Manuscript V@1 · V.tex · figure · e7ddf37d

Polynomial closure and coherent witness

The one-oscillator mixed insertion › Same-scalar coherent displacement · lines 2979–3010

V@1#fig:analytic-coherent-witness

metricscalarrepresented quantumdefect / unresolvedsupport / scope

Number graph

common core

\mathscr D_{\mathrm{mix}}

Finite occupied sectors. No universal particle ceiling.

\|\mathcal R_{g,f}\Psi\|\le C_{g,f}\|(N+2)\Psi\|

Shifts of at most two.

dense analytic vectors

essential self-adjointness

Applies to $\mathcal R_{g,f}$ and $A_f^g(\lambda)$ , not the mixed cubic vertex.

Coherent witness

\Psi_0=\mathsf E(0)\otimes\mathsf E(z),\quad \Psi_s=\mathsf E(s/\sqrt2)\otimes\mathsf E(z)

z=v_e,\quad d=2\|v_e\|^2>0

\Delta\langle A_f^g(\kappa)\rangle=\kappa s\, d

Same scalar factor. Represented polynomial witness. No measurement instrument.

Polynomial closure and coherent witness. Number-graph bounds and shifts of at most two give analytic vectors on the common core, yielding essential self-adjointness by Nelson's criterion Nelson. Coherent vectors lie in the number graph domain and exhibit a nonzero same-scalar displacement witness. Neither conclusion constructs an interacting Hamiltonian or a measurement implementation.

Polynomial closure and coherent witness

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