Derivation of \( h_{\rm eff} \) Scaling in the C.O.R.E. Framework

David Barbeau, Independent Researcher

david@bigbadaboom.ca | www.bigbadaboom.ca

May 04, 2026

License: arXiv.org perpetual, non-exclusive license 1.0. Non-commercial use (e.g., education, videos) encouraged with attribution to David Barbeau. Commercial use requires permission.

Abstract

The original C.O.R.E. papers contain a systematic algebraic error in the composite frequency scaling step and conflate two distinct physical regimes: (1) internal atomic transitions (defining local clocks and rulers) and (2) external photoelectric thresholds. The erroneous step (“\( \nu = E/h_{\rm eff} \propto 1/\varepsilon^2 \times 1/\varepsilon = 1/\varepsilon \)” ) appears unchanged across CUGE v3 (Appendix A), the material-dependent \( h \) note, REFORM, ZEUS §2, ASH, and every local-\( c \)-invariance claim. The corrected derivation below uses only the stated C.O.R.E. postulates, SI base units, dimensionless refractive index \( n(r) \equiv \sqrt{\varepsilon_r(r)\mu_r(r)} \), and phase continuity. No new parameters are introduced.

Local \( c \)-invariance is restored exactly for internal standards (universal \( h \)). Photoelectric detection yields a material-dependent effective Planck constant \( h_{\rm eff}(r) \propto 1/\varepsilon(r) \), consistent with the Poynting theorem in the impedance-invariant CUGE vacuum. All prior C.O.R.E. results (redshift, half-effect, CMB, Bell correlations, etc.) remain intact.

1. Responsive Vacuum (CUGE)

Mass induces symmetric variations:

\varepsilon(r) = \varepsilon_0 \left(1 + \frac{\Phi(r)}{2c^2}\right), \qquad \mu(r) = \mu_0 \left(1 + \frac{\Phi(r)}{2c^2}\right), \tag{1}

n(r) \approx 1 + \frac{\Phi(r)}{2c^2} \quad (\text{dimensionless}), \tag{2}

where \( \Phi(r) > 0 \) is the gravitational potential magnitude (\( [\Phi] = \rm m^2 s^{-2} \)), \( c = 299\,792\,458~\rm m\,s^{-1} \) (exact, invariant), and impedance \( Z_0 = \sqrt{\mu/\varepsilon} = \sqrt{\mu_0/\varepsilon_0} \) remains globally constant (no reflections).

2. Internal Atomic Transitions (Clocks & Rulers) — Universal \( h \)

Atomic energy levels are screened by the local permittivity:

V \propto \frac{1}{\varepsilon(r)} \implies |E_n| \propto \frac{1}{\varepsilon(r)^2}. \tag{3}

(Units: J, SI-consistent via dielectric scaling.)

Coordinate transition frequency (universal bare \( h \)):

\nu_{\rm coord} = \frac{|E_i - E_f|}{h} \propto \frac{1}{\varepsilon(r)^2}. \tag{4}

Bohr radius (local ruler):

a_0(r) \propto \varepsilon(r). \tag{5}

Coordinate wavelength:

\lambda_{\rm coord} = \frac{c_{\rm coord}}{\nu_{\rm coord}} \propto \frac{c/\varepsilon}{{\rm const}/\varepsilon^2} \propto \varepsilon \, c. \tag{6}

Local measurements (observer uses own atomic standards):

Measured wavelength (in local ruler units):

\lambda_{\rm meas} = \frac{\lambda_{\rm coord}}{a_0(r)} \propto \frac{\varepsilon}{\varepsilon} = \text{constant}. \tag{7}

Local clock rate \( f_{\rm clock} \propto \nu_{\rm coord} \propto 1/\varepsilon^2 \) (clocks run slower in deeper potentials).
Measured frequency (accounting for slower local time):

f_{\rm meas} = \nu_{\rm coord} \times \Bigl( \frac{\rm dt}{\rm d\tau_{\rm local}} \Bigr) \propto \nu_{\rm coord} \times \varepsilon^2 \propto \text{constant}. \tag{8}

Thus the locally measured speed of light is

c_{\rm local} = \lambda_{\rm meas} \cdot f_{\rm meas} = c = 299\,792\,458~\rm m\,s^{-1}, \tag{9}

independent of \( \Phi(r) \) or direction. No effective \( h_{\rm eff} \) is required for internal standards; local \( c \)-invariance follows automatically from dielectric scaling and co-varying rulers/clocks.

3. Photoelectric Thresholds (ASH) — Material-Dependent \( h_{\rm eff} \)

This is a separate external-detection regime. In the CUGE vacuum (constant \( Z_0 \)) the time-averaged intensity is

I = \frac{1}{2} \frac{E_0^2}{Z_0} \implies E_0 = \sqrt{2 I Z_0}, \tag{10}

independent of \( \varepsilon(r) \) (impedance invariance).

Work function scales as

\phi(r) \propto \frac{1}{\varepsilon(r)^2}. \tag{11}

Energy gain per optical cycle for a driven electron:

\Delta E_{\rm cycle} \propto \frac{e^2 E_0^2}{m \omega^2} \propto \frac{E_0^2}{\nu^2}. \tag{12}

At threshold (\( \Delta E_{\rm cycle} \approx \phi \)):

\frac{E_0^2}{\nu_{\rm th}^2} \approx \phi \implies I_{\rm th} \propto \phi \cdot \nu_{\rm th}^2 \propto \frac{1}{\varepsilon^2} \cdot \nu_{\rm th}^2. \tag{13}

The phenomenological threshold relation is

I_{\rm th} \approx \frac{h_{\rm eff} \nu_{\rm th} - \phi}{A_{\rm eff}}, \tag{14}

where \( A_{\rm eff} \) is an effective atomic area (\( \propto a_0^2 \propto \varepsilon^2 \); cancels in leading scaling). Near cutoff the dominant term yields

h_{\rm eff} \approx \frac{\phi}{\nu_{\rm th}} \propto \frac{1}{\varepsilon(r)}. \tag{15}

Thus \( h_{\rm eff}(r) \propto 1/\varepsilon(r) \) for photoelectric detection (opposite the original claim). The power-law detection probability \( P_{\rm det} \propto |\cos(2(\phi - \theta))|^\nu \) (ASH/Bell extensions) is unaffected; only the vacuum-response scaling is corrected.

4. Resolution of the Original Inconsistency

The erroneous Poynting step used the non-CUGE form \( I = \frac12 c \varepsilon E_0^2 \) (valid only for fixed \( \mu = \mu_0 \)), leading to incorrect \( E_0 \propto 1/\sqrt{\varepsilon} \). True CUGE (constant \( Z_0 \)) eliminates this suppression.
Internal transitions (universal \( h \)) and external photoelectric detection were conflated.
The algebraic step was written incorrectly (\( \times \) instead of division).

With the corrections, local \( c \)-invariance is restored from first principles, photoelectric \( h_{\rm eff} \) scales as \( 1/\varepsilon(r) \), and every prior C.O.R.E. result (redshift, half-effect, CMB, Bell correlations, etc.) remains fully intact.

5. Summary of Scalings

Regime	Scaling of \( \varepsilon(r) \)	\( h \)	Measured \( c_{\rm local} \)	Notes
Internal transitions (clocks/rulers)	\( + \)	Universal \( h \)	\( = c \) (exact)	Dielectric + co-scaling
Photoelectric thresholds (ASH)	\( + \)	\( h_{\rm eff} \propto 1/\varepsilon \)	—	External detection only

All derivations satisfy SI base units, dimensionless \( n(r) \), local gradients (m⁻¹) distinguished from integrated phase (dimensionless), and phase continuity. The framework is now internally consistent on this axis.

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.
Barbeau, D. (2025). The ZigZag Eternal Universe System (ZEUS v3). rxiverse:2508.0003.