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Entry 1457 · 2026-08-21 · marici.Benincasa · research

1457 — Exponential FRW Produces the First Source-Defined Radial Translation

Status

First source-defined falsifier beyond Entry 1456's fixed-power-law class. Derived from the exceptional cosmology stated in footnote 4 of Benincasa, arXiv:1909.02517v1.

Frozen background

The source relates the flat-space toy-model coupling to the FRW warp factor by

λk(η)=λk[a(η)],2+(d−1)(2−k)/2.\lambda_k(\eta) = \lambda_k[a(\eta)]^{,2+(d-1)(2-k)/2}.

For d=1d=1, it identifies the non-power-law solution

a(η)=a0eA0η,A0>0,a(\eta)=a_0e^{A_0\eta}, \qquad A_0>0,

which vanishes as η→−∞\eta\to-\infty. The coupling becomes, independently of the valency kk,

λk(η)=λka02e2A0η.\lambda_k(\eta)=\lambda_ka_0^2e^{2A_0\eta}.

This normalization is frozen from the source before comparison with the Marici carrier.

The endpoint is a finite-proper-time Big Bang

Cosmic proper time satisfies

dt=a(η)dη.dt=a(\eta)d\eta.

Choosing the endpoint origin gives

t=a0A0eA0η,a=A0t.t=\frac{a_0}{A_0}e^{A_0\eta}, \qquad a=A_0t.

Thus η=−∞\eta=-\infty is the finite proper-time endpoint t=0t=0, with a=0a=0. Unlike the generic power-law discussion, this source example is a literal finite-time Big-Bang boundary.

Exact site kernel

At one interaction site, let EsE_s denote the sum of incident energies in the time-evolution exponential. The source time factor is

∫−∞0e2A0ηeiEsη,dη=12A0+iEs=1i(Es−2iA0).\int_{-\infty}^{0} e^{2A_0\eta}e^{iE_s\eta},d\eta = \boxed{\frac{1}{2A_0+iE_s}} = \frac{1}{i(E_s-2iA_0)}.

Therefore the usual linear site-energy wall is translated to

Es=2iA0.E_s=2iA_0.

For fixed nonzero A0A_0, scaling only the physical energies does not act homogeneously:

12A0+iREs≠R−112A0+iEs.\frac{1}{2A_0+iRE_s} \neq R^{-1}\frac{1}{2A_0+iE_s}.

The simultaneous spurionic scaling (Es,A0)↦R(Es,A0)(E_s,A_0)\mapsto R(E_s,A_0) restores homogeneity, but that is not the physical all-soft normal scaling with the background held fixed.

Classification

The source introduces no new incidence label: each translated denominator is attached to the same labelled site occurrence as the original partial-energy factor. What changes is its coefficient/kernel geometry over the background parameter A0A_0.

The narrow surviving hypothesis is therefore

existing labelled energy/Cut carrier+an A0-dependent translated coefficient object.\boxed{ \text{existing labelled energy/Cut carrier} + \text{an }A_0\text{-dependent translated coefficient object}. }

This is evidence for H2, not H1. Calling Es−2iA0=0E_s-2iA_0=0 a new carrier wall would be premature: it is a source-derived deformation of an existing labelled wall and collapses to it at A0=0A_0=0.

First radial correction

Around the power-law/conical limit A0=0A_0=0,

12A0+iEs=1iEs−2A0(iEs)2+O(A02).\frac{1}{2A_0+iE_s} = \frac{1}{iE_s} - \frac{2A_0}{(iE_s)^2} +O(A_0^2).

The first nonhomogeneous radial datum is a double pole on the existing site wall. It is coefficient/Rees data unless sewing forces an additional incidence object.

Next finite falsifier

For a connected graph, attach the translated kernel occurrencewise at every site and test resolved Cut sewing:

  1. preserve each labelled site and its valency-independent shift 2iA02iA_0;
  2. compare transform-before-Cut with Cut-before-transform;
  3. compute whether the first A0A_0-Rees grade is generated by differentiated existing partial-energy factors;
  4. classify any failure as a coefficient coherence defect or a genuinely new carrier incidence.

Provenance

View source on GitHub