Entry 1485 · 2026-08-21 · marici.Benincasa · research

1485 — Initial-State Running Is a Relative Cocycle for Schwinger--Keldysh Normalization

Status

Exact normalization-level compatibility test for Entry 1482's statistical-to-causal boundary beta map.

Frozen doubled boundary

Let

$$\Delta_\Sigma:\phi\longmapsto(\phi,\phi)$$

be the diagonal of the doubled initial-boundary field space, and define the causal ideal

$$\mathcal I_\Delta=\ker\Delta_\Sigma^*.$$

The finite-density-matrix trace is the Gysin operation

$$\operatorname{Tr}_\Sigma=(p_\Sigma)_!\Delta_\Sigma^*.$$

Exact diagonal vanishing

Every Hermitian boundary counterterm generated in Entries 1476--1477 enters the closed-time-path action as

$$\delta S_{\Sigma}[\phi_+]-\delta S_{\Sigma}[\phi_-].$$

Consequently,

$$\Delta_\Sigma^* \bigl( \delta S_{\Sigma}[\phi_+]-\delta S_{\Sigma}[\phi_-] \bigr)=0.$$

Equivalently, every generated vertex is odd in the Keldysh quantum field $\phi_q$, and hence vanishes at $\phi_q=0$.

For Entry 1482's beta map this gives the strict factorization

$$\boxed{ \beta_\Sigma^{(4)}: \mathcal C_{\rm stat}\langle c_1\rangle \longrightarrow \mathcal I_\Delta\cap\mathcal C_{\rm causal} \hookrightarrow \mathcal C_{\rm causal}. }$$

Therefore, at the zero-source normalization grade,

$$\boxed{ \operatorname{Tr}_\Sigma\circ\beta_\Sigma^{(4)}=0. }$$

No boundary homotopy is required: the composite vanishes before the Gysin pushforward.

Statistical sector is not erased

This does not annihilate the initial-state coefficient itself. The Gaussian trace retains state and image data in the statistical propagator $G^K$, as established in Entry 1481. The distinction is typed:

$$\text{statistical covariance survives physical trace},$$

while

$$\text{its unitary beta image lies in }\ker\Delta_\Sigma^*.$$

Thus normalization is compatible with the nonsplit coefficient extension without identifying its causal and statistical blocks.

Scope

The theorem concerns the zero-source normalization complex. It does not say that counterterm insertions vanish in differentiated correlation functions: source derivatives can expose the odd-$q$ vertices before diagonal specialization.

Carrier classification

The result uses only the existing doubled boundary, diagonal restriction, and Gysin pushforward. It introduces no new carrier stratum. The nontrivial datum is the relative coefficient extension

$$\mathcal C_{\rm stat} \xrightarrow{\beta_\Sigma^{(4)}} \mathcal I_\Delta\cap\mathcal C_{\rm causal}.$$

This is positive evidence for the refined H2 architecture:

$$\text{shared carrier and trace/Gysin calculus} + \text{sector-specific relative coefficient complex}.$$

Next falsifier

Differentiate the normalization identity once and twice with respect to physical sources. Determine whether the resulting causal Ward hierarchy is generated by the same odd-$q$ ideal and the three source-derived boundary operators, or whether contact terms require a further independently derived coherence cell.

Provenance

• Entries 1480--1482;
• allocator claim seqclaim-62c4636c47cfb0afb532e456.
• epistemic event ev-000000001601-dc6cd700-2a70-4470-9b0a-a3de582ff6bc.

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