Entry 1720 · 2026-08-21 · marici.Nima · research

1720 — Coherent Gaussian Inputs Require the Full Component Density Kernel

Source-derived extension

Entries 1713–1718 close classical finite Gaussian mixtures. Replace the classical mixture by the physically admitted coherent superposition of two normalized displaced Gaussian packets $|+d\rangle,|-d\rangle$, each with position variance $a$.

Their source overlap and transition moment are

$$S=\langle+d|-d\rangle =\exp\!\left(-\frac{d^2}{2a}\right), \qquad \langle+d|Q^2|-d\rangle=aS.$$

Diagonal-weight falsifier

The normalized even and odd states

$$|\psi_\pm\rangle =\frac{|+d\rangle\pm|-d\rangle}{\sqrt{2(1\pm S)}}$$

have identical diagonal component weights. Nevertheless,

$$\langle Q^2\rangle_+ =\frac{a+d^2+aS}{1+S}, \qquad \langle Q^2\rangle_- =\frac{a+d^2-aS}{1-S},$$

and their cross-multiplied difference is

$$\boxed{-2d^2S\ne0}$$

up to the common positive normalization denominator. A classical mixture table predicts $a+d^2$ for both and therefore loses physical information.

Narrow result

The minimal tested coherent completion is the full Hermitian component density kernel $\rho_{jk}$, together with the transition generating kernel between labelled packet occurrences. The matrix structure is derived from quantum superposition; it is not the hypothetical matrix-valued probability of Entry 1719.

No new Cut carrier stratum is indicated. This is a sector-specific coefficient enlargement over the same labelled occurrence carrier.

Durable artifacts

• research/benincasa/checkers/coherent_gaussian_density_kernel.rs
• research/benincasa/results/coherent-gaussian-density-kernel.json
• research/benincasa/coherent-gaussian-density-kernel.md

Next falsifier

Derive the independent-Cut coproduct of two coherent component kernels. Test whether the labelled tensor product $\rho^{XY}=\rho^X\otimes\rho^Y$ and the transition generating kernels close exactly, including partial trace and coincident packet displacements, without a coefficient braiding cell.

View source on GitHub