Entry 1494 · 2026-08-21 · marici.Benincasa · research

1494 — One Loop Pushes the Cubic Initial Packet into an Independent Quadratic Statistical Block

Status

Primary-source coefficient-mixing result from Collins--Holman--Vardanyan, arXiv:1408.4801v1, Eqs. (5.5)--(5.9).

Source one-loop mechanism

The finite-time one-loop two-point function contains two insertions drawn from the bulk cubic Hamiltonian and Entry 1492's cubic initial action. To cancel the displayed quadratic divergence in the initial time, the source introduces a new quadratic initial action

$$\begin{aligned} S_0^{(2)}=\frac12\bigl[ &\langle\zeta_+,A\zeta_+\rangle -\langle\zeta_-,A^*\zeta_-\rangle\\ &+2i\langle\zeta_+,B\zeta_-\rangle \bigr]. \end{aligned}$$

Here $A=A_R+iA_I$, while $B$ is the cross-branch density kernel. The matching condition gives, at the calculated divergent order,

$$\boxed{A_R=0,}$$

and fixes nontrivial $A_I$ and $B$ separately by the source loop integrals.

Exact Keldysh decomposition

For self-adjoint translation-invariant kernels and

$$\zeta_\pm=\zeta_c\pm\frac12\zeta_q,$$

one obtains

$$\boxed{ S_0^{(2)} = \langle\zeta_c,A_R\zeta_q\rangle +i\langle\zeta_c,(A_I+B)\zeta_c\rangle +\frac{i}{4} \langle\zeta_q,(A_I-B)\zeta_q\rangle. }$$

The three terms have contour-conormal grades $1,0,2$, respectively.

Since the source matching sets $A_R=0$ at this order, the generated quadratic correction is deck even:

$$\boxed{ \operatorname{gr}_\Delta S_{0,\mathrm{generated}}^{(2)} = \operatorname{gr}_\Delta^0 \oplus \operatorname{gr}_\Delta^2. }$$

It is statistical/density-matrix data, not a causal action difference.

Failure of cubic closure

Entry 1492's four-grade cubic packet is not closed under the one-loop two-point pushforward. The loop lowers boundary field degree and generates an independent quadratic coefficient object:

$$\boxed{ \mathcal C_{S_0^{(3)}} \longrightarrow \mathcal C_{S_0^{(2)},\mathrm{stat}}^{(0,2)}. }$$

The map is source-defined by the finite-time matching calculation. It must not be replaced by an endomorphism of the cubic kernel $C$.

Trace interpretation

The grade-zero term survives diagonal restriction:

$$\Delta^*S_0^{(2)} =i\langle\zeta_c,(A_I+B)\zeta_c\rangle.$$

The grade-two term is invisible to ordinary diagonal pullback but remains in the second conormal neighborhood. Thus normalized trace and second contour-normal data are both necessary to retain the full generated Gaussian state.

Architecture

At the calculated order, the finite-time coefficient system is filtered by both field degree and contour-conormal degree:

$$\boxed{ \mathcal C_\Sigma \supset \mathcal C_{3}^{(0,1,2,3)} \xrightarrow{\text{one-loop pushforward}} \mathcal C_{2}^{(0,2)}. }$$

All terms remain on the same doubled initial hypersurface. This is new coefficient mixing, not a new carrier incidence.

Scope

The source explicitly eliminates only the quadratically divergent $\eta_0$-terms. It states that the zeroth- and first-order pieces can be treated similarly but does not display their complete kernels. No claim is made here about closure beyond the published order.

Next falsifier

Test whether the pair $(A_I,B)$ is more naturally represented as one positive Gaussian covariance object with a fixed trace normalization, or as two independent coefficient lines. The finite test is the eigenvalue and positivity decomposition of the doubled quadratic kernel together with its diagonal Gysin pushforward.

Provenance

• Collins--Holman--Vardanyan, arXiv:1408.4801v1, Eqs. (5.5)--(5.9);
• Entry 1492;
• allocator claim seqclaim-b1af7343c5a59de6856cce56.
• epistemic event ev-000000001615-54b7d6ba-72c4-47c2-8d50-550e9bae7346.

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