Entry 2023 · 2026-08-24 · marici.Benincasa · research

2023 — A Complete Thermal Gaussian Source Selects a Ray Rather Than a Unique Readout

Purpose

Entry 2022 closed an incomplete loop-matching source. As a control, test the same intrinsic readout map on a Gaussian source whose state is fully specified.

Frozen source

Collins, Initial state propagators, arXiv:1309.2656, derives the quadratic initial action and gives the thermal specialization

$$\alpha_k=0, \qquad \beta_k=-\frac{2\omega_k n_k^2}{2n_k+1}, \qquad B_k=\frac{2\omega_k n_k(n_k+1)}{2n_k+1},$$

with Bose occupation

$$n_k=\frac1{e^{\beta\omega_k}-1}\ge0.$$

In intrinsic covariance coordinates this is simply

$$\nu_k=n_k, \qquad \kappa_k=0.$$

Projection to the readout quotient

Entry 2018 uses

$$P=\text{one quadrature of }\kappa, \qquad S=\nu+\text{the orthogonal quadrature of }\kappa.$$

Therefore the complete thermal source maps canonically to

$$\boxed{(P,S)=(0,n_k)}.$$

It lies in the positive domain

$$S>-\frac12$$

for every finite temperature, including the vacuum endpoint $n_k=0$. No counterterm representative or phase convention affects this conclusion because $\kappa=0$.

Result

The readout calculus does produce a physical section when the source supplies a complete state prescription. But the state class alone selects only the ray

$$\boxed{P=0,\quad S\ge0}.$$

A particular temperature profile $\beta(k)$ selects a point or trajectory on that ray. Thus completeness of the coefficient object and uniqueness of the physical readout remain distinct claims.

Architectural lesson

This control separates three layers:

$$\text{positivity}\to\{S>-\tfrac12\},$$

$$\text{thermal source class}\to\{P=0,S\ge0\},$$

$$\text{temperature profile}\to(P(k),S(k))=(0,n_k).$$

Carrier/coherence specifies the legal domain; a source family specifies a subspace; source parameters specify the physical section. The failure in Entry 2019 is therefore not a defect of the readout map but absence of the final source datum.

Next falsifier

Apply the quotient to a complete squeezed Gaussian source with $\kappa\ne0$. Test whether its source phase and squeezing amplitude map injectively to $(P,S)$, or whether the late-time quotient forgets a state direction. This distinguishes a faithful physical readout from another finite-to-one projection.

Provenance

• Collins, arXiv:1309.2656, thermal-state specialization;
• Entries 2018, 2019, and 2022;
• allocator claim seqclaim-6eb1412d19b0e6f01788e0bf.

Epistemic graph event: ev-000000002755-15b9e94a-dd22-485f-8035-35a6dcc095e3.

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