A 3D Shannon--Nyquist Measurement Geometry Foundation

We build a \emph{measurement-first} 3D Shannon--Nyquist geometry in which a repeatable comparison protocol declares exactly which spatial degrees of freedom are \emph{resolved}, i.e.\ faithfully retainable (alias-free reconstructible) from Nyquist samples on the Nyquist lattice forced by Theorem~1.1. Repeatability forces the resolution rule to be an idempotent \(P\in\mathrm{vN}(L_{\mathrm{meas}})\), hence (a.e.) a sharp spectral projector; with ``resolved'' defined in the Shannon--Nyquist sense used here (maximal alias-free retention at the forced Nyquist density),this fixes \(P\) to be the full alias-free Nyquist retention projector (Proposition~2.3),so \(\mathcal W\) is only a \emph{derived label} for \(P\) (unique up to null sets), not an additional protocol choice. We show that two protocol probes—edge-transport response and closed-plaquette response—collapse to a single locked scalar invariant determined intrinsically by this resolved-mode calculus. On the edge side, Shannon-faithfulness reduces 3D resolution to three orthogonal 1D Shannon channels, forcing a Nyquist sampling lattice in \(U^\perp\) (Theorem~1.1). This forcing step yields a canonical discrete representative of \(L_{\mathrm{meas}}\): the unique 19-point Mehrstellen Laplacian \(L_{19}\) (Definition~1.10, Proposition~1.9).In particular, its one-shell (finite-range) character is \emph{derived} from the forced local-rank and symmetry constraints (see Definition~1.6).Its symbol determines a transverse--traceless projected susceptibility \(\kappa_\infty\) (unique \(SO(3)\)-equivariant TT projector) and a Fej\'er-locked spectral shift \(L_{\mathrm{eff}}\). On the loop side the plaquette operator is a closed word in the same forced neighbor transports. A first-order expansion shows the holonomy itself satisfies \(H_{\gamma_\varepsilon}=\Id+\mathcal O(\varepsilon^2)\) at the operator level (Proposition~4.16), so every conjugacy-invariant protocol statistic has forced \(\mathcal O(\varepsilon^2)\) onset; trace-cyclicity appears only as a convenient corollary for trace-based functionals. After quotienting by the universal isotropic contraction fixed by protocol-kernel normalisation, the loop sector exports a loop-independent phase-cumulant scalar locked to the edge invariant. All derivations are internal to the Shannon--Nyquist sampling/information calculus and standard harmonic/spectral analysis.As a demonstration of how the framework can export dimensionless constants from measurement geometry, a specific external identification hypothesis is recorded separately (Appendix~C); under that hypothesis, the canonical evaluation coincides with the fine-structure constant.The public repository~\cite{Stieger2025FejerYork} implements the internal pipeline and reproduces the canonical locked output for the forced one-shell avatar. 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.