Spectral Geometry of Framed Lattice Quotients

We distinguish local, spectral, metric, and curvature limits for finite framed Cayley quotients. Their flat Laplacians have exact Fourier spectra and a three-regime spectral dimension; normalized heat traces are stable under a vanishing density of edge modifications, uniformly on compact positive-time intervals. A finite translation stencil has a polyhedral stable path norm: Euclidean distortion \(1+\varepsilon\) requires \(\Theta_d(\varepsilon^{-(d-1)/2})\) oriented directions. In two dimensions the polygon count and integer-stencil constant \(\pi/\sqrt2\) are sharp; at exact polygon thresholds, rational lattice directions realize the continuous optimum only for four or eight oriented steps. Positive weighted Dirichlet forms nevertheless Mosco-converge to uniformly elliptic Laplace--Beltrami operators. For fixed endpoints on a refining lattice we establish the complete mesh--time crossover: when \(a/t\to\rho\), the scaled heat-kernel logarithm converges to an explicit Legendre action \(\mathcal A_\rho\). For nonzero displacement the Euclidean action occurs precisely at \(\rho=0\); finite positive ratios retain nonquadratic actions, and the infinite-ratio action collapses. A tilted local limit theorem extends the crossover and prefactor to symmetric positive generating stencils. Separate scale windows yield frame-holonomy and heat-kernel curvature estimators. No Lorentzian or quantum Einstein limit is claimed.