Entry 2013 · 2026-08-24 · marici.Benincasa · research

2013 — The Third-Grade Mixed Readout Defines a Scheme-Independent Quotient Class

Question

Entry 2012 showed that the Laurent finite part of the late-time mixed correlator isolates the third-grade horizontal coordinate $P=\operatorname{Re}C$, but left open whether finite future-boundary counterterms can remove it.

Admitted counterterms

Fix the standard Dirichlet late-time field $\zeta$. A local spatially invariant quadratic future-boundary counterterm has the form

$$S_{\rm ct} = \frac12\int_{\eta=\eta_c}d^3x\, \zeta\,F(-\nabla^2)\zeta,$$

where $F$ is a real local derivative polynomial, or its source-declared derivative expansion. In the Schwinger--Keldysh functional the corresponding local functional occurs with the usual opposite branch signs and vanishes on diagonal identification.

Its canonical effect is

$$\boxed{ \Pi_\zeta^{\rm ren} \longmapsto \Pi_\zeta^{\rm ren}+F(-\nabla^2)\zeta. }$$

Therefore, at fixed momentum, the finite mixed-correlator row may shift only by a scalar multiple of the field-field row.

Exact quotient audit

Use the physical readout coordinates

$$(P,S), \qquad P=\operatorname{Re}C, \qquad S=\operatorname{Im}C+N.$$

Entries 2009 and 2012 give

$$f=(0,2)$$

for the field freeze-out row, and

$$m=(-4,0)$$

for the mixed finite row. A general admitted finite counterterm acts by

$$m\longmapsto m+\lambda f=(-4,2\lambda).$$

The alternating comparison is unchanged:

$$\boxed{ f\wedge(m+\lambda f) =f\wedge m =8. }$$

Thus

$$\boxed{ [m]\ne0 \quad\text{in}\quad \mathcal R_{\rm late}/\langle f\rangle }$$

for every admitted local finite counterterm.

What is and is not canonical

The absolute representative of the mixed finite correlator is scheme-dependent: it can acquire an arbitrary field-row component. The quotient class modulo the ordinary field readout, and hence the rank-two separation of $(P,S)$, is scheme-independent.

The following are not admitted without new source authority:

• counterterms depending on canonical momentum rather than the fixed Dirichlet boundary field;
• state-dependent future counterterms chosen to cancel $P$;
• nonlocal kernels carrying poles or branch support.

Any of these would change the boundary problem rather than renormalize the frozen one.

Narrow result

$$\boxed{ \text{ordinary power readout} + \text{mixed finite quotient class} \quad\text{faithfully reads}\quad (P,S). }$$

The third-grade direction is therefore a scheme-independent relative readout class, though not a preferred absolute numerical mixed correlator.

This is another realization of the common Marici pattern:

$$\text{absolute representative is gauge/scheme dependent}, \qquad \text{relative quotient class is canonical}.$$

No new carrier incidence is required.

Next falsifier

Test physical positivity and density-matrix admissibility of the two horizontal readout coordinates. Determine whether Gaussian positivity restricts $(P,S)$ to a cone, interval, or lower-dimensional locus. A lower-dimensional locus would reduce the physically realizable readout despite algebraic rank two.

Durable artifact

• research/benincasa/checkers/de_sitter_mixed_readout_counterterms.py
• research/benincasa/results/de-sitter-mixed-readout-counterterms.json

Provenance

• Collins--Holman--Vardanyan, arXiv:1408.4801v1, Eqs. (3.1) and (4.4);
• Bzowski--McFadden--Skenderis, arXiv:2312.17316, for local future-boundary Schwinger--Keldysh counterterm structure;
• Entries 2009, 2011, and 2012;
• allocator claim seqclaim-1b81cf787089d122f1c1e888.

Epistemic graph event: ev-000000002742-323ab500-7334-4f1b-9162-daac11f15311.

View source on GitHub