Finite-Channel Dirac Tomography on Flat Tori: Nonlinear Recovery, Genuine Blindness, and a Matching Resolution Edge

Abstract

We study what geometric information about an unknown conformal metric on a marked flat spin torus can be recovered from a fixed low-energy compression of its Dirac operator using a fixed finite collection of operator-valued observation channels. After the nonlinear reparameterization $w = e^{-u}$, the compressed commutator data become an exactly Fourier-diagonal linear observation map in $w$, with singular values determined by a lattice covariogram. A uniform lattice-lens estimate gives the edge conditioning exponent $(d+1)/4$ and yields global nonlinear recovery on bounded band-limited conformal classes. Beyond the same frequency edge we construct explicit conformal metrics that are genuinely non-isometric to the flat metric yet exactly invisible to the fixed commutator channels. For any fixed finite family of compressed resolvent channels, the response has no linear term and obeys a normalized Hilbert--Schmidt bound $E_R(\tau) \le C(\tau^2 \beta_d(\Lambda) + \tau^4)$. Combining recovery and blindness gives a noise- and regularity-aware matching resolution edge at $2\Lambda$. Here finite-channel means a fixed finite collection of finite-rank operator-valued channels at each cutoff, not a cutoff-independent number of scalar matrix entries.