Foundations from Minimal Change

Research article · Jinku Guo

The Spectrum of a Compact Internal Space II: Effective Couplings and Mass Scales

arXiv SSRN Zenodo Read manuscript PDF Files and citation

Author: Jinku Guo

Work type: Research article

arXiv: 2607.24329 · DOI: 10.48550/arXiv.2607.24329

SSRN: Abstract and download page

Zenodo: 10.5281/zenodo.22067468

Repository copy: Current article directory

Updated: 27 October 2026.

The common spectral calculation selects three fermion generations and links particle masses, gauge couplings and cosmological quantities. The paper reports 41 comparison entries and seven groups of observable tests.

Research overview

Turning generated structure into quantitative relations

Once a compact geometry, a matter carrier and a scalar connector have been constructed, the next question is quantitative. Their interactions must determine normalized couplings, transfer amplitudes and mass scales. This paper develops that calculation across the particle, gravitational and cosmological sectors.

The organizing principle is a common spectral response. The internal geometry and representation content determine matrix elements, kinetic norms and transmission factors. Eliminating their shared parameters produces relations between observables. This creates correlations between sectors that would be independent in a model with a separate adjustable coefficient for each quantity.

Discrete selection and the common endpoint

The calculation first determines which matter and spectral sectors survive the source and screening conditions. Joint source and color screening select three matter copies. Retention of the required shell support and three positive Dirac positions gives an admissible interval for the integer spectral domain. Its unique integer is $R=5$.

The tensor endpoint is determined by a positive response with an exact scaling structure. After the spectral variable is rescaled, its dependence on the dimensionless endpoint $c=kL$ takes the form $A_R/c$. Matching this response to the prescribed endpoint value gives one positive root. The computed center is approximately

$$ kL\simeq 2.4935343. $$

Existence, uniqueness and the numerical enclosure are treated separately from the displayed center. This endpoint enters the subsequent transfer relations repeatedly. The paper thereby links discrete sector selection to a continuous normalization used throughout the calculations.

Mass relations and normalized interactions

Repeated transmission through the compact geometry generates hierarchical mass factors. The paper derives a geometric relation among six fermion masses by eliminating common transfer parameters. It also obtains a shared bandwidth relation between the electron and electroweak scales. Their significance lies in the restricted pattern: changing a common spectral quantity affects several predicted relations at once.

The analysis distinguishes geometric masses, pole masses and running masses. Conversion factors carry the renormalization scale and prescription required for comparison with measured particle data. This makes the relations testable at the correct level and identifies which finite action and two-point calculations determine their physical readout.

Canonical normalization also gives the gauge and scalar interactions used in finite matching. The paper develops the associated family mixing and a one-loop, tadpole-free electroweak matching calculation. Its results include a Born-level top width near 1.53445 GeV. These calculations extend the construction from representation labels to amplitudes and observables with a stated normalization.

Gravitational and cosmological consequences

The same source covariance fixes a nonlinear gravitational flux and its local potential. The resulting Bekenstein-Milgrom-type field problem has a unique static weak solution under the specified boundary data. Coupling this response to pressureless matter gives a smooth pre-crossing evolution and an energy balance. Thus the gravitational sector contains a defined field equation and dynamics in addition to a dimensional scale estimate.

The occupied homogeneous cosmological realization includes a relational matter clock and positive constrained propagation forms on the finite interval under study. A common neutrino normalization cancels from a relation between a vacuum-density fraction and a combination of the cosmic temperature, Planck scale and Hubble rate. This produces a cross-sector test whose structure follows from the shared normalization.

Finite cosmological comparisons are reported with their actual statistical outcome. In the adopted DESI DR2 mapping, thirteen components give a chi-square of about 29.51 and a tail probability near 0.00554. The discrepancy is a useful constraint on that realization because the comparison is made after the common spectral quantities have been fixed. Nuclear and primordial-abundance calculations supply additional checks at different physical scales.

The contribution of the common generating route

The principal results are the selection of three matter copies and the unique integer domain, the unique tensor endpoint, linked mass relations, normalized interaction calculations, and explicit gravitational and cosmological realizations. Their connection is the central advance. A single spectral construction imposes dependencies across quantities normally introduced in separate phenomenological sectors.

The accompanying computational framework records those dependencies and propagates the available numerical enclosures. An interval due to the tensor endpoint, a perturbative matching correction and a discrepancy with reference data each carry a distinct meaning. This separation makes it possible to assess the construction as a predictive system. Its strongest quantitative claims can be traced to a specific spectral response, while comparisons identify which physical readouts are most sensitive to further calculation.

Quantitative results and observational tests

The calculation selects three fermion generations, the unique admissible integer spectral domain R = 5, and a common tensor endpoint kL ≈ 2.49353433. These quantities enter linked calculations of gauge couplings, fermion mass hierarchies, electroweak scales and cosmological observables. Eliminating shared transfer parameters gives a relation among six geometric fermion masses and connects the electron and electroweak scales.

The paper's principal comparison table contains 41 entries, including 37 entries with explicit signed percentage residuals. Selected results are listed below. Residuals are calculated against the reference values adopted in that table; the corresponding mass definitions and approximation orders are retained.

Quantity Calculated value Reference used in the paper Relative residual
Weak coupling g₂ at the emergence scale 0.508848 0.508845 +0.00054%
Strong coupling αₛ(MZ) 0.118041 0.1179 +0.120%
Electroweak scale v 246.190 GeV 246.22 GeV −0.012%
Z-boson mass 91.12484 GeV 91.1876 GeV −0.069%
W-boson mass, leading ρ approximation 80.371745 GeV 80.369 GeV +0.0034%
Higgs mass, tree-level relation 124.98344 GeV 125.20 GeV −0.173%
Atmospheric neutrino mass-squared splitting 0.002511207 eV² 0.002511 eV² +0.0082%
Vacuum-density fraction ΩΛ 0.685044 0.6847 +0.050%
Baryon-density fraction Ωb 0.0494082 0.0493 +0.219%
Baryon-to-photon ratio 6.12823 × 10⁻¹⁰ 6.1 × 10⁻¹⁰ +0.463%

At the emergence scale, the three matched gauge couplings each differ by less than 0.003% from the adopted Standard Model extrapolations. Low-energy comparisons include the strong coupling, electroweak masses, Z widths and hadronic observables. The neutrino mass branch gives 0.0026074, 0.0086914 and 0.0501797 eV, with a total of 0.0614785 eV.

The paper also specifies seven groups of observable tests: a tensor-to-scalar ratio r = 0.0253; normal neutrino ordering; an atmospheric mixing value sin²θ₂₃ = 0.551; a neutrino mass sum 0.0615 eV; a PMNS phase near 8π/7 on the stated mixing ansatz; an unmixed pseudoscalar glueball target of 2.426 GeV; and a low-acceleration scale a₀ = 1.204 × 10⁻¹⁰ m s⁻² with its specified nonlinear response.

The same comparison programme records where the present readouts differ appreciably from their references. These include the solar rest-mass splitting, the neutron lifetime and the adopted DESI DR2 distance realization. For the latter, 13 correlated components give χ² ≈ 29.51 and p ≈ 0.00554. Coupling the successful comparisons and the discrepancies to one generating record makes the cross-sector constraints explicit.

Research subjects and results

Paper keywords

Three fermion generations; Fermion mass hierarchies; Fermion mass relations; Gauge coupling constants; Higgs boson mass; Neutrino masses; Cosmological parameters.

Primary subjects

Three fermion generations; Fermion mass hierarchies; Fermion mass relations; Gauge coupling constants; Higgs boson mass; Neutrino masses; Cosmological parameters; Fermion generations; Standard Model parameters; Electroweak symmetry breaking; Higgs boson; Particle cosmology; One-loop matching; Charged lepton masses; Spectral geometry.

Research subjects

Fermion masses; Quark masses; Lepton masses; Electron mass; Top quark mass; Top quark decay; Higgs mass; Fermi constant; Electroweak scale; Family mixing; Neutrino mass normalization; Vacuum energy; Cosmological expansion; Primordial abundances; Nonlinear gravitational fields.

Methods

Spectral analysis; Canonical normalization; Effective couplings; Renormalization schemes; Finite matching; Perturbative quantum field theory; One-loop calculations; Self-energy calculations; Spectral transfer; Parameter elimination; Nonlinear elliptic equations; Weak solutions; Cosmological parameter estimation; Numerical quadrature; Spectral tail bounds.

Results and research questions

Selection of three fermion generations; Geometric relation among six fermion masses; Correlated electron and electroweak mass scales; Unique transverse-traceless spectral endpoint; Gauge and Higgs matching from a common action; Born-level top quark decay width; Cosmological ratios independent of neutrino mass normalization; Nonlinear gravitational flux from source covariance.

Chinese subject terms

三代费米子; 费米子质量层级; 六费米子质量关系; 规范耦合常数; 标准模型参数; 带电轻子质量; 电弱对称性破缺; 一圈匹配; 希格斯质量; 顶夸克衰变; 粒子宇宙学; 真空能.

Public file

This directory provides the public PDF. LaTeX sources and editable research files remain outside this repository.

Companion article

The Spectrum of a Compact Internal Space I: Gauge Structure and a Scalar Mass Gap gives the source, gauge–matter construction and scalar response used here.

Citation and archival records

Jinku Guo. The Spectrum of a Compact Internal Space II: Effective Couplings and Mass Scales. 2026. arXiv:2607.24329. doi:10.48550/arXiv.2607.24329.

The repository PDF is identified by its Git history and checksum. The public article records identify the work or their respective deposited versions.

Research context

Related research and results · Programme overview · Proofs and calculations