Cosmology Without Dark Matter Using
Spatial-Causal Geometry (SCG) and the γ_cause Invariant

D. J. Hallman — SCG@azfn.com
2026 (v2) — DOI: 10.5281/zenodo.19164227
CC BY 4.0 © 2026 D. J. Hallman

Abstract

Space is a physical medium whose local state is completely described by two scalars: the electric permittivity $\varepsilon_0$ and the magnetic permeability $\mu_0$. Their product $\varepsilon_0\mu_0$ is the medium's causal density. Every dynamical phenomenon in the universe — motion, radiation, gravity, structure — follows from gradients of this field.

The governing equation is:

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

It has no time coordinate. Cosmological dynamics are spatial, not temporal.

From this equation, and from the geometric invariant $\gamma_{\rm cause} \approx 1.216$ — the arc-to-closure ratio of any oscillation at the speed limit of any medium — the following follow without additional postulates.

Redshift is a frequency ratio with four physically distinct components. Only one has ever been removed from cosmological measurements. What remains is gravitational Doppler plus path loss — both distance-proportional, neither separated. The Hubble parameter is their sum per unit distance. It is not a recession rate. It is not a constant.

$G$ varies inversely with the square root of $\varepsilon_0\mu_0$. Confirmed at laboratory scale by Pound and Rebka and operationally by GPS. Every galaxy redshift survey already encodes a $G$ history. The decomposition has not been done.

As $\varepsilon_0\mu_0$ rarifies, $G_{\rm eff}$ intensifies. Dark matter is the artifact of assuming the diluent is constant. Rotation curves and gravitational lensing are demonstrated with zero free parameters. The Bullet Cluster separates lensing from baryonic mass because the $\varepsilon_0\mu_0$ field decouples from matter at collision.

Black holes are collective $\varepsilon_0\mu_0$ sinks. They drain the intergalactic medium irreversibly over cosmological time. The field thins. $G_{\rm eff}$ rises globally. Both photons and gravitational waves elongate as they traverse the thinning field — the same mechanism, the same medium. The cosmic microwave background is the photon survival threshold: the distance at which path loss has accumulated enough that only microwave-wavelength closures still reach us intact. No hot origin is required. No cooling is required. No recombination epoch is required.

Without the kinematic time dilation assumption — shown algebraically inconsistent with Special Relativity's own postulates — there is no universal time axis, no scale factor, no backward integration, and no $T_0$. The Big Bang is not a physical event. It is what happens when you run a broken assumption to its conclusion.

1. The Medium

Space is not empty. It is a physical medium with two measurable properties at every point: electric permittivity $\varepsilon_0$ and magnetic permeability $\mu_0$. Maxwell identified both in 1865 and derived from them the propagation speed of electromagnetic disturbances:

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

This is not a definition of $c$. It is a measurement of the medium. $c$ is local: the propagation speed set by the local state of the field at each point in space. 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 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.

The product $\varepsilon_0\mu_0$, treated as a single scalar field, is the load-bearing primitive of Spatial-Causal Geometry. All dynamics follow from its gradients. The governing acceleration law is the barotropic Euler equation for the $\varepsilon_0\mu_0$ medium:

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

Acceleration at any point in space is the logarithmic gradient of the causal density field, scaled by $c^2$. No mass term appears. No $G$ appears. No time coordinate appears.

This equation is not a proposal. It is the barotropic Euler equation applied to the $\varepsilon_0\mu_0$ scalar field, with $c^2 \equiv dp/d\rho$ fixed by the medium's own definition of propagation speed. $G$ enters only when translating this result into Newtonian language. It is a units bridge: the conversion factor between the $\varepsilon_0\mu_0$ field description and the mass-based description Newton had available. It is not a fundamental constant of nature. It varies with the field.

2. Light in the Medium

A photon is the $\varepsilon_0\mu_0$ medium's smallest propagating product closure. It is a transverse oscillation of the field that propagates at the local $c = 1/\sqrt{\varepsilon_0\mu_0}$ and carries no charge, no rest mass in the conventional sense, and no ratio perturbation of the field. Both $\varepsilon_0$ and $\mu_0$ are displaced in the same sense throughout the photon's cycle — it is a product perturbation, not a ratio perturbation. In free propagation the photon is purely product.

The photon's geometry is fixed by a single invariant. Any oscillation propagating at the speed limit of any medium satisfies a self-referential closure condition: its transverse amplitude equals its own radian length scale, $A = \lambda/2\pi$. This forces the arc-to-closure ratio to a unique value:

$$\boxed{\gamma_{\rm cause} = \frac{L}{\lambda} = \frac{2}{\pi}\,E(-1) \approx 1.2160}$$

where $E(-1)$ is the complete elliptic integral of the second kind evaluated at $m = -1$. Three independent arguments — causal arc-length equality, least action with no external parameter, and speed-limit energy partition at the 45-degree zero crossing — all demand this condition.

$\gamma_{\rm cause}$ is substrate-independent. It is not a property of the $\varepsilon_0\mu_0$ medium. It is a property of the geometry of bounded oscillatory closure at any speed limit. The $\varepsilon_0\mu_0$ medium found it first, in the photon. The geometry predates the medium.

From the closure condition, 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 the photon's momentum and its geometrically fixed transverse radius:

$$\hbar = p \cdot r_{\rm ph} = p \cdot \frac{\lambda}{2\pi}$$

$E = h\nu$ is not a postulate. It is the energy of an oscillation whose radius is fixed by the closure condition. The photon carries three properties from emission to absorption, unchanged in free propagation: frequency, polarity axis, and transverse radius $r_{\rm ph} = \lambda/2\pi$. It accumulates nothing in transit. The medium carries the history; the photon is the ruler.

3. Redshift

Every observed astronomical redshift is a frequency ratio. The photon left the source at one frequency and arrived at the receiver at another. That ratio has four physically distinct causes, which have never been separated in any cosmological measurement.

3.1 The Four Components

Reception Doppler

The receiver is in motion. A moving receiver encounters photon oscillations at a rate different from their rate in the medium — higher if approaching, lower if receding. This is a property of the receiver's motion, not of the photon or the source. It is removed via the seasonal stellar frequency shift: Earth's known orbital velocity produces a measurable annual redshift cycle for every stellar source. The seasonal signature isolates reception Doppler exactly and leaves the remaining three contributions as a residual.

Emission Doppler

The source has a peculiar velocity. A source moving along the line of sight emits photons into the medium at a frequency shifted by its motion — lower if receding, higher if approaching. Across a large population of sources, emission Doppler averages to zero — peculiar velocities point in all directions with no systematic preference. The Andromeda galaxy is blueshifted — it is approaching us — confirming that the distribution of peculiar velocities across the local group is centered on zero, not systematically positive. For cosmological averages, emission Doppler is not the dominant term.

Gravitational Doppler

The $\varepsilon_0\mu_0$ field differs between source and receiver. The photon is born at a frequency set by the local $\varepsilon_0\mu_0$ at the source. It arrives at a frequency set by the local $\varepsilon_0\mu_0$ at the receiver. The ratio of the two propagation speeds is the gravitational frequency shift, 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

The observed redshift-distance relation is smooth and continuous from the nearest galaxies to the maximum observed distances, with no onset distance and no change in slope. Whatever mechanism produces path loss has been operating at all scales continuously.

The key distinction from gravitational Doppler is stationarity. A static $\varepsilon_0\mu_0$ density difference between source and receiver produces reversible gravitational Doppler: a photon going one way loses energy, a photon going the other way gains it back exactly. Pound and Rebka confirmed this. Path loss is irreversible. The energy is transferred into the field and does not return to the photon. The candidate mechanism — black holes as collective $\varepsilon_0\mu_0$ sinks — is developed in Section 6.

3.2 What H Actually Is

After removing reception Doppler, orthodoxy attributes the remaining redshift entirely to emission Doppler interpreted as recession velocity. But 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 from the others. $H$ is their sum per unit distance, wearing a velocity label it was never entitled to.

In SCG, the combined redshift per unit distance is:

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

where the first term is the gravitational Doppler rate from the local $\varepsilon_0\mu_0$ gradient and $\ell$ is the path loss coefficient — the fractional energy lost per unit distance to the global $\varepsilon_0\mu_0$ thinning. Neither term has been isolated in any cosmological measurement. $H$ is their sum.

$H$ is not a velocity. It is not a constant. It is a field property that varies with position as mass distribution and field thinning rate vary.

3.3 The Hubble Tension

Both components of $H$ vary with position and field history. The $\varepsilon_0\mu_0$ gradient varies because mass distribution is not uniform. The path loss coefficient varies because the black hole distribution is not uniform — filaments and clusters have more black holes draining the intergalactic medium than voids do.

A measurement sampling a dense region returns a different $H$ than one sampling a void. Different methods probing different volumes at different scales return different values — not because of experimental error or missing physics, but because the field is genuinely non-uniform and $H$ is a local property of that field.

The Hubble tension is the field speaking. Early-universe methods sample one region and scale. Late-universe methods sample another. The field is not obligated to be uniform. Calling $H$ a constant was the mistake.

4. G Varies with the Field

$G$ is not a fundamental constant of nature. It is a units bridge: the translation coefficient that appears when the $\varepsilon_0\mu_0$ field equation is expressed in Newtonian mass language. 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.

From the field equation and the identification of mass as the integrated $\varepsilon_0\mu_0$ field over a closure volume, $G$ emerges as a local ratio:

$$G \propto \frac{1}{\sqrt{\varepsilon_0\mu_0}}$$

Where the field is denser, $G$ is smaller. Where the field is thinner, $G$ is larger. $G$ is not a universal constant. It is locally stable in the environments where it has always been measured — laboratories, the inner solar system, planetary surfaces — because $\varepsilon_0\mu_0$ gradients are shallow there. The stability is local, not universal.

4.1 Laboratory and Operational Confirmation

Pound and Rebka confirmed the $G \propto 1/\sqrt{\varepsilon_0\mu_0}$ relationship at 22 metres. The top of the tower has lower $\varepsilon_0\mu_0$ density than the bottom. Therefore $G_{\rm eff}$ is higher at the top. Clocks run faster at the top by exactly the ratio that $\varepsilon_0\mu_0$ predicts. The tower is not growing. The field is thinner at the top.

GPS confirms this operationally every day. The satellite clocks are corrected for the $\varepsilon_0\mu_0$ field difference between orbital altitude and the Earth's surface. The correction is not a relativistic curiosity. It is an engineering requirement. Remove it and GPS fails within minutes.

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. Different laboratories sit in slightly different field environments and measure slightly different values of $G$. The scatter is signal.

4.2 G as a Function of Field Depth

The gravitational Doppler component of a galaxy's observed redshift encodes the $\varepsilon_0\mu_0$ ratio between the source environment and the receiver. That ratio gives $G$ at the source directly:

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

where $z_{\rm grav}$ is specifically the gravitational Doppler component of the observed redshift — not the total observed redshift, which also contains emission Doppler and path loss.

Every galaxy in every redshift survey is already a $G$ measurement at that field depth. The Sloan Digital Sky Survey, DESI, and every future photometric catalog encode $G$'s history continuously from the local neighbourhood to the CMB. The prediction is falsifiable: $G(z_{\rm grav})$ should be a smooth monotonic function rising with $z_{\rm grav}$, with the same proportionality constant confirmed at laboratory scale. Every redshift survey is already a $G$ history. The decomposition has not previously been posed in these terms.

5. The Field Thins: Dark Matter Dissolves

As $\varepsilon_0\mu_0$ rarifies, $G_{\rm eff}$ intensifies. This is not a separate postulate. It follows directly from $G \propto 1/\sqrt{\varepsilon_0\mu_0}$. Where the field is thinner, gravitational coupling is stronger.

Every phenomenon that orthodoxy attributes to dark matter is a measurement of this effect in a context where the $\varepsilon_0\mu_0$ gradient was not accounted for. Dark matter is not a particle. It is the artifact of assuming the diluent is constant. The following five demonstrations each resolve to the same field.

5.1 Rotation Curves

The outer disk of a galaxy sits in lower $\varepsilon_0\mu_0$ density than the inner disk. Lower $\varepsilon_0\mu_0$ means higher $G_{\rm eff}$. Higher $G_{\rm eff}$ means stronger gravitational coupling for the same baryonic mass. The orbital velocities in the outer disk do not fall off as Newtonian gravity from the visible mass distribution predicts — not because there is additional mass, but because $G_{\rm eff}$ is higher there than the inner-disk value assumed in the Newtonian calculation.

The $\varepsilon_0\mu_0$ field naturally segments into discrete causal domains with boundaries set by the $\gamma_{\rm cause}$ spacing law:

$$\Delta r_i = \gamma_{\rm cause}\sqrt{r_i}$$

where $r_i$ is the inner radius of domain $i$ and $\Delta r_i$ is its radial width. $\gamma_{\rm cause}$ partitions the galaxy into a sequence of nested shells, each with its own gravitational coupling strength. The rotation curve is a readout of that sequence — each flat segment is one domain, each transition is a $\gamma_{\rm cause}$-spaced boundary. The flat outer curve is the asymptotic behavior of the outer domain sequence: a consequence of the $\sqrt{r}$ spacing law producing progressively wider domains at large radius.

This prediction was tested against 175 galaxies from the SPARC catalog with zero free parameters. The $\gamma_{\rm cause}$ domain-spacing rule predicts kinematic transition locations with a median RMSD of 1.06 km/s before any velocity data is consulted. No dark matter halo was fitted. No NFW profile. No Burkert profile. No isothermal sphere. The geometry predicts the transitions. The transitions are where the data says they are.

5.2 Gravitational Lensing

Light follows gradients of $\varepsilon_0\mu_0$. Where the field curves, the photon path curves. The deflection angle for a photon passing a mass concentration is enhanced by $\gamma_{\rm cause}$ relative to the GR prediction from baryonic mass alone:

$$\theta_{\rm SCG} = \gamma_{\rm cause} \cdot \theta_{\rm GR}$$

because the photon's causal arc length per wavelength exceeds its forward propagation distance by exactly $\gamma_{\rm cause}$. GR's temporal dimension absorbs a share of the total curvature budget that belongs to the spatial field. The $\gamma_{\rm cause}$ factor recovers it.

This prediction was tested against 186 strong gravitational lenses from the SLACS and CASTLES catalogs with zero free parameters. The $\gamma_{\rm cause}$ scaling removes the systematic 16–31% deficit of unscaled GR without introducing any additional mass component. One number, two completely independent datasets, zero parameters.

5.3 The Bullet Cluster

The Bullet Cluster is two galaxy clusters that have passed through each other. The baryonic matter — gas and plasma — is electromagnetically coupled. It collides, heats, decelerates, and piles up between the two clusters. Its position is visible in X-ray.

The $\varepsilon_0\mu_0$ field is not electromagnetically coupled. It has no electromagnetic cross-section. It does not attend the collision. It continues on the original trajectory of each cluster.

Lensing follows the $\varepsilon_0\mu_0$ field. X-ray follows the baryons. After the collision, the lensing centroid and the baryonic mass centroid are in different locations — not because there is a dark matter halo that passed through itself, but because the field and the baryons respond to the collision differently. The field ignores it. The baryons do not.

Throwing litter from a car window and then having an accident does not affect the litter. The litter has its own trajectory the moment it leaves your hand. The accident is a baryon event.

The lensing centroid offset should decrease over time as the $\varepsilon_0\mu_0$ field relaxes toward equilibrium with the new baryon distribution, on a timescale set by the light-crossing time of the system — approximately 10 million years for the Bullet Cluster. The archival Hubble and Chandra data already contains this prediction.

5.4 CMB Peak Structure

The cosmic microwave background angular power spectrum shows a harmonic sequence of peaks. In the standard model these are acoustic oscillations of the photon-baryon plasma in the early universe, and dark matter is required to supply additional gravitational depth. In the $\varepsilon_0\mu_0$ framework the CMB is not a thermal relic. Its angular structure reflects the interference geometry of $\gamma_{\rm cause}$-spaced causal shells at the coherence horizon, not plasma oscillations. The peak sequence arises from the discrete nature of causal shell spacing. The odd/even asymmetry emerges from the parity of the shell interference function. No additional gravitational source is required.

5.5 Large-Scale Structure

The cosmic web of filaments, voids, and clusters is the interference pattern of $\varepsilon_0\mu_0$ causal shells. Where shells intersect constructively, their gradients reinforce — these are the filaments and clusters. Where interference is destructive, the field relaxes and voids form. Structure does not grow in time through gravitational instability seeded by dark matter fluctuations. It unfolds spatially as the $\varepsilon_0\mu_0$ field projects its coherence outward through successive shell intersections.

The quasi-periodic filament spacing near 150 Mpc follows from the average causal shell spacing at cosmological scales — the same $\Delta r_i = \gamma_{\rm cause}\sqrt{r_i}$ law that governs galactic domain boundaries, extended to the largest scales. One law operating across fifteen orders of magnitude in distance.

6. Black Holes as ε₀μ₀ Sinks

6.1 The Sink Picture

A black hole is a region where the $\varepsilon_0\mu_0$ field has been compressed to the point where $c_{\rm local} = 1/\sqrt{\varepsilon_0\mu_0}$ approaches zero. At the event horizon, no closure geometry can form that reaches the outside. Nothing escapes not because of infinite curvature but because the local propagation speed has dropped to the point where outward propagation is geometrically impossible.

The field inside the event horizon is not returned to the intergalactic medium. Each black hole permanently removes a quantity of $\varepsilon_0\mu_0$ from the surrounding field and sequesters it. Over cosmological time — as black holes form, grow by accretion, and merge — they collectively and irreversibly drain $\varepsilon_0\mu_0$ from the intergalactic medium. The field thins. $G_{\rm eff}$ rises smoothly and globally.

This is the candidate mechanism for path loss. It is stated as a candidate. The derivation from first principles has not been completed. What is established is the consequence: if black holes collectively thin the $\varepsilon_0\mu_0$ medium, then photons and gravitational waves traversing the thinning field arrive with stretched closures — lower frequency, longer wavelength.

6.2 Field Thinning Is Asymmetric

The $\varepsilon_0\mu_0$ thinning produced by black holes is not spatially uniform. Filaments and clusters concentrate black holes. Voids have almost none. The consequence is that $G_{\rm eff}$ is highest in voids — where the field has been drained least — and lowest in dense regions. This is the inverse of the naive expectation, and it is the key to understanding why rotation curve anomalies are strongest in the outer disks of galaxies: those outer disks extend into lower $\varepsilon_0\mu_0$ environments where the intergalactic field has been thinned by the surrounding large-scale structure.

A prediction follows directly: void galaxies — galaxies sitting in the most underdense large-scale environments — should show the maximum rotation curve anomaly, because they sit in the most rarified $\varepsilon_0\mu_0$ environment and therefore experience the highest $G_{\rm eff}$. This is testable against existing survey data.

6.3 No Singularity

The singularity at the center of a black hole is an artifact of running the GR field equations past their own domain of validity into a regime they were never designed to describe.

In the $\varepsilon_0\mu_0$ picture, the interior of a black hole is not a breakdown of physics. It is a geometric saturation. As the $\varepsilon_0\mu_0$ density increases toward the center, the Planck length grows — the quantum floor rises in deep wells. At the point where the Planck length equals the horizon scale, the quantum floor and the classical ceiling meet. This is not infinite density. It is the wave equation meeting itself at its own scale.

The apparent singularity is the geometry meeting itself. It is not a catastrophe. It is a boundary condition.

7. Gravitational Waves

Gravitational waves are disturbances in the $\varepsilon_0\mu_0$ field. They propagate at $c_{\rm local} = 1/\sqrt{\varepsilon_0\mu_0}$ by definition — they are disturbances in the same field that defines $c$. The LIGO detection of gravitational waves traveling at $c$ was an accidental confirmation that gravity and electromagnetism share the same medium. Both photons and gravitational waves propagate through the same medium at the same speed because they are both disturbances in the same field.

7.1 Wavelength Extension in a Thinning Medium

A water wave loses amplitude as it travels. Gravity continuously acts on the water surface, flattening the peaks. The wavelength grows. The amplitude drops. The total energy deposited by the pebble is still in the water — spread over a larger geometry at lower concentration per unit length. Eventually the amplitude falls below the threshold where surface tension can maintain a coherent wave structure. The wave does not vanish. It dilutes below the coherence threshold. The ocean is very slightly warmer.

A gravitational wave traversing a thinning $\varepsilon_0\mu_0$ medium behaves the same way. The merger event deposits a fixed energy into the field as a propagating disturbance. As the disturbance travels through a medium whose $\varepsilon_0\mu_0$ is progressively lower — because black holes have been collectively draining it — the closure geometry that fit the denser medium now fits the thinner medium at a longer wavelength. The wave elongates.

This is not amplitude reduction. It is wavelength extension — the gravitational analog of cosmological redshift. The same field thinning that stretches photon closures stretches gravitational wave closures. One mechanism, both consequences.

The chirp from a compact binary merger at cosmological distance arrives with its wavelength extended by the accumulated field thinning along the line of sight. This is a different observational signature from any kinematic or expansion-based prediction, and it is already present in the LIGO catalog waiting for the right analysis.

7.2 The NANOGrav Background

The NANOGrav collaboration has detected a stochastic gravitational wave background — a persistent low-frequency signal consistent with gravitational waves arriving from all directions. In the $\varepsilon_0\mu_0$ picture, this is the accumulated history of merger events across cosmological distances, all arriving through a thinning medium. Each event contributes a gravitational wave disturbance whose wavelength has been extended by the field thinning along its path. The superposition of these extended, slightly-shifted signals from all directions produces the stochastic background.

The two backgrounds — microwave and gravitational wave — are the same phenomenon in two different field modes. Both represent the far end of their respective propagation curves through a thinning $\varepsilon_0\mu_0$ field.

8. The Cosmic Microwave Background

The cosmic microwave background is the far end of the redshift curve. Path loss accumulates continuously with distance through a thinning $\varepsilon_0\mu_0$ field. There is no onset distance and no change in slope across the full observed range. The same mechanism that redshifts a nearby galaxy redshifts a source at any distance.

8.1 The Blackbody Spectrum

The CMB is not a destination for all light. It is a filter. At the maximum thinning depth, microwave wavelengths are what the field still permits to propagate and arrive as detectable photons. Photons whose closures were stretched below the microwave threshold were absorbed into the field before reaching us — their energy deposited into the medium, not into the arriving photon distribution. Photons at shorter wavelengths came from closer in, where less thinning had accumulated. Microwave is what gets through at that depth. Not because all light becomes microwave — but because microwave is the survival threshold at maximum thinning distance.

The observed 2.725 K is the name we give the blackbody curve that fits the surviving photon flux density spectrum at that threshold. It is not a temperature in the thermodynamic sense. It is a frequency measurement — the characteristic closure geometry of photons at the edge of detectability through a fully thinned field — expressed in temperature units because the blackbody formalism fits the shape. No primordial plasma. No relic cooling. The field has been thinning. 2.725 K is where the surviving photons run out.

8.2 Angular Anisotropies

The small temperature fluctuations across the CMB sky reflect the spatial structure of the $\varepsilon_0\mu_0$ field at the maximum thinning depth. The $\gamma_{\rm cause}$-spaced causal shell structure of the field produces preferred spatial frequencies that project onto preferred angular scales. The harmonic peak sequence in the angular power spectrum is the interference pattern of these shells — not acoustic oscillations of a primordial plasma.

The isotropy of the CMB on large scales follows directly from the filter picture. Path loss accumulates with distance in all directions from any observer. The maximum thinning depth is approximately the same distance in every direction, because the black hole distribution is approximately isotropic on the largest scales. The filter operates at the same depth everywhere we look. Of course it appears isotropic.

8.3 Every Observer Has Their Own CMB

The CMB is not a special shell around a special origin. It is the maximum thinning depth as seen from a particular location in the field.

An observer sufficiently distant from our galaxy sees our local group, our filament, our Sun — all of it — as microwave sources. From that distance, our galaxy is part of their CMB. Every observer in the field is surrounded by their own microwave background, defined by their own maximum thinning depth in every direction.

The universe has no center and no edge because every point is the center of its own maximum thinning depth. The CMB's apparent isotropy and universality are not evidence of a special origin event. They are the inevitable geometry of a path-integrated effect observed from inside the field that produces it.

8.4 The JWST Confirmation

JWST observations of massive, morphologically mature galaxies at high redshift present a direct problem for the expansion model: its lookback time interpretation requires those galaxies to have assembled within a few hundred million years of the Big Bang. They are too structurally complete, too massive, too old for the time available under hierarchical formation.

In the $\varepsilon_0\mu_0$ picture there is no age constraint from redshift. A high redshift means the light traversed a great distance through a thinning field and lost energy in proportion to that distance and thinning. The galaxy is as old as it is. JWST's "impossible" galaxies are not impossible. They are old. The expansion model was reading distance as time.

9. No T₀ Without KTD

The Big Bang is not derived from observation. It is derived from an assumption.

The assumption is kinematic time dilation (KTD): the postulate that a moving clock runs slow by the factor $\gamma = 1/\sqrt{1-v^2/c^2}$, independently of any field interaction. This assumption has been shown to be algebraically inconsistent with Special Relativity's own second postulate. The derivation of the Lorentz factor from SR's postulates produces an equation in which both sides describe the same physical situation — an algebraic contradiction, not a derivation. KTD was never established. It was assumed and propagated.

The chain from KTD to the Big Bang is direct:

$$\text{KTD} \;\to\; \text{universal time axis} \;\to\; a(t) \;\to\; H(t) \;\to\; T_0 \;\to\; \text{Big Bang}$$

KTD establishes a universal time axis — a cosmic time that runs the same way for all observers regardless of field environment. Without KTD there is no universal time axis. The field equation has no time coordinate and requires none.

A universal time axis is required to define the scale factor $a(t)$. Without a universal time axis there is no scale factor. Without a scale factor there is no time-dependent Hubble parameter $H(t) = \dot{a}/a$. $H$ is a local field property. It is not the time derivative of anything. Without $H(t)$ there is no backward integration to a $T_0$. Without $T_0$ there is no Big Bang.

The first link is broken. The chain does not connect.

10. Observational Program

The following predictions follow directly from the $\varepsilon_0\mu_0$ field equation and are falsifiable with existing or near-term instruments. None requires free parameters beyond the confirmed Pound-Rebka proportionality constant and $\gamma_{\rm cause}$.

10.1 G from the Four-Component Decomposition

Every galaxy redshift survey encodes a $G$ history. The program: apply the seasonal frequency shift pipeline to remove reception Doppler; confirm emission Doppler averages to zero; isolate the gravitational Doppler component; apply $G(z_{\rm grav})/G_{\rm here} = \sqrt{(\varepsilon_0\mu_0)_{\rm here}/(\varepsilon_0\mu_0)(z_{\rm grav})}$ anchored by Pound-Rebka. The result is $G$ as a function of field depth, zero free parameters. The prediction: a smooth monotonic function rising with $z_{\rm grav}$, continuous from the local neighbourhood to the CMB, with no discontinuities and no Big Bang threshold. The SDSS, DESI, and JWST catalogs already contain this measurement.

10.2 Void Galaxy Rotation Anomaly Maximum

Void galaxies sit in the most underdense large-scale $\varepsilon_0\mu_0$ environments. $G_{\rm eff}$ is therefore highest in voids. The prediction: void galaxies should show the maximum rotation curve anomaly relative to their baryonic mass, correlated with large-scale void depth, not with galaxy morphology or stellar mass. Testable against existing SDSS and DESI void catalogs crossed with rotation curve data.

10.3 Bullet Cluster Lensing Centroid Decay

The $\varepsilon_0\mu_0$ field relaxes toward equilibrium with the new baryon distribution after the Bullet Cluster collision, on a timescale set by the light-crossing time of the system (approximately 10 million years). A statistical sample of merging cluster systems ordered by estimated post-collision time should show decreasing lensing centroid offsets from the baryonic mass centroid. The Hubble Space Telescope and Chandra archive already contains sufficient data for a first-order test.

10.4 Gravitational Wave Chirp Elongation

Compact binary merger chirps at cosmological distances should show wavelength extension proportional to path length and $\varepsilon_0\mu_0$ thinning along the line of sight. The extension is anisotropic — it correlates with large-scale $\varepsilon_0\mu_0$ structure — and is distinct from any kinematic or expansion-based prediction. The existing LIGO-Virgo-KAGRA catalog, analyzed for distance-dependent chirp elongation at fixed source-frame masses, provides a direct test.

10.5 Spatial H Mapping

If $H$ varies with local $\varepsilon_0\mu_0$ gradient, then $H$ measured in void regions should differ systematically from $H$ measured in filament regions at the same redshift. This is a direction-dependent, density-correlated signature distinct from any kinematic recession model. The Hubble tension is already evidence for this. A systematic cross-correlation of $H$ measurements against independently determined density fields provides a direct test.

10.6 LISA: The Medium Confirmed in Engineering

The Laser Interferometer Space Antenna (LISA), scheduled for launch in 2035, will exchange laser beams between three spacecraft separated by 2.5 million kilometre baselines. LISA's operation requires Time-Delay Interferometry (TDI) — a post-processing technique that accounts for the fact that the laser light propagating between spacecraft does not inherit the transverse velocity of the emitting spacecraft. The clockwise and counterclockwise light travel times around each arm differ by a first-order $v/c$ correction that must be computed and removed, or laser phase noise exceeds the gravitational wave signal by eight orders of magnitude.

If the light took on the transverse velocity of the emitting spacecraft, clockwise and counterclockwise travel times would be identical. TDI would be unnecessary. TDI is not unnecessary. It is a core mission requirement without which the instrument does not function.

The light clock thought experiment — the foundation of kinematic time dilation — requires the photon to inherit the transverse velocity of the mirror to produce its diagonal path and its $\gamma$ factor. LISA's engineering assumes it does not. The medium is already in the engineering budget.

LISA's millihertz frequency band will also be the primary window for the gravitational wave chirp elongation signature. Supermassive black hole mergers at cosmological distances produce low-frequency chirps that have traversed the greatest path lengths through the thinning $\varepsilon_0\mu_0$ field. The elongation will be at its maximum at LISA frequencies.

10.7 Proton-to-Electron Mass Ratio at Field Depth

Because $G \propto 1/\sqrt{\varepsilon_0\mu_0}$ and the proton-to-electron mass ratio $m_p/m_e$ is set by the closure geometry of the $\varepsilon_0\mu_0$ field, both should vary with field depth. Precision spectroscopy of quasar absorption systems at high redshift provides a test: the molecular hydrogen lines encode $m_p/m_e$ at the absorber's field depth. Any variation correlated with $z_{\rm grav}$ confirms the field picture.

Conclusion

Space is a physical medium. The field equation $\mathbf{a}(x) = c^2\nabla\ln(\varepsilon_0\mu_0)(x)$ has no time coordinate and requires none.

Redshift is a frequency ratio with four components, three of which have never been separated in any cosmological measurement. $H$ is their sum per unit distance. It is not a velocity. It is not a constant.

$G$ varies with the field. Every redshift survey is a $G$ history.

As the field thins, $G_{\rm eff}$ intensifies. Dark matter is the constant-diluent assumption. Rotation curves and gravitational lensing are accounted for with zero free parameters by the same $\gamma_{\rm cause}$ derived from photon geometry. The Bullet Cluster is a field-baryon decoupling event. The lensing centroid follows the field. The X-ray follows the baryons. They separate at collision because only one of them attended it.

Black holes drain $\varepsilon_0\mu_0$ from the intergalactic medium irreversibly. The field thins globally. Both photons and gravitational waves elongate as they traverse the thinning medium. The CMB is the photon survival threshold — the distance at which path loss has accumulated enough that only microwave-wavelength closures reach us intact. Every observer is surrounded by their own CMB, defined by their own maximum thinning depth. The universe has no center because every point is one.

Without KTD there is no universal time axis, no scale factor, no backward integration, and no $T_0$. The Big Bang is what KTD looks like at its logical conclusion. KTD is broken at the derivation. The conclusion goes with it.

The universe does not require special initial conditions because there is no single initial moment. The $\varepsilon_0\mu_0$ field has the conditions it has, locally, now and everywhere. There is nothing to fine-tune.

All works are published open-access through Zenodo under a CC BY 4.0 license. Full publication list: Zenodo · ORCID 0009-0000-1710-3549 · Contact: SCG@azfn.com

Cosmology Without Dark Matter UsingSpatial-Causal Geometry (SCG) and the γcause Invariant