In the C.O.R.E. framework (CUGE + ASH + REFORM), mass induces symmetric variations in vacuum permittivity and permeability while preserving global impedance \(Z_0 = \sqrt{\mu/\varepsilon}\). We prove that this invariance forces the Coulomb coupling parameter \(C(\omega)\) in the hydrodynamic dispersion relation \(k_{\rm eff}^2(r,\omega) = \omega^2[A(\omega) + C(\omega)/r]\) to be exactly frequency-independent across all hydrogenic eigenstates. The derivation uses only the constitutive relations of the responsive vacuum, the definition of local speed of light, and the requirement of observed spectral stability (Rydberg formula accurate to 9+ significant figures). Three corollaries follow: absence of vacuum-boundary reflections, exact local \(c\)-invariance for co-scaled instruments, and perfect hydrogenic degeneracy. The result is falsifiable and closes a key open question in the framework.
The Atomic Statistical Hypothesis (ASH) combined with the Classical Unification of Gravity and Electromagnetism (CUGE) gives the symmetric vacuum response:
where \(\Phi(r)\) is the positive gravitational potential magnitude (SI: m² s⁻²) and \(c = 299\,792\,458\) m s⁻¹ (exact). The vacuum impedance is therefore
with \(f = 1 + \Phi/(2c^2)\). \(Z_0\) is globally constant, independent of position, mass distribution, or frequency. This invariance is a direct consequence of the symmetric scaling and is preserved under all C.O.R.E. transformations.
The local coordinate speed follows immediately:
Even though \(\varepsilon(r)\) and \(\mu(r)\) vary, their product remains fixed, so \(c(r)\) is invariant. Atomic clocks and rulers co-scale with \(\varepsilon(r)\) (ASH), ensuring measured two-way light speed equals \(c\) in every local frame.
From the responsive vacuum (CUGE/REFORM), the time-harmonic wave equation for the effective medium yields the dispersion relation
where \(A(\omega)\) encodes the reactive stop-band from vacuum strain energy \(u_{\rm vac}\) and \(C(\omega)\) is the Coulombic \(1/r\) core.
The Coulomb term arises from the interaction of charge with the vacuum admittance \(Y_0 = 1/Z_0\). The effective contribution is
Substituting the definitions \(Z_0 = \sqrt{\mu_0/\varepsilon_0}\) and \(c_0 = 1/\sqrt{\mu_0\varepsilon_0}\) gives
Because \(Z_0\) is globally invariant (Section 1), \(C(\omega)\) cannot depend on \(\omega\) or position.
Assume, for contradiction, that \(C(\omega)\) varies with frequency. Then the fine-structure-related coupling strength
would also vary. But \(\beta\) is experimentally constant across all hydrogenic eigenstates (Rydberg formula holds to 9+ figures). This contradicts observation. Therefore \(C(\omega)\) is independent of \(\omega\).
Corollary 1: No Reflections at Mass Boundaries
Global impedance invariance \(Z(r) = Z_0\) eliminates any mismatch when electromagnetic waves cross regions of differing mass density. No vacuum-boundary reflections are expected or observed.
Corollary 2: Local \(c\)-Invariance Across Eigenstates
The local sound speed in the hydrodynamic picture is
With \(C(\omega)\) constant and the impedance constraint already guaranteeing \(c_{\rm coord}(r) = c_0\), every eigenstate experiences identical vacuum impedance structure.
Corollary 3: Hydrogenic Spectrum Stability
Energy levels derive from \(\alpha(\omega_n) = -\kappa_n^2\) with \(\kappa_n \propto \beta/(2n)\). Since \(\beta\) (containing \(C(\omega)\)) is constant, the Rydberg formula \(E_n \propto 1/n^2\) follows exactly.
| Quantity | Expression | Depends on \(Z_0\)? | Status |
|---|---|---|---|
| Impedance | \(Z = \sqrt{\mu/\varepsilon} = Z_0\) | No (by definition) | Invariant |
| Local speed of light | \(c(r) = 1/\sqrt{\mu\varepsilon} = c_0\) | No | Invariant |
| Coulomb coupling | \(C(\omega) \propto e^2/(Z_0 c_0 \hbar)\) | Only if \(Z_0\) varies | Constant (proven) |
| Energy levels | \(E_n \propto \beta^2/n^2\) (\(\beta \propto C(\omega)\)) | — | Stable (observed) |
| Condition | Observable Consequence | Current Status |
|---|---|---|
| \(Z(r,\omega)\) varies | Frequency-dependent \(\beta\) or broken degeneracy | Not observed |
| Different \(\beta\) for different \(n\) | Deviations from Rydberg formula | Not observed |
| Impedance mismatch at boundaries | Vacuum-boundary reflections in EM waves | Not observed |
Vacuum impedance invariance, a direct consequence of the symmetric CUGE response, rigorously forces \(C(\omega)\) to be frequency-independent. This supplies the missing first-principles link between the responsive vacuum and the observed stability of the hydrogenic spectrum. The derivation uses only existing C.O.R.E. mechanisms—no new postulates, no free parameters, and full SI-unit consistency with dimensionless refractive index \(n(r)\). The result strengthens the entire framework and provides clean, testable signatures for future precision spectroscopy.
References (selected)
Barbeau, D. (2025). Classical Unification of Gravity and Electromagnetism (CUGE v3). viXra:2507.0112.
Barbeau, D. (2025). Experimental Validation of the Atomic Statistical Hypothesis (ASH). viXra:2507.0123.
Barbeau, D. (2025). REfractive Foundation of Relativity and Mechanics (REFORM v3). rxiverse:2508.0021.
White, H. et al. (2026). Emergent quantization from a dynamic vacuum. Phys. Rev. Research 8, 013264.