Part II: Extended mathematical development
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Multiresolution Signal Calculus, Singular Packets, and Stable Reconstruction This part preserves the complete standalone mathematical branch, with the explicit audit repairs recorded in the accompanying source ledger. Its sixty-three proofs are prose proofs; no additional Lean coverage is asserted. Section and result numbers carry the prefix E to distinguish this scope from Part I.
Abstract
Signals on a lattice admit exact coordinates indexed by the residue classes of every finite index sublattice. Convolution becomes a matrix of packet transfers, while filter banks and packet trees become exact changes of coordinates. We develop this structure on sequence spaces, prove quantitative stability for imperfect and perturbed banks, and give an exact dynamic recursion for adaptive packet trees. A singular signal is written as a smooth component plus step or truncated power packets that record jumps in the signal and its derivatives. Retaining the correct packets leaves only the smooth approximation error. Estimated packets are governed by exact formulas involving value error, region overlap, boundary displacement, and the norm of the smooth approximation operator. The main recovery result treats separated jumps sampled on a uniform grid. An explicit contrast gap gives exact recovery of every jump cell, grid scale location accuracy, amplitude control, and finite norm consistency. Probability bounds cover sub Gaussian noise and repeated measurements. Haar coefficients, two shifted grids, and annihilating differences give complementary localization results. The final sections treat Gibbs oscillation, nonlinear residue interactions, numerical illustrations, and cyclic Cayley coordinates.