seed bL₁=73 of this triple is doubly certified: the GTE cascade formula G (Nc) =Nc⁴- (Nc²+1) /2-Nc and the Frobenius prime chain F (n) =n⁴-n²+1 independently yield 73 at n=Nc=3, with agreement provably unique at n=3 (fgci\ᵤnique\ₐt\ₙc, UgpLean. Universality. FrobeniusChain, 17 theorems, zero sorry). Preregistered falsification tests for JUNO, DUNE, CMB-S4, and LEGEND-1000 are listed in: falsifiability.

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