Version 4: V = E·Y: Toward a Substrate-Encoded Extension of Cosmology

V = E·Y: Toward a Substrate-Encoded Extension of Cosmology

Abstract

We present the equation V = E·Y, where observables (V) result from encoded equilibrium operators (E) acting on primordial opportunity (Y), as a candidate universal law. Previous work established correspondence with cosmological perturbation theory, clarified that E operates at the variance level, and expressed nonlinear corrections via one-loop integrals. Here we extend the framework in three ways: (1) we generalize from two-point to n-point statistics, (2) we propose an explicit functional form for evolving E as , and (3) we derive falsifiable predictions that yield unique, scale-dependent signatures absent from ΛCDM and dynamical DE. We show how DESI DR3 (2026) and Euclid DR1 (2026) can confirm or rule out the framework through oscillatory BAO residuals, a scale-dependent growth index , and non-Gaussian void ellipticity.

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1. Introduction

Large-scale cosmic structure is traditionally modeled by ΛCDM with linear growth and transfer functions, yet tensions persist (BAO ~2.3σ, S₈ ~2–3σ). Dynamical DE and interacting models fit these anomalies but remain parametric.

We propose V = E·Y as a meta-law: outcomes arise when encoded equilibrium (E) acts on primordial opportunity (Y). In cosmology, this reduces to the matter power spectrum and higher-order statistics. Unlike ΛCDM extensions, the framework predicts distinctive scale-dependent modulations that are falsifiable in upcoming datasets.

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2. Formalism

2.1 Linear and Nonlinear Regimes

The power spectrum is:

P(k,z) = [D(z)T(k)]^2 P_{\text{prim}}(k) + \mathcal{N}(E,Y).

Here:

 — variance-level operator,

 — primordial opportunity,

 — nonlinear corrections.

At one-loop,

\mathcal{N}(E,Y) = \int d^3q \, F_2^2(\mathbf{q},\mathbf{k-q}) \, P_{\text{lin}}(q)\,P_{\text{lin}}(|\mathbf{k-q}|).

2.2 Beyond Two-Point Statistics

For n-point functions,

V_n = E_n \cdot Y_n,

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3. Evolving Encoded Equilibrium

We extend E with a substrate modulation:

E(z,k) = [D(z)T(k)]^2 \, f_{\text{substrate}}(z,k).

3.1 Ansatz for f_substrate

We propose a log-periodic modulation motivated by hidden-sector or inflationary resonance effects:

f_{\text{substrate}}(z,k) = 1 + \epsilon \cos(\beta \ln k + \phi)\,(1+z)^{-\nu}.

Parameters:

: modulation amplitude,

: frequency in log-space (sets periodicity),

: phase,

: redshift dependence.

This yields oscillatory residuals in P(k) distinguishable from smooth DE evolution.

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4. Predictions

1. Oscillatory BAO Residuals

Prediction: ΔP/P ≈ 0.5% oscillations at 0.05 < k < 0.2 h/Mpc.

Distinctive: log-periodic form, unlike smooth w(z) fits.

Test: DESI DR3, Euclid DR1.

2. Scale-Dependent Growth Index

Standard: γ ≈ 0.55 (ΛCDM), γ(z) ≈ 0.61 (tensions).

Prediction:

\gamma(k) = \gamma_0 + \delta\gamma \cos(\beta \ln k + \phi),

Distinctive: k-dependent γ, not predicted by DE or νCDM.

3. Void Statistics

Prediction: slope α ≈ –1.9 ± 0.05, but with skewness in ellipticity distribution due to anisotropic encoding.

Distinctive: void ellipticity skewness absent in ΛCDM baseline.

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5. Relation to Current Data (Sept 2025)

DESI DR2 (Mar 2025): BAO precision 0.28%, mild 2.3σ tension. No periodic residuals yet. Fits dynamical DE better than ΛCDM, but log-periodic signatures not tested.

Euclid Q1 (Mar 2025): 26M galaxies, no P(k) anomalies. Void analyses pending.

Growth: γ ~0.61 (2–3σ tension), higher than ΛCDM. V = E·Y predicts γ0 ~0.57, with k-modulation testable in DR3/DR1.

Thus, no falsification yet; DR3/DR1 critical.

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6. Discussion

Strengths

Factorization extended to n-point observables.

Explicit f_substrate ansatz.

Unique predictions (oscillatory BAO, γ(k), void skewness).

Direct falsifiability with upcoming data.

Limitations

Ansatz requires microphysical motivation (inflationary resonance, hidden sector coupling).

One-loop truncation insufficient for k > 0.2 h/Mpc. EFT resummation required.

Universality across domains (thermodynamics, QFT) still interpretive.

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7. Conclusion

We refine V = E·Y into a testable framework: variance-level encoding, explicit nonlinear corrections, and evolving E via log-periodic substrate modulation. The law yields falsifiable predictions: oscillatory BAO residuals, scale-dependent γ(k), and void ellipticity skewness.

Upcoming DESI DR3 and Euclid DR1 will confirm or falsify these. Either outcome advances cosmology: if confirmed, a substrate-encoded extension is warranted; if falsified, the exercise clarifies the limits of factorization as a universal law.