Version 6: The Swygert Theory of Everything AO / V = E·Y: Substrate Modulation with Inflationary Priors, EFT Implementation, and n-Point Forecasts

V = E·Y: Substrate Modulation with Inflationary Priors, EFT Implementation, and n-Point Forecasts

Abstract

We refine V = E·Y—observables  arise when encoded equilibrium operators  act on primordial opportunity —into a testable cosmological model with microphysical priors and concrete analysis paths. Building on variance-level encoding and one-loop/EFT nonlinearities, we (1) derive inflation-based priors for the log-periodic modulation , (2) specify IR-resummed EFT-of-LSS implementation for , (3) extend to bispectrum forecasts for resonant non-Gaussianity, and (4) provide model selection criteria that distinguish substrate modulation from smooth CPL dark energy and CDM. We state hard falsifiability thresholds aligned with DESI DR3 / Euclid DR1.

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1. Setup and Notation

Core law (variance level):

P(k,z)=E(z,k)\,Y(k)+\mathcal{N}(E,Y),\quad

E(z,k)=[D(z)T(k)]^2\,f_{\text{sub}}(z,k),\quad Y(k)=P_{\rm prim}(k).

Nonlinear term (schematic):

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

Substrate modulation (log-periodic):

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

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2. Inflationary Microphysics → Parameter Priors

We motivate  via resonant inflation/axion-monodromy–type dynamics:

V(\phi)=\tfrac{1}{2}m^2\phi^2+\Lambda^4\cos(\phi/f)\;\Rightarrow\;

P_{\rm prim}(k)\propto k^{n_s-1}\!\left[1+\epsilon_\star\cos\!\big(\beta\ln(k/k_\star)+\phi\big)\right].

Amplitude : small-feature regime ⇒  (broad, theory-driven to avoid strong CMB constraints; tightened by joint CMB+LSS fits).

Frequency :  at horizon crossing; take  (log-space oscillations resolvable in LSS; very high  damped by transfer/EFT).

Phase : uninformative prior .

Redshift exponent : slow dilution of modulation in late growth ⇒  (prior allows modest time evolution).

> These are theory priors, not data-informed posteriors; analyses should report prior→posterior flow explicitly.

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3. Nonlinear Accuracy: IR-Resummed EFT Implementation

To achieve sub-percent predictions in :

(a) Wiggle/no-wiggle split:

P_{\rm lin}(k)=P_{\rm nw}(k)+P_{\rm w}(k),\quad

P_{\rm w}(k)\to e^{-k^2\Sigma^2/2}P_{\rm w}(k),

(b) EFT counterterms:

\mathcal{N}_{\rm EFT}=2\,c_s^2\,k^2\,P_{\rm lin}(k) + \text{higher orders},

(c) Substrate insertion: apply  to the wiggle channel (most sensitive to oscillations):

P_{\rm w}(k)\;\mapsto\; e^{-k^2\Sigma^2/2}\,f_{\text{sub}}(z,k)\,P_{\rm w}(k),

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4. Beyond 2-Point: Resonant Bispectrum Forecast

Tree-level matter bispectrum:

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

B_{\rm res}(k_1,k_2,k_3)\propto f_{\rm NL}^{\rm res}\,

\frac{\sin\!\big[\beta\ln\!(k_t/k_\star)+\phi\big]}{(k_1k_2k_3)^2}\;,\quad k_t\!=\!k_1\!+\!k_2\!+\!k_3.

SNR forecast (schematic):

\mathrm{SNR}^2=\sum_{\{k\}}\frac{\big[B_{\rm res}(\vec k)\big]^2}{\mathrm{Var}[B(\vec k)]}\;\;\text{with}\;\;\mathrm{Var}[B]\approx s\,\frac{(2\pi)^3}{V}\,P_{\rm tot}(k_1)P_{\rm tot}(k_2)P_{\rm tot}(k_3),

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5. Distinguishing from CPL DE and CDM

Key difference: smooth  or extra neutrino mass cannot create log-periodic features in . We therefore propose nested model comparison:

Models:

: ΛCDM(+EFT);

: CPL  (+EFT);

: ΛCDM+ (+EFT);

: substrate  (+EFT).

Evidence & Bayes factors: compute .

Require  (decisive) for claim of substrate detection;  to falsify .

Frequentist cross-check: look for periodogram peaks in residuals of  and oscillatory templates in , controlling look-elsewhere effects via  prior.

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6. Concrete Analysis Recipe (CLASS/CAMB + EFT)

1. Primordial spectrum: enable oscillatory  with .

2. Transfer/growth: standard Boltzmann output .

3. IR resummation: BAO wiggle/no-wiggle split; apply .

4. EFT counterterms: fit  per bin.

5. Substrate insertion: multiply wiggle term by .

6. Likelihoods: DESI DR2/3 , RSD; Euclid DR1 lensing + bispectrum (mock covariance).

7. Parameters: .

8. Outputs: posteriors +  against , , .

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7. Updated Predictions with Priors

Oscillatory BAO residuals:

 in ;

with  and damping  from IR/EFT.

Scale-dependent growth:

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

  \gamma_0\approx 0.57,\; |\delta\gamma|\lesssim \mathcal{O}(10^{-2}),

Void statistics: slope  plus ellipticity skewness correlated with  of the surrounding spectrum (distinct from log-normal ΛCDM).

Bispectrum: resonant  with  oscillations; forecast SNR  for  few, given Euclid volume and  (to be validated in mocks).

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8. Hard Falsifiability (unchanged, now tied to )

Falsify substrate modulation  if any holds:

DESI DR3 P(k): no periodic peak in  residuals with amplitude  over  and .

Euclid DR1 growth:  consistent with constant  (no k-dependence) and .

Euclid voids: no ellipticity skewness beyond ΛCDM mocks and .

Euclid bispectrum: no  oscillatory template detection at  and .

Passing any one “no-signal + decisive Bayes” test rules out the substrate model under the stated priors.

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9. Discussion and Scope

Ansatz specificity: now tied to a concrete inflation sector; priors restrict flexibility.

Small-scale fidelity: IR-resummed EFT specifies damping/counterterms; extendable to two-loop if needed.

Universality: cosmology is fully worked; cross-domain mappings (thermo/QFT) remain heuristic and are not used for inference here.

Distinctiveness:  periodicity in  and -dependent  are not produced by smooth CPL or CDM.

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10. Conclusion

Version 6 pins V = E·Y to: inflation-motivated  with priors, an IR-resummed EFT pipeline usable today, and n-point forecasts (bispectrum) that yield unique, falsifiable signatures. With decisive Bayes thresholds pre-declared, DESI DR3 and Euclid DR1 will either validate a substrate-encoded extension or falsify it cleanly.