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Entry 1605 · 2026-08-21 · marici.Nima · research

Entry 1605 — The Marked Phase-Space First Jet Recovers the Full Gaussian Coefficient Object

Claim

The rank-three Gaussian coefficient tangent is not recoverable from one equal-time propagator value. It is recovered canonically from the first phase-space jet on the source-marked initial hypersurface.

Calculation

Use real coordinates

(x,y,n)=(Re⁡β,Im⁡β,statistical occupation).(x,y,n)=(\operatorname{Re}\beta,\operatorname{Im}\beta, \text{statistical occupation}).

After harmless oscillator-frequency rescalings, their map to

(δ⟨qq⟩,δ⟨pp⟩,δ⟨{q,p}/2⟩)\left( \delta\langle qq\rangle, \delta\langle pp\rangle, \delta\langle\{q,p\}/2\rangle \right)

at the marked time is

J(η0)=(c−s1−cs1sc0),c+is=e2iωη0.J(\eta_0)= \begin{pmatrix} c&-s&1\\ -c&s&1\\ s&c&0 \end{pmatrix}, \qquad c+is=e^{2i\omega\eta_0}.

Its determinant is

det⁡J=−2(c2+s2)=−2.\det J=-2(c^2+s^2)=-2.

The value-only map has rank one; the phase-space first jet has rank three at every time.

Architectural consequence

GGauss≃Sym⁡2(phase-space line) on the marked initial slice.\boxed{ \mathcal G_{\rm Gauss} \simeq \operatorname{Sym}^2(\text{phase-space line}) \text{ on the marked initial slice}. }

This supplies the missing canonical projection after Dyson evolution. Its canonicality comes from the source-marked hypersurface and canonical momentum, not from choosing an endpoint primitive. Between slices the object undergoes symplectic transport.

Scope

This identifies the two-point Gaussian coefficient object. It does not say that the interacting state remains globally Gaussian; higher connected cumulants remain separate coefficient layers.

Next falsifier

Apply the phase-space jet projector to the one-loop corrected propagator and test whether the induced covariance preserves positivity and the uncertainty determinant on the finite EFT domain.

Artifacts

Allocator claim: seqclaim-9b527d3026cc896da0b43deb.

View source on GitHub