Curvature-Domain Wireless Communications: Gauge-Fixed Signal Spaces, Fredholm Capacity, and Differentiation-Limited Scaling for Continuous Apertures

We develop a curvature-domain formulation for continuous-aperture signaling, in which the transmit phase is represented through its second spatial derivative after quotienting out affine piston-and-tilt gauge freedom. The resulting gauge-fixed synthesis operator is bounded and compact, with sharp Poincare-Wirtinger constant $C_L=L^2/β_1^2$, and its modal Gram spectrum is available in closed form, $ρ_m=(L/β_m)^4$, with $β_m$ the roots of $\cosβ\coshβ=1$. Under a bounded-support square-integrable propagation kernel the tangent operator is Hilbert-Schmidt, so the infinite-dimensional capacity is a well-defined Fredholm-determinant supremum. The optimal signaling law is a dual-budget generalized water-filling with one Lagrange multiplier for curvature power and one for phase excursion. From the exact nonlinear phase-only aperture law we derive the coherent tangent channel with an explicit Frechet remainder bound and a multi-chart atlas for large excursions. At the receiver, curvature inferred from noisy phase samples by second differences has a pentadiagonal noise covariance with spectral norm $Θ(Δx^{-4})$. A deterministic diagnostic suite measures each mechanism against its closed form, with a null run beside every claim: the computed spectrum matches $(L/β_m)^4$ to relative error $3.78\times10^{-15}$; the tangent remainder has fitted slope 1.0000; the dual-budget law is solved with both multipliers strictly active to a KKT residual of $3.90\times10^{-15}$; the derivative-noise bound is approached to 0.999981; and the differentiation-limited branch is observed at exponent 0.2175, then collapses to 0.0576 once the mode count saturates at the Shannon number, while a flat-propagation null run holds at 0.2019. Curvature is thus a well-posed, gauge-invariant coordinate, and below the Shannon number the gauge, rather than the medium, governs the scaling.