Learnable Entire-Kernel Spectral Layers: Discrete Frames, De-Aliasing, BM Regularization, and Beyond-Density MIT Injectivity

We study learnable spectral layers whose feature family is generated by an entire function, motivated by the Master Integral Transform (MIT) of [1]. In a periodic discrete model on T, we define an oversampled multi-β analysis operator A(M) g,K built from an entire generator g and show that, on the one-sided bandlimited subspace PWK (T), it is a tight frame with an explicit inverse, sharp frame bounds, and noise-stability constants governed solely by the Taylor coefficients of g. For general (non-bandlimited) signals we derive exact aliasing identities and quantify two de-aliasing mechanisms: a deterministic multi-resolution cancellation scheme and a randomized rotation estimator with unbiasedness, MSE = O(1 /m), and high-probability bounds. We extend the discrete theory to Td, allowing general multivariate entire generators G(z) = ∑α∈Nd cαzα, and obtain exact inversion and conditioning bounds on tensor bands PW(d) K (Td) with explicit constants. To connect discrete layers to continuum MIT injectivity, we formalize density control of active Taylor indices. We prove that bounded gaps imply positive one-sided interior Beurling–Malliavin density (hence, in particular, positive lower counting density), closing an end-to-end bridge from gap-regularized learning to the injectivity theorem of [1]. For bounded-gap sequences we also give a weighted-series characterization of strong a-regularity, yielding computable surrogate penalties. Finally, we prove two injectivity mechanisms that do not rely on density: (i) a β-analytic injectivity theorem (access to multiple β-channels near 0) for any nonconstant entire kernel, and (ii) a finite-band generic-shift result ensuring invertibility on PWK for nonpolynomial generators. Full-data experiments illustrate conditioning collapse without coefficient floors and confirm the predicted 1 /m de-aliasing variance decay. Contributions (take-home). (i) Certified band-invertible spectral layers: exact inversion + frame bounds + noise stability on PWK (T) and PW(d) K (Td) with constants controlled by Taylor coefficients; (ii) Provable de-aliasing: deterministic multi-resolution cancellation and randomized rotation Monte Carlo with unbiasedness, MSE = O(1 /m), and high-probability bounds; (iii) A closed discrete-to-continuum bridge: bounded-gap activity ⇒ DBM > 0 and hence the lower counting density required by MIT injectivity; (iv) Beyond-density injectivity + nonharmonic structure: two density-free injectivity mechanisms and a monomial rigidity principle. Applications. These results yield MIT-certified modules for machine learning and scientific computing: invertible FFT-like embeddings for grid data, learnable positional encodings, and drop-in replacements for Fourier blocks in neural operators with explicit conditioning control and provable de-aliasing.