The first twenty-six pages follow the corrected twenty-result formal core. The traditional paper's second part restores the broader mathematical developments of the earlier expanded manuscript. The next thirteen worked pages make their additional hypotheses and conclusions visible.
Four transitions, with the missing input made explicit
| Transition | Additional input | Resulting control |
|---|---|---|
| Residues to lattices | A finite-index sublattice and one representative per coset | Exact addresses in several dimensions |
| Invertibility to stability | Positive lower frame bound and bounded perturbation | Quantified coefficient and reconstruction error |
| Known jumps to recovery | Separated jumps, sampling, a Lipschitz background, and a strict contrast gap | Exact jump cells and bounded location/amplitude error |
| Recovered packets to approximation | A bounded linear approximation operator and its norm | A complete two-term error estimate |
The two parts have different formal evidence
The inherited Lean map covers the twenty designated results of Part I through its 225 supporting targets. The additional traditional proofs in Part II are not silently included in that map. The new pages below are mathematical explanations and exact worked specializations of those developments, with a separate finite-example checker. The original manuscript's broad edge-recovery questions remain broader than separated one-dimensional jumps. Conditions such as one jump per sample cell, an available wavelet support, and a positive detection margin do real work. In particular, the ambiguous observation on V19 still rules out an unrestricted recovery guarantee at its stated noise radius.
How to read the extension
V28 to V30 extend coordinates and stability. V31 to V34 identify what a singular packet contains and measure its error. V35 to V39 give three detection mechanisms. V40 closes the route by combining packet accuracy with a specified approximation operator.
Read this as
A positive recovery theorem can coexist with an impossibility example. Their sampling models, noise bounds, and signal classes decide which conclusion applies.