Entry 1492 · 2026-08-21 · marici.Benincasa · research

1492 — A Source-Derived Cubic Initial Action Occupies Four Keldysh Grades

Status

Primary-source nonlinear finite-time boundary action from Collins--Holman-- Vardanyan, arXiv:1408.4801v1, Eqs. (4.7)--(4.8).

This is a distinct inflationary $\zeta$ sector. It is not retroactively identified with the linearized tadpole block corrected in Entry 1491.

Frozen cubic initial action

The source introduces a cubic initial action at finite $t_0$:

$$S_0^{(3)} =\mathcal N \left[ T_C(\zeta_+,\zeta_+,\zeta_+) -T_{C^*}(\zeta_-,\zeta_-,\zeta_-) \right],$$

where $T_C$ is the source trilinear kernel including the displayed spatial derivatives, $\mathcal N$ is the real background normalization, and

$$\boxed{ C(\mathbf k_1,\mathbf k_2,\mathbf k_3) =\frac1K\left(\frac1{K\eta_0}-i\right), \qquad K=k_1+k_2+k_3. }$$

Thus, generically,

$$C_R=\frac1{K^2\eta_0}, \qquad C_I=-\frac1K$$

are both nonzero.

Keldysh polarization

Set

$$\zeta_\pm=\zeta_c\pm\frac12\zeta_q, \qquad C=C_R+iC_I.$$

By trilinearity, the real-kernel part is deck anti-invariant:

$$\begin{aligned} T_{C_R}(\zeta_+^3)-T_{C_R}(\zeta_-^3) ={}& \sum_{\text{one }q} T_{C_R}(\zeta_c,\zeta_c,\zeta_q)\\ &+\frac14T_{C_R}(\zeta_q,\zeta_q,\zeta_q), \end{aligned}$$

where the sum retains the three labelled placements of the quantum field.

The imaginary-kernel part is deck invariant:

$$\begin{aligned} i\left[ T_{C_I}(\zeta_+^3)+T_{C_I}(\zeta_-^3) \right] ={}& 2iT_{C_I}(\zeta_c,\zeta_c,\zeta_c)\\ &+\frac{i}{2} \sum_{\text{two }q} T_{C_I}(\zeta_c,\zeta_q,\zeta_q). \end{aligned}$$

The derivative placements are occurrence labels and are not collapsed in these sums.

Four associated grades

The source action therefore has the exact contour-conormal support

$$\boxed{ \operatorname{gr}_\Delta^m S_0^{(3)}\neq0 \quad\text{for}\quad m=0,1,2,3, }$$

with character decomposition

$$\boxed{ \begin{array}{c|c|c} \text{kernel part}&\text{deck character}&\text{grades}\\ \hline C_R&-1&1,3\\ iC_I&+1&0,2. \end{array} }$$

The odd pair is causal/action-like. The even pair is statistical/density- matrix data. Both are fixed by one complex source kernel.

Trace typing

On the contour diagonal $\zeta_q=0$, the odd part vanishes, while

$$\Delta^*S_0^{(3)} =2i\mathcal N T_{C_I}(\zeta_c,\zeta_c,\zeta_c)$$

survives. This is compatible with density-matrix normalization through the overall factor $Z^{-1}$; it is not a unitary action difference and should not be placed in $\ker\Delta^*$.

Architectural consequence

The nonlinear finite-time initial state is not merely a statistical line feeding a causal block. At cubic order, one source kernel already packages a deck-graded coefficient object with linked even and odd conormal grades:

$$\boxed{ \mathcal C_{S_0^{(3)}} = \mathcal C_{+}^{(0,2)} \oplus \mathcal C_{-}^{(1,3)}. }$$

This is sector-specific coefficient complexity on the existing doubled initial-boundary carrier. No new support stratum is introduced.

Narrow relation to Entry 1491

Entry 1491 remains correct: the 2005 one-loop tadpole source itself fixes only first contour grade. The present 2014 source independently demonstrates that higher odd and even grades occur when a nonlinear initial action is actually derived. Similar algebra does not establish identity of the two coefficient objects.

Next falsifier

Test the finite-time matching condition that fixes $C$ as a morphism of the trace/Gysin complex. Determine whether the linked four-grade packet is preserved by the one-loop two-point correction in Sec. 5.2 or whether an independent quadratic initial kernel is generated.

Provenance

• Collins--Holman--Vardanyan, arXiv:1408.4801v1, Eqs. (4.7)--(4.8);
• Entries 1480--1481, 1485, 1487, and 1491;
• allocator claim seqclaim-8094f482b58f8d325664b3ad.
• epistemic event ev-000000001612-6b39582c-a788-45dc-8cc2-1b595df816f2.

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