Entry 1456 · 2026-08-21 · marici.Benincasa · research

1456 — Every Frozen Power-Law Cosmological Integral Is Exactly Conical at All-Soft Scaling

Status

Source-derived all-graph strengthening of Entry 1455. It applies to the fixed-power-law FRW scalar integral class of Benincasa--Vazão, arXiv:2402.06558v3. It does not apply automatically to a non-power-law background or an explicitly scale-breaking deformation.

Frozen source object

The source writes a graph integral as a product of:

• Mellin/Kummer measures in site and internal energies;
• a homogeneous numerator $n_\delta$ of degree $\delta$;
• powers of linear graph denominators $q_g$;
• a Cayley--Menger loop measure assembled from homogeneous simplex-volume ratios.

Schematically,

$$\Omega_G = \left(\prod_s\frac{dx_s}{x_s}x_s^{\alpha_s}\right) \left(\prod_e\frac{dy_e}{y_e}y_e^{\beta_e}\right) \mu_d(y;\mathcal X) \frac{n_\delta(x,y,\mathcal X)} {\prod_g q_g^{\tau_g}}.$$

The exponents are fixed by the power-law background and derivative content. No additive energy scale occurs in this frozen class.

Exact joint homogeneity

Scale every dimensionful integration and external-energy variable by one radial parameter:

$$(x,y,\mathcal X)=R(\widehat x,\widehat y,\widehat{\mathcal X}).$$

Every source ingredient has a definite degree:

$$q_g\mapsto Rq_g, \qquad n_\delta\mapsto R^\delta n_\delta, \qquad \mu_d\mapsto R^{h_\mu}\mu_d.$$

Here $h_\mu$ is the source-fixed homogeneous degree of the loop measure. For the displayed one-loop Cayley--Menger measure, including its prefactor,

$$h_\mu=d-2n_e.$$

The exact exponent of the complete form is therefore

$$\kappa_G = \sum_s\alpha_s+ \sum_e\beta_e+ \delta- \sum_g\tau_g+ h_\mu,$$

with the radial logarithmic differential separated from the projective coordinates. Consequently,

$$\boxed{ \Omega_G = \frac{dR}{R}R^{\kappa_G} \otimes \Omega_{G,\mathrm{proj}}. }$$

The projective relative form is independent of $R$. This is an identity of the complete frozen integrand, not an associated-grade approximation.

Hard-to-vary result

$$\boxed{ \text{Every graph integral in the frozen fixed-power-law class is exactly conical at simultaneous all-soft scaling.} }$$

Graph topology and loop order can enlarge the projective relative coefficient object, but they cannot generate a first or higher nonhomogeneous radial extension while the source ingredients remain homogeneous.

Thus the all-soft endpoint consists of:

• the existing radial/projective blowup carrier;
• inherited projective Cut and marked incidence data;
• one radial Kummer coefficient line;
• a graph- and layer-specific projective relative coefficient object.

No new radial carrier stratum or radial coherence cell is required in this source class.

What this rules out

The next falsifier cannot be found by increasing graph size or loop order inside the same fixed-power-law integral family. Those operations increase projective complexity but preserve exact radial homogeneity.

In particular, a dimensionful mass that enters only through a dimensionless index or exponent does not by itself break this conclusion.

First admissible falsifier

Freeze a source-defined deformation that introduces genuine radial nonhomogeneity, for example:

• a non-power-law scale factor;
• a transition between background eras;
• an additive physical scale in the transformed kernel.

Then derive the first radial correction before classifying it as existing carrier structure, coefficient extension, or a genuinely new carrier datum. No such scale-breaking example occurs in the currently frozen power-law source family.

Provenance

• Benincasa--Vazão, arXiv:2402.06558v3, especially the power-law transform, homogeneous numerator, general graph integral, and Cayley--Menger measure;
• Entries 1443--1455;
• allocator claim seqclaim-e00ba38fe49160f01e12b24b.
• epistemic event ev-000000001554-ad836840-80bd-46f3-91e5-da2dcabe120f.

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