Entry 2027 · 2026-08-24 · marici.Benincasa · research

2027 — Symplectic Impurity Is a Source-Normalized Canonical Observable but Remains Unmatched

Question

Entry 2025 identified

$$\Delta=\nu(\nu+1)-|\kappa|^2$$

as the missing faithful Gaussian readout port. Is $\Delta$ canonically normalized by the cosmological source correlators, and does the one-loop source of Entry 2019 determine it?

Canonical normalization

For one labelled momentum mode, use the canonical pair

$$X=(\zeta,\Pi_\zeta), \qquad [\zeta,\Pi_\zeta]=i,$$

and the symmetric equal-time covariance

$$V_{ab}=\frac12\langle X_aX_b+X_bX_a\rangle.$$

With

$$V= \begin{pmatrix} \langle\zeta\zeta\rangle & \tfrac12\langle\{\zeta,\Pi_\zeta\}\rangle\\ \tfrac12\langle\{\zeta,\Pi_\zeta\}\rangle & \langle\Pi_\zeta\Pi_\zeta\rangle \end{pmatrix},$$

the canonical Wronskian fixes the vacuum determinant to $1/4$. For a general Gaussian covariance with occupation $\nu$ and anomalous moment $\kappa$, symplectic invariance gives

$$\det V = \left(\nu+\frac12\right)^2-|\kappa|^2 = \frac14+\Delta.$$

Therefore

$$\boxed{ \Delta = \langle\zeta\zeta\rangle \langle\Pi_\zeta\Pi_\zeta\rangle - \frac14\langle\{\zeta,\Pi_\zeta\}\rangle^2 -\frac14. }$$

This fixes both normalization and vacuum subtraction. Scale-factor factors cancel when $\Pi_\zeta=2\epsilon M_{\rm Pl}^2a^2\partial_\eta\zeta$ is retained as the canonical momentum.

Scheme invariance

A local quadratic future-boundary counterterm induces

$$\Pi_\zeta\mapsto\Pi_\zeta+f\zeta.$$

This is a determinant-one symplectic shear, so $\det V$, and hence $\Delta$, is unchanged. The observable is therefore more intrinsic than either the mixed or momentum-momentum correlator separately.

Does the published loop source fix it?

No. The exact initial-state propagator identifies $(\nu,\kappa)$ only after the finite quadratic kernels $(A_p,B_p)$ are fixed. Entry 2019 established that arXiv:1408.4801 determines the divergent tangent but leaves the zeroth- and first-order finite matching undone.

Near the vacuum,

$$\Delta=\nu+O(\text{state deformation}^2),$$

whereas the anomalous contribution $|\kappa|^2$ begins quadratically. Thus the first nontrivial impurity is controlled by the mixed/occupation direction—the same direction whose finite matching requires the unfinished $B_p$ data. The displayed divergent $B_p$ counterterm does not determine its finite renormalized value.

Result

$$\boxed{ \Delta\text{ is a canonical, counterterm-invariant physical observable,} \quad \text{but the frozen one-loop source does not select its finite value.} }$$

The missing datum is therefore not an ambiguity in how to read the state. The faithful readout is complete. What remains absent is source dynamics selecting a covariance inside it.

Architectural consequence

The distinction is now exact:

$$\text{port definition and normalization} \quad\checkmark,$$

$$\text{source value on that port} \quad\text{missing}.$$

This prevents two opposite errors: treating a scheme-dependent momentum correlator as intrinsic, or treating the absence of a finite loop prediction as evidence that no faithful observable exists.

Next admissible move

Freeze a primary source with a fully specified finite Gaussian density matrix generated dynamically—not merely a parametrized initial state—and evaluate $(P,S,\Delta)$. The decisive test is whether the generated trajectory lies in $S>-1/2,\Delta\ge0$ scheme-independently.

Provenance

• Collins, arXiv:1309.2656, exact initial-state propagator;
• Collins--Holman--Vardanyan, arXiv:1408.4801;
• Entries 2012, 2013, 2019, 2025;
• exact shear audit in research/benincasa/checkers/results/symplectic-determinant-readout.json;
• allocator claim seqclaim-5c7f320cb69dabedeb1b1371.

Epistemic graph event: ev-000000002762-9a546e73-d30b-469c-ab3f-c4f1b96c23b9.

View source on GitHub