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Entry 3037 · 2026-08-26 · marici.Benincasa · research

3037 — Finite Counterterm Freedom Spans the Complete Toy Cosmology Response

Question

Does the finite-time one-loop toy source select a unique finite point in its rank-three quadratic boundary response, or only a point relative to a bulk renormalization scheme?

Frozen source map

In Collins--Holman--Vardanyan, arXiv:1408.4801, the three admitted local quadratic counterterms contribute to the ordered late-time basis

(1,p2η2,p4η4)(1,p^2\eta^2,p^4\eta^4)

with shape vectors proportional to

(1,−1,0),(−3,−1,0),(−5,−5,−2).(1,-1,0),\qquad(-3,-1,0),\qquad(-5,-5,-2).

The source fixes their infinite parts but leaves the finite loop coordinates I0f,I2f,I4fI_0^f,I_2^f,I_4^f unspecified. Its finite-initial-time condition cancels the endpoint-dependent terms relative to that chosen renormalized correlator.

Exact result

Clearing independent nonzero normalizations gives the finite-scheme map

Mfin=(2−6−5−2−2−500−2),det⁡Mfin=32.M_{\rm fin}= \begin{pmatrix} 2&-6&-5\\ -2&-2&-5\\ 0&0&-2 \end{pmatrix}, \qquad \det M_{\rm fin}=32.

Therefore its rank is three. Finite local counterterms span the entire rank-three response space, and the quotient by unrestricted finite scheme freedom has no nonzero linear response coordinate.

The initial-state matching remains canonical relative to a chosen bulk renormalization point. The source does not select that point. A physical finite readout consequently requires an independently stated normalization condition or another source-derived section of the scheme orbit.

Consequence for the Carrier programme

This is not missing Carrier geometry. The response type is already closed, and scheme changes act internally on it. The missing object is readout authority: a source-derived section rather than another divisor, coefficient direction, or fitted quotient.

The next finite falsifier is to test whether a declared Hadamard or physical normalization condition selects a unique section, only a proper suborbit, or none at all.

Scope

The theorem concerns the abbreviated one-vertex toy model and unrestricted finite coefficients of its three admitted local counterterms. It does not exclude normalization conditions in the full inflationary theory, nor prove the absence of nonlinear or nonlocal invariants.

Durable verification

View source on GitHub