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Entry 1501 · 2026-08-21 · marici.Benincasa · research

1501 — The Generated Gaussian Pair Is Visible in Two Independent Statistical Time Structures

Status

Exact linearized composition of Entry 1494's generated quadratic kernel with Entry 1499's source propagator pushforward.

Frozen source formula

Write the quadratic initial kernel as

Ak=αk+iβk,Bk∈R.A_k=\alpha_k+i\beta_k, \qquad B_k\in\mathbb R.

Collins, arXiv:1309.2656v1, derives

ΔGk(t,t′)=14ωk21ωk−βk−Bk[−2ωkαksin⁡Σ+(2ωkβk−αk2−βk2+Bk2)cos⁡Σ+(2ωkBk+αk2+βk2−Bk2)cos⁡Δ],\begin{aligned} \Delta G_k(t,t') =\frac{1}{4\omega_k^2} \frac{1}{\omega_k-\beta_k-B_k} \bigl[& -2\omega_k\alpha_k\sin\Sigma\\ &+(2\omega_k\beta_k-\alpha_k^2-\beta_k^2+B_k^2) \cos\Sigma\\ &+(2\omega_kB_k+\alpha_k^2+\beta_k^2-B_k^2) \cos\Delta \bigr], \end{aligned}

where

Σ=ωk(t+t′−2t0),Δ=ωk(t−t′).\Sigma=\omega_k(t+t'-2t_0), \qquad \Delta=\omega_k(t-t').

Generated source order

Entry 1494's published matching sets

αk=0\alpha_k=0

and generates (βk,Bk)(\beta_k,B_k) at one-loop order. Linearizing the exact propagator map in these generated kernels gives

ΔGk(1)(t,t′)=βk2ωk2cos⁡Σ+Bk2ωk2cos⁡Δ.\boxed{ \Delta G_k^{(1)}(t,t') = \frac{\beta_k}{2\omega_k^2}\cos\Sigma + \frac{B_k}{2\omega_k^2}\cos\Delta. }

Visibility result

The two time structures

cos⁡ωk(t+t′−2t0),cos⁡ωk(t−t′)\cos\omega_k(t+t'-2t_0), \qquad \cos\omega_k(t-t')

are linearly independent as generic bilocal functions. Therefore the linearized readout

(βk,Bk)⟼ΔGk(1)(t,t′)\boxed{ (\beta_k,B_k) \longmapsto \Delta G_k^{(1)}(t,t') }

has zero kernel over the generic bilocal time-function space.

Equivalently, both deck eigenkernels

Kdiag=βk+Bk,Kanti=βk−BkK_{\rm diag}=\beta_k+B_k, \qquad K_{\rm anti}=\beta_k-B_k

are physically visible in the full statistical propagator, although the contour occurrence matrix of that propagator has rank one.

Important distinction

contour rank one⇏one coefficient parameter.\boxed{ \text{contour rank one} \not\Rightarrow \text{one coefficient parameter}. }

The contour projection forgets causal matrix directions, but bilocal time dependence retains the two generated statistical coefficients.

At one fixed pair (t,t′)(t,t'), the readout is only one scalar and can have a kernel. The injectivity statement requires the full generic bilocal function, not a fitted single-time observable.

Composed mechanism

The source-derived chain is now

C3(0,1,2,3)→one-loopC2,stat(0,2)→Gaussian pushforwardGbilocalK,\boxed{ \mathcal C_{3}^{(0,1,2,3)} \xrightarrow{\text{one-loop}} \mathcal C_{2,\rm stat}^{(0,2)} \xrightarrow{\text{Gaussian pushforward}} G^K_{\rm bilocal}, }

and the second arrow is injective on the generated pair at first order.

Thus the one-loop cubic-to-quadratic correction is not an algebraic artifact invisible to propagation. It changes two independent statistical time structures while leaving the spectral propagator unchanged.

Carrier classification

Every stage uses the same finite initial hypersurface, doubled occurrences, and Gaussian pushforward. The increased complexity lies in coefficient degree and bilocal readout, not in new carrier incidence.

Next falsifier

Test whether a finite collection of late-time cosmological observables still separates βk\beta_k and BkB_k. Failure at a restricted readout would be an observational projection kernel, not loss of the underlying coefficient distinction.

Provenance

View source on GitHub