This work explores a computational approach to understanding
force hierarchy through geometric compatibility analysis between scalar field
dynamics and gauge symmetry groups. Building upon the Unified Scalar Resonance
Model (USRM) [Hall & Covington, 2024], which established empirical
constants α=0.211, β=4.73, and γ=4.81 from cosmic scaling laws (R²=0.97, N=40+,
p<10⁻³⁷), we investigate whether group character integral calculations can
predict relative force coupling strengths. Our framework treats the scalar
field as a universal "source" with impedance Z_source = α(1+iβ/γ) =
0.211+0.207i, while force symmetries act as geometric "loads" with
impedances determined by group structure.
Computational Results: We present calculations for four
fundamental interactions with varying degrees of validation: (1) U(1)
Electromagnetic - Geometric derivation yields η_EM = 23.2±2.5%, consistent with
empirical α=21.1% within uncertainties; provides framework self-consistency
check though uses α as input. (2) SU(2) Weak - Triangular mesh calculation on
3-sphere yields η=60.57% (fundamental) and η_adj=58.4% (adjoint), representing
genuine predictions from group topology with hierarchy preserved across
representations. (3) SU(3) Strong - Monte Carlo integration yields η=64.24% (fundamental)
and η_adj=61.9% (adjoint), maintaining Strong>Weak ordering across
representation pairs. (4) ISO(3,1) Poincaré - Lorentz sampling suggests η≈0%
with phase opposition; full diffeomorphism group not calculated, results
provide an estimated mechanism (~70-85% confidence) for gravity's relative
weakness. The observed pattern—Strong (64%) > Weak (61%) > EM (23%)
>> Gravity (~0%)—spans the empirically observed range through what
appears to be a topological compatibility mechanism.
Parameter Robustness: Analysis across α∈[0.19,0.23]×(β/γ)∈[0.93,1.05]
(9 parameter combinations) indicates hierarchy preservation with no ordering
reversals. Adjoint representation calculations yield couplings 2-3 percentage
points weaker than fundamentals while maintaining hierarchy ordering,
suggesting the mechanism may be intrinsic to group topology rather than
representation-dependent. Log-linear analysis reveals an empirical scaling
relationship η_i ∝ α_i^B with B=0.374±0.208 for fundamental representations, though with notably
different behaviour (B_adj=0.046) for adjoint representations, indicating
representation-dependent geometric coupling modes.
Physical Interpretation: The computational results suggest a
possible geometric compatibility mechanism: Strong force (toroidal SU(3) Cartan
torus) may resonate with toroidal scalar field structure, Weak force (helical
SU(2) 3-sphere) exhibits controlled phase mismatch (~80°), electromagnetic
force (spherical U(1)) shows dimensional incompatibility, while gravity
(hyperbolic spacetime) exhibits phase opposition. The categorical distinction
between compact internal symmetries and non-compact spacetime symmetry may
offer insight into the observed hierarchy without requiring fine-tuning, though
complete theoretical derivation remains an open question.
Theoretical Framework: We present the calculations within a
unified field action: S_unified = ∫d⁴x√(-g)[R/16πG + ατ²/16πG + (1/2)∂Φ∂Φ -
V(Φ) + Σ η_i(G_i)·ℒ_i^gauge], where coupling efficiencies η_i(G_i)=1-|Γ_i|² emerge from character integrals. Section 2.11 explores a
possible connection to quantum field theory through the Wilson effective
action, suggesting character integrals might arise naturally from scalar-gauge
field coupling, though a complete first-principles derivation remains to be
established.
Experimental Validation: A noteworthy development emerged
from analysis of T2K neutrino oscillation data. We tested a prediction arising
from a "bulk hypothesis" interpretation (Section 6.5) suggesting
scalar field Φ might represent displacement into higher-dimensional space with
3:1 time differential, predicting energy-dependent baseline modification
L_eff(E) = L × (1 - 0.13 E/2.5 GeV). Analysis of 113M Normal Ordering samples
from T2K MCMC release (arXiv:2506.05889v2) yielded -15.84% deviation at 2.5 GeV
compared to the predicted -13% ± 4.5% (0.62σ agreement), with pattern
validation across 9 energy bins showing crossover at 0.6 GeV and approximately
linear energy scaling as anticipated. A critical entropy correction—separating
Normal and Inverted mass orderings in the MCMC chains—proved essential,
resolving parameter extraction to Δm²₃₂ = 2.4986×10⁻³ eV² (0.3% from published
values). This represents a second independent neutrino-sector test alongside
the CP violation prediction δ_CP = 260° (T2K: 234°±19°, 0.94σ agreement),
though we emphasise these should be regarded as preliminary validations pending
additional experimental confirmation.
Assessment and Limitations: This work shows that
computational group theory can produce force coupling estimates within 2-10% of
target values for gauge forces using three universal constants and group
structure. The electromagnetic calculation provides internal consistency
(though it uses α as input), while SU(2) and SU(3) represent genuine a priori
predictions. Gravity results are estimates based on the ISO (3,1) local
approximation rather than the complete diffeomorphism group calculation.
We emphasise several important caveats: The framework remains
phenomenological, comparable to Fermi's effective theory of weak
interactions—operationally useful but not yet derived from complete first
principles. Section 2.11 suggests a possible connection to QFT through Wilson's
effective action, indicating character integrals might emerge from scalar-gauge
coupling, though rigorous proof awaits further theoretical development. The
power law relationship (η ∝ α^B)
is empirically observed but lacks a complete theoretical derivation. Geometric
correction factors (5-10% adjustments) are phenomenological rather than
first-principles. The representation-dependent exponent difference (B_fund vs
B_adj) may indicate interesting physics regarding field oscillation modes, but
requires careful theoretical investigation.
The neutrino predictions warrant particular caution: while
the agreement between prediction and T2K data is encouraging (both observables
within 1σ), this should be viewed as preliminary pending independent
experimental confirmation from NOvA, DUNE, and other facilities during
2025-2030. The framework makes specific, falsifiable predictions that will be
tested: if DUNE measures δ_CP ≠ 260°±20°, if force hierarchy reverses under
measurement, if SU(2) deviates >15% from 61%, or if gravitational wave-scalar
correlations differ from predicted anti-correlation, the framework would be
ruled out.
Author(s) Details
Derrick Covington
US Department of Veterans Affairs, United States.
Please see the book here :- https://doi.org/10.9734/bpi/mono/978-93-88417-91-4
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