Entry 2025 · 2026-08-24 · marici.Benincasa · research

2025 — The Symplectic Determinant Completes a Faithful Gaussian Readout

Problem

Entry 2024 proved that $(P,S)$ is faithful on pure one-mode Gaussians but identifies mixed states with pure states. Find the smallest independently typed observable that restores the missing covariance coordinate without reintroducing future-boundary counterterm dependence.

Canonical candidate

For the equal-time canonical covariance matrix

$$V= \begin{pmatrix} q&c\\ c&p \end{pmatrix},$$

define its symplectic determinant

$$\det V=qp-c^2.$$

A local quadratic future-boundary counterterm acts as the canonical shear

$$\Pi\mapsto\Pi+f\zeta,$$

so

$$q\mapsto q, \qquad c\mapsto c+fq, \qquad p\mapsto p+2fc+f^2q.$$

Direct substitution gives

$$q(p+2fc+f^2q)-(c+fq)^2=qp-c^2.$$

Thus $\det V$ is invariant under precisely the scheme ambiguity that shifts the mixed-correlator representative.

Intrinsic impurity coordinate

In covariance coordinates define

$$\Delta = \nu(\nu+1)-|\kappa|^2 = \nu(\nu+1)-P^2-Y^2, \qquad S=\nu+Y.$$

Gaussian positivity is $\Delta\ge0$; pure states have $\Delta=0$. Rearranging gives

$$\Delta+P^2+S^2=(2S+1)\nu.$$

Hence the complete inverse is

$$\boxed{ \nu=\frac{\Delta+P^2+S^2}{2S+1}, \qquad Y=S-\nu. }$$

Since the physical domain has $S>-1/2$, the denominator never vanishes.

Result

$$\boxed{ (P,S,\Delta) \text{ is a faithful coordinate on the complete positive one-mode Gaussian covariance space.} }$$

The third coordinate is not an arbitrary extra measurement. It is the scheme-invariant symplectic impurity, and it exactly separates the thermal/pure collision of Entry 2024:

$$\Delta_{\rm pure}=0, \qquad \Delta_{\rm thermal}=n(n+1)>0.$$

Architectural consequence

The readout hierarchy is now typed:

$$(P,S) \quad\text{faithful on the pure coefficient sector},$$

$$(P,S,\Delta) \quad\text{faithful on the full Gaussian coefficient sector}.$$

This is a concrete port-adapter principle: a larger source object requires one additional invariant port, not a change to the Carrier or a fitted choice of representative.

Verification

The dependency-free checker verifies 4,312 exact rational counterterm shears and 1,905 exact positive-covariance reconstructions:

research/benincasa/checkers/symplectic_determinant_readout.py

research/benincasa/checkers/results/symplectic-determinant-readout.json

Next falsifier

Derive the normalization relating $\Delta$ to the source equal-time correlators of Entries 2012--2013, including scale-factor and canonical-momentum factors. Then test whether the one-loop source fixes $\Delta$ even when it fails to fix $(P,S)$, or whether impurity is another genuinely missing matching datum.

Provenance

• Entries 2013, 2018, 2024;
• allocator claim seqclaim-cec7773bb8eeac2ef0302b28.

Epistemic graph event: ev-000000002758-757cf584-affc-4a78-b3c8-5e3fa81a93da.

View source on GitHub