Cosmology in Entropy-Time¶

April 2026 · Addendum to Stage 0 Gravity and Master Reference Section VI

The Σ-reparameterization transforms RTSG cosmology from a set of conjectures into a framework with falsifiable predictions.

Gravity = Stage 0 Entropy Production¶

\[\kappa_{\text{grav}} = \lim_{\dim(CS) \to 1} \kappa\]

Gravity is the lowest-complexity form of instantiation. In entropy language: gravity is the minimum-complexity mode of entropy production. Every system that produces entropy — at any scale — gravitates.

By the Action-Entropy Identity, the Einstein-Hilbert action at Stage 0 is:

\[S_{EH} = -\Sigma_{\text{Stage 0}} + \text{const}\]

This connects RTSG to:

Jacobson (1995): Einstein's equations from thermodynamic entropy on local Rindler horizons
Verlinde (2010): Gravity as entropic force
Padmanabhan (2010): Equipartition of energy on horizons

RTSG provides the mechanism these approaches lack: the Will Field GL theory gives the explicit entropy functional, and the bisimulation quotient gives the state space on which entropy is defined.

Dynamical Dark Energy — A Falsifiable Prediction¶

The v3 Cosmological Constant¶

\[\Lambda_{\text{eff}} \sim \langle\rho_W\rangle_{PS}\]

The v4 Entropy-Time Cosmological "Constant"¶

\[\Lambda_{\text{eff}}(\Sigma) \sim \langle\dot\Sigma^2|\partial_\Sigma W|^2 + |\nabla W|^2 + \alpha|W|^2 + (\beta/2)|W|^4\rangle_{PS}\]

The key difference: \(\dot\Sigma\) appears in the kinetic term. As the universe ages, \(\dot\Sigma\) evolves — and \(\Lambda_{\text{eff}}\) evolves with it.

The Prediction¶

Epoch	\(\dot\Sigma\)	\(\Lambda_{\text{eff}}\)	Physics
Inflation	Extremely high	Very large	Rapid entropy production drives exponential expansion
Radiation era	High, decreasing	Decreasing	Entropy production slowing as structure forms
Matter era	Moderate	Small	Structure well-formed, entropy production steady
Late acceleration	Low, approaching floor	Small positive floor	Entropy production approaching asymptotic rate
Heat death	\(\to 0\)	\(\to\) residual constant	Maximum entropy reached

Falsifiable prediction: \(\Lambda_{\text{eff}}\) has a specific time-dependence determined by \(\dot\Sigma(t)\). The equation of state parameter \(w\) is NOT exactly \(-1\) — it has a slow evolution:

\[w(z) = -1 + \frac{d\ln\dot\Sigma}{d\ln a}\]

where \(a\) is the scale factor. This is testable with DESI, Euclid, and Rubin Observatory data.

Dark Matter = Maximum Entropy Phase¶

Dark matter is the uncondensed (disordered) phase of the Will Field: \(\langle W \rangle = 0\), \(\alpha > 0\). This IS the maximum entropy configuration — structure uniformly distributed, no condensation.

Entropy interpretation:

Baryonic matter (5.4%) = condensed phase (\(\alpha < 0\) locally). Entropy concentrated in structure.
Dark matter (26.8%) = uncondensed phase (\(\alpha > 0\)). Maximum entropy. Gravitates via Stage 0 CS but no electromagnetic interaction (requires condensation).
Dark energy (67.8%) = kinetic entropy \(\dot\Sigma^2|\partial_\Sigma W|^2\) — the cosmic entropy production rate itself.

Falsifiable prediction: Dark matter should exhibit GL critical exponents in structure formation. The DM halo profile should follow from the GL correlation function, not from collisionless Boltzmann dynamics.

The BBN Freeze (Safety Caveat)¶

Any net \(Q \to B\) conversion (QS to PS at cosmic scale) must freeze before Big Bang Nucleosynthesis (\(a < a_{\text{BBN}}\)). The baryonic fraction integral is constrained:

\[f_b = \int_0^{\Sigma_{\text{BBN}}} \frac{d\Sigma_{\text{condensed}}}{d\Sigma}\,d\Sigma \leq 0.054\]

Black Hole Information in Entropy-Time¶

Unitarity¶

\[\pi \circ U_\Sigma = \bar{U}_\Sigma \circ \pi\]

In entropy-time, black hole evolution is unitary per entropy-step, not per clock-step. Information is preserved in the bisimulation quotient.

Hawking Radiation = Entropy Equilibration¶

By the Action-Entropy Identity, the black hole interior and exterior approach the same \(\Sigma\) maximum. Hawking radiation is the mechanism of entropy equilibration — structure flows from the high-entropy interior to the lower-entropy exterior until the quotient is uniform.

The Information "Paradox" Dissolves¶

There is no paradox because there are no two times to compare — there is only one entropy parameter \(\Sigma\) along which evolution is unitary. The apparent paradox arose from comparing early-time and late-time states in clock-time, which is the derived quantity. In entropy-time, the evolution is continuous and unitary throughout.

Confidence update: BH Information: 72% → 75%.

Open Questions¶

Question	Status	Impact
Is \(w \neq -1\) detectable with current surveys?	Calculable	Direct DESI/Euclid test
DM halo profile from GL correlation function?	Open	Would distinguish RTSG DM from CDM
Does \(\dot\Sigma\) have a cosmic floor \(> 0\)?	Open	Determines whether heat death is reachable
Inflation = GL phase transition?	Conjecture	\(\alpha\) sign flip at Planck scale