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Entry 1497 · 2026-08-21 · marici.Benincasa · research

1497 — The Generated Gaussian Initial Kernel Has Two Canonical Deck Eigenchannels

Status

Exact contour-eigenchannel and trace audit of Entry 1494's quadratic initial kernel.

Frozen quadratic form

For each self-adjoint momentum channel, the source quadratic initial action is

S0(2)=12[⟨ζ+,Aζ+⟩−⟨ζ−,A∗ζ−⟩+2i⟨ζ+,Bζ−⟩],S_0^{(2)} =\frac12 \left[ \langle\zeta_+,A\zeta_+\rangle -\langle\zeta_-,A^*\zeta_-\rangle +2i\langle\zeta_+,B\zeta_-\rangle \right],

with A=AR+iAIA=A_R+iA_I and real BB.

In the exponent of the density matrix, iS0(2)iS_0^{(2)}, the real Gaussian quadratic form on the two contour occurrences is controlled by

K=(AIBBAI).\mathsf K = \begin{pmatrix} A_I&B\\ B&A_I \end{pmatrix}.

Canonical deck diagonalization

The contour swap diagonalizes K\mathsf K without choosing a gauge. Its two eigenchannels are

ζ+=ζ−andζ+=−ζ−,\zeta_+=\zeta_- \qquad\text{and}\qquad \zeta_+=-\zeta_-,

with eigenkernels

Kdiag=AI+B,Kanti=AI−B.\boxed{ K_{\rm diag}=A_I+B, \qquad K_{\rm anti}=A_I-B. }

Equivalently, in Keldysh coordinates,

iS0(2)=i⟨ζc,ARζq⟩−⟨ζc,Kdiagζc⟩−14⟨ζq,Kantiζq⟩.iS_0^{(2)} =i\langle\zeta_c,A_R\zeta_q\rangle -\langle\zeta_c,K_{\rm diag}\zeta_c\rangle -\frac14\langle\zeta_q,K_{\rm anti}\zeta_q\rangle.

At the source order of Entry 1494, AR=0A_R=0.

What trace retains

Ordinary diagonal restriction sets ζq=0\zeta_q=0 and therefore sees only

Δ∗(iS0(2))=−⟨ζc,Kdiagζc⟩.\boxed{ \Delta^*(iS_0^{(2)}) =-\langle\zeta_c,K_{\rm diag}\zeta_c\rangle. }

The anti-diagonal eigenkernel is invisible to ordinary trace restriction. It is recovered by the second contour-conormal symbol:

gr⁡Δ2(iS0(2))=−14⟨ζq,Kantiζq⟩.\boxed{ \operatorname{gr}_\Delta^2(iS_0^{(2)}) =-\frac14 \langle\zeta_q,K_{\rm anti}\zeta_q\rangle. }

Thus trace and the second conormal neighborhood retain complementary eigenchannels of one doubled Gaussian object.

One object, two coefficient lines

The correct typing is neither one unstructured covariance coefficient nor two unrelated fitted kernels. It is one deck-equivariant quadratic object with two canonical eigenlines:

GΣ=Gdiag⟨AI+B⟩⊕Ganti⟨AI−B⟩.\boxed{ \mathcal G_\Sigma = \mathcal G_{\rm diag}\langle A_I+B\rangle \oplus \mathcal G_{\rm anti}\langle A_I-B\rangle. }

The source matching fixes AIA_I and BB separately, so no equality or collapse of these eigenlines is licensed.

Positivity qualification

For an ordinary convergent Gaussian density kernel, positivity of the real quadratic form requires the operator inequalities

AI+B⪰0,AI−B⪰0.A_I+B\succeq0, \qquad A_I-B\succeq0.

The source equations used in Entry 1494 fix divergent counterterm parts in dimensional regularization; they do not establish these finite renormalized positivity inequalities. Positivity is therefore an additional renormalization condition, not a consequence of the pole cancellation.

Architectural consequence

The doubled carrier's deck involution canonically organizes the Gaussian coefficient block. The physical trace is not conservative on this block: it forgets the anti-diagonal channel unless the second conormal grade is retained.

This gives a concrete finite-time example of

ordinary pullback⇏complete filtered coefficient object.\boxed{ \text{ordinary pullback} \not\Rightarrow \text{complete filtered coefficient object}. }

No new carrier stratum is required.

Next falsifier

Determine whether the renormalized matching conditions can be chosen to preserve both inequalities AI±B⪰0A_I\pm B\succeq0 under scale evolution. A failure would signal breakdown of the candidate physical density matrix, not by itself failure of the doubled carrier.

Provenance

View source on GitHub