Version 5: V = E·Y: A Substrate-Encoded Extension of Cosmic Structure

V = E·Y: A Substrate-Encoded Extension of Cosmic Structure

Abstract

We propose V = E·Y as a candidate universal law, where observables (V) arise when encoded equilibrium operators (E) act upon primordial opportunity (Y). Earlier formulations clarified variance-level operators, one-loop nonlinear corrections, and extensions to n-point statistics. Here we advance the framework by: (1) deriving a log-periodic modulation  from inflationary resonance physics, (2) extending nonlinear corrections with EFT resummation, (3) applying the principle to the bispectrum as a worked n-point example, and (4) issuing explicit falsifiability criteria tied to DESI DR3 (2026) and Euclid DR1 (2026). These refinements shift V = E·Y from abstraction to a testable cosmological model.

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1. Introduction

ΛCDM explains much of cosmic structure but faces persistent tensions (H₀, S₈, BAO). Extensions such as dynamical dark energy (DE) and interacting models fit anomalies parametrically but lack mechanism.

We present V = E·Y: outcomes emerge when encoded equilibrium (E) acts on primordial opportunity (Y). Applied to cosmology, this structure generalizes perturbation theory, introduces substrate-driven modulations, and yields falsifiable predictions distinct from ΛCDM or wCDM.

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2. Formalism

2.1 Power Spectrum

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

 — primordial opportunity.

 — nonlinear corrections.

2.2 Nonlinear Extension

At one-loop:

\mathcal{N}_{1\text{-loop}} = \int d^3q \, F_2^2(\mathbf{q},\mathbf{k-q}) P_{\text{lin}}(q) P_{\text{lin}}(|\mathbf{k-q}|).

For , effective field theory (EFT) counterterms and resummation yield:

\mathcal{N}(E,Y) = \mathcal{N}_{1\text{-loop}} + c_s^2 k^2 P_{\text{lin}}(k) + \cdots

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3. Microphysical Origin of f_substrate

We extend E with a modulation:

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

3.1 Inflationary Resonance Toy Model

Consider an axion-like inflaton  with potential:

V(\phi) = \frac{1}{2}m^2\phi^2 + \Lambda^4 \cos\!\left(\frac{\phi}{f}\right).

P_{\text{prim}}(k) \propto k^{n_s-1}\left[1 + \epsilon \cos(\beta \ln k + \phi)\right],

After transfer through D(z) and T(k), this yields:

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

Thus, the ansatz is not arbitrary — it follows from resonant inflationary physics.

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4. Extension to n-Point Statistics

4.1 Bispectrum Example

The tree-level bispectrum is:

B(k_1,k_2,k_3) = 2F_2(k_1,k_2) P_{\text{lin}}(k_1) P_{\text{lin}}(k_2) + \text{cyc}.

In V = E·Y form:

,

 convolution of  kernels and growth factors.

Adding substrate modulation:

E_3 \to E_3 f_{\text{substrate}}(z,k_{\rm eff}),

Prediction: oscillatory bispectrum residuals, not captured by smooth DE or neutrino models.

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5. Predictions

1. Oscillatory BAO Residuals

Prediction: ΔP/P ≈ 0.5% log-periodic oscillations in 0.05 < k < 0.2 h/Mpc.

Distinctive: periodic structure in ln k, unlike smooth w(z).

Test: DESI DR3, Euclid DR1.

2. Scale-Dependent Growth Index

Prediction:

\gamma(k) = 0.57 + \delta\gamma \cos(\beta \ln k + \phi), \quad |\delta\gamma|\lesssim 0.02.

Test: Euclid DR1 lensing + DESI RSD.

3. Void Statistics

Prediction: slope α ≈ –1.9 ± 0.05 with measurable skewness in ellipticity.

Distinctive: skewness not expected in ΛCDM log-normal void models.

Test: Euclid void catalogs.

4. Oscillatory Bispectrum Residuals

Prediction: log-periodic modulations in equilateral and squeezed bispectrum configurations.

Distinctive: non-Gaussian signature absent in smooth DE models.

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6. Relation to Current Data (Sept 2025)

DESI DR2: BAO 2.3σ tension with Planck, no oscillatory residuals detected at 0.28% precision.

Euclid Q1: 26M galaxies catalogued; no P(k) anomalies yet. Void/bispectrum analyses pending.

Growth: γ ~0.61 in some reconstructions, above ΛCDM’s 0.55; no k-dependence reported.

Thus, no falsification yet, but no unique confirmation. DR3/DR1 critical.

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7. Falsifiability

This framework is falsifiable. Specifically:

If DESI DR3 does not detect oscillatory residuals (ΔP/P > 0.5%) in 0.05 < k < 0.2 h/Mpc, the proposed  modulation is ruled out.

If Euclid DR1 finds γ consistent with a constant (γ ≈ 0.55 ± 0.01) and no scale dependence, the k-modulated growth prediction is falsified.

If Euclid void catalogs show Gaussian ellipticity distributions without skewness, the void prediction fails.

These conditions would invalidate the substrate-encoded model.

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8. Conclusion

By deriving log-periodic modulation from inflationary resonance, extending nonlinear corrections with EFT, and applying V = E·Y to the bispectrum, we have strengthened the framework into a testable cosmological model. Unique predictions — oscillatory BAO, γ(k), void skewness, and bispectrum modulations — will be confirmed or falsified by DESI DR3 and Euclid DR1.

V = E·Y is no longer a metaphor: it is a falsifiable proposal for a substrate-encoded extension of cosmic structure.