A 3D Shannon--Nyquist Measurement Geometry Foundation

We build a three-dimensional Shannon--Nyquist measurement geometry in which \emph{two independent protocol probes}---open Nyquist edges (transport) and closedNyquist plaquettes (loop holonomy)---export \emph{one locked, dimensionless scalar} determined functorially by the measurement pair $(L_{\mathrm{meas}},\mathcal W)$. The resolved (bandlimited) sector is defined by a spectral window projector for a self-adjoint Laplacian-like operator $L_{\mathrm{meas}}$. A Shannon-faithfulness argument fixes the cubic Nyquist lattice; within a declared one-shell finite-range avatar class the discrete representative is forced to be the 19-point Mehrstellen Laplacian $L_{19}$. From its Brillouin-zone symbol, using the canonical sharp window pullback, we extract intrinsic spectral data (local rank, York--TTsusceptibility, and a Fej\'er-lock localisation shift) that control extensive Gaussian multi-mode channels and combine into a single geometric capacity scale. The loop probe lives in the same resolved-mode calculus: loop insertions act by inner conjugation in windowed trace functionals, so the linear response vanishes and the leading closed-plaquette signal is quadratic; after quotienting by a protocol kernel absorbing the universal isotropic contraction, the loop sector exports a loop-independent Wilson--cumulant scalar locked to the edge invariant. An optional appendix records a \emph{conditional} bridge hypothesis identifying the protocol output with the electromagnetic fine-structure constant via a plaquetteholonomy; it is not used in the internal derivation. Reproducibility is supported by a cited script repository that evaluates the pipeline and reproduces the quoted canonical value (Eq.~(4.29)) under the conventions of Appendix~A.5. Note to readers. This work is part of an independent research programme on 3D Shannon–Nyquist measurement geometry, emergent curvature, and projection cosmology. I am not currently affiliated with an institution and therefore do not have a straightforward path to arXiv endorsement. If you are familiar with arXiv policies and think this work might merit a presence there (or have suggestions for improving it to that standard), I would be grateful to hear from you. You can contact me at: shannon@ground-truths.org.