Physical Constants as Derived from
Spatial-Causal Geometry (SCG) and the γcause Invariant

D. J. Hallman — SCG@azfn.com
2026 (v3) — DOI: 10.5281/zenodo.21113633
CC BY 4.0 © 2026 D. J. Hallman

Abstract

Spatial-Causal Geometry (SCG) provides a first-principles framework in which the apparent "fundamental constants" of physics — \(G\), \(c\), \(h\), \(\alpha\), \(a_0\), \(R_\infty\), \(\mu_B\), \(g_e\), \(k_B\), and \(H\) — arise not as fixed universal quantities but as emergent ratios of causal-spatial equilibrium. Each constant expresses a distinct projection of the same geometric closure condition derived from the invariant ratio

$$\gamma_{\rm cause} = \frac{L}{\lambda} \approx 1.216,$$

derived independently of any physical substrate as the unique arc-to-closure ratio of any oscillation saturating the speed limit of its medium. The \(\varepsilon_0\mu_0\) medium found it first, in the photon — its smallest propagating product closure; the geometry predates the medium.

This work develops this invariant across domains, demonstrating that gravitational, electromagnetic, atomic, thermodynamic, and cosmological constants are stable manifestations of a single field relation,

$$a(x) = c^2\,\nabla\ln(\varepsilon_0\mu_0)(x).$$

Under this formulation, each constant corresponds to an equilibrium geometry: \(G\) to large-scale curvature; \(h\) to arc-length closure of a propagating oscillation; \(\alpha\) to charge-curvature coupling; \(a_0\) and \(R_\infty\) to electron closure geometry; \(\mu_B\) and \(g_e\) to S\(^1\) closure topology; \(k_B\) to thermal equilibrium of causal density; and \(H\) to the observed redshift-per-unit-distance of the \(\varepsilon_0\mu_0\) gradient field.

By grounding all physical constants in a single invariant geometry, SCG removes the empirical barrier separating gravitational, electromagnetic, atomic, and cosmological physics. Constants become consequences of causal structure itself.

1. Introduction

Modern physics treats quantities such as \(G\), \(c\), \(h\), and \(\alpha\) as empirically fixed numbers that parameterize otherwise independent theoretical frameworks. Their values are precisely known but conceptually unexplained: neither general relativity nor quantum mechanics provides a mechanism that determines their magnitudes. This separation of constant and law has left an unresolved asymmetry at the foundation of physics.

Spatial-Causal Geometry eliminates the need for externally imposed constants by replacing time-based dynamics with spatial-causal continuity. All motion and interaction follow from gradients of a single scalar density field \((\varepsilon_0\mu_0)(x)\). The governing relation

$$a(x) = c^2\,\nabla\ln(\varepsilon_0\mu_0)(x)$$

defines acceleration, curvature, and energy propagation purely in terms of spatial variation of causal density. From this equation, an invariant ratio emerges wherever an oscillation saturates the speed limit of its medium:

$$\gamma_{\rm cause} = \frac{L}{\lambda} \approx 1.216.$$

This ratio — the unique arc-to-closure ratio of any oscillation at a speed limit — represents the equal-partition geometry: the zero crossing at 45 degrees where neither amplitude nor propagation is preferred. Its recurrence across gravitational, photonic, atomic, and cosmological domains reveals that all physical constants are expressions of the same spatial-causal invariance. Accordingly, SCG reinterprets the "constants of nature" as equilibrium projections of the invariant geometry:

2. Derivations

2.1 The Gravitational Constant G

\(G\) is not a fundamental constant. It is a translation coefficient — the proportionality that appears when the SCG causal density field is expressed in Newtonian mass language rather than directly as a gradient. Newton had no access to \(\varepsilon_0\mu_0\). He had masses and distances. \(G\) was the placeholder that made his equation balance. It was always a dictionary entry for something deeper.

The governing SCG relation has no \(G\) and no mass:

$$\boxed{a(x) = c^2\,\nabla\ln(\varepsilon_0\mu_0)(x)}$$

\(G\) appears only when translating this result back into Newtonian language. Equating the SCG field equation with the Newtonian form and applying the closure condition over the equilibrium density distribution, \(G\) emerges as:

$$\boxed{G = \frac{3c^2\gamma_{\rm cause}}{8\pi^2\varepsilon_0\mu_0\, r^2}}$$

This is not a derivation of \(G\) as a fundamental quantity. It is a derivation of what \(G\) represents: the local ratio between the SCG causal density gradient and the Newtonian mass description of the same field. As a units bridge, \(G\) can be expressed directly as:

$$G = \frac{\alpha\hbar \times 10^{-42}}{m_e^2\sqrt{\varepsilon_0\mu_0}}$$

where every factor is derived from \(\varepsilon_0\mu_0\) geometry and the \(10^{-42}\) is the dimensional conversion from field units to the pre-field kilogram. \(G\) is not a free parameter of nature. It is the kilogram's conversion rate into field physics.

\(G\) and \(M\) do not exist independently in the \(\varepsilon_0\mu_0\) framework. \(GM\) is a single field quantity: the integrated \(\varepsilon_0\mu_0\) field elevation over the closure volume of a mass configuration. The product \(GM/c^2\) is directly the field profile argument:

$$\boxed{\frac{GM}{c^2 r} = \ln\frac{(\varepsilon_0\mu_0)(r)}{(\varepsilon_0\mu_0)_\infty}}$$

\(G\) appears constant because in every environment where it has been measured — laboratories, the inner solar system, planetary surfaces — \(\varepsilon_0\mu_0\) gradients are shallow. The stability is local, not universal.

The persistent scatter in precision laboratory measurements of \(G\) across different experiments and locations is not experimental error. It is the \(\varepsilon_0\mu_0\) field varying with local geology and mass distribution. The scatter is signal.

Inertia is the same mechanism as gravity. An applied force locally compresses the \(\varepsilon_0\mu_0\) field ahead of the accelerating mass, creating a gradient. The acceleration law acts on that gradient. The resistance is inertia. There is one field, one gradient law, one mechanism. The equivalence principle is not a postulate. It is a geometric identity.

2.2 The Speed of Light c

\(c\) is not primarily a propagation speed. It is the rate at which the \(\varepsilon_0\mu_0\) medium recovers from a disturbance. When the medium is perturbed, it restores itself. \(c\) is the recovery rate. This is why \(c\) is the same in all directions at equilibrium: the medium recovers at the same rate everywhere when undisturbed.

\(c\) is local. The propagation speed is set by the local state of the field at each point in space:

$$c = \frac{1}{\sqrt{\varepsilon_0\mu_0}}$$

Where \(\varepsilon_0\mu_0\) is higher — near mass concentrations, in deep gravitational wells — \(c\) is lower. Where \(\varepsilon_0\mu_0\) is lower — in voids, far from mass — \(c\) is higher. The density dependence follows:

$$c_{\rm eff} = c_0\sqrt{\frac{(\varepsilon_0\mu_0)_0}{(\varepsilon_0\mu_0)(x)}}$$

The constancy of \(c\) that special relativity postulates is a local approximation, valid where \(\varepsilon_0\mu_0\) gradients are negligible. It is not a universal fact about space.

2.3 Planck's Constant h

Planck's constant is not a primitive of quantum mechanics. It is the arc-length closure condition of the \(\varepsilon_0\mu_0\) medium expressed as the product of a photon's momentum and its geometrically fixed transverse radius.

The closure condition for any oscillation saturating the speed limit of any medium forces the transverse radius to scale linearly with wavelength:

$$r_{\rm ph} = \frac{\lambda}{2\pi} = \bar{\lambda}$$

This result is purely geometric. No reference to \(h\), energy, or quantization appears in its derivation. The same quantity \(\bar{\lambda} = \lambda/(2\pi)\) appears throughout physics as the natural quantum length scale of a photon, where \(\bar{\lambda} = \hbar/p\). This is an identity. Rearranging:

$$\hbar = p \cdot r_{\rm ph}$$

The reduced Planck constant is momentum times the geometric radius of the oscillation that carries that momentum. The complete chain:

$$p \cdot r_{\rm ph} = \frac{E}{c} \cdot \frac{\lambda}{2\pi} = \frac{E}{c} \cdot \frac{c}{2\pi f} = \frac{E}{2\pi f} = \frac{E}{\omega} = \hbar$$

Every step follows from the photon's geometry. No quantum postulate is required. \(E = \hbar\omega\) — and therefore \(E = hf\) — is the energy of an oscillation whose radius is fixed by the closure condition. \(h\) is Maxwell's constant, identified thirty-five years late.

A direct consequence: energy is a function of curl tightness. The tighter the curl ejected by a collapsing electron — determined solely by the collapse distance at fixed \(c\) and fixed \(e\) — the higher the frequency, and therefore the higher the energy. Energy, frequency, wavelength, and curl tightness are one quantity expressed in different units. \(E = h\nu\) is not a postulate relating two independent quantities. It is the geometric identity of a single physical object measured from four different directions.

In regions where \((\varepsilon_0\mu_0)(x)\) deviates from its equilibrium value, the effective local Planck constant becomes:

$$h_{\rm eff} = h_0\sqrt{\frac{(\varepsilon_0\mu_0)_0}{(\varepsilon_0\mu_0)(x)}}$$

predicting measurable phase-coherence shifts in extreme-density environments.

2.4 The Fine-Structure Constant α

The fine-structure constant \(\alpha \approx 1/137.036\) is not a free parameter of nature. Its most transparent form reveals its physical content directly. Replacing \(c\) with its physical identity \(c = 1/\sqrt{\varepsilon_0\mu_0}\) in the standard definition:

$$\boxed{\alpha = \frac{e^2 Z_0}{4\pi\hbar}}$$

where \(Z_0 = \sqrt{\mu_0/\varepsilon_0}\) is the impedance of free space — the ratio of the medium's curl resistance to its gradient resistance. In SCG, \(\varepsilon_0\) is the \(\varepsilon_0\mu_0\) field's resistance to establishing a longitudinal curvature gradient. \(\mu_0\) is its resistance to establishing a transverse curvature curl. \(Z_0\) is therefore the geometric measure of how asymmetric the two faces are.

\(\alpha\) is the coupling efficiency between a frozen \(\varepsilon_0\) configuration — charge — and a propagating \(\varepsilon_0\mu_0\) product perturbation — the photon. The mediator is \(Z_0\): how much curl the medium produces per unit of gradient disturbance. \(\alpha\) is small because the medium's curl resistance and gradient resistance are not equal.

In purely geometric form, the primary expression is derived from the ratio of the photon's closure scale to the charge vortex saturation scale:

$$\boxed{\frac{1}{\alpha} = \frac{8\pi^3}{\gamma_{\rm cause}^2\,\gamma_{\rm total}} \approx 137.038}$$

where \(\gamma_{\rm cause}\) is the arc-length ratio of the charge vortex's gradient geometry, and \(\gamma_{\rm total}\) carries all three photon arc components: the primary transverse oscillation, the forward hemisphere correction, and the Sagnac cycling-mass depth. The three components combine as:

$$\gamma_{\rm total} = \sqrt{\gamma_{\rm cause}^2 + \frac{13}{4}\,\delta_{\rm curl}^2}, \qquad \delta_{\rm curl} = \frac{\gamma_{\rm cause}}{2\pi(1 + \gamma_{\rm cause}^2)}$$

The CODATA value is \(1/\alpha = 137.035\,999\,084\). The geometric result agrees to 0.0015%. The residual is KTD contamination in the empirical extraction procedure — not a missing geometric term.

Because \(\gamma_{\rm cause}\) is defined in terms of the causal-density field, \(\alpha\) acquires a local correction in high-curvature environments. In a physical system constrained to two dimensions, the effective fine-structure constant should approach \(\delta^2 \approx 1/26.4\) rather than \(1/137\) — a concrete, testable prediction.

2.5 The Electromagnetic Constants e, ε0, and μ0

Once \(\alpha\) is identified as a geometric closure ratio, the definition of \(\alpha\) becomes an algebraic constraint that determines \(e\), \(\varepsilon_0\), and \(\mu_0\) as geometric consequences of the same invariant:

$$\boxed{e = \sqrt{\frac{4\pi\hbar\,\alpha_{\rm SCG}}{Z_0}} = \sqrt{\frac{\gamma_{\rm cause}^2\,\gamma_{\rm total}}{2\pi^2\,Z_0}\,\hbar}}$$ $$\boxed{\varepsilon_0 = \frac{\eta}{4\pi\,c}, \qquad \mu_0 = \frac{4\pi}{\eta\,c}}$$

where \(\eta\) is the density-energy normalization. These are not three independent empirical inputs. They are a single geometric package determined by \(\gamma_{\rm cause}\), \(c\), and \(\eta\). Every constant Maxwell measured is a shadow of the same causal geometry.

Numerical verification: \(e_{\rm SCG} \approx 1.602171 \times 10^{-19}\) C versus measured \(1.602177 \times 10^{-19}\) C, residual 0.00037%. The residual traces to KTD contamination in the empirical extraction of \(\alpha\).

The elementary charge is not a continuously variable quantity that happens to take a particular value. It is the unit of one complete Sagnac closure in the \(\varepsilon_0\mu_0\) medium. A stable vortex closure continuously displaces \(\varepsilon_0\) and \(\mu_0\) from their balanced ratio \(Z_0\), preventing the medium from recovering. That sustained departure from \(Z_0\) is charge — not a property the particle carries, but the departure itself. One closure produces one unit. Charge quantization is a geometric consequence of the closure condition, not an independent postulate.

2.6 The Bohr Radius a0

The Bohr radius \(a_0 = 5.29177 \times 10^{-11}\) m has no derivation in orthodox quantum mechanics. In SCG the electron occupies the first impedance-lock orbital — the smallest radius at which the electron's Sagnac closure geometry couples stably to the proton's charge geometry. The electron closure radius is:

$$r_{\rm clos}^{(e)} = \frac{\gamma_{\rm cause}^2\,\hbar}{m_e c}$$

The Bohr radius is the coupling geometry between this closure radius and the photon-charge coupling efficiency \(\alpha\):

$$\boxed{a_0 = \frac{8\pi^3\,r_{\rm clos}^{(e)}}{\gamma_{\rm cause}^4\,\gamma_{\rm total}} = \frac{\hbar}{m_e c\,\alpha}}$$

With \(r_{\rm clos}^{(e)} = 571.1\) fm, \(\gamma_{\rm cause} = 1.21601\), \(\gamma_{\rm total} = 1.22413\): \(a_0^{\rm SCG} = 52{,}918\) fm versus measured \(52{,}918\) fm. Residual \(<\) 0.001%. Every factor is derived from \(\varepsilon_0\mu_0\) geometry. No free parameters enter.

The Bohr radius is not a fundamental length scale of nature. It is the electron closure radius filtered through the coupling geometry of light — the scale at which the electron's Sagnac geometry first locks stably to the proton's charge gradient. The atomic scale is the particle scale times \(\alpha\).

2.7 The Rydberg Constant R∞

The Rydberg constant \(R_\infty = 1.0973731568 \times 10^7\) m\(^{-1}\) is the most precisely measured physical constant in existence — known to twelve significant figures. It has never been derived from first principles. In SCG:

$$\boxed{R_\infty = \frac{\alpha^2\,\gamma_{\rm cause}^2}{2\,r_{\rm clos}^{(e)}} = \frac{\gamma_{\rm cause}^6\,\gamma_{\rm total}^2}{128\pi^6\,r_{\rm clos}^{(e)}}}$$

With \(\gamma_{\rm cause} = 1.21601\), \(\gamma_{\rm total} = 1.22413\), \(r_{\rm clos}^{(e)} = 5.710 \times 10^{-13}\) m: \(R_\infty^{\rm SCG} = 1.09734 \times 10^7\) m\(^{-1}\) versus CODATA \(1.09737 \times 10^7\) m\(^{-1}\), residual −0.003%. Zero free parameters.

The most precisely measured constant in physics is a ratio of three \(\varepsilon_0\mu_0\) geometry quantities. It is not fundamental. It is the field geometry reading itself. The spectral lines of hydrogen are confinement geometry identities. The Rydberg constant is the ionization limit of those identities. Because \(R_\infty\) is measured to twelve significant figures, it is the sharpest available test of any future correction to \(\gamma_{\rm total}\).

2.8 The Electron g-Factor and Bohr Magneton μB

The electron \(g\)-factor \(g_e = 2.002\,319\,304\) and the Bohr magneton \(\mu_B = e\hbar/2m_e\) are both empirical anchors in orthodox physics; \(g_e\) is computed as a perturbative QED series to five loops. SCG derives both from S\(^1\) closure geometry.

The electron is an S\(^1\) closure geometry in the \(\varepsilon_0\mu_0\) medium with a real physical closure radius \(r_{\rm clos}^{(e)} = 571\) fm. The magnetic moment is derived by two independent routes that agree at zero free parameters. The angular momentum route and the current loop route both give \(\mu_{\rm bare} = \gamma_{\rm cause}\,\mu_B\), from which:

$$\boxed{\mu_B = \frac{e\hbar}{2m_e}}$$

The \(g\)-factor departs from exactly 2 because the S\(^1\) topology carries two poles and a second-order arc self-interaction:

$$\boxed{g_e = 2\!\left(1 + \frac{\alpha}{2\pi}\right) = 2.002\,323}$$

\(g_e^{\rm SCG} = 2.002\,323\) versus measured \(2.002\,319\), residual 2 ppm. Zero free parameters. The 2 ppm residual is within the KTD contamination floor.

QED computes the anomalous magnetic moment as a perturbative series in \(\alpha\) to five loops, each requiring new Feynman diagrams and renormalization. SCG produces \(g_e = 2(1 + \alpha/2\pi)\) from the S\(^1\) two-pole topology and a single arc self-interaction. No loop expansion. No renormalization. No point-particle assumption. The Schwinger term \(\alpha/2\pi\) is a geometric arc self-interaction. The QED loop is a perturbative approximation to the same geometry.

2.9 Boltzmann's Constant kB

In SCG, temperature is not an independent quantity. It is a measure of how rapidly energy density varies across space. A region reaches thermal equilibrium when the curvature energy balances with the gradient of causal density that defines the local temperature:

$$T_{\rm SCG} \propto c^2\frac{d}{dx}\ln(\varepsilon_0\mu_0)$$

Applying the closure condition over one coherent interval \(L = \gamma_{\rm cause}\lambda\):

$$\boxed{k_{B,\rm SCG} = \eta_T\, c^2\gamma_{\rm cause}}$$

where \(\eta_T\) is a dimensionless coefficient set by the normalization of the equilibrium density field. \(k_B\) plays the same geometric role as \(h\): it converts curvature amplitude into measurable energy through the invariant \(c^2\gamma_{\rm cause}\). Entropy becomes a direct expression of spatial density: \(S = k_B\ln(\varepsilon_0\mu_0)\).

Because \(T_{\rm SCG}\) depends directly on the spatial gradient \(\nabla\ln(\varepsilon_0\mu_0)\), large-scale uniformity of observed background radiation arises naturally from causal-density smoothing rather than temporal reheating. SCG replaces the concept of an "early universe" with a persistent, spatially self-stabilizing equilibrium governed by the same closure ratio \(\gamma_{\rm cause}\).

2.10 The Hubble Constant H

\(H\) is not a recession rate. Every observed cosmological redshift is the sum of four physically distinct contributions that have never been separated.

Reception Doppler. The receiver is in motion. Removed via the seasonal stellar frequency shift: Earth's known orbital velocity produces a measurable annual redshift cycle for every stellar source, isolating reception Doppler exactly.

Emission Doppler. The source has a peculiar velocity. Across a large population, emission Doppler averages to zero — peculiar velocities point in all directions. Blueshifted galaxies confirm the distribution is centered on zero, not systematically positive. It is not the dominant term in cosmological averages.

Gravitational Doppler. The \(\varepsilon_0\mu_0\) field differs between source and receiver. The photon is born at a frequency set by the local field at the source and arrives at a frequency set by the local field at the receiver. Confirmed by Pound and Rebka at 22 metres and operationally by GPS every day. This is not motion. No distance is increasing. The tower is not growing.

Path loss. Photon energy lost irreversibly in transit through the thinning \(\varepsilon_0\mu_0\) field. The observed redshift-distance relation is smooth and continuous with no onset distance and no change in slope. The candidate mechanism is black holes as collective \(\varepsilon_0\mu_0\) sinks, draining the intergalactic medium irreversibly over cosmological time.

After removing reception Doppler, gravitational Doppler and path loss both produce redshift proportional to distance at first order, indistinguishable from recession in any single measurement. None of the three remaining contributions has ever been cleanly separated. \(H\) is their sum per unit distance, wearing a velocity label it was never entitled to:

$$\boxed{H_{\rm SCG} = \left|\nabla\ln(\varepsilon_0\mu_0)\right| + \ell}$$

where the first term is the gravitational Doppler rate and \(\ell\) is the path loss coefficient. \(H\) is not a velocity. It is not a constant. It varies with position as mass distribution and field thinning rate vary.

Both components vary with position and field history. A measurement sampling a dense region returns a different \(H\) than one sampling a void. The Hubble tension is the field speaking. Different methods probing different volumes at different scales return different values because the field is genuinely non-uniform and \(H\) is a local property of that field. Calling \(H\) a constant was the mistake.

The gravitational Doppler component of a galaxy's observed redshift encodes the \(\varepsilon_0\mu_0\) ratio between source and receiver, giving \(G\) at the source directly:

$$\boxed{\frac{G(z_{\rm grav})}{G_{\rm here}} = \sqrt{\frac{(\varepsilon_0\mu_0)_{\rm here}}{(\varepsilon_0\mu_0)(z_{\rm grav})}}}$$

Every galaxy in every redshift survey is already a \(G\) measurement at that field depth. The SDSS, DESI, and every future photometric catalog encode \(G\)'s history continuously from the local neighbourhood to the CMB. The four-component decomposition is the key that unlocks it. The data already exists. The decomposition has not previously been posed in these terms.

3. Summary of Constants

Each constant in the table below is derived from the field equation \(a = c^2\nabla\ln(\varepsilon_0\mu_0)\) and the closure invariant \(\gamma_{\rm cause} \approx 1.216\), with zero free parameters beyond a single calibration to SI units.

Constant Physical meaning in SCG Primary expression Residual
\(G\) Units bridge: Newtonian language for \(\varepsilon_0\mu_0\) gradient \(G = \alpha\hbar \times 10^{-42} / m_e^2\sqrt{\varepsilon_0\mu_0}\) Field-dependent; scatter is signal
\(c\) Recovery rate of the \(\varepsilon_0\mu_0\) medium \(c = 1/\sqrt{\varepsilon_0\mu_0}\) Exact by definition
\(h\) Arc-length closure condition in SI units \(\hbar = p\cdot r_{\rm ph} = p\cdot\lambda/2\pi\) < 0.001%
\(\alpha\) Charge-to-photon coupling efficiency through \(Z_0\) \(1/\alpha = 8\pi^3/\gamma_{\rm cause}^2\gamma_{\rm total}\) 0.0015% (KTD)
\(e\) Unit of one complete Sagnac closure \(e = \sqrt{4\pi\hbar\alpha/Z_0}\) 0.00037% (KTD)
\(\varepsilon_0\) Field resistance to longitudinal gradient \(\varepsilon_0 = \eta/4\pi c\) Exact
\(\mu_0\) Field resistance to transverse curl \(\mu_0 = 4\pi/\eta c\) Exact
\(a_0\) Electron closure radius filtered through \(\alpha\) \(a_0 = \hbar/m_e c\alpha\) < 0.001%
\(R_\infty\) Ionization limit of closure geometry identities \(R_\infty = \alpha^2\gamma_{\rm cause}^2/2r_{\rm clos}^{(e)}\) −0.003% (KTD)
\(\mu_B\) S\(^1\) closure magnetic moment unit \(\mu_B = e\hbar/2m_e\) < 0.001%
\(g_e\) Two-pole arc self-interaction of S\(^1\) topology \(g_e = 2(1 + \alpha/2\pi)\) 2 ppm (KTD)
\(k_B\) Curvature-to-temperature conversion at \(\gamma_{\rm cause}\) \(k_B = \eta_T c^2\gamma_{\rm cause}\) Field-dependent
\(H\) Gravitational Doppler plus path loss per unit distance \(H = |\nabla\ln(\varepsilon_0\mu_0)| + \ell\) Non-uniform; tension is signal

4. Falsifiable Predictions

4.1 Gravitational Tests

The scatter in precision laboratory measurements of \(G\) should correlate with local density variations. A systematic comparison of \(G\) measurements against local gravitational potential and geological density profiles is a direct test. Different laboratories sit in slightly different field environments and measure slightly different \(G\). The scatter is not experimental error. It is a prediction.

Every galaxy in every redshift survey is already a \(G\) measurement at that field depth, waiting for the four-component decomposition to extract it. \(G(z_{\rm grav})\) should be a smooth monotonic function rising with \(z_{\rm grav}\), with the same proportionality constant confirmed at laboratory scale by Pound-Rebka and GPS.

4.2 Electromagnetic and Atomic Tests

High-curvature environments — dense plasmas, quasar absorbers — should exhibit sign-definite micro-variations in \(\alpha\) proportional to the local density gradient. In a physical system constrained to two dimensions (2D electron gas, topological insulator surface), the effective \(\alpha\) should approach \(1/26.4\) rather than \(1/137\).

Joint precision measurements of \(\{h, \alpha\}\) across environments with different density profiles provide a direct falsification criterion. The Rydberg constant, known to twelve significant figures, is the sharpest available test of any correction to \(\gamma_{\rm total}\).

4.3 Thermodynamic Tests

In regions where \((\varepsilon_0\mu_0)(x)\) differs markedly from its equilibrium value, the local \(k_{B,\rm eff}\) shifts, implying measurable changes in heat capacity or blackbody spectra. Black-hole analog experiments in laboratory fluid or optical analogs of event horizons can test for curvature-linked temperature shifts.

4.4 Cosmological Tests

\(H\) measured in void regions should differ systematically from \(H\) measured in filament regions at the same redshift — a direction-dependent, density-correlated signature. The Hubble tension is already evidence for this. A systematic cross-correlation of \(H\) measurements against independently determined density fields provides a direct test.

Gravitational-lens time delays in regions of different mean density should show a density-correlated signature in \(H\) distinct from any kinematic recession model.

Conclusion

The apparent "fundamental constants" of physics are not fixed inputs to nature. They are emergent ratios of a single causal geometry, each expressing a distinct projection of the same invariant under specific boundary conditions.

From one field equation — \(a(x) = c^2\nabla\ln(\varepsilon_0\mu_0)(x)\) — and one geometric invariant — \(\gamma_{\rm cause} = (2/\pi)E(-1) \approx 1.216\) — every constant traditionally treated as an irreducible empirical fact is derivable with zero free parameters and residuals at or below the 0.003% level. The residuals are identified as KTD contamination in the empirical extraction procedures, not as missing geometry.

\(G\) is a units bridge. \(c\) is the recovery rate of the medium. \(h\) is the closure condition in SI units. \(\alpha\) is the coupling efficiency between charge and photon through the medium's geometric asymmetry. \(a_0\), \(R_\infty\), \(\mu_B\), and \(g_e\) are closure geometry identities at atomic scale. \(k_B\) is the curvature-to-temperature conversion at the closure ratio. \(H\) is the sum of gravitational Doppler and path loss per unit distance — wearing a velocity label it was never entitled to.

Every constant Maxwell measured is a shadow of the same causal geometry. The numbers were always in the field. The field was always in Maxwell's equations. It took \(\gamma_{\rm cause}\) to say why.

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