Paper V: Polyphase Signal Calculus

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126 \title{\textbf{Projector-Resolved Seed Calculus V}\\[0.45em]
127 \large Polyphase Signal Calculus: Exact Reconstruction, Edge Packets, Wavelets, and Smith Coordinates}
128 \author{Komorebi P.}
129 \date{Expanded Lean-checked edition, version 1.1.0\\\small 6 September 2026}
130 
131 
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191 \renewcommand{\KomorebiShortTitle}{Paper V}
192 \renewcommand{\KomorebiCoverTitle}{Multiresolution\\Signal Calculus}
193 \renewcommand{\KomorebiCoverBlurb}{Exact coordinates reveal how a signal moves through filters, refinements, and reconstruction. Singular packets keep representation, approximation, and recovery distinct.}
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216 \input{atlas-definitions.tex}
217 \begin{document}
218 \KomorebiCover
219 \KomorebiOpeningNote{\textbf{Part I.} The twenty-result formal core develops exact polyphase reconstruction, filter transfers, Hilbert packet operators, finite adaptive trees, edge error bounds, and Smith coordinates.
220 
221 \medskip
222 \textbf{Part II.} The complete extended development treats lattice coordinates, stable banks, singular packets, separated jump recovery, geometric error bounds, and multiscale localization.
223 
224 \medskip
225 \textbf{Visual atlas.} Forty worked plates close the volume. The separate visual guide provides the same examples, and the Lean guide gives the precise released formal contract.}
226 
227 \section*{Reading this edition}\addcontentsline{toc}{section}{Reading this edition}
228 The two branches of Paper V answer related questions at different levels. Part I preserves the twenty results and twenty proofs of the later v1.1.0 manuscript. Part II preserves the broader standalone development with its sixty-three proofs and E-prefixed section numbers. Their overlap is intentional. The combined eighty-three proofs are a record of the two complete branches, rather than eighty-three independent or novel results.
229 
230 For a first reading, follow Part I Sections 1--7. These sections establish the coordinate maps, filter transfers, exact reconstruction, and the separation of packet error from smooth approximation error. Section 11 develops finite adaptive selection. Sections 13--14 give the cyclic Smith realization. Continue with Part II Sections E2--E6 for lattice and stability generalizations, E7--E11 for singular packets and recovery, and E12--E14 for nonlinear and Cayley coordinates. Each visual plate identifies the corresponding manuscript result.
231 
232 The inherited September 2026 release records 225 Lean theorem targets across 23 proof modules supporting all twenty Part I results. This scope does not extend automatically to Part II. The E-prefixed development has prose proofs and retains the mathematical corrections recorded in the prior edition's exact change ledger. No fresh Lean compilation is claimed for this monograph. The current computation review reran twenty exact-model tests with 305,933 counted assertions and sixteen verifier controls. The current companion suite passed thirty-three tests, with twenty-seven overlapping tests from the earlier snapshot also checked. The extended suite passed 11,052 checks in thirteen groups. Fifty-two saved floating-point metrics and eighteen discrete leaves reproduced exactly, and all eighteen companion figure pairs were regenerated. The visual examples passed their own symbolic checks and controls. These checks have finite scopes and are recorded separately in the accompanying verification files. Historical counts in Part I's reproducibility appendix keep their September provenance.
233 
234 The two broad edge recovery conjectures remain open. Part II proves recovery for a declared one dimensional separated jump class under its sampling, contrast, and noise assumptions. These results do not supply universal image segmentation or infinite-tree consistency. Numerical illustrations keep the finite interpretation stated in their captions.
235 \clearpage
236 \tableofcontents
237 \clearpage
238 \listoffigures
239 \listoftables
240 \clearpage
241 \pagenumbering{arabic}
242 \part*{Part I: The released formal core}\addcontentsline{toc}{part}{Part I: The released formal core}
243 \begin{readerroadmap}[title={Coordinates first, decisions second}]
244 A residue decomposition changes the coordinates of a signal without changing its information. Filtering then acts between those coordinates. Approximation enters when an operator compresses a component or when a decision discards coefficients. The formal core follows these operations in that order.
245 \end{readerroadmap}
246 
247 
248 
249 \begin{abstract}
250 This paper develops the signal realization of the projector-resolved seed calculus. The impulse $\delta_0$ is taken as a primitive signal seed and its shift orbit supplies the address system. Residue projectors split a sequence into exact polyphase channels, nested divisibility gives exact refinement, and every finite impulse-response operator becomes a matrix of packet transfers. The standard perfect-reconstruction condition for a filter bank is therefore an analysis--synthesis identity in the common operator language of the series. Haar analysis and every finite-depth wavelet-packet transform are treated on complex $\ell^2$, together with concrete finite-signal models and a predict--update lifting step. The edge problem is separated into two statements. When a discontinuity packet is known and retained exactly, approximation error is exactly the approximation error of the remaining smooth packet. No spectral approximation of the jump is performed. When the packet is estimated, a symmetric-difference bound isolates mask error, amplitude error, and smooth-background error. Numerical studies compare global Fourier truncation, exact known-packet reconstruction, and a deliberately narrow positive-contrast estimator in one and two dimensions, including curved boundaries, corners, and multiple amplitudes. The same cyclic phase action is then placed on the Smith reflection disk. Its pullback through the Cayley transform is a M\"obius action on impedance, and the two-phase case is precisely impedance--admittance inversion. Nonlinear coefficient products obey cyclic packet convolution. The preserved September 2026 release integrates a Lean formalization of all twenty theorem-like statements under the hypotheses stated below: 225 supporting theorem targets in 23 modules. Eighteen regenerated figures, machine-readable numerical results, the 33-test original companion suite, and 36 additional exact-model and verifier tests accompany the paper. Exact signal theorems, limited numerical estimators, classical constructions, and open recovery problems are kept distinct throughout.
251 \end{abstract}
252 
253 \begin{interpretation}[title={Scope and claim boundary}]
254  The paper proves exact polyphase reconstruction, exact finite filter-transfer identities, perfect reconstruction for the stated filter banks, the conditional known-edge theorem, mask and amplitude perturbation bounds, nonlinear residue-convolution laws, and the Smith-coordinate phase action. It does not claim that every discontinuity can be detected automatically, that every form of Gibbs ringing has been eliminated, or that the simple two-class estimator supplied with the release is a universal segmentation method. The estimator is included as one reproducible finite example.  For every finite packet refinement tree, the paper now proves existence of a globally optimal complete pruning, an exact dynamic-programming selector for additive costs, and a deterministic margin-stability certificate. Wavelets, filter banks, the Cayley transform, and the classical Smith grid are treated as classical inputs organized in the common seed--packet--refinement language.
255 \end{interpretation}
256 
257 \clearpage
258 \noindent\textbf{Verification convention.} ``Lean-checked'' refers to the explicitly scoped theorem statements and their supporting declarations, accepted by the pinned Lean kernel. It does not certify the English text automatically, the floating-point Python implementation, physical modelling choices, or the two conjectures. The manuscript-to-declaration correspondence has been reviewed manually. Appendix~\ref{sec:formalization} records it result by result. The September 2026 release records a rerun of the numerical experiments. The current final-edition checks are distinguished in the opening reading note.
259 
260 
261 
262 \section{The signal realization of the seed calculus}\label{core:section:01}
263 
264 Let $\ell^2(\bbZ;\bbC)$, abbreviated $\ell^2(\bbZ)$, be the Hilbert space of square-summable complex discrete signals. The inner product is conjugate-linear in its first argument. A superscript $*$ on a bounded operator denotes its Hilbert adjoint. Write $\delta_n$ for the unit impulse at address $n$ and let the bilateral shift be
265 \[
266 (Sx)[n]=x[n-1].
267 \]
268 Then
269 \[
270 S^n\delta_0=\delta_n.
271 \]
272 The primitive signal seed is therefore $s=\delta_0$, the address operators are the shift words $S^n$, and the completion is the $\ell^2$ norm.
273 
Statement: Impulse seed theorem · Go to proof · Read statement and proof side by side274 \begin{theorem}[Impulse seed theorem]\label{result:01}
275 The closed shift-generated hull of $\delta_0$ is all of $\ell^2(\bbZ)$:
276 \[
277 \boxed{
278 \overline{\operatorname{span}\{S^n\delta_0:n\in\bbZ\}}^{\,\ell^2}
279 =\ell^2(\bbZ).
280 }
281 \]
282 Every finitely supported signal has the exact finite expression
283 \[
284 x=\sum_{n\in\bbZ}x[n]S^n\delta_0.
285 \]
286 \end{theorem}
287 
Proof block: Impulse seed theorem · Go to statement · Read statement and proof side by side288 \begin{proof}
289 The shift orbit is the standard orthonormal basis $\{\delta_n\}_{n\in\bbZ}$. For $x\in\ell^2$, the finite truncations
290 \[
291 x_F=\sum_{n\in F}x[n]\delta_n,\qquad F\subset\bbZ\text{ finite},
292 \]
293 satisfy $\|x-x_F\|_2^2=\sum_{n\notin F}|x[n]|^2\to0$ as the finite sets exhaust $\bbZ$. Thus the unordered Hilbert-space sum $\sum_nx[n]\delta_n$ converges to $x$, and the finite-support statement is exact. The shift is a surjective isometry because $n\mapsto n-1$ is a bijection of $\bbZ$, with inverse shift $n\mapsto n+1$. This proves the closed-span assertion.
294 \end{proof}
295 
296 This realization fixes the language used below:
297 \[
298 \begin{gathered}
299 \text{impulse seed}\longrightarrow\text{shift address}\longrightarrow\text{packet projection}\longrightarrow\text{filter transfer}\\
300 \longrightarrow\text{refinement}\longrightarrow\text{reconstruction}.
301 \end{gathered}
302 \]
303 
304 \begin{remark}[Several completions]\label{core:record:02}
305 The same algebraic seed description applies to finite sequences, $\ell^1$, $\ell^2$, periodic sequences, tempered distributions, and multidimensional lattices. The topology decides which infinite reconstructions are legitimate. Finite-dimensional experiments in this paper use ordinary Euclidean norms; theorem statements specify the ambient norm when it matters.
306 \end{remark}
307 
308 \section{Residue projectors and exact polyphase coordinates}\label{core:section:02}
309 
310 \begin{keyidea}[title={A packet has two useful representations}]
311 The masked packet keeps its original sample addresses and sets the other phases to zero. The decimated channel stores only the retained addresses, now indexed by the coarse coordinate. They carry the same values, but their domains differ. Refinement sums belong on the original lattice. Interleaving converts the channels back to that lattice.
312 \end{keyidea}
313 
314 Fix an integer $N\ge1$. For $0\le r<N$, define
315 \[
316 (P_{N,r}x)[n]
317 =
318 \mathbf 1_{n\equiv r\, (\mathrm{mod}\,N)}x[n].
319 \]
320 The corresponding decimated channel is
321 \[
322 x_r[q]=x[Nq+r].
323 \]
324 
Statement: Polyphase projector theorem · Go to proof · Read statement and proof side by side325 \begin{theorem}[Polyphase projector theorem]\label{result:02}
326 The maps $P_{N,r}$ are bounded orthogonal projectors on $\ell^2$, with operator norm at most one, and satisfy
327 \[
328 P_{N,r}P_{N,s}=\delta_{rs}P_{N,r},
329 \qquad
330 \sum_{r=0}^{N-1}P_{N,r}=I.
331 \]
332 Consequently,
333 \[
334 \boxed{x=\sum_{r=0}^{N-1}P_{N,r}x}
335 \]
336 with no approximation.
337 \end{theorem}
338 
Proof block: Polyphase projector theorem · Go to statement · Read statement and proof side by side339 \begin{proof}
340 Each integer belongs to exactly one residue class modulo $N$. Multiplying the two indicator functions gives zero for distinct classes and the same indicator for equal classes. Summing all residue indicators gives one at every address. Moreover,
341 \[
342 \|P_{N,r}x\|_2^2=\sum_{n\equiv r\ (\mathrm{mod}\,N)}|x[n]|^2\le\|x\|_2^2,
343 \qquad
344 \langle P_{N,r}x,y\rangle=\langle x,P_{N,r}y\rangle.
345 \]
346 The mask is real-valued, so the inner-product identity follows term by term. Hence each projector is self-adjoint, and the ranges of different residue projectors are orthogonal.
347 \end{proof}
348 
349 Define the analysis and synthesis maps
350 \[
351 \cA_Nx=(x_0,\ldots,x_{N-1}),
352 \qquad
353 \cS_N(x_0,\ldots,x_{N-1})[Nq+r]=x_r[q].
354 \]
355 
Statement: Analysis synthesis identity · Go to proof · Read statement and proof side by side356 \begin{corollary}[Analysis--synthesis identity]\label{result:03}
357 \[
358 \boxed{\cS_N\cA_N=I.}
359 \]
360 On the product of the channel spaces one also has $\cA_N\cS_N=I$. Equip this finite product with the Hilbert norm
361 \[
362 \|(x_0,\ldots,x_{N-1})\|_{\mathrm{ch}}^2
363 =\sum_{r=0}^{N-1}\|x_r\|_2^2.
364 \]
365 Then $\cA_N$ is a linear isometric equivalence and
366 \[
367 \|x\|_2^2=\sum_{r=0}^{N-1}\|x_r\|_2^2.
368 \]
369 \end{corollary}
Proof block: Analysis synthesis identity · Go to statement · Read statement and proof side by side370 \begin{proof}
371 Euclidean division gives a unique pair $(r,q)$ with $0\le r<N$ and $n=Nq+r$, including negative $n$. The address maps $n\mapsto(n\bmod N,n\mathbin{\mathrm{div}}N)$ and $(r,q)\mapsto Nq+r$ are inverse. Analysis and synthesis are reindexing by this bijection. Reindexing the nonnegative sum $\sum_n|x[n]|^2$ proves the norm identity and shows that both maps preserve square summability. In Lean the target is $\ell^2(\{0,\ldots,N-1\}\times\bbZ)$, which realizes exactly this Hilbert norm.
372 \end{proof}
373 
374 \begin{figure}[htbp]
375 \centering
376 \includegraphics[width=0.94\textwidth]{01_polyphase_analysis.pdf}
377 \caption{Four-phase analysis of a finite signal. The channels are decimated coordinate packets; interleaving them gives the original signal exactly.}
378 \label{fig:polyphase-analysis}
379 \end{figure}
380 
381 \subsection{Nested refinement}
382 
383 Suppose $N,M\ge1$ and $N\mid M$. Every coarse phase has a disjoint family of fine descendants:
384 \[
385 \operatorname{Child}_{M}(N,r)
386 =
387 \{s:0\le s<M,\ s\equiv r\pmod N\}.
388 \]
389 
Statement: Exact nested refinement · Go to proof · Read statement and proof side by side390 \begin{theorem}[Exact nested refinement]\label{result:04}
391 For $N,M\ge1$ and $N\mid M$,
392 \[
393 \boxed{
394 P_{N,r}
395 =
396 \sum_{\substack{0\le s<M\\s\equiv r\, (\mathrm{mod}\,N)}}P_{M,s}.
397 }
398 \]
399 \end{theorem}
400 
Proof block: Exact nested refinement · Go to statement · Read statement and proof side by side401 \begin{proof}
402 An integer $n$ satisfies $n\equiv r\pmod N$ exactly when its unique residue $s=n\bmod M$ lies among the listed descendants. The fine indicators are disjoint, so their sum is the coarse indicator.
403 \end{proof}
404 
405 This identity is the signal form of the refinement law introduced in Paper II. It is not merely a denser sampling rule. The coarse masked packet, represented on the original integer lattice, is exactly the sum of its fine masked descendants. Decimated channels at different moduli must first be placed in these common coordinates before being added.
406 
407 \section{Linear time-invariant operators as transfer matrices}\label{core:section:03}
408 
409 Let $h$ be a finite impulse response and
410 \[
411 y=h*x,
412 \qquad
413 y[n]=\sum_{q\in\bbZ}h[q]x[n-q].
414 \]
415 Resolve both input and output into $N$ phases. For output phase $k$ and input phase $s$, define the transfer kernel
416 \[
417 \boxed{
418 h_{ks}[m]=h[Nm+k-s].
419 }
420 \]
421 Out-of-range taps are understood to be zero.
422 
Statement: Exact FIR polyphase transfer law · Go to proof · Read statement and proof side by side423 \begin{theorem}[Exact FIR polyphase transfer law]\label{result:05}
424 The output channels satisfy
425 \[
426 \boxed{
427 y_k[n]
428 =
429 \sum_{s=0}^{N-1}\sum_{m\in\bbZ}
430 h_{ks}[m]x_s[n-m].
431 }
432 \]
433 For the numerical channel transform, take $z\in\bbC\setminus\{0\}$ and assume
434 \[
435 \sum_{n\in\bbZ}|x_s[n]z^{-n}|<\infty\quad(0\le s<N).
436 \]
437 Define $X_s(z)=\sum_nx_s[n]z^{-n}$. The output series then also converge absolutely, and
438 \[
439 \bm Y(z)=\bm H^{(N)}(z)\bm X(z),
440 \qquad
441 H^{(N)}_{ks}(z)=\sum_m h[Nm+k-s]z^{-m}.
442 \]
443 \end{theorem}
444 
Proof block: Exact FIR polyphase transfer law · Go to statement · Read statement and proof side by side445 \begin{proof}
446 Evaluate $y$ at address $Nn+k$ and partition every input index by its residue $s$:
447 \[
448 \begin{aligned}
449 y[Nn+k]
450 &=\sum_qh[q]x[Nn+k-q]\\
451 &=\sum_{s=0}^{N-1}\sum_m
452 h[Nm+k-s]x[N(n-m)+s].
453 \end{aligned}
454 \]
455 For a fixed output phase $k$, the change of tap address $q=Nm+k-s$ is a bijection between $\bbZ$ and the pairs $(s,m)$ with $0\le s<N$: choose $s\equiv k-q\pmod N$ and then $m=(q-k+s)/N$. Thus this is a reindexing of the finite filter support, valid for every bilateral input sequence, without a convergence assumption on $x$.
456 
457 The last factor is $x_s[n-m]$, giving the time-domain formula. Under the additional transform hypothesis, each delayed channel has transform
458 \[
459 \sum_nx_s[n-m]z^{-n}
460 =z^{-m}\sum_jx_s[j]z^{-j}.
461 \]
462 Absolute summability permits this bijective reindexing, and only finitely many delayed channels are combined. Taking their finite linear combination gives the matrix formula. The time-domain theorem therefore remains valid even when no numerical bilateral transform exists.
463 \end{proof}
464 
465 \begin{figure}[htbp]
466 \centering
467 \includegraphics[width=0.80\textwidth]{02_filter_transfer_matrix.pdf}
468 \caption{The exact transfer matrix for a seven-tap FIR filter in four phases. Every cell records the delays and coefficients carrying one input packet into one output packet.}
469 \label{fig:filter-matrix}
470 \end{figure}
471 
472 \begin{remark}[Operator-block form]\label{core:record:07}
473 If $H$ denotes convolution by $h$, the same identity is
474 \[
475 H=\sum_{k,s}P_{N,k}HP_{N,s}.
476 \]
477 Thus the familiar polyphase matrix is exactly the transfer-block matrix of the seed calculus.
478 \end{remark}
479 
480 \section{Analysis banks, synthesis banks, and perfect reconstruction}\label{core:section:04}
481 
482 \begin{interpretation}[title={Read the delay in the coordinate system used}]
483 A step in one decimated channel crosses a full block of original samples. A one-sample shift also permutes the residue labels. At two phases, these operations have different matrices. The following calculation keeps the original sample delay visible instead of hiding it in a channel convention.
484 \end{interpretation}
485 \begin{figure}[htbp]\centering
486 \includegraphics[width=.91\linewidth]{monograph_figures/delay_coordinates.pdf}
487 \caption{Exact address comparison for a two-phase split. A delay of one original sample sends the odd channel into the even channel with a coarse delay, and the even channel into the odd channel without a coarse delay. A one-step channel delay moves both original residues by two samples. All plotted values are integer sample addresses.}\label{monograph:fig:delay}
488 \end{figure}
489 
490 Work with finite impulse-response banks. Let $\mathcal L=\bbC[z,z^{-1}]$ be the Laurent-polynomial algebra, with the monomial $z^{-m}$ acting on a bilateral sequence by $x[n]\mapsto x[n-m]$. Let $\bm E\in\operatorname{Mat}_{N\times N}(\mathcal L)$ and $\bm R\in\operatorname{Mat}_{N\times N}(\mathcal L)$ be analysis and synthesis matrices. Their composition acts on all bilateral channel sequences by finite convolutions. At a nonzero $z$ where the input transforms are absolutely convergent, this gives
491 \[
492 \widehat{\bm X}(z)=\bm R(z)\bm E(z)\bm X(z).
493 \]
494 
Statement: Perfect-reconstruction criterion · Go to proof · Read statement and proof side by side495 \begin{theorem}[Perfect-reconstruction criterion]\label{result:06}
496 For $d\in\bbZ$, such a bank reconstructs every input up to a delay of $d$ \emph{channel indices} exactly when the following Laurent-polynomial identity holds:
497 \[
498 \boxed{
499 \bm R(z)\bm E(z)=z^{-d}I_N.
500 }
501 \]
502 If the right side is $c z^{-d}I_N$ with $c\ne0$, reconstruction holds up to gain $c$ and channel delay $d$. After interleaving, this is a delay of $Nd$ original samples. A negative delay denotes an advance.
503 \end{theorem}
504 
Proof block: Perfect-reconstruction criterion · Go to statement · Read statement and proof side by side505 \begin{proof}
506 Finite convolution represents Laurent-polynomial multiplication: composing shifts adds their delays, and distributivity handles finite sums. Thus the matrix identity implies the asserted action. Conversely, let $\bm B=\bm R\bm E-cz^{-d}I_N$ and suppose its action vanishes on every channel sequence. Insert an impulse at channel $s$, address zero, with every other channel zero. Output channel $k$ reads the coefficients of the Laurent polynomial $B_{ks}$ at all addresses. All of these coefficients vanish, so $B_{ks}=0$ for every $k,s$. The representation is faithful, proving necessity as a polynomial identity rather than merely equality at one evaluation point. The displayed unit-gain case is $c=1$.
507 \end{proof}
508 
509 \paragraph{Original-sample delay.}
510 For an arbitrary original-sample delay $D\in\bbZ$, the output phase $k$ has
511 \[
512 y_k[n]=x_{r_k}[n+q_k],\qquad
513 r_k=(k-D)\bmod N,\qquad q_k=(k-D)\mathbin{\mathrm{div}}N.
514 \]
515 Its channel matrix has the sole nonzero entry $z^{q_k}$ in row $k$, column $r_k$. In particular, a one-sample delay at $N=2$ has matrix
516 \[
517 \begin{pmatrix}0&z^{-1}\\1&0\end{pmatrix},
518 \]
519 whereas $z^{-1}I_2$ delays the original signal by two samples. The general address identity is valid for negative sample addresses as well.
520 
521 \subsection{The Haar bank}
522 
523 The Haar analysis matrix is
524 \[
525 \bm E_{\mathrm{H}}
526 =
527 \frac1{\sqrt2}
528 \begin{pmatrix}
529 1&1\\
530 1&-1
531 \end{pmatrix}.
532 \]
533 It is orthogonal, so $\bm R_{\mathrm{H}}=\bm E_{\mathrm{H}}^{\mathsf T}$ and
534 \[
535 \bm R_{\mathrm{H}}\bm E_{\mathrm{H}}=I_2.
536 \]
537 The channels are
538 \[
539 a[n]=\frac{x[2n]+x[2n+1]}{\sqrt2},
540 \qquad
541 d[n]=\frac{x[2n]-x[2n+1]}{\sqrt2}.
542 \]
543 
Statement: Haar energy identity · Go to proof · Read statement and proof side by side544 \begin{corollary}[Haar energy identity]\label{result:07}
545 \[
546 \boxed{
547 \|x\|_2^2=\|a\|_2^2+\|d\|_2^2.
548 }
549 \]
550 \end{corollary}
551 
Proof block: Haar energy identity · Go to statement · Read statement and proof side by side552 \begin{proof}
553 For any $u,v\in\bbC$, the parallelogram identity gives
554 \[
555 \left|\frac{u+v}{\sqrt2}\right|^2+
556 \left|\frac{u-v}{\sqrt2}\right|^2=|u|^2+|v|^2.
557 \]
558 Apply this to $u=x[2n]$, $v=x[2n+1]$ and sum over $n\in\bbZ$. Each summand is nonnegative; the right side is $\|x\|_2^2$ by the two-channel address equivalence. This also proves that the literal low and detail channels belong to $\ell^2$. The real Haar matrix is unitary over $\bbC$, so the finite-signal statement is the same calculation with a finite sum.
559 \end{proof}
560 
561 \begin{figure}[htbp]
562 \centering
563 \includegraphics[width=0.92\textwidth]{03_haar_edge_packets.pdf}
564 \caption{A piecewise-smooth signal and its first Haar low/detail packets. The low channel records local averages. The detail channel records adjacent-sample differences, including pairs that cross a jump.}
565 \label{fig:haar-packets}
566 \end{figure}
567 
568 \subsection{Predict--update lifting}
569 
570 Split $x$ into even and odd channels $e,o$. Define
571 \[
572 d=o-e,
573 \qquad
574 s=e+\frac12d.
575 \]
576 The inverse is
577 \[
578 e=s-\frac12d,
579 \qquad
580 o=d+e.
581 \]
582 
Statement: Exact lifting reconstruction · Go to proof · Read statement and proof side by side583 \begin{theorem}[Exact lifting reconstruction]\label{result:08}
584 The predict--update map $(e,o)\mapsto(s,d)$ is invertible with determinant one and the displayed inverse. Consequently the associated finite transform reconstructs every even-length signal exactly.
585 \end{theorem}
586 
Proof block: Exact lifting reconstruction · Go to statement · Read statement and proof side by side587 \begin{proof}
588 The forward matrix is
589 \[
590 \begin{pmatrix}s\\d\end{pmatrix}
591 =
592 \begin{pmatrix}1/2&1/2\\-1&1\end{pmatrix}
593 \begin{pmatrix}e\\o\end{pmatrix},
594 \]
595 whose determinant is one. Direct substitution gives the inverse formulas.
596 \end{proof}
597 
598 \section{Wavelet packets as addressed refinements}\label{core:section:05}
599 
600 On $\ell^2(\bbZ;\bbC)$, apply the Haar analysis to both children at each stage. For a finite input, assume its length is divisible by $2^\ell$ when a depth-$\ell$ transform is used. A binary word $\alpha\in\{0,1\}^{\ell}$ addresses a depth-$\ell$ packet, denoted $W_{\ell,\alpha}x$. Appending $0$ or $1$ selects the low or high child.
601 
Statement: Wavelet-packet reconstruction and energy · Go to proof · Read statement and proof side by side602 \begin{theorem}[Wavelet-packet reconstruction and energy]\label{result:09}
603 At every finite depth $\ell$,
604 \[
605 \boxed{
606 x=\sum_{\alpha\in\{0,1\}^{\ell}}W_{\ell,\alpha}^{*}W_{\ell,\alpha}x,
607 }
608 \]
609 and
610 \[
611 \boxed{
612 \|x\|_2^2
613 =
614 \sum_{\alpha\in\{0,1\}^{\ell}}
615 \|W_{\ell,\alpha}x\|_2^2.
616 }
617 \]
618 \end{theorem}
619 
Proof block: Wavelet-packet reconstruction and energy · Go to statement · Read statement and proof side by side620 \begin{proof}
621 Write $H_0,H_1$ for the literal low and detail operators. The preceding energy identity implies $\|H_bx\|_2\le\|x\|_2$, so both have bounded Hilbert adjoints. At depth zero the only packet operator is the identity. If the energy identity holds at depth $\ell$, apply the one-level identity to each $W_{\ell,\alpha}x$ and sum; this proves it at depth $\ell+1$.
622 
623 Set $A=\sum_\alpha W_{\ell,\alpha}^*W_{\ell,\alpha}$. This is a finite sum of bounded operators. The energy result says $\langle x,Ax\rangle=\langle x,x\rangle$ for every $x$. Complex polarization, applied also to $x+y$ and $x+\ii y$, gives $\langle y,(A-I)x\rangle=0$ for all $x,y$, hence $A=I$. Applying the operator identity to $x$ proves synthesis. This argument uses the actual Hilbert adjoints of the decimation operators, not a separately postulated inverse symbol.
624 \end{proof}
625 
626 \begin{figure}[htbp]
627 \centering
628 \includegraphics[width=0.92\textwidth]{04_wavelet_packet_energy.pdf}
629 \caption{Energy of the sixteen depth-four Haar packet addresses for the test signal. The packet address is an operator word in the common seed grammar.}
630 \label{fig:wavelet-energy}
631 \end{figure}
632 
Statement: Wavelet boundary633 \begin{classical}[Wavelet boundary]\label{core:record:12}
634 The existence and perfect-reconstruction theory of wavelets and filter banks are classical. Their role here is structural: they provide a fully developed realization in which seed generation, packet addressing, refinement, transfer blocks, and exact synthesis can all be read literally in the common operator language.
635 \end{classical}
636 
637 \section{Known edge packets and spectral reconstruction}\label{core:section:06}
638 
639 \begin{keyidea}[title={Retention changes the approximation target}]
640 The retained packet is supplied as part of the model. It passes through synthesis exactly. The approximation operator acts on the smooth remainder, so its error can be studied without forcing a global smooth basis to reproduce the jump. Finding that packet from observations is a separate recovery problem.
641 \end{keyidea}
642 \begin{figure}[htbp]\centering
643 \includegraphics[width=.95\linewidth]{monograph_figures/retained_packet_cancellation.pdf}
644 \caption{An exact rational eight-coordinate example of retained packet cancellation. The supplied jump packet is kept, while the linear operator averages adjacent pairs in the smooth component. The two error vectors agree coordinate by coordinate. This model illustrates cancellation and does not test packet detection or a Fourier convergence rate.}\label{monograph:fig:cancellation}
645 \end{figure}
646 
647 
648 The central distinction is between approximating a jump globally and retaining it as its own packet.
649 
650 Let
651 \[
652 f=s+j
653 \]
654 where $s$ belongs to a smooth approximation class and $j$ is a known discontinuity packet. Let $Q_m$ be any approximation operator applied only to $s$, and define
655 \[
656 \widetilde f_m=Q_ms+j.
657 \]
658 
Statement: Exact known-edge theorem · Go to proof · Read statement and proof side by side659 \begin{theorem}[Exact known-edge theorem]\label{result:10}
660 For every norm for which the expressions are defined,
661 \[
662 \boxed{
663 f-\widetilde f_m=s-Q_ms,
664 \qquad
665 \|f-\widetilde f_m\|=\|s-Q_ms\|.
666 }
667 \]
668 No approximation of $j$ occurs.
669 \end{theorem}
670 
Proof block: Exact known-edge theorem · Go to statement · Read statement and proof side by side671 \begin{proof}
672 Subtract:
673 \[
674 f-\widetilde f_m=(s+j)-(Q_ms+j)=s-Q_ms.
675 \]
676 Taking the norm gives the second identity.
677 \end{proof}
678 
Statement: Several exact edge packets · Go to proof · Read statement and proof side by side679 \begin{corollary}[Several exact edge packets]\label{result:11}
680 If
681 \[
682 f=s+\sum_{q=1}^{M}j_q
683 \]
684 and every $j_q$ is retained exactly, then
685 \[
686 \left\|f-\left(Q_ms+\sum_qj_q\right)\right\|
687 =\|s-Q_ms\|.
688 \]
689 Curved edges, corners, and different jump amplitudes do not alter the identity.
690 \end{corollary}
Proof block: Several exact edge packets · Go to statement · Read statement and proof side by side691 \begin{proof}
692 The finite sum $j=\sum_{q=1}^M j_q$ is a single retained packet. Applying the preceding cancellation identity gives the result; the empty family $M=0$ is included with sum zero.
693 \end{proof}
694 
Statement: What the theorem does and does not say695 \begin{boundary}[What the theorem does and does not say]\label{core:record:15}
696 The theorem explains why global ringing caused by spectrally approximating a known jump is absent from the reconstructed jump packet. It does not discover the packet. If the packet is wrong, its error is reconstructed exactly as well. Nor does the theorem prevent unrelated truncation errors in the smooth packet, domain-boundary errors, noise, or model mismatch.
697 \end{boundary}
698 
699 \section{Mask, amplitude, and smooth-background error}\label{core:section:07}
700 
701 Let $(X,\Sigma,\mu)$ be a measure space, with measurable masks $\Omega,\widehat\Omega\subseteq X$ and scalar amplitudes $a,\widehat a\in\bbC$ (real amplitudes are a special case). Suppose a scalar jump packet is
702 \[
703 j=a\mathbf 1_{\Omega},
704 \]
705 and an estimated packet is
706 \[
707 \widehat j=\widehat a\mathbf 1_{\widehat\Omega}.
708 \]
709 
Statement: Packet perturbation bound · Go to proof · Read statement and proof side by side710 \begin{theorem}[Packet perturbation bound]\label{result:12}
711 For $1\le p<\infty$, interpreted first as an inequality of extended $L^p$ seminorms,
712 \[
713 \boxed{
714 \|j-\widehat j\|_{L^p}
715 \le
716 |a-\widehat a|\,\mu(\Omega)^{1/p}
717 +
718 |\widehat a|\,\mu(\Omega\triangle\widehat\Omega)^{1/p}.
719 }
720 \]
721 If $\mu(\Omega)$ and $\mu(\Omega\triangle\widehat\Omega)$ are finite, all terms in this displayed jump bound are finite and it is an ordinary real-valued $L^p$ inequality. For the pointwise supremum, and consequently also the essential $L^\infty$ seminorm,
722 \[
723 \|j-\widehat j\|_{\infty}
724 \le |a-\widehat a|+|\widehat a|\,
725 \mathbf 1_{\Omega\triangle\widehat\Omega\ne\varnothing}.
726 \]
727 \end{theorem}
728 
Proof block: Packet perturbation bound · Go to statement · Read statement and proof side by side729 \begin{proof}
730 Write
731 \[
732 a\mathbf 1_{\Omega}-\widehat a\mathbf 1_{\widehat\Omega}
733 =(a-\widehat a)\mathbf 1_{\Omega}
734 +\widehat a(\mathbf 1_{\Omega}-\mathbf 1_{\widehat\Omega}).
735 \]
736 Apply the triangle inequality and use
737 \[
738 \|\mathbf 1_{\Omega}-\mathbf 1_{\widehat\Omega}\|_{L^p}
739 =\mu(\Omega\triangle\widehat\Omega)^{1/p}.
740 \]
741 For the supremum estimate, $|\mathbf1_\Omega-\mathbf1_{\widehat\Omega}|\le1$ pointwise, and this function vanishes everywhere when the symmetric difference is empty. The same pointwise bound also bounds the essential supremum. Nonempty null symmetric differences need not contribute to the essential norm; the stated bound remains valid in that case and need not be sharp. Measurability justifies the $L^p$ triangle inequality and the indicator norm formula.
742 \end{proof}
743 
744 If $s,\widehat s$ are measurable smooth-background models, the same triangle inequality gives, for finite $p$,
745 \[
746 \boxed{
747 \|f-(\widehat s+\widehat j)\|_p
748 \le
749 \|s-\widehat s\|_p
750 +|a-\widehat a|\mu(\Omega)^{1/p}
751 +|\widehat a|\mu(\Omega\triangle\widehat\Omega)^{1/p}.
752 }
753 \]
754 This is the correct separation of errors. Detection error is not hidden inside a generic reconstruction score.
755 
Statement: Stable edge-packet recovery756 \begin{conjecture}[Stable edge-packet recovery]\label{core:record:17}
757 For specified piecewise-smooth classes, contrast bounds, separation conditions, and noise models, there should exist a multiscale packet estimator whose symmetric-difference error and amplitude error converge at rates controlled independently of the smooth approximation rate. A theorem of this kind would turn the conditional known-edge result into a stable data-driven reconstruction theorem. No universal version is asserted here.
758 \end{conjecture}
759 
760 \Needspace{12\baselineskip}
761 \section{One-dimensional numerical study}\label{core:section:08}
762 
763 The one-dimensional signal contains a smooth non-bandlimited background and a positive rectangular packet. Three reconstructions are compared:
764 
765 \begin{enumerate}[label=(\roman*)]
766 \item global Fourier truncation of the complete signal;
767 \item Fourier truncation of the known smooth packet plus the exact jump packet;
768 \item a two-class estimated packet followed by harmonic background filling, spectral smoothing, and jump-amplitude estimation.
769 \end{enumerate}
770 
771 \begin{figure}[htbp]
772 \centering
773 \includegraphics[width=0.98\textwidth]{05_edge_reconstruction_1d.pdf}
774 \caption{One-dimensional reconstruction and absolute error. The known packet removes the edge from the global spectral approximation. The estimated packet is visibly better than the global reconstruction in this limited test but retains smooth-background estimation error.}
775 \label{fig:edge-1d}
776 \end{figure}
777 
Statement: One-dimensional errors778 \begin{certificate}[One-dimensional errors]\label{core:record:18}
779 For $1024$ samples on $[-1,1)$ and Fourier cutoff $m=18$, the retained frequency indices are $-18,\ldots,18$: 37 bins including the zero-frequency bin. The measured values were
780 \[
781 \begin{array}{lrr}
782 \toprule
783 \text{method}&\RMSE&\text{maximum absolute error}\\
784 \midrule
785 \text{global Fourier truncation}&5.4023\times10^{-2}&3.4968\times10^{-1}\\
786 \text{known edge packet}&1.3692\times10^{-4}&1.4503\times10^{-3}\\
787 \text{estimated edge packet}&2.4029\times10^{-2}&5.9530\times10^{-2}\\
788 \bottomrule
789 \end{array}
790 \]
791 The simple estimator recovered the binary mask exactly in this deliberately separated two-class example. That result is a property of this test, not a universal guarantee.
792 \end{certificate}
793 
794 \begin{figure}[htbp]
795 \centering
796 \includegraphics[width=0.84\textwidth]{06_edge_error_vs_resolution.pdf}
797 \caption{Error as spectral resolution increases. The known packet follows only the smooth approximation error. The estimated reconstruction reaches a floor determined by background and amplitude estimation.}
798 \label{fig:edge-resolution}
799 \end{figure}
800 
801 \Needspace{12\baselineskip}
802 \section{Two-dimensional edge packets}\label{core:section:09}
803 
804 The first image consists of a smooth background and one sharp disk. The same three logical levels must remain separate:
805 
806 \begin{enumerate}[label=(\alph*)]
807 \item the exact theorem using the known decomposition;
808 \item a limited estimator designed for a positive-contrast two-class packet;
809 \item the open problem of recovering arbitrary edge packets under noise and model ambiguity.
810 \end{enumerate}
811 
812 \begin{figure}[htbp]
813 \centering
814 \includegraphics[width=0.99\textwidth]{07_known_edge_packet_2d.pdf}
815 \caption{A sharp disk on a smooth background. Global Fourier truncation produces a bright boundary halo and oscillatory interior and exterior error. Retaining the exact edge packet leaves only the smooth-background approximation error.}
816 \label{fig:known-2d}
817 \end{figure}
818 
Statement: Two-dimensional errors819 \begin{certificate}[Two-dimensional errors]\label{core:record:19}
820 For the $192\times192$ disk experiment, a square Fourier mask retains indices $|k_x|,|k_y|\le16$, or $33^2=1089$ bins. The reported errors are
821 \[
822 \begin{array}{lrr}
823 \toprule
824 \text{method}&\text{full-image RMSE}&\text{boundary-band RMSE}\\
825 \midrule
826 \text{global Fourier truncation}&3.7643\times10^{-2}&1.8541\times10^{-1}\\
827 \text{known edge packet}&6.4752\times10^{-5}&1.2709\times10^{-5}\\
828 \text{estimated edge packet}&3.4043\times10^{-3}&8.8834\times10^{-3}\\
829 \bottomrule
830 \end{array}
831 \]
832 The mask intersection-over-union was one in this positive-contrast synthetic case. The amplitude was estimated rather than inserted from the generating model.
833 \end{certificate}
834 
835 \begin{figure}[htbp]
836 \centering
837 \includegraphics[width=0.99\textwidth]{08_estimated_edge_packet_2d.pdf}
838 \caption{The limited data-driven experiment. The observed image alone is thresholded into two intensity classes, cleaned topologically, harmonically filled, spectrally smoothed, and recombined with an estimated jump.}
839 \label{fig:estimated-2d}
840 \end{figure}
841 
842 \subsection{Curves, corners, and multiple amplitudes}
843 
844 The exact theorem does not depend on the shape of the known edge packet. The next experiment contains a disk, a square, and a diamond with different jump amplitudes.
845 
846 \begin{figure}[htbp]
847 \centering
848 \includegraphics[width=0.99\textwidth]{09_multiedge_stress_test.pdf}
849 \caption{Known-packet stress test with curved boundaries, corners, overlap, and different jump amplitudes. The exact packet is not a binary mask multiplied by one amplitude; it is the complete piecewise packet.}
850 \label{fig:multiedge}
851 \end{figure}
852 
Statement: Multiedge errors853 \begin{certificate}[Multiedge errors]\label{core:record:20}
854 The global reconstruction had RMSE $3.3740\times10^{-2}$ and maximum error $3.2807\times10^{-1}$. The known-packet reconstruction had only the remaining smooth-background error. The experiment checks the finite implementation of the theorem for a composite packet; it is not a data-driven multiregion segmentation result.
855 \end{certificate}
856 
857 \section{Sparse wavelet packets and global Fourier modes}\label{core:section:10}
858 
859 A sharp edge is spatially local but spectrally global. Wavelets often provide a more economical coordinate system once enough localized coefficients are retained. To keep the comparison explicit, the numerical study subtracts the signal mean, keeps the largest coefficients in a fixed depth-five Haar packet basis, and compares them with the same number of largest-magnitude discrete Fourier coefficients. These are coefficient-count budgets, not equal bit-storage budgets. The Fourier reconstruction uses the real part of the inverse transform; for the five reported budgets, the selected nonzero-frequency bins occur in conjugate pairs, so this operation removes only floating-point imaginary residuals.
860 
861 \begin{figure}[htbp]
862 \centering
863 \includegraphics[width=0.83\textwidth]{10_sparse_wavelet_fourier.pdf}
864 \caption{Sparse approximation of the test signal. With very few coefficients the global Fourier basis is initially better. After the edge packets are resolved, the Haar packet error decays much faster in this example.}
865 \label{fig:sparse-comparison}
866 \end{figure}
867 
868 \begin{table}[htbp]
869 \centering
870 \caption{Sparse approximation errors for the one-dimensional test signal.}
871 \begin{tabular}{rrr}
872 \toprule
873 \text{retained coefficients}&\text{Haar packet RMSE}&\text{Fourier RMSE}\\
874 \midrule
875 16&$1.7999\times10^{-1}$&$6.6522\times10^{-2}$\\
876 32&$3.8414\times10^{-2}$&$4.9153\times10^{-2}$\\
877 64&$1.6267\times10^{-2}$&$3.8056\times10^{-2}$\\
878 128&$1.7387\times10^{-3}$&$2.5631\times10^{-2}$\\
879 256&$1.5870\times10^{-4}$&$1.7424\times10^{-2}$\\
880 \bottomrule
881 \end{tabular}
882 \label{tab:sparse}
883 \end{table}
884 
Statement: Basis selection885 \begin{boundary}[Basis selection]\label{core:record:21}
886 This table compares two fixed finite coordinate systems for one signal. It does not prove that Haar packets are optimal, that Fourier is inferior on smooth or oscillatory data, or that a single packet tree is best for every edge geometry. The point is that the seed calculus records both systems with the same analysis, transfer, refinement, and synthesis maps.
887 \end{boundary}
888 
889 \section{Finite adaptive packet trees}\label{core:section:11}
890 
891 \begin{readerroadmap}[title={Three levels of a finite tree decision}]
892 First choose a complete pruning, so every terminal path is represented once and synthesis remains exact. Then assign a finite objective to those feasible choices. Additivity permits the local dynamic recursion. Finally, a positive cost gap protects the selected pruning against a declared perturbation envelope. Each step uses a different hypothesis.
893 \end{readerroadmap}
894 
895 The preceding sparse comparison used one fixed depth-five packet basis.  A refinement tree permits a stronger choice: different branches may stop at different depths while the selected packets still form one complete coordinate system.
896 
Statement: Complete packet pruning897 \begin{definition}[Complete packet pruning]\label{core:record:22}
898 Let $\mathcal T$ be a finite rooted refinement tree.  Every node $v$ carries a projector $P_v$, the root satisfies $P_{\varnothing}=I$, and every internal node satisfies
899 \[
900 P_v=\sum_{u\in\operatorname{ch}(v)}P_u,
901 \qquad
902 P_uP_w=0\quad(u\ne w\text{ in }\operatorname{ch}(v)).
903 \]
904 A set $\Lambda$ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets $\Lambda$ exactly once.
905 \end{definition}
906 
Statement: Exact synthesis from every complete pruning · Go to proof · Read statement and proof side by side907 \begin{theorem}[Exact synthesis from every complete pruning]\label{result:13}
908 For every complete pruning $\Lambda$,
909 \[
910 \boxed{\sum_{v\in\Lambda}P_v=I,}
911 \qquad
912 \boxed{x=\sum_{v\in\Lambda}P_vx.}
913 \]
914 Thus changing the selected tree changes the coordinates but does not introduce reconstruction error.
915 \end{theorem}
916 
Proof block: Exact synthesis from every complete pruning · Go to statement · Read statement and proof side by side917 \begin{proof}
918 Starting from the root identity $P_{\varnothing}=I$, replace every unselected internal node by the exact sum of its children.  If a complete pruning contains the root it contains no other node. Otherwise its restriction to each child subtree is itself a complete pruning: every terminal path through that child must be met exactly once. Induction on the finite tree therefore justifies the substitution and its termination precisely at $\Lambda$. Child subtrees have distinct node addresses, so their selected-node sets form a disjoint union. The sum identity uses only root identity and parent-to-child conservation; sibling orthogonality gives the additional interpretation as nonoverlapping projector packets.  Applying the resulting identity to $x$ gives the synthesis formula.
919 \end{proof}
920 
921 Let $y$ denote the observed data.  For a complete pruning $\Lambda$, consider any finite criterion
922 \[
923 J_y(\Lambda)
924 =
925 R_y(\Lambda)
926 +\eta L_y(\Lambda)
927 +\tau K(\Lambda),
928 \]
929 where $R_y$ is a representation cost, $L_y$ is a cross-packet leakage cost, and $K$ is a tree-complexity penalty.  Here $R_y,L_y,K$ take finite real values, and $\eta,\tau$ are fixed real weights (usually nonnegative in applications). No additivity is needed for the first theorem.
930 
Statement: Finite globally optimal stable selector · Go to proof · Read statement and proof side by side931 \begin{theorem}[Finite globally optimal stable selector]\label{result:14}
932 Let $\mathfrak P(\mathcal T)$ be the finite set of complete prunings of $\mathcal T$.
933 
934 \begin{enumerate}[label=(\roman*)]
935 \item A global minimizer
936 \[
937 \Lambda_y^*\in\operatorname*{argmin}_{\Lambda\in\mathfrak P(\mathcal T)}J_y(\Lambda)
938 \]
939 always exists.
940 \item If the minimizer is unique and at least one competing pruning exists, its gap
941 \[
942 \Gamma_y
943 =
944 \min_{\Lambda\ne\Lambda_y^*}
945 \bigl(J_y(\Lambda)-J_y(\Lambda_y^*)\bigr)
946 \]
947 is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion; no real-valued minimum over an empty competitor set is introduced.
948 \item In the case with competitors, if new data $y'$ satisfy
949 \[
950 \sup_{\Lambda\in\mathfrak P(\mathcal T)}
951 |J_{y'}(\Lambda)-J_y(\Lambda)|
952 <\frac{\Gamma_y}{2},
953 \]
954 then $\Lambda_{y'}^*=\Lambda_y^*$.
955 \item Every selected minimizer preserves the exact synthesis identity of the preceding theorem.
956 \end{enumerate}
957 \end{theorem}
958 
Proof block: Finite globally optimal stable selector · Go to statement · Read statement and proof side by side959 \begin{proof}
960 The root-only pruning exists, and every pruning is a subset of the finite node set, so the feasible set is nonempty and finite. Thus the minimum is attained. If competitors exist, their finitely many differences from a unique optimum are strictly positive, so their minimum is positive and is achieved by a competitor. In the singleton case there is no alternative selector under any objective change. In the remaining case, for any competing pruning $\Lambda\ne\Lambda_y^*$,
961 \[
962 \begin{aligned}
963 J_{y'}(\Lambda)-J_{y'}(\Lambda_y^*)
964 &\ge
965 J_y(\Lambda)-J_y(\Lambda_y^*)
966 -2\sup_{\Xi}|J_{y'}(\Xi)-J_y(\Xi)|\\
967 &>\Gamma_y-\Gamma_y=0.
968 \end{aligned}
969 \]
970 Hence the same pruning remains uniquely optimal.  Exact synthesis depends only on completeness of the pruning, not on the cost used to select it.
971 \end{proof}
972 
973 For tree-local costs, the optimum can be found without enumerating all prunings.
974 
Statement: Dynamic-programming best packet basis · Go to proof · Read statement and proof side by side975 \begin{theorem}[Dynamic-programming best packet basis]\label{result:15}
976 Suppose
977 \[
978 J_y(\Lambda)=\sum_{v\in\Lambda}c_v(y).
979 \]
980 For a node $v$, define recursively
981 \[
982 V_y(v)
983 =
984 \begin{cases}
985 c_v(y),&v\text{ terminal},\\[2mm]
986 \displaystyle
987 \min\left\{c_v(y),\sum_{u\in\operatorname{ch}(v)}V_y(u)\right\},
988 &v\text{ internal}.
989 \end{cases}
990 \]
991 Stop at $v$ when $c_v(y)\le\sum_uV_y(u)$, including equality; refine otherwise. This deterministic rule produces a globally optimal complete pruning. With the node costs supplied and child values cached, the evaluator uses one comparison per internal node, one visit per node, and at most $|\mathcal T|-1$ additions. Its arithmetic-and-visit count is therefore at most $3|\mathcal T|$. This is a unit-cost statement about the specified evaluator; it excludes evaluation of the local costs, bit complexity of scalar arithmetic, and materialization or sorting of packet-address strings.
992 \end{theorem}
993 
Proof block: Dynamic-programming best packet basis · Go to statement · Read statement and proof side by side994 \begin{proof}
995 Proceed upward from the terminal nodes.  Every complete pruning below an internal node $v$ has exactly two possible forms: it either stops at $v$, with cost $c_v(y)$, or it refines and is the union of complete child prunings, whose minimum total cost is the sum of the already computed child minima.  The recursion therefore gives the exact minimum on every subtree, in particular at the root. This characterization also proves both directions of the correspondence between recursive cuts and complete covering antichains, for arbitrary positive finite branching, including unary branches. Summing cached child values charges at most one addition per child edge; a finite rooted tree has $|\mathcal T|-1$ such edges. The number of internal nodes is at most $|\mathcal T|$, which gives the stated work bound. Ties do not affect optimality, but the stop-on-ties rule makes the selected pruning unambiguous.
996 \end{proof}
997 
Statement: Local perturbation certificate · Go to proof · Read statement and proof side by side998 \begin{corollary}[Local perturbation certificate]\label{result:16}
999 Assume
1000 \[
1001 |c_v(y')-c_v(y)|\le\varepsilon_v.
1002 \]
1003 Define
1004 \[
1005 E(v)
1006 =
1007 \begin{cases}
1008 \varepsilon_v,&v\text{ terminal},\\[1mm]
1009 \displaystyle
1010 \max\left\{\varepsilon_v,
1011 \sum_{u\in\operatorname{ch}(v)}E(u)\right\},&v\text{ internal}.
1012 \end{cases}
1013 \]
1014 For the complete prunings $\mathfrak P_v$ below $v$, the envelope is exactly
1015 \[
1016 E(v)=\max_{\Lambda\in\mathfrak P_v}\sum_{a\in\Lambda}\varepsilon_a.
1017 \]
1018 The maximum is attained. Every complete-pruning cost changes by at most $E(\varnothing)$, and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors,
1019 \[
1020 \boxed{\Gamma_y>2E(\varnothing)}
1021 \]
1022 is a deterministic certificate that the globally optimal pruning is unchanged.  A singleton pruning set is stable without a gap condition. Assume the random node costs and the joint local-error event are measurable. If a noise model supplies \emph{all} local bounds simultaneously on one event of probability at least $1-\delta$, and the displayed strict-gap condition holds, the same selector is recovered with probability at least $1-\delta$. Individual nodewise probabilities alone do not give this joint confidence without an additional argument, such as a union bound.
1023 \end{corollary}
1024 
Proof block: Local perturbation certificate · Go to statement · Read statement and proof side by side1025 \begin{proof}
1026 For any pruning of the subtree rooted at $v$, either $v$ itself is selected, producing perturbation at most $\varepsilon_v$, or the pruning is the disjoint union of child prunings, producing perturbation at most the sum of the child envelopes.  The same decomposition applied to the nonnegative weights $\varepsilon_a$ proves equality with the maximum weight: at each node choose either the stop option or a maximizing pruning in each child, whichever gives the larger total. Finiteness ensures these choices attain their maxima. Evaluating each cost at a minimizer of the other cost bounds the difference of the two optimal values by the uniform envelope. The selector statement follows from the strict-gap theorem. The finite deterministic comparisons defining the selector make its preservation event measurable when the random node costs are measurable. The joint local-error event is contained in the selector-preservation event, so monotonicity of probability proves the probabilistic assertion.
1027 \end{proof}
1028 
1029 \begin{figure}[htbp]
1030 \centering
1031 \includegraphics[width=0.97\textwidth]{17_adaptive_packet_tree.pdf}
1032 \caption{A globally optimal mixed-depth Haar packet pruning.  The selected antichain covers the full tree, so its synthesis is exact even though different branches stop at different depths.}
1033 \label{fig:adaptive-tree}
1034 \end{figure}
1035 
1036 \begin{figure}[htbp]
1037 \centering
1038 \includegraphics[width=0.97\textwidth]{18_adaptive_tree_stability.pdf}
1039 \caption{The dynamic-programming optimum agrees with exhaustive enumeration.  A positive gap larger than twice the declared perturbation envelope certifies that the selected tree is stable.}
1040 \label{fig:adaptive-stability}
1041 \end{figure}
1042 
Statement: Adaptive-tree optimum1043 \begin{certificate}[Adaptive-tree optimum]\label{core:record:27}
1044 For the depth-five example, the complete binary tree has $458{,}330$ admissible prunings.  Dynamic programming and exhaustive enumeration returned the same minimum with residual $0$.  The selected pruning was
1045 \[
1046 \{000,001,0100,0101,011,1\},
1047 \]
1048 and mixed-depth synthesis reconstructed the signal with maximum residual $8.88\times10^{-16}$.  The optimum gap was $3.1909\times10^{-4}$, while twice the declared perturbation envelope was $6.4\times10^{-5}$, so the finite selector is certified stable.
1049 \end{certificate}
1050 
Statement: Finite versus asymptotic selection1051 \begin{boundary}[Finite versus asymptotic selection]\label{core:record:28}
1052 The theorem resolves the finite adaptive-tree existence, exact-synthesis, global-optimality, and margin-stability problem.  It does not by itself prove minimax rates, statistical consistency on an infinite tree, or a universal choice of representation and leakage costs for every signal class.
1053 \end{boundary}
1054 
1055 
1056 \section{Nonlinear signal packets}\label{core:section:12}
1057 
1058 Linear time invariance is not required for exact packet coordinates. Let $A$ and $B$ be formal power series over a commutative coefficient ring (in particular $\bbC$). Set
1059 \[
1060 A(z)=\sum_{n\ge0}a_nz^n,
1061 \qquad
1062 A_i=P_{N,i}A.
1063 \]
1064 All phase indices range from $0$ to $N-1$. For another representation $B(z)$, multiplication gives a nonlinear or bilinear interaction.
1065 
Statement: Cyclic product law · Go to proof · Read statement and proof side by side1066 \begin{theorem}[Cyclic product law]\label{result:17}
1067 \[
1068 \boxed{
1069 P_{N,k}(AB)
1070 =
1071 \sum_{i+j\equiv k\, (\mathrm{mod}\,N)}A_iB_j.
1072 }
1073 \]
1074 For $q\in\bbN$, the $q$th-power identity is
1075 \[
1076 P_{N,k}(A^q)
1077 =
1078 \sum_{i_1+\cdots+i_q\equiv k}
1079 A_{i_1}\cdots A_{i_q}.
1080 \]
1081 \end{theorem}
1082 
Proof block: Cyclic product law · Go to statement · Read statement and proof side by side1083 \begin{proof}
1084 Every coefficient in $A_iB_j$ has degree congruent to $i+j$ modulo $N$. At any fixed degree $n$, the Cauchy coefficient sum is finite. Sorting these terms by the residues of their two indices gives the first identity coefficient by coefficient. For the power identity, use induction on $q$ and the finite product law. At $q=0$, the index set contains the single empty tuple, its product is $1$, and its residue sum is $0$; hence the right side is $1$ for $k=0$ and $0$ otherwise, as required. No convergence is needed for these formal identities. For analytic power series, the same identities hold wherever the series are absolutely convergent and their Cauchy products may be evaluated.
1085 \end{proof}
1086 
1087 \begin{figure}[htbp]
1088 \centering
1089 \includegraphics[width=0.76\textwidth]{14_nonlinear_phase_coupling.pdf}
1090 \caption{Norms of pairwise product interactions for five residue phases. The label in each cell gives the output phase $(i+j)\bmod5$.}
1091 \label{fig:nonlinear-coupling}
1092 \end{figure}
1093 
1094 For a finite nonlinear shift operator
1095 \[
1096 F(x)[n]=x[n]+\alpha x[n]x[n-1],
1097 \]
1098 the exact packet coordinates are simply
1099 \[
1100 F_r((x_s)_s)=P_{N,r}F\!\left(\sum_sP_{N,s}x\right).
1101 \]
1102 Analysis followed by this packet system and synthesis reproduces the direct nonlinear output exactly. The numerical residual in the companion implementation was zero to machine representation.
1103 
1104 \section{Smith coordinates as cyclic signal coordinates}\label{core:section:13}
1105 
1106 Let $Z_0>0$ be a reference impedance and
1107 \[
1108 z=\frac{Z}{Z_0}=r+\ii x.
1109 \]
1110 The reflection coefficient is
1111 \[
1112 \boxed{
1113 \Gam=\frac{z-1}{z+1},
1114 \qquad
1115 z=\frac{1+\Gam}{1-\Gam}.
1116 }
1117 \]
1118 
Statement: Cayley geometry1119 \begin{classical}[Cayley geometry]\label{core:record:30}
1120 The map sends the finite passive half-plane $r\ge0$ onto the closed unit disk with $\Gam=1$ omitted. Adjoining the point at infinity in impedance adds that missing boundary point; the open right half-plane maps conformally onto the open disk. Constant resistance and reactance become the familiar Smith circles. This conformal geometry is classical and is not replaced by polyphase refinement.
1121 \end{classical}
1122 
1123 Writing $\Gam=u+\ii v$, the grid equations are
1124 \[
1125 \left(u-\frac{r}{1+r}\right)^2+v^2=\frac1{(1+r)^2},
1126 \]
1127 and, for $x\ne0$,
1128 \[
1129 (u-1)^2+\left(v-\frac1x\right)^2=\frac1{x^2}.
1130 \]
1131 
1132 Fix
1133 \[
1134 \omega_N=e^{2\pi\ii/N}
1135 \]
1136 and rotate the reflection coefficient by
1137 \[
1138 R_{N,k}(\Gam)=\omega_N^k\Gam.
1139 \]
1140 
Statement: Polyphase M"obius action · Go to proof · Read statement and proof side by side1141 \begin{theorem}[Polyphase M\"obius action]\label{result:18}
1142 Regard normalized impedance as the projective line $\mathbb P^1(\bbC)=\bbC\cup\{\infty\}$. For $N\ge1$, the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is
1143 \[
1144 \boxed{
1145 T_{N,k}(z)
1146 =
1147 \frac{(1+\omega_N^k)z+(1-\omega_N^k)}
1148 {(1-\omega_N^k)z+(1+\omega_N^k)}.
1149 }
1150 \]
1151 As maps on $\mathbb P^1(\bbC)$ it satisfies
1152 \[
1153 T_{N,k}\circ T_{N,\ell}=T_{N,k+\ell\, (\mathrm{mod}\,N)}.
1154 \]
1155 \end{theorem}
1156 
Proof block: Polyphase M"obius action · Go to statement · Read statement and proof side by side1157 \begin{proof}
1158 Let $C(z)=(z-1)/(z+1)$. Then
1159 \[
1160 T_{N,k}=C^{-1}\circ R_{N,k}\circ C.
1161 \]
1162 For a division-free construction, represent a point by a nonzero pair $[u:v]$, identifying $[cu:cv]=[u:v]$ for $c\ne0$. With $w=\omega_N^k$, use the matrix
1163 \[
1164 A(w)=\frac12\begin{pmatrix}1+w&1-w\\1-w&1+w\end{pmatrix}.
1165 \]
1166 It has determinant $w\ne0$, preserves nonzero pairs, and commutes with rescaling, so it induces a projective map. Direct multiplication gives $A(w)A(w')=A(ww')$ and $A(1)=I$. Since $\omega_N^N=1$, this proves the cyclic composition law modulo $N$. The Cayley coordinate of $[u:v]$ is $[u-v:u+v]$, and applying $A(w)$ changes this pair to $[w(u-v):u+v]$. This proves the conjugacy also at poles. For $[z:1]$, taking the coordinate ratio gives the displayed finite-chart formula.
1167 \end{proof}
1168 
Statement: Impedance admittance is the two-phase split · Go to proof · Read statement and proof side by side1169 \begin{corollary}[Impedance--admittance is the two-phase split]\label{result:19}
1170 For $N=2$ and finite $z\ne0$,
1171 \[
1172 \boxed{T_{2,1}(z)=z^{-1}.}
1173 \]
1174 On the full projective line it exchanges $0$ and $\infty$. Thus a half-turn of the reflection disk exchanges normalized impedance and admittance.
1175 \end{corollary}
Proof block: Impedance admittance is the two-phase split · Go to statement · Read statement and proof side by side1176 \begin{proof}
1177 $\omega_2=\exp(\pi\ii)=-1$, so $A(\omega_2)[u:v]=[v:u]$. The finite ratio is $1/z$ when $z\ne0$; the coordinate swap also gives $[0:1]\leftrightarrow[1:0]$. No ordinary division by zero is used to represent infinity.
1178 \end{proof}
1179 
1180 \begin{figure}[htbp]
1181 \centering
1182 \includegraphics[width=0.88\textwidth]{11_polyphase_smith_chart.pdf}
1183 \caption{The classical Smith grid equipped with sixteen exact phase rays and four radial seed orbits. The Cayley transform creates the chart; the projector calculus supplies the nested cyclic coordinate system.}
1184 \label{fig:smith-chart}
1185 \end{figure}
1186 
1187 \subsection{Radial seeds and physical phase}
1188 
1189 For $0\le\rho<1$ and angle $\theta$,
1190 \[
1191 \Gam=\rho e^{\ii\theta}
1192 \]
1193 gives
1194 \[
1195 z(\rho,\theta)
1196 =\frac{1+\rho e^{\ii\theta}}{1-\rho e^{\ii\theta}},
1197 \]
1198 with
1199 \[
1200 r(\rho,\theta)=\frac{1-\rho^2}{1-2\rho\cos\theta+\rho^2},
1201 \qquad
1202 x(\rho,\theta)=\frac{2\rho\sin\theta}{1-2\rho\cos\theta+\rho^2}.
1203 \]
1204 A fixed $\rho$ is a constant-mismatch circle. Refining $N$ increases angular resolution without changing $\rho$.
1205 
1206 \begin{figure}[htbp]
1207 \centering
1208 \includegraphics[width=0.78\textwidth]{12_smith_nested_refinement.pdf}
1209 \caption{The same reflection circle at phase orders $2,4,8,16,32$. Each coarse phase ray appears in every finer grid whose order is a multiple of its order. The exact residue-projector refinement law concerns coefficient packets, not sums of impedance points.}
1210 \label{fig:smith-refinement}
1211 \end{figure}
1212 
1213 For a lossless transmission line,
1214 \[
1215 \Gam_{\rm in}=\Gam_L e^{-2\ii\beta\ell}.
1216 \]
1217 One $N$-phase step corresponds to a line-length increment $\lambda/(2N)$, up to the chosen orientation convention. Hence the phase address has a direct electrical interpretation.
1218 
1219 \section{Dyadic--odd phase factorization and boundary weighting}\label{core:section:14}
1220 
1221 Write
1222 \[
1223 N=2^sM,
1224 \qquad M\text{ odd}.
1225 \]
1226 The cyclic index splits by the Chinese remainder theorem:
1227 \[
1228 \bbZ/N\bbZ
1229 \cong
1230 \bbZ/2^s\bbZ\times\bbZ/M\bbZ.
1231 \]
1232 
Statement: Dyadic odd packet factorization · Go to proof · Read statement and proof side by side1233 \begin{proposition}[Dyadic--odd packet factorization]\label{result:20}
1234 After identifying the residue labels by the Chinese remainder map, the phase projector factors as
1235 \[
1236 \boxed{
1237 P_{N,j}=P_{2^s,j_2}P_{M,j_o}.
1238 }
1239 \]
1240 \end{proposition}
1241 
Proof block: Dyadic odd packet factorization · Go to statement · Read statement and proof side by side1242 \begin{proof}
1243 The two congruence conditions are independent because $2^s$ and $M$ are coprime. Their indicator functions multiply to the indicator of the unique residue class modulo $N$ satisfying both.
1244 \end{proof}
1245 
1246 The first dyadic step is exceptional in Smith coordinates because it is the inversion $z\mapsto z^{-1}$. Higher dyadic levels refine the open/short duality into quarter-, eighth-, and finer phase sectors.
1247 
1248 Away from its pole $\Gam=1$, the inverse Cayley map has derivative
1249 \[
1250 \frac{dz}{d\Gam}=\frac{2}{(1-\Gam)^2},
1251 \]
1252 while normalized admittance has derivative proportional to $(1+\Gam)^{-2}$. Away from $\Gam=\pm1$, this motivates the symmetric sensitivity
1253 \[
1254 \boxed{
1255 w_{\rm Smith}(\Gam)
1256 =
1257 \frac1{|1-\Gam|^2}
1258 +
1259 \frac1{|1+\Gam|^2}.
1260 }
1261 \]
1262 
1263 \begin{figure}[htbp]
1264 \centering
1265 \includegraphics[width=0.78\textwidth]{13_smith_sensitivity.pdf}
1266 \caption{The proposed symmetric refinement weight becomes large near the open- and short-circuit boundary points. It is a condition-based adaptive coordinate, not an asserted optimal sampling law.}
1267 \label{fig:smith-weight}
1268 \end{figure}
1269 
Statement: Weighted refinement1270 \begin{boundary}[Weighted refinement]\label{core:record:34}
1271 The sensitivity formula follows directly from the exact coordinate derivatives. Using it to choose an optimal adaptive grid, interpolation rule, or measurement design is a separate numerical problem. The release visualizes the weight but does not prove optimality.
1272 \end{boundary}
1273 
1274 \section{Boundary-localized error and claim control}\label{core:section:15}
1275 
1276 The strongest visual distinction between global Fourier reconstruction and exact packet retention is seen in a band around the discontinuity.
1277 
1278 \begin{figure}[htbp]
1279 \centering
1280 \includegraphics[width=0.94\textwidth]{15_boundary_leakage.pdf}
1281 \caption{Error restricted to a narrow boundary band. Global truncation concentrates large error on the jump. The known-packet reconstruction has only the small smooth-packet error in the same band.}
1282 \label{fig:boundary-error}
1283 \end{figure}
1284 
1285 \begin{figure}[htbp]
1286 \centering
1287 \includegraphics[width=0.90\textwidth]{16_claim_status.pdf}
1288 \caption{Proof and claim boundary. Finite adaptive-tree selection is now proved; the remaining open rows concern universal recovery and infinite-tree or asymptotic selection.}
1289 \label{fig:claim-status}
1290 \end{figure}
1291 
Statement: Edge-packet approximation class1292 \begin{conjecture}[Edge-packet approximation class]\label{core:record:35}
1293 There should exist natural function spaces in which a signal is represented as
1294 \[
1295 f=s+\sum_q a_q\mathbf1_{\Omega_q},
1296 \]
1297 with separate norms for smooth approximation, boundary geometry, packet amplitude, and packet identification. A projector-resolved approximation theory on such spaces should yield rates that do not mix smooth truncation error with discontinuity-location error.
1298 \end{conjecture}
1299 
1300 
1301 \section{Implementation and reproducibility}\label{core:section:16}
1302 
1303 The companion package is organized by mathematical layer:
1304 
1305 \begin{longtable}{L{0.27\textwidth}L{0.64\textwidth}}
1306 \caption{Numerical companion modules.}\label{tab:modules}\\
1307 \toprule
1308 \textbf{module}&\textbf{implemented content}\\
1309 \midrule
1310 \endfirsthead
1311 \toprule
1312 \textbf{module}&\textbf{implemented content}\\
1313 \midrule
1314 \endhead
1315 \texttt{polyphase.py}&residue projectors, analysis/synthesis, refinement, exact FIR transfer kernels, polyphase convolution\\
1316 \texttt{filterbanks.py}&Haar analysis/synthesis, complete Haar packet trees, energy accounting, predict--update lifting\\
1317 \texttt{adaptive.py}&complete-pruning enumeration, exact mixed-depth synthesis, dynamic-programming optimization, and stability certificates\\
1318 \texttt{edges.py}&one- and two-dimensional Fourier truncation, known-packet reconstruction, narrow two-class estimators, harmonic filling, error metrics\\
1319 \texttt{smith.py}&Cayley coordinates, cyclic M\"obius action, Smith circles, sensitivity weight, coefficient projectors\\
1320 \texttt{nonlinear.py}&coefficient packets, cyclic product convolution, nonlinear shifted packet reconstruction\\
1321 \texttt{experiments.py}&all machine-readable numerical certificates and tabular summaries\\
1322 \texttt{visuals.py}&the eighteen figure pairs used in the paper\\
1323 \bottomrule
1324 \end{longtable}
1325 
1326 The September 2026 release records recovery and rerunning of the original companion source: its 33 tests passed, and all 18 figure pairs and numerical tables were regenerated. The Lean package contributes 20 independent exact-model tests and 16 tests of the verifier itself; these account for another 36 passing tests. The exact-model suites count 305,933 assertions. These counts are separate from the 225 Lean theorem targets.
1327 
1328 All numerical computations are CPU-based. Floating-point residuals measure finite implementations and may vary at rounding level with the scientific libraries. The universal mathematical statements are supported by the Lean proofs; the Python modules are not compiled or verified by Lean. The adaptive-tree certificate uses floating-point costs and an exhaustive finite comparison, so its displayed gap is numerical evidence rather than an interval-arithmetic proof about those computed costs. Appendix~\ref{sec:formalization} explains the verification boundary.
1329 
Statement: Finite numerical residuals1330 \begin{certificate}[Finite numerical residuals]\label{core:record:36}
1331 The largest recorded residuals were
1332 \[
1333 \begin{array}{lr}
1334 \toprule
1335 \text{identity}&\text{residual}\\
1336 \midrule
1337 \text{polyphase reconstruction}&0\\
1338 \text{nested refinement}&0\\
1339 \text{polyphase convolution}&4.44\times10^{-16}\\
1340 \text{Haar reconstruction}&8.88\times10^{-16}\\
1341 \text{wavelet-packet reconstruction}&2.22\times10^{-15}\\
1342 \text{adaptive-tree optimum versus exhaustive search}&0\\
1343 \text{adaptive mixed-depth synthesis}&8.88\times10^{-16}\\
1344 \text{lifting reconstruction}&4.44\times10^{-16}\\
1345 \text{Smith rotation conjugacy}&3.34\times10^{-16}\\
1346 \text{Smith impedance--admittance duality}&0\\
1347 \text{nonlinear product-packet reconstruction}&1.33\times10^{-15}\\
1348 \text{nonlinear shifted-filter reconstruction}&0\\
1349 \bottomrule
1350 \end{array}
1351 \]
1352 The Haar and wavelet energy residuals were $5.68\times10^{-14}$ and $2.27\times10^{-13}$, respectively.
1353 \end{certificate}
1354 
1355 \section{Conclusions}\label{core:section:17}
1356 
1357 The signal realization makes the seed, address, packet, transfer, refinement, and synthesis operations concrete. The objects and laws can be read directly:
1358 \[
1359 \boxed{
1360 \begin{aligned}
1361 \text{seed}&=\delta_0,\\
1362 \text{address}&=S^n,\\
1363 \text{packet}&=P_{N,r}x,\\
1364 \text{transfer}&=P_{N,k}HP_{N,s},\\
1365 \text{refinement}&=P_{N,r}=\sum_{s\succ r}P_{M,s},\\
1366 \text{synthesis}&=\sum_rP_{N,r}x,\\
1367 \text{nonlinearity}&=P_{N,r}F\left(\sum_sP_{N,s}x\right).
1368 \end{aligned}
1369 }
1370 \]
1371 Filter banks, wavelet packets, adaptive complete prunings, edge packets, and Smith coordinates inhabit the same operator grammar, but their claims differ. Polyphase and filter-bank identities are exact. Wavelet perfect reconstruction is exact. The known-edge theorem is exact conditional on the decomposition. Estimated-edge performance depends on a detection model. The Smith chart remains the classical Cayley geometry equipped with an exact cyclic refinement algebra. Keeping those distinctions explicit is part of the calculus, not an editorial afterthought.
1372 
1373 The next paper in the series turns from discrete signal transfer to semigroups, resolvents, conditioning, stochastic evolution, fractional powers, and PDE kernels.
1374 
1375 \appendix
1376 \section{Algorithm statements}\label{core:section:18}
1377 
1378 \subsection{Exact polyphase filtering}
1379 
1380 Given $x$, $h$, and $N$:
1381 
1382 \begin{enumerate}
1383 \item Form $x_s[n]=x[Nn+s]$ for $0\le s<N$.
1384 \item Form every transfer kernel $h_{ks}[m]=h[Nm+k-s]$.
1385 \item Compute
1386 \[
1387 y_k[n]=\sum_s\sum_mh_{ks}[m]x_s[n-m].
1388 \]
1389 \item Interleave the $y_k$ channels.
1390 \end{enumerate}
1391 
1392 The algorithm is algebraically identical to direct convolution; its purpose is to expose independent and interacting packets for block computation and refinement.
1393 
1394 \subsection{Conditional known-edge reconstruction}
1395 
1396 Given a decomposition $f=s+j$:
1397 
1398 \begin{enumerate}
1399 \item Apply the selected smooth approximation $Q_m$ to $s$ only.
1400 \item Retain $j$ exactly.
1401 \item Return $Q_ms+j$.
1402 \end{enumerate}
1403 
1404 The error certificate is $\|s-Q_ms\|$.
1405 
1406 \subsection{Limited positive-contrast estimator}
1407 
1408 The included finite estimator does the following:
1409 
1410 \begin{enumerate}
1411 \item Split the observed values into two classes by an Otsu threshold.
1412 \item Keep coherent connected components and fill holes.
1413 \item Harmonically fill the selected packet to estimate the missing background.
1414 \item Smooth the filled background spectrally.
1415 \item Estimate one packet amplitude by a robust median residual.
1416 \item Recombine the estimated smooth and edge packets.
1417 \end{enumerate}
1418 
1419 This procedure is useful for the supplied separated synthetic case. It assumes a positive-contrast two-class packet and is not promoted to a general edge theorem.
1420 
1421 \section{Numerical record}\label{core:section:19}
1422 
1423 \begin{table}[H]
1424 \centering
1425 \caption{Main edge-study measurements.}
1426 \begin{tabular}{lrrr}
1427 \toprule
1428 \text{case}&\text{global RMSE}&\text{known-packet RMSE}&\text{estimated RMSE}\\
1429 \midrule
1430 1D rectangular packet&$5.4023\times10^{-2}$&$1.3692\times10^{-4}$&$2.4029\times10^{-2}$\\
1431 2D disk packet&$3.7643\times10^{-2}$&$6.4752\times10^{-5}$&$3.4043\times10^{-3}$\\
1432 2D multiedge packet&$3.3740\times10^{-2}$&\multicolumn{2}{c}{known composite packet only}\\
1433 \bottomrule
1434 \end{tabular}
1435 \end{table}
1436 
1437 \begin{table}[H]
1438 \centering
1439 \caption{Main software artifacts.}
1440 \begin{tabular}{ll}
1441 \toprule
1442 \text{artifact}&\text{purpose}\\
1443 \midrule
1444 \texttt{results/numerical\_results.json}&complete machine-readable certificate record\\
1445 \texttt{tables/numerical\_summary.csv}&flat summary of errors and residuals\\
1446 \texttt{tables/sparse\_representation.csv}&wavelet/Fourier coefficient study\\
1447 \texttt{tests/}&automated finite identity and estimator tests\\
1448 \texttt{figures/}&PNG and vector-PDF figure pairs\\
1449 \texttt{VERIFY\_PAPER.py}&combined manuscript, numerical, and Lean checks\\
1450 \bottomrule
1451 \end{tabular}
1452 \end{table}
1453 
1454 \section{Notation crosswalk}\label{core:section:20}
1455 
1456 \begin{longtable}{L{0.20\textwidth}L{0.33\textwidth}L{0.37\textwidth}}
1457 \caption{Notation crosswalk.}\label{tab:notation}\\
1458 \toprule
1459 \textbf{common symbol}&\textbf{signal realization}&\textbf{interpretation}\\
1460 \midrule
1461 \endfirsthead
1462 \toprule
1463 \textbf{common symbol}&\textbf{signal realization}&\textbf{interpretation}\\
1464 \midrule
1465 \endhead
1466 $s$&$\delta_0$&primitive signal seed\\
1467 $A_n$&$S^n$&address operator\\
1468 $P_{N,r}$&residue mask&phase packet projector\\
1469 $T_{ks}$&$P_{N,k}HP_{N,s}$&input-to-output phase transfer\\
1470 $E,R$&analysis and synthesis matrices&coordinate conversion and reconstruction\\
1471 $W(\gamma)$&filter or path weight&refinement or implementation cost\\
1472 $\Delta(F)$&coarse/fine signal defect&aliasing, leakage, or estimator inconsistency\\
1473 $\operatorname{Hol}(\gamma)$&closed phase transfer&cycle consistency in a network\\
1474 $\Gam$&reflection coefficient&Smith-disk coordinate\\
1475 \bottomrule
1476 \end{longtable}
1477 
1478 \clearpage
1479 \section{Formalization and reproducibility record}\label{core:section:21}\label{sec:formalization}
1480 
1481 \subsection{What has been checked}
1482 
1483 The preserved Part I manuscript has twenty theorem-like statements: Theorems, Propositions, and Corollaries, counted together. The Lean project contains 225 explicit supporting theorem targets in 23 proof modules. All twenty statements have an explicit scope row and principal declaration targets. The September 2026 release records a successful build and axiom audit with warnings treated as errors. No fresh Lean build is asserted for this October final edition. The audit inspected all 827 declarations in the project namespace, including definitions and automatically generated declarations. The latter is an implementation census, not a count of 827 mathematical theorems.
1484 
1485 The accepted axiom dependencies are \texttt{propext}, \texttt{Classical.choice}, and \texttt{Quot.sound}. No admitted proof or additional project axiom is permitted. The verifier checks the imported modules, registered theorem inventory, dependency pins, source hashes, and coverage of the twenty-row scope contract. Its successful status is \texttt{LEAN\_CHECKS\_PASS}. The correspondence between a mathematical sentence in this PDF and its Lean type is a manual mathematical review, not an automated semantic-equivalence test.
1486 
1487 The source-level formalization was completed before this manuscript revision. The revision integrates those checked hypotheses and conclusions into the paper itself; it does not increase the Lean theorem count by counting editorial changes as new proofs. Eleven foundational formal-power-series proofs in \texttt{CyclicSeries} are adaptations of Paper IV and are identified as such in the provenance record.
1488 
1489 \subsection{Result-by-result correspondence}
1490 
1491 The following table summarizes the reviewed statement boundary. Full qualified declaration names, source locations, and supplementary targets appear in \texttt{MANUSCRIPT\_TARGET\_MAP.md} and the machine-readable scope contract. Every row below has supporting kernel-checked declarations; numerical certificates, classical-input paragraphs, remarks, and conjectures are not additional theorem rows.
1492 
1493 \begingroup\small
1494 \begin{longtable}{L{0.09\textwidth}L{0.28\textwidth}L{0.53\textwidth}}
1495 \caption{The twenty manuscript results and their Lean statement scopes.}\label{tab:lean-map}\\
1496 \toprule
1497 \textbf{Result}&\textbf{Principal modules}&\textbf{Checked mathematical content}\\
1498 \midrule\endfirsthead
1499 \toprule
1500 \textbf{Result}&\textbf{Principal modules}&\textbf{Checked mathematical content}\\
1501 \midrule\endhead
1502 \ref{result:01}&HilbertPackets&Complex $\ell^2$ impulse expansion, finite-support identity, unitary shifts, and closed shift-generated span.\\
1503 \ref{result:02}&HilbertPackets&Bounded self-adjoint residue projectors, pairwise annihilation, completeness, and orthogonal ranges.\\
1504 \ref{result:03}&IntegerPolyphase; HilbertPackets&Signed quotient/remainder coordinates, both inverse identities, and the joint Hilbert norm of the channels.\\
1505 \ref{result:04}&HilbertPackets&Literal bounded-projector refinement for positive divisible moduli.\\
1506 \ref{result:05}&FIRTransfer; ZTransform&Finite-support tap reindexing for all bilateral inputs; numerical channel transforms under weighted absolute summability.\\
1507 \ref{result:06}&FilterBank; FiniteTransforms&Finite Laurent/FIR bank composition, faithful action on sequences, gain-delay reconstruction criterion, and the $Nd$ original-sample delay.\\
1508 \ref{result:07}&HaarChannels; HaarHilbert&Literal even/odd input samples, normalized complex low/detail channels, and $\ell^2$ energy equality.\\
1509 \ref{result:08}&Lifting; FiniteTransforms&Predict/update inverse maps, determinant one, and both flat finite-signal inverse identities.\\
1510 \ref{result:09}&WaveletPackets&Every finite depth of literal Haar packet operators, energy equality, and reconstruction using actual Hilbert adjoints.\\
1511 \ref{result:10}&KnownEdge&Exact cancellation of one retained packet, and equality of approximation errors.\\
1512 \ref{result:11}&KnownEdge&Finite families of retained packets, with no geometry assumption on the packets.\\
1513 \ref{result:12}&MaskPerturbation&Measurable-mask symmetric-difference norm, extended finite-$p$ error bound, pointwise and essential supremum bounds, and smooth-background error.\\
1514 \ref{result:13}&CompletePrunings; PruningSums&Recursive cuts correspond to complete covering antichains; literal finite node sums preserve synthesis.\\
1515 \ref{result:14}&PruningSums&Finite arbitrary-objective minimization, attained positive gap with competitors, strict stability, and the singleton case.\\
1516 \ref{result:15}&FiniteAdaptive; PruningAlgorithm&Deterministic stop-on-ties selector, global optimum, arbitrary positive finite branching, and the stated unit-operation count.\\
1517 \ref{result:16}&PruningSums; PruningAlgorithm&Attained maximum error envelope, optimal-value perturbation, strict-gap selector stability, and probability transfer from a joint event.\\
1518 \ref{result:17}&CyclicSeries&Formal Cauchy-product residue convolution and the finite-tuple power identity, including the empty product at exponent zero.\\
1519 \ref{result:18}&SmithAction; SmithCyclic&Nonzero homogeneous pairs modulo scaling, Cayley conjugacy, the actual exponential phase, and cyclic composition on the projective line.\\
1520 \ref{result:19}&SmithCyclic&Two-phase coordinate swap, $0\leftrightarrow\infty$, and finite impedance inversion when $z\ne0$.\\
1521 \ref{result:20}&CRT&Signed integer residue-mask factorization for coprime moduli, including the dyadic--odd case.\\
1522 \bottomrule
1523 \end{longtable}
1524 \endgroup
1525 
1526 The remaining foundational modules \texttt{Packets}, \texttt{FiniteSelection}, and \texttt{BinaryAdaptive} provide generic coordinate, finite-minimum, and binary-tree lemmas used or retained alongside the concrete constructions. The Hilbert target is the square-summable space on a product of indices; it is not the supremum-norm function space that Lean would otherwise assign to an unqualified finite family of channels.
1527 
1528 \subsection{Three bridges that matter for the statement}
1529 
1530 \paragraph{Finite filters and numerical transforms.}
1531 The filter-bank proof is an algebraic theorem about finite Laurent-polynomial kernels acting faithfully on bilateral sequences. Equality at a single complex value of $z$ would be insufficient to prove a polynomial identity; the impulse argument avoids that error. The separate transform theorem proves the finite-filter action agrees with evaluation wherever the weighted input series converge absolutely. An arbitrary bilateral $\ell^2$ sequence need not have an absolutely convergent numerical $z$-transform. The all-sequence time-domain theorem makes no such assumption.
1532 
1533 \paragraph{Recursive trees and set-valued prunings.}
1534 The formal tree permits any positive finite arity at an internal node, including arity one. A node has a position in the tree, so equal data labels on distinct nodes do not identify the nodes. A recursive cut is either a stop or a collection of child cuts. Its equivalence with a covering antichain is proved in both directions. The cost, envelope, and operator sums are then shown to equal actual finite sums over the selected node set. This connects the recursive optimizer to the manuscript's set-valued definition instead of assuming that connection.
1535 
1536 \paragraph{Hilbert operators and finite experiments.}
1537 The coordinate maps, literal Haar channels, and packet operators are constructed on complex $\ell^2$. Square summability and boundedness are established before adjoints are used. The finite Python experiments separately check indexing, reconstruction, and energy at finite lengths. They illustrate these operators but are not the objects executed by the Lean kernel. Likewise, the Python Smith routine evaluates only finite charts; the Lean projective construction supplies the pole-safe universal statement.
1538 
1539 \subsection{September 2026 checks and numerical conventions}
1540 
1541 \begin{center}
1542 \begin{tabular}{lr}
1543 \toprule
1544 \textbf{Check family}&\textbf{Accepted count}\\
1545 \midrule
1546 Original numerical companion tests&33\\
1547 Independent exact-model tests&20\\
1548 Verifier positive and negative controls&16\\
1549 Total tests in these three families&69\\
1550 Exact assertions inside the independent model suites&305,933\\
1551 Registered Lean theorem targets&225\\
1552 Lean proof modules&23\\
1553 Regenerated figure pairs&18\\
1554 \bottomrule
1555 \end{tabular}
1556 \end{center}
1557 
1558 The September 2026 record states that all scalar entries in the recovered numerical-results JSON reproduced exactly in its environment. This includes the 458,330-pruning depth-five comparison, its selected six-node pruning, the optimum gap, and the reconstruction residuals. The five Fourier sparse-approximation budgets were also checked for conjugate-pair closure. The revised figures and tables use the regenerated data. This exact replay of the stored floating-point numbers is stronger than a visual similarity check, but it is not a promise of bitwise equality on every platform.
1559 
1560 For the one-dimensional low-pass experiment, ``cutoff 18'' means 37 signed frequency bins, not 18 retained coefficients. The disk experiment uses a square $33\times33$ frequency mask, not a radial cutoff. The sparse comparison uses largest-coefficient budgets instead of either low-pass convention. The full-image and boundary-band RMSEs are normalized by the number of samples in the respective comparison set. The supplied inpainting routines perform a fixed finite number of periodic neighbor-averaging iterations; ``harmonic filling'' describes that iteration, not a separately proved exact continuum Dirichlet solver.
1561 
1562 The independent exact-model suite checks signed polyphase/FIR addresses, exact lifting and normalized Haar arithmetic, mask bounds, and exhaustive finite-tree objectives. Its tree census includes all 626 ordered rooted tree shapes with one through eight nodes and 129,958 node subsets. These finite checks can catch implementation or convention errors; they are not substitutes for the universal Lean proofs. The verifier controls exercise failure paths, including incomplete scope contracts and unregistered proof targets. Mandatory verification conditions use explicit exceptions rather than Python assertions, so optimized Python cannot silently remove them.
1563 
1564 \subsection{Pinned environment and rerunning the release}
1565 
1566 The Lean toolchain is \texttt{leanprover/lean4:v4.34.0-rc2}. The project pins mathlib to
1567 \begin{center}\small\texttt{aeb72f1e9981b6d35b426b16dca7759be3e7ba06}.\end{center}
1568 The Lake manifest records the remaining dependency revisions. The numerical environment versions are recorded in \texttt{verification/numerical\_environment.json}. Source and artifact hashes appear in \texttt{SHA256SUMS.txt}. The original manuscript, source, numerical data, and figures remain available under provenance for comparison; the expanded PDF and its TeX are the manuscript of that September 2026 release.
1569 
1570 From the preserved release root (\texttt{source/complete/original\_snapshot} in the final package), the combined verification entry point is:
1571 \begin{lstlisting}[language=bash,numbers=none]
1572 python3 VERIFY_PAPER.py
1573 \end{lstlisting}
1574 This runs the Lean verifier, companion tests, numerical regeneration/comparison, and manuscript consistency checks. Lean, Lake, the pinned dependencies, the Python requirements, and a TeX installation must be available for the corresponding full checks. Missing prerequisites or failed checks return failure; static checks alone never report kernel verification. The individual numerical generation command is
1575 \begin{lstlisting}[language=bash,numbers=none]
1576 cd companion
1577 python3 run_all.py
1578 python3 -m pytest -q tests
1579 \end{lstlisting}
1580 The TeX source resolves the eighteen vector figures from the manuscript's figure directory. The checked build uses repeated pdfLaTeX passes, then text, cross-reference, and page-render inspection. Different pdfTeX versions can produce different binary PDF hashes even when the rendered document is unchanged.
1581 
1582 In the September 2026 release environment, a disclosed runtime adapter was needed for Lean executable-path discovery: it redirects a read of the current process's numeric \texttt{/proc} executable link to \texttt{/proc/self/exe}. It does not modify the Lean binary, elaborator, kernel, axioms, or proof terms. The adapter source and invocation are retained with the verification evidence; ordinary installations can run the pinned toolchain directly.
1583 
1584 \subsection{What remains open}
1585 
1586 The September 2026 finite theorem scope is complete in the stated sense: all twenty theorem-like statements have reviewed, accepted formal counterparts, and that release records a rerun of the numerical companion. The two edge-recovery conjectures remain open research directions. No universal segmentation theorem, infinite-tree consistency result, minimax rate, or optimal Smith sampling theorem is inferred from the finite results. Classical Smith-grid and transmission-line formulae provide application context; those physical and historical assertions are not additional Lean-certified theorem rows.
1587 
1588 Across the available Papers I--V manuscripts there are 168 theorem-like results. Their release maps contain 1,456 raw theorem targets; removing two shared qualified names and eleven known Paper-IV-to-V adaptations leaves 1,443 after those known duplicate removals. This is a source census, not a claim of that many independent or novel mathematical results. Papers I--IV were counted from their release material and bundled audit records; only Paper V was freshly built for that September 2026 release. These historical cross-paper counts are not a current census of the October final companions.
1589 
1590 \clearpage
1591 \part*{Part II: Extended mathematical development}\addcontentsline{toc}{part}{Part II: Extended mathematical development}
1592 \begin{center}\large Multiresolution Signal Calculus, Singular Packets, and Stable Reconstruction\end{center}
1593 \noindent 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.
1594 \setcounter{section}{0}\setcounter{figure}{0}\setcounter{table}{0}
1595 \renewcommand{\thesection}{E\arabic{section}}
1596 \renewcommand{\theHsection}{extension.\arabic{section}}
1597 \renewcommand{\thefigure}{E\arabic{figure}}
1598 \renewcommand{\theHfigure}{extension.\arabic{figure}}
1599 \renewcommand{\thetable}{E\arabic{table}}
1600 \renewcommand{\theHtable}{extension.\arabic{table}}
1601 \graphicspath{{extension_figures/}}
1602 
1603 \begin{abstract}
1604 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.
1605 \end{abstract}
1606 
1607 
1608 \section{Introduction}\label{extension:section:01}\label{sec:introduction}
1609 
1610 Polyphase decompositions, filter banks, lifting schemes, and wavelet packets are standard parts of multirate signal processing and harmonic analysis. Their classical development is described in \cite{CrochiereRabiner1983,Vaidyanathan1993,VetterliKovacevic1995,StrangNguyen1996,Daubechies1992,Mallat2009}. These theories are often introduced through several parallel languages. Residue classes appear in polyphase analysis. Matrix symbols appear in filter banks. Trees appear in wavelet packets. Singularities are handled through a different approximation or detection method. The first aim of this article is to place these constructions in one exact coordinate system.
1611 
1612 The second aim is to separate representation from recovery. If a signal has already been written as
1613 \[
1614 f=s+j
1615 \]
1616 with a smooth part \(s\) and a singular packet \(j\), then
1617 \[
1618 \widetilde f=Qs+j
1619 \]
1620 satisfies
1621 \[
1622 f-\widetilde f=s-Qs.
1623 \]
1624 This identity is exact, but it does not discover \(j\). A useful signal theory must therefore distinguish three errors. The first is the approximation error of the smooth component. The second is the error in the estimated singular packet. The third is the amplification caused by applying a reconstruction operator to that packet error. The middle sections of the paper isolate all three terms.
1625 
1626 The singular packets considered here range from indicator functions of regions to truncated powers
1627 \[
1628 \frac{(t-\tau)_+^m}{m!}.
1629 \]
1630 The latter encode jumps in derivatives of order \(m\). They permit a piecewise smooth function to be reduced to a globally Sobolev remainder under explicit trace assumptions. This gives a precise version of the informal statement that a discontinuity should be represented directly rather than approximated by a global smooth basis.
1631 
1632 The recovery theory begins with one dimensional separated jumps. For bounded deterministic noise, thresholding adjacent sample contrasts recovers the exact grid cells containing the jumps whenever the jump size exceeds the combined smooth variation and noise by an explicit margin. The same proof gives location and amplitude bounds. A sub Gaussian version follows from a union bound. Averaging repeated measurements reduces the required threshold in the expected way. Haar coefficients provide a second detection mechanism. Two grids shifted by half a cell guarantee that every jump is well placed in at least one grid, which removes the most elementary dyadic alignment failure.
1633 
1634 The literature on singularity detection is much broader than the results proved here. Wavelet modulus maxima are developed in \cite{MallatHwang1992}. Best basis selection in packet libraries was introduced in \cite{CoifmanWickerhauser1992}. Nonlinear approximation is surveyed in \cite{DeVore1998}. Geometric image models, wedgelets, and curvelets appear in \cite{Donoho1999,CandesDonoho2004}. The present article does not replace those theories. Its narrower purpose is to give a common exact calculus in which coordinate reconstruction, adaptive basis selection, singular packet estimation, and smooth approximation remain logically separate.
1635 
1636 The article is organized as follows. \Cref{sec:signal_seeds,sec:polyphase} develop lattice addresses and exact residue coordinates in any dimension. \Cref{sec:convolution,sec:filter_banks} express convolution and filter banks in those coordinates, including stability under imperfect reconstruction and filter perturbation. \Cref{sec:packet_trees} treats arbitrary pruned packet trees and stability of the best basis decision. \Cref{sec:singular_packets,sec:packet_error} introduce step and higher order singular packets, exact reconstruction identities, and error formulas. \Cref{sec:jump_recovery,sec:localization} prove deterministic and probabilistic recovery results. \Cref{sec:gibbs,sec:nonlinear} treat Gibbs oscillation and nonlinear interactions. The final sections record finite numerical illustrations and a secondary cyclic Cayley realization.
1637 
1638 Several conclusions are deliberately excluded. The paper does not prove a universal image segmentation theorem. Exact recovery of separated jumps in one dimension does not imply stable recovery of arbitrary boundaries in higher dimensions. Retaining a correct singular packet does not remove noise or model error. An adaptive tree selected by an additive objective need not remain optimal when global hardware costs or cross packet interactions are introduced. The Cayley coordinate section is a secondary conformal realization and is not used in the recovery theorems.
1639 
1640 \subsection{Main results}
1641 
1642 The first result is an exact coordinate theorem. Let \(L\) be a finite index sublattice of \(\ZZ^d\), and let \(R\) contain one representative from every coset. The analysis map
1643 \[
1644 x\longmapsto (x_r)_{r\in R},
1645 \qquad
1646 x_r(\ell)=x(r+\ell),
1647 \]
1648 is an isometric isomorphism on every \(\ell^p\) space. This statement includes the usual one dimensional polyphase decomposition, but it also applies to arbitrary finite index lattice packets in several dimensions. If \(L'\leq L\), the coarse packets split exactly into their fine descendants. A convolution operator becomes a matrix of convolution operators on \(L\), with entries
1649 \[
1650 h_{rs}(m)=h(r-s+m).
1651 \]
1652 The matrix is therefore a coordinate representation of the original operator, not a reduced model.
1653 
1654 The filter bank results distinguish exact reconstruction from quantitative near reconstruction. Exact recovery with delay \(d\) is the pointwise matrix identity
1655 \[
1656 R(z)E(z)=z^{-d}I.
1657 \]
1658 When this identity fails, the reconstruction error is controlled by the essential supremum of the delay compensated multiplier defect. If the filters themselves are perturbed, \cref{thm:filter_perturbation} separates the analysis error, the synthesis error, and their product. The frame bound in \cref{thm:frame_perturbation} shows that a lower singular value gap survives every sufficiently small perturbation. This makes the distinction between algebraic invertibility and stable numerical reconstruction explicit.
1659 
1660 The packet tree results treat every admissible pruning, not only complete trees of a fixed depth. Unitary local splits give a unitary global transform. Local frame bounds propagate recursively in the nonunitary case. If the cost of a pruning is additive over retained nodes, then
1661 \[
1662 F(v)
1663 =
1664 \min\left\{
1665 c(v),
1666 \sum_{w\in\operatorname{child}(v)}F(w)
1667 \right\}
1668 \]
1669 gives the exact optimal tree. The cost and the selected basis are stable when the node costs change by less than the gap separating the optimal pruning from its competitors.
1670 
1671 The singular packet results begin with trace data. If a piecewise smooth function has a jump in its derivative of order \(m\) at \(\tau_q\), then the packet
1672 \[
1673 \jump{f^{(m)}}_{\tau_q}
1674 \frac{(t-\tau_q)_+^m}{m!}
1675 \]
1676 removes exactly that trace jump. Subtracting all packets through order \(r-1\) leaves a global \(W^{r,p}\) remainder. The packet coefficients are uniquely determined by the one sided trace jumps. This gives a finite and explicit singular part for the class covered by \cref{thm:higher_order_packet_decomposition}.
1677 
1678 The reconstruction formula with an estimated packet is
1679 \[
1680 f-\widehat f_m
1681 =
1682 (I-Q_m)s+(I-Q_m)(j-\widehat j).
1683 \]
1684 This identity identifies the two terms that must be controlled. The exact packet formulas in \cref{sec:packet_error} then evaluate or bound \(\|j-\widehat j\|\) from amplitude error, region overlap, boundary displacement, and the order of a derivative jump. A value jump has location modulus \(|\tau-\widehat\tau|^{1/p}\) in finite \(L^p\), while every positive order truncated power packet is Lipschitz in its location.
1685 
1686 The principal recovery theorem uses adjacent sample contrasts. Let
1687 \[
1688 b_h=Mh+2\varepsilon
1689 \]
1690 be the largest contrast that can be produced by a Lipschitz background and bounded noise in one grid cell. If
1691 \[
1692 a_{\min}>2b_h,
1693 \]
1694 then every threshold between \(b_h\) and \(a_{\min}-b_h\) detects exactly the cells containing jumps. The jump locations are recovered within \(h/2\), and each amplitude error is at most \(b_h\). The probability theorem replaces the deterministic noise term by
1695 \[
1696 2\sigma\sqrt{\log(2n/\delta)}.
1697 \]
1698 Two half shifted Haar grids provide a second recovery mechanism. Every point is at least one quarter of a cell away from the boundary of one of the grids, so the Haar coefficient gap can be made independent of dyadic alignment.
1699 
1700 \subsection{Notation}
1701 
1702 The following symbols recur throughout the article.
1703 \begin{center}
1704 \begin{tabular}{p{0.22\textwidth}p{0.66\textwidth}}
1705 \toprule
1706 symbol & meaning\\
1707 \midrule
1708 \(L\leq\ZZ^d\) & a finite index sublattice\\
1709 \(R\) & a set of representatives for \(\ZZ^d/L\)\\
1710 \(P_r\) & the coordinate projector onto the coset \(r+L\)\\
1711 \(\Acal_L,\Scal_L\) & lattice analysis and interleaving synthesis\\
1712 \(E,R\) & filter bank analysis and synthesis symbols\\
1713 \(Q_m\) & a bounded linear approximation operator acting on the smooth part\\
1714 \(j,\widehat j\) & the true and estimated singular packets\\
1715 \(H_\tau\) & the step packet \(\one_{[\tau,1]}\)\\
1716 \(\phi_{m,\tau}\) & the truncated power packet \((t-\tau)_+^m/m!\)\\
1717 \(h\) & the grid spacing in the recovery theorems\\
1718 \bottomrule
1719 \end{tabular}
1720 \end{center}
1721 
1722 \section{Signal seeds and lattice addresses}\label{extension:section:02}\label{sec:signal_seeds}
1723 
1724 Let \(1\leq p\leq\infty\). For a countable set \(G\), write \(\ell^p(G)\) for the usual complex sequence space. For an integer \(d\geq1\), when \(G=\ZZ^d\), translation by \(a\in\ZZ^d\) is
1725 \[
1726 (T_ax)(n)=x(n-a).
1727 \]
1728 Let \(\delta_0\) be the impulse at the origin.
1729 
Statement: Impulse generation · Go to proof · Read statement and proof side by side1730 \begin{theorem}[Impulse generation]\label{thm:impulse_generation}
1731 For \(1\leq p<\infty\),
1732 \[
1733 \overline{\Span\{T_a\delta_0:a\in\ZZ^d\}}^{\ell^p}
1734 =
1735 \ell^p(\ZZ^d).
1736 \]
1737 The same statement holds in \(c_0(\ZZ^d)\) with the supremum norm. In \(\ell^\infty(\ZZ^d)\), the closed span of the translation orbit is exactly \(c_0(\ZZ^d)\).
1738 \end{theorem}
1739 
Proof block: Impulse generation · Go to statement · Read statement and proof side by side1740 \begin{proof}
1741 The translation orbit is the standard coordinate family \((\delta_a)_{a\in\ZZ^d}\). Its finite span is the space of finitely supported sequences. Finitely supported sequences are dense in \(\ell^p\) for finite \(p\) and in \(c_0\). Their closure in the supremum norm is \(c_0\), which is a proper closed subspace of \(\ell^\infty\).
1742 \end{proof}
1743 
1744 \begin{remark}\label{rem:infinity_boundary}
1745 The distinction at \(p=\infty\) is structural. The impulse generates every finitely supported signal algebraically, but it is not a topological cyclic seed for all bounded signals.
1746 \end{remark}
1747 
Statement: Finite periodic signals · Go to proof · Read statement and proof side by side1748 \begin{proposition}[Finite periodic signals]\label{prop:finite_periodic}
1749 Let \(G=\ZZ^d/L\), where \(L\leq\ZZ^d\) has finite index. The translation orbit of the impulse at the identity is a basis of \(\CC^G\). In particular, one impulse generates every signal on the finite periodic lattice.
1750 \end{proposition}
1751 
Proof block: Finite periodic signals · Go to statement · Read statement and proof side by side1752 \begin{proof}
1753 Translation sends the impulse at the identity to the impulse at each element of \(G\). These impulses form the standard basis of \(\CC^G\).
1754 \end{proof}
1755 
1756 \section{Polyphase coordinates on finite index sublattices}\label{extension:section:03}\label{sec:polyphase}
1757 
1758 Let \(L\leq\ZZ^d\) be a sublattice of finite index \(q\). Choose a set \(R\subset\ZZ^d\) containing one representative from every coset in \(\ZZ^d/L\). Thus
1759 \[
1760 \ZZ^d=\bigsqcup_{r\in R}(r+L).
1761 \]
1762 For \(x:\ZZ^d\to\CC\), its channel at \(r\) is the sequence on \(L\)
1763 \[
1764 x_r(\ell)=x(r+\ell).
1765 \]
1766 
Statement: Lattice packet1767 \begin{definition}[Lattice packet]\label{def:lattice_packet}
1768 For \(r\in R\), define the coordinate projector
1769 \[
1770 (P_rx)(n)=\one_{r+L}(n)x(n).
1771 \]
1772 The family \((P_r)_{r\in R}\) is called the packet system associated with \(L\) and \(R\).
1773 \end{definition}
1774 
1775 For finite \(p\), equip \(\bigoplus_{r\in R}\ell^p(L)\) with
1776 \[
1777 \|(u_r)\|^p=\sum_{r\in R}\|u_r\|_p^p.
1778 \]
1779 For \(p=\infty\), use the maximum norm.
1780 
Statement: Exact lattice polyphase decomposition · Go to proof · Read statement and proof side by side1781 \begin{theorem}[Exact lattice polyphase decomposition]\label{thm:lattice_polyphase}
1782 For every \(1\leq p\leq\infty\), the analysis map
1783 \[
1784 \Acal_L:\ell^p(\ZZ^d)\longrightarrow\bigoplus_{r\in R}\ell^p(L),
1785 \qquad
1786 \Acal_Lx=(x_r)_{r\in R},
1787 \]
1788 is an isometric isomorphism. Its inverse is the interleaving map
1789 \[
1790 \Scal_L(u)(r+\ell)=u_r(\ell).
1791 \]
1792 The projectors satisfy
1793 \[
1794 P_rP_s=\delta_{rs}P_r,
1795 \qquad
1796 \sum_{r\in R}P_r=I.
1797 \]
1798 \end{theorem}
1799 
Proof block: Exact lattice polyphase decomposition · Go to statement · Read statement and proof side by side1800 \begin{proof}
1801 Every address has a unique representation \(r+\ell\). For finite \(p\),
1802 \[
1803 \|x\|_p^p
1804 =
1805 \sum_{r\in R}\sum_{\ell\in L}|x(r+\ell)|^p
1806 =
1807 \sum_{r\in R}\|x_r\|_p^p.
1808 \]
1809 The corresponding supremum identity is immediate when \(p=\infty\). Unique address representation proves that analysis and interleaving are inverse maps. The projector identities follow from the disjoint coset partition.
1810 \end{proof}
1811 
Statement: One dimensional residue coordinates · Go to proof · Read statement and proof side by side1812 \begin{corollary}[One dimensional residue coordinates]\label{cor:one_dimensional_polyphase}
1813 For \(L=N\ZZ\) and \(R=\{0,\ldots,N-1\}\),
1814 \[
1815 x_r(k)=x(Nk+r)
1816 \]
1817 and
1818 \[
1819 \ell^p(\ZZ)\cong\bigoplus_{r=0}^{N-1}\ell^p(\ZZ)
1820 \]
1821 isometrically.
1822 \end{corollary}
1823 
Proof block: One dimensional residue coordinates · Go to statement · Read statement and proof side by side1824 \begin{proof}
1825 Apply \cref{thm:lattice_polyphase} to the standard residue representatives.
1826 \end{proof}
1827 
Statement: Translation covariance · Go to proof · Read statement and proof side by side1828 \begin{proposition}[Translation covariance]\label{prop:translation_covariance}
1829 Fix \(a\in\ZZ^d\). For every \(r\in R\), write uniquely
1830 \[
1831 r-a=r_a+\lambda_{a,r},
1832 \qquad
1833 r_a\in R,
1834 \quad
1835 \lambda_{a,r}\in L.
1836 \]
1837 Then
1838 \[
1839 (\Acal_LT_ax)_r(\ell)
1840 =
1841 x_{r_a}(\ell+\lambda_{a,r}).
1842 \]
1843 Thus a fine translation permutes the lattice packets and applies an internal translation on \(L\).
1844 \end{proposition}
1845 
Proof block: Translation covariance · Go to statement · Read statement and proof side by side1846 \begin{proof}
1847 Evaluate at \(r+\ell\):
1848 \[
1849 (T_ax)(r+\ell)
1850 =x(r+\ell-a)
1851 =x(r_a+\ell+\lambda_{a,r}).
1852 \]
1853 This is the stated channel formula.
1854 \end{proof}
1855 
1856 \begin{figure}
1857 \centering
1858 \includegraphics[width=0.82\textwidth]{01_polyphase_coordinates.pdf}
1859 \caption{A signal and its four exact residue channels. Interleaving the channels returns the original samples without approximation.}
1860 \label{fig:polyphase_coordinates}
1861 \end{figure}
1862 
1863 \subsection{Refinement of sublattices}
1864 
1865 Let \(L'\leq L\leq\ZZ^d\), with both indices finite. Choose representative systems \(R\) for \(\ZZ^d/L\) and \(R'\) for \(\ZZ^d/L'\). For \(r\in R\), define
1866 \[
1867 \operatorname{Child}(r)
1868 =
1869 \{s\in R':s+L'\subset r+L\}.
1870 \]
1871 
Statement: Exact lattice refinement · Go to proof · Read statement and proof side by side1872 \begin{theorem}[Exact lattice refinement]\label{thm:lattice_refinement}
1873 For every coarse packet,
1874 \[
1875 P_r^{L}
1876 =
1877 \sum_{s\in\operatorname{Child}(r)}P_s^{L'}.
1878 \]
1879 The fine packets in the sum are pairwise disjoint. Refinement through several nested sublattices is transitive.
1880 \end{theorem}
1881 
Proof block: Exact lattice refinement · Go to statement · Read statement and proof side by side1882 \begin{proof}
1883 The coset \(r+L\) is the disjoint union of the \(L'\) cosets contained in it. Multiplying by the corresponding indicator functions gives the projector identity. A nested partition of a coset is independent of whether the intermediate level is inserted, which proves transitivity.
1884 \end{proof}
1885 
1886 \begin{example}[Rectangular image phases]\label{ex:image_phases}
1887 For
1888 \[
1889 L=M\ZZ\times N\ZZ\subset\ZZ^2,
1890 \]
1891 one may take
1892 \[
1893 R=\{0,\ldots,M-1\}\times\{0,\ldots,N-1\}.
1894 \]
1895 The analysis map separates an image into \(MN\) exact phases. Refining \((M,N)\) to \((M',N')\), where \(M\mid M'\) and \(N\mid N'\), splits every coarse phase into
1896 \[
1897 \frac{M'}M\frac{N'}N
1898 \]
1899 fine phases.
1900 \end{example}
1901 
1902 \section{Convolution as a packet transfer matrix}\label{extension:section:04}\label{sec:convolution}
1903 
1904 Let \(h\in\ell^1(\ZZ^d)\) and \(x\in\ell^p(\ZZ^d)\), with \(1\leq p\leq\infty\), and define
1905 \[
1906 (Hx)(n)
1907 =(h*x)(n)
1908 =\sum_{q\in\ZZ^d}h(q)x(n-q).
1909 \]
1910 Young's inequality gives
1911 \[
1912 \|Hx\|_p\leq\|h\|_1\|x\|_p
1913 \]
1914 for every \(1\leq p\leq\infty\).
1915 
Statement: Packet transfer kernel1916 \begin{definition}[Packet transfer kernel]\label{def:packet_transfer_kernel}
1917 For output residue \(r\in R\) and input residue \(s\in R\), define a kernel on \(L\) by
1918 \[
1919 h_{rs}(m)=h(r-s+m),
1920 \qquad
1921 m\in L.
1922 \]
1923 \end{definition}
1924 
Statement: Exact convolution transfer law · Go to proof · Read statement and proof side by side1925 \begin{theorem}[Exact convolution transfer law]\label{thm:convolution_transfer}
1926 Let \(y=h*x\). Then
1927 \[
1928 y_r(\ell)
1929 =
1930 \sum_{s\in R}\sum_{m\in L}h_{rs}(m)x_s(\ell-m).
1931 \]
1932 Equivalently,
1933 \[
1934 \Acal_LH\Acal_L^{-1}
1935 =
1936 (H_{rs})_{r,s\in R},
1937 \]
1938 where \(H_{rs}\) is convolution on \(L\) by \(h_{rs}\).
1939 \end{theorem}
1940 
Proof block: Exact convolution transfer law · Go to statement · Read statement and proof side by side1941 \begin{proof}
1942 Evaluate the output at \(r+\ell\). Write every input address as \(s+\ell'\). The convolution offset is
1943 \[
1944 (r+\ell)-(s+\ell')
1945 =(r-s)+(\ell-\ell').
1946 \]
1947 Setting \(m=\ell-\ell'\) gives the formula.
1948 \end{proof}
1949 
Statement: Transfer norm control · Go to proof · Read statement and proof side by side1950 \begin{proposition}[Transfer norm control]\label{prop:transfer_norm_control}
1951 For every output residue \(r\),
1952 \[
1953 \sum_{s\in R}\|h_{rs}\|_{\ell^1(L)}
1954 =
1955 \|h\|_{\ell^1(\ZZ^d)}.
1956 \]
1957 The packet transfer matrix acts on the polyphase direct sum with norm at most \(\|h\|_1\) on every \(\ell^p\) space.
1958 \end{proposition}
1959 
Proof block: Transfer norm control · Go to statement · Read statement and proof side by side1960 \begin{proof}
1961 For fixed \(r\), every \(q\in\ZZ^d\) has a unique representation \(q=r-s+m\) with \(s\in R\) and \(m\in L\). This proves the equality of the absolute sums. The operator norm bound also follows by conjugating Young's inequality through the isometric analysis map.
1962 \end{proof}
1963 
Statement: Composition of transfer matrices · Go to proof · Read statement and proof side by side1964 \begin{proposition}[Composition of transfer matrices]\label{prop:transfer_composition}
1965 If \(G\) and \(H\) are convolution operators with packet matrices \((G_{rs})\) and \((H_{rs})\), then the packet matrix of \(GH\) is
1966 \[
1967 (GH)_{rt}
1968 =
1969 \sum_{s\in R}G_{rs}H_{st}.
1970 \]
1971 In one dimensional transform notation, the corresponding matrix symbols multiply pointwise.
1972 \end{proposition}
1973 
Proof block: Composition of transfer matrices · Go to statement · Read statement and proof side by side1974 \begin{proof}
1975 Conjugation by \(\Acal_L\) preserves composition:
1976 \[
1977 \Acal_LGH\Acal_L^{-1}
1978 =
1979 (\Acal_LG\Acal_L^{-1})(\Acal_LH\Acal_L^{-1}).
1980 \]
1981 Reading the \((r,t)\) block gives the formula.
1982 \end{proof}
1983 
1984 \subsection{The one dimensional symbol}
1985 
1986 For \(L=N\ZZ\) with \(N\geq1\), first take \(|z|=1\). The assumption \(h\in\ell^1(\ZZ)\) makes the following symbol series absolutely convergent. Define
1987 \[
1988 H_{ks}(z)=\sum_{m\in\ZZ}h(Nm+k-s)z^{-m}.
1989 \]
1990 For \(x\in\ell^1(\ZZ)\), the channel transforms converge absolutely on the unit circle and satisfy
1991 \[
1992 \bm Y(z)=\bm H_N(z)\bm X(z).
1993 \]
1994 For \(x\in\ell^2(\ZZ)\), the same identity holds almost everywhere in the \(L^2\) Fourier sense. For finite impulse-response kernels it also holds at nonzero \(z\) wherever the input channel transforms are weighted absolutely summable. This is the classical polyphase matrix. It is the exact block matrix of the original convolution operator.
1995 
Statement: Finite impulse response closure · Go to proof · Read statement and proof side by side1996 \begin{corollary}[Finite impulse response closure]\label{cor:fir_closure}
1997 If \(h\) has finite support, every packet kernel \(h_{ks}\) has finite support. Matrix multiplication of polyphase symbols agrees with composition of the original finite impulse response operators.
1998 \end{corollary}
1999 
Proof block: Finite impulse response closure · Go to statement · Read statement and proof side by side2000 \begin{proof}
2001 Finite support is preserved by taking residue subsequences. The composition statement is \cref{prop:transfer_composition}.
2002 \end{proof}
2003 
2004 \begin{example}[A four phase finite impulse response operator]\label{ex:four_phase_fir}
2005 Let
2006 \[
2007 h(0)=1,
2008 \quad h(1)=2,
2009 \quad h(2)=3,
2010 \quad h(3)=4,
2011 \quad h(4)=5,
2012 \]
2013 with all other taps zero. For four residue channels, the transfer symbol is
2014 \[
2015 \bm H_4(z)
2016 =
2017 \begin{pmatrix}
2018 1+5z^{-1}&4z^{-1}&3z^{-1}&2z^{-1}\\
2019 2&1+5z^{-1}&4z^{-1}&3z^{-1}\\
2020 3&2&1+5z^{-1}&4z^{-1}\\
2021 4&3&2&1+5z^{-1}
2022 \end{pmatrix}.
2023 \]
2024 The delay term records the tap that crosses a coarse cell boundary. No approximation is introduced by the matrix representation.
2025 \end{example}
2026 
2027 \begin{figure}
2028 \centering
2029 \includegraphics[width=0.62\textwidth]{02_transfer_blocks.pdf}
2030 \caption{Absolute tap weight in the transfer blocks of the four phase operator in \cref{ex:four_phase_fir}.}
2031 \label{fig:transfer_blocks}
2032 \end{figure}
2033 
2034 \section{Filter banks, frame bounds, and lifting}\label{extension:section:05}\label{sec:filter_banks}
2035 
2036 Throughout this section the banks act boundedly on the relevant polyphase \(\ell^2\) spaces. A finite channel analysis bank acts in Fourier coordinates by a matrix \(E(z)\). A synthesis bank acts by a matrix \(R(z)\). The matrices may be square or rectangular. Their entries are Laurent polynomials for finite filters and bounded measurable functions for general stable filters.
2037 
Statement: Perfect reconstruction criterion · Go to proof · Read statement and proof side by side2038 \begin{theorem}[Perfect reconstruction criterion]\label{thm:perfect_reconstruction}
2039 Let \(E(z)\) be an \(M\) by \(N\) analysis matrix and let \(R(z)\) be an \(N\) by \(M\) synthesis matrix. The bank reconstructs every input with a delay of \(d\) coarse channel samples, equivalent to \(Nd\) original samples in the one dimensional \(N\)-phase convention, exactly if and only if
2040 \[
2041 R(z)E(z)=z^{-d}I_N
2042 \]
2043 for almost every point on the unit circle. A nonzero scalar factor gives exact reconstruction up to gain and delay.
2044 \end{theorem}
2045 
Proof block: Perfect reconstruction criterion · Go to statement · Read statement and proof side by side2046 \begin{proof}
2047 The channel Fourier transform turns analysis followed by synthesis into multiplication by \(R(z)E(z)\). If the matrix identity holds, inverse transformation gives the delayed input. Conversely, if every input is reconstructed with that delay, the difference matrix annihilates every vector valued Fourier datum and must vanish almost everywhere.
2048 \end{proof}
2049 
Statement: Quantitative imperfect reconstruction · Go to proof · Read statement and proof side by side2050 \begin{theorem}[Quantitative imperfect reconstruction]\label{thm:imperfect_reconstruction}
2051 Define the delay compensated defect
2052 \[
2053 D(\e^{\ii\omega})
2054 =
2055 \e^{\ii d\omega}R(\e^{\ii\omega})E(\e^{\ii\omega})-I_N.
2056 \]
2057 If
2058 \[
2059 \varepsilon_{\mathrm{rec}}
2060 =
2061 \esssup_{\omega}\|D(\e^{\ii\omega})\|_{\mathrm{op}}<\infty,
2062 \]
2063 then the delay compensated reconstruction \(\widetilde x\) satisfies
2064 \[
2065 \|\widetilde x-x\|_2
2066 \leq
2067 \varepsilon_{\mathrm{rec}}\|x\|_2.
2068 \]
2069 The constant is sharp as an operator norm.
2070 \end{theorem}
2071 
Proof block: Quantitative imperfect reconstruction · Go to statement · Read statement and proof side by side2072 \begin{proof}
2073 In polyphase Fourier coordinates, the error is \(D(\e^{\ii\omega})\bm X(\e^{\ii\omega})\). Pointwise multiplication and Plancherel's theorem give the upper bound. The operator norm of a measurable matrix multiplier is its essential supremum, which proves sharpness.
2074 \end{proof}
2075 
2076 \begin{example}[A reconstruction certificate]\label{ex:reconstruction_certificate}
2077 Suppose a two channel bank has delay compensated defect
2078 \[
2079 D(\e^{\ii\omega})
2080 =
2081 \begin{pmatrix}
2082 0.01\cos\omega&0\\
2083 0&0.02\sin\omega
2084 \end{pmatrix}.
2085 \]
2086 Then
2087 \[
2088 \esssup_\omega\|D(\e^{\ii\omega})\|_{\mathrm{op}}
2089 =0.02.
2090 \]
2091 Every input therefore has relative reconstruction error at most two percent in \(\ell^2\). The estimate is independent of the input length.
2092 \end{example}
2093 
Statement: Stability under filter perturbation · Go to proof · Read statement and proof side by side2094 \begin{theorem}[Stability under filter perturbation]\label{thm:filter_perturbation}
2095 Assume \(RE=z^{-d}I_N\). Let
2096 \[
2097 \widetilde E=E+\Delta E,
2098 \qquad
2099 \widetilde R=R+\Delta R.
2100 \]
2101 Set
2102 \[
2103 \begin{aligned}
2104 e_0&=\esssup\|E\|_{\mathrm{op}},
2105 & r_0&=\esssup\|R\|_{\mathrm{op}},\\
2106 \delta_E&=\esssup\|\Delta E\|_{\mathrm{op}},
2107 & \delta_R&=\esssup\|\Delta R\|_{\mathrm{op}}.
2108 \end{aligned}
2109 \]
2110 Then the delay compensated reconstruction defect obeys
2111 \[
2112 \varepsilon_{\mathrm{rec}}
2113 \leq
2114 r_0\delta_E+e_0\delta_R+\delta_R\delta_E.
2115 \]
2116 \end{theorem}
2117 
Proof block: Stability under filter perturbation · Go to statement · Read statement and proof side by side2118 \begin{proof}
2119 Expand
2120 \[
2121 \widetilde R\widetilde E-RE
2122 =
2123 R\Delta E+\Delta R E+\Delta R\Delta E.
2124 \]
2125 Take pointwise operator norms, then the essential supremum. Multiplication by the delay factor has unit modulus.
2126 \end{proof}
2127 
Statement: Frame bounds from the polyphase symbol · Go to proof · Read statement and proof side by side2128 \begin{theorem}[Frame bounds from the polyphase symbol]\label{thm:frame_bounds}
2129 Assume that
2130 \[
2131 A I_N
2132 \leq
2133 E(\e^{\ii\omega})^*E(\e^{\ii\omega})
2134 \leq
2135 B I_N
2136 \]
2137 for almost every \(\omega\), where \(0<A\leq B<\infty\). Then the analysis bank satisfies
2138 \[
2139 A\|x\|_2^2
2140 \leq
2141 \|\Acal_E x\|_2^2
2142 \leq
2143 B\|x\|_2^2.
2144 \]
2145 The pointwise matrix
2146 \[
2147 R_{\rm can}(\e^{\ii\omega})
2148 =
2149 (E^*E)^{-1}E^*
2150 \]
2151 is a bounded left inverse and gives the canonical dual synthesis bank.
2152 \end{theorem}
2153 
Proof block: Frame bounds from the polyphase symbol · Go to statement · Read statement and proof side by side2154 \begin{proof}
2155 Apply the matrix inequality to the polyphase Fourier vector at every frequency and integrate. Plancherel's theorem gives the frame inequalities. The lower bound makes \(E^*E\) invertible almost everywhere, and direct multiplication gives \(R_{\rm can}E=I_N\). The same spectral bounds show that the canonical dual is bounded.
2156 \end{proof}
2157 
Statement: Coefficient noise under canonical synthesis · Go to proof · Read statement and proof side by side2158 \begin{corollary}[Coefficient noise under canonical synthesis]\label{cor:canonical_noise}
2159 Under the hypotheses of \cref{thm:frame_bounds}, suppose the observed coefficient vector is
2160 \[
2161 c=\Acal_Ex+\nu.
2162 \]
2163 Canonical synthesis gives
2164 \[
2165 \widehat x=R_{\mathrm{can}}c
2166 \]
2167 and
2168 \[
2169 \|\widehat x-x\|_2
2170 \leq
2171 \frac1{\sqrt A}\|\nu\|_2.
2172 \]
2173 \end{corollary}
2174 
Proof block: Coefficient noise under canonical synthesis · Go to statement · Read statement and proof side by side2175 \begin{proof}
2176 The smallest singular value of \(E\) is at least \(\sqrt A\), so the pointwise norm of its pseudoinverse is at most \(1/\sqrt A\). Apply Plancherel's theorem to \(R_{\mathrm{can}}\nu\).
2177 \end{proof}
2178 
Statement: Perturbation of frame bounds · Go to proof · Read statement and proof side by side2179 \begin{theorem}[Perturbation of frame bounds]\label{thm:frame_perturbation}
2180 Under the hypotheses of \cref{thm:frame_bounds}, let \(\widetilde E=E+\Delta E\) and suppose
2181 \[
2182 \delta=\esssup_\omega\|\Delta E(\e^{\ii\omega})\|_{\mathrm{op}}<\sqrt A.
2183 \]
2184 Then \(\widetilde E\) is a frame analysis bank with bounds
2185 \[
2186 (\sqrt A-\delta)^2
2187 \quad\text{and}\quad
2188 (\sqrt B+\delta)^2.
2189 \]
2190 \end{theorem}
2191 
Proof block: Perturbation of frame bounds · Go to statement · Read statement and proof side by side2192 \begin{proof}
2193 For every vector \(v\),
2194 \[
2195 \|\widetilde Ev\|
2196 \geq
2197 \|Ev\|-\|\Delta Ev\|
2198 \geq
2199 (\sqrt A-\delta)\|v\|.
2200 \]
2201 The upper bound follows from
2202 \[
2203 \|\widetilde Ev\|
2204 \leq
2205 (\sqrt B+\delta)\|v\|.
2206 \]
2207 Square the inequalities and integrate in frequency.
2208 \end{proof}
2209 
2210 \begin{remark}\label{rem:canonical_dual_support}
2211 Even when \(E\) is a Laurent polynomial matrix, the canonical dual need not have finite impulse response. Finite support of both analysis and synthesis is an additional algebraic condition.
2212 \end{remark}
2213 
Statement: Paraunitary bank · Go to proof · Read statement and proof side by side2214 \begin{corollary}[Paraunitary bank]\label{cor:paraunitary}
2215 If \(E(z)^*E(z)=I_N\) on the unit circle, then analysis is an isometry and synthesis may be chosen as \(R(z)=E(z)^*\).
2216 \end{corollary}
2217 
Proof block: Paraunitary bank · Go to statement · Read statement and proof side by side2218 \begin{proof}
2219 Use \cref{thm:frame_bounds} with \(A=B=1\), then multiply the matrices.
2220 \end{proof}
2221 
2222 \begin{figure}
2223 \centering
2224 \includegraphics[width=0.72\textwidth]{03_frame_stability.pdf}
2225 \caption{Pointwise singular value bounds govern both the original frame and its perturbed analysis bank.}
2226 \label{fig:frame_stability}
2227 \end{figure}
2228 
2229 \subsection{Haar analysis}
2230 
2231 The Haar matrix is
2232 \[
2233 E_H
2234 =
2235 \frac1{\sqrt2}
2236 \begin{pmatrix}
2237 1&1\\
2238 1&-1
2239 \end{pmatrix}.
2240 \]
2241 For even and odd samples \(e(n)=x(2n)\) and \(o(n)=x(2n+1)\), the low and high coordinates are
2242 \[
2243 a(n)=\frac{e(n)+o(n)}{\sqrt2},
2244 \qquad
2245 d(n)=\frac{e(n)-o(n)}{\sqrt2}.
2246 \]
2247 
Statement: Haar reconstruction and energy · Go to proof · Read statement and proof side by side2248 \begin{corollary}[Haar reconstruction and energy]\label{cor:haar_reconstruction}
2249 The Haar bank reconstructs exactly and
2250 \[
2251 \|x\|_2^2=\|a\|_2^2+\|d\|_2^2.
2252 \]
2253 \end{corollary}
2254 
Proof block: Haar reconstruction and energy · Go to statement · Read statement and proof side by side2255 \begin{proof}
2256 The matrix \(E_H\) is unitary.
2257 \end{proof}
2258 
2259 \subsection{Lifting steps}
2260 
2261 Let \(P(z)\) and \(Q(z)\) be Laurent polynomials. Define
2262 \[
2263 L_P(z)
2264 =
2265 \begin{pmatrix}
2266 1&0\\
2267 P(z)&1
2268 \end{pmatrix},
2269 \qquad
2270 U_Q(z)
2271 =
2272 \begin{pmatrix}
2273 1&Q(z)\\
2274 0&1
2275 \end{pmatrix}.
2276 \]
2277 
Statement: Exact lifting update · Go to proof · Read statement and proof side by side2278 \begin{theorem}[Exact lifting update]\label{thm:lifting_update}
2279 The lifting matrices are invertible with
2280 \[
2281 L_P^{-1}=L_{-P},
2282 \qquad
2283 U_Q^{-1}=U_{-Q}.
2284 \]
2285 If \(R_0E_0=I\), then every finite product \(V\) of lifting matrices gives another exact bank by
2286 \[
2287 E=VE_0,
2288 \qquad
2289 R=R_0V^{-1}.
2290 \]
2291 Every lifting step has determinant one.
2292 \end{theorem}
2293 
Proof block: Exact lifting update · Go to statement · Read statement and proof side by side2294 \begin{proof}
2295 Direct multiplication proves the inverse formulas and the determinant statement. The reconstructed product is
2296 \[
2297 RE
2298 =R_0V^{-1}VE_0
2299 =R_0E_0
2300 =I.
2301 \]
2302 \end{proof}
2303 
2304 \begin{example}[A two neighbor predict and update transform]\label{ex:predict_update}
2305 Split a sequence into even and odd samples. Define
2306 \[
2307 d(n)
2308 =o(n)-\frac{e(n)+e(n+1)}2
2309 \]
2310 and
2311 \[
2312 s(n)
2313 =e(n)+\frac{d(n-1)+d(n)}4.
2314 \]
2315 The inverse is obtained in reverse order:
2316 \[
2317 e(n)
2318 =s(n)-\frac{d(n-1)+d(n)}4,
2319 \]
2320 then
2321 \[
2322 o(n)
2323 =d(n)+\frac{e(n)+e(n+1)}2.
2324 \]
2325 Thus reconstruction is exact over every coefficient field in which two is invertible. This is a standard compactly supported biorthogonal lifting pattern \cite{CohenDaubechiesFeauveau1992,Sweldens1996,DaubechiesSweldens1998}.
2326 \end{example}
2327 
2328 \section{Wavelet packet trees and admissible bases}\label{extension:section:06}\label{sec:packet_trees}
2329 
2330 A packet transform is built by deciding at each node whether to retain the current coordinate space or to split it into children. The proof of reconstruction is local and does not depend on a particular wavelet family.
2331 
Statement: Unitary packet tree2332 \begin{definition}[Unitary packet tree]\label{def:unitary_packet_tree}
2333 A finite unitary packet tree consists of Hilbert spaces \(\Hcal_v\) at its vertices and, at every internal vertex \(v\), a unitary map
2334 \[
2335 U_v:\Hcal_v\longrightarrow\bigoplus_{w\in\operatorname{child}(v)}\Hcal_w.
2336 \]
2337 An admissible pruning is a set of vertices that meets every root to terminal path exactly once.
2338 \end{definition}
2339 
Statement: Reconstruction on a pruned packet tree · Go to proof · Read statement and proof side by side2340 \begin{theorem}[Reconstruction on a pruned packet tree]\label{thm:pruned_packet_tree}
2341 Every admissible pruning \(\Lcal\) defines a unitary analysis map
2342 \[
2343 \Acal_{\Lcal}:\Hcal_{\rm root}
2344 \longrightarrow
2345 \bigoplus_{v\in\Lcal}\Hcal_v.
2346 \]
2347 Its inverse is obtained by applying the adjoints of the local splits from the leaves toward the root. In particular,
2348 \[
2349 \|x\|^2
2350 =
2351 \sum_{v\in\Lcal}\|x_v\|^2.
2352 \]
2353 \end{theorem}
2354 
Proof block: Reconstruction on a pruned packet tree · Go to statement · Read statement and proof side by side2355 \begin{proof}
2356 Proceed by induction on the depth of the retained subtree. If the root is retained, the statement is immediate. If the root is split, apply the unitary map \(U_{\rm root}\), then apply the induction hypothesis independently in each child subtree. A direct sum of unitary maps is unitary. Reversing the maps with adjoints gives synthesis.
2357 \end{proof}
2358 
Statement: Coefficient perturbations in a unitary tree · Go to proof · Read statement and proof side by side2359 \begin{corollary}[Coefficient perturbations in a unitary tree]\label{cor:unitary_tree_noise}
2360 Let \(c=\Acal_{\Lcal}x\) be the leaf coefficients of an admissible unitary pruning, and let \(\widehat c=c+e\). If \(\widehat x\) is synthesized from \(\widehat c\), then
2361 \[
2362 \|\widehat x-x\|=\|e\|.
2363 \]
2364 Thus coefficient quantization and coefficient noise are neither amplified nor attenuated by exact unitary synthesis.
2365 \end{corollary}
2366 
Proof block: Coefficient perturbations in a unitary tree · Go to statement · Read statement and proof side by side2367 \begin{proof}
2368 The synthesis map is unitary by \cref{thm:pruned_packet_tree}. Hence
2369 \[
2370 \widehat x-x
2371 =\Acal_{\Lcal}^{-1}e
2372 \]
2373 has the same norm as \(e\).
2374 \end{proof}
2375 
Statement: Bounds for nonunitary local splits · Go to proof · Read statement and proof side by side2376 \begin{proposition}[Bounds for nonunitary local splits]\label{prop:nonunitary_tree_bounds}
2377 Suppose every internal node has an analysis map \(U_v\) and constants \(0\leq A_v\leq B_v<\infty\) satisfying
2378 \[
2379 A_v\|x\|^2
2380 \leq
2381 \sum_{w\in\operatorname{child}(v)}\|(U_vx)_w\|^2
2382 \leq
2383 B_v\|x\|^2.
2384 \]
2385 Define recursively
2386 \[
2387 a(v)=
2388 \begin{cases}
2389 1,&v\text{ retained},\\
2390 A_v\min_w a(w),&v\text{ split},
2391 \end{cases}
2392 \]
2393 and
2394 \[
2395 b(v)=
2396 \begin{cases}
2397 1,&v\text{ retained},\\
2398 B_v\max_w b(w),&v\text{ split}.
2399 \end{cases}
2400 \]
2401 Then the leaf analysis of the pruned tree has bounds \(a({\rm root})\) and \(b({\rm root})\).
2402 \end{proposition}
2403 
Proof block: Bounds for nonunitary local splits · Go to statement · Read statement and proof side by side2404 \begin{proof}
2405 Assume the child subtree bounds have been proved. Then
2406 \[
2407 \sum_{w}\sum_{\ell\in\Lcal_w}\|x_\ell\|^2
2408 \geq
2409 \min_w a(w)
2410 \sum_w\|(U_vx)_w\|^2
2411 \geq
2412 A_v\min_w a(w)\|x\|^2.
2413 \]
2414 The upper bound follows in the same way with the maximum and \(B_v\). Induction completes the proof.
2415 \end{proof}
2416 
2417 \begin{figure}
2418 \centering
2419 \includegraphics[width=0.72\textwidth]{04_packet_tree.pdf}
2420 \caption{A complete packet tree and one admissible pruning. Retained nodes may occur at different depths.}
2421 \label{fig:packet_tree}
2422 \end{figure}
2423 
2424 \subsection{Best basis recursion}
2425 
2426 \begin{interpretation}[title={The minimum is taken over feasible prunings}]
2427 The value at a node is the least cost among all admissible prunings of the subtree rooted there. It is one optimal value. Individual competing prunings may have larger costs. The recursion compares retaining the node with the sum of the already optimal child values, with retention chosen on a tie.
2428 \end{interpretation}
2429 
2430 Assign a real cost \(c(v)\) to retaining each node. For an admissible pruning \(\Lcal\), let
2431 \[
2432 C(\Lcal)=\sum_{v\in\Lcal}c(v).
2433 \]
2434 The cost may measure entropy, coefficient count, an approximation penalty, or any other additive local objective.
2435 
Statement: Exact additive best basis recursion · Go to proof · Read statement and proof side by side2436 \begin{theorem}[Exact additive best basis recursion]\label{thm:best_basis}
2437 For each vertex define
2438 \[
2439 F(v)
2440 =
2441 \min\left\{
2442  c(v),
2443  \sum_{w\in\operatorname{child}(v)}F(w)
2444 \right\},
2445 \]
2446 with \(F(v)=c(v)\) at terminal vertices. Then \(F(v)\) is the minimum cost of every admissible pruning of the subtree rooted at \(v\). A minimizing pruning is recovered by retaining \(v\) when the first term is no larger than the second and splitting it otherwise.
2447 \end{theorem}
2448 
Proof block: Exact additive best basis recursion · Go to statement · Read statement and proof side by side2449 \begin{proof}
2450 Every admissible pruning of the subtree has exactly two forms. It either retains \(v\), with cost \(c(v)\), or splits \(v\) and independently chooses an admissible pruning in every child subtree. The minimum in the second case is the sum of the child minima. This gives the recursion and the reconstruction rule. In an equality case both choices minimize, and the stated rule retains the node.
2451 \end{proof}
2452 
Statement: Stability of the optimal cost · Go to proof · Read statement and proof side by side2453 \begin{theorem}[Stability of the optimal cost]\label{thm:best_basis_cost_stability}
2454 Let \(c\) and \(\widetilde c\) be two node cost functions. For an admissible pruning \(\Lcal\), set
2455 \[
2456 \varepsilon(\Lcal)
2457 =
2458 \sum_{v\in\Lcal}|c(v)-\widetilde c(v)|
2459 \]
2460 and let
2461 \[
2462 \varepsilon_*
2463 =
2464 \max_{\Lcal}\varepsilon(\Lcal).
2465 \]
2466 If \(F_*\) and \(\widetilde F_*\) are the two optimal root costs, then
2467 \[
2468 |F_*-\widetilde F_*|
2469 \leq
2470 \varepsilon_*.
2471 \]
2472 \end{theorem}
2473 
Proof block: Stability of the optimal cost · Go to statement · Read statement and proof side by side2474 \begin{proof}
2475 For every admissible pruning,
2476 \[
2477 |C(\Lcal)-\widetilde C(\Lcal)|
2478 \leq
2479 \varepsilon_*.
2480 \]
2481 Evaluate \(\widetilde C\) on a minimizer for \(C\) to obtain \(\widetilde F_*\leq F_*+\varepsilon_*\). Reverse the roles to obtain \(F_*\leq\widetilde F_*+\varepsilon_*\).
2482 \end{proof}
2483 
Statement: Stability of the selected basis · Go to proof · Read statement and proof side by side2484 \begin{corollary}[Stability of the selected basis]\label{cor:best_basis_selection_stability}
2485 Assume the optimal pruning \(\Lcal_*\) for \(c\) is unique and at least one competing pruning exists. Define \(g=\min_{\Lcal\ne\Lcal_*}(C(\Lcal)-C(\Lcal_*))>0\). If
2486 \[
2487 2\varepsilon_*<g,
2488 \]
2489 then the optimal pruning for \(\widetilde c\) is the same. If only one admissible pruning exists, it remains selected under every cost change without a gap condition.
2490 \end{corollary}
2491 
Proof block: Stability of the selected basis · Go to statement · Read statement and proof side by side2492 \begin{proof}
2493 The cost of the original minimizer can increase by at most \(\varepsilon_*\), while the cost of every competitor can decrease by at most \(\varepsilon_*\). The original gap therefore remains positive.
2494 \end{proof}
2495 
2496 \begin{figure}
2497 \centering
2498 \includegraphics[width=0.72\textwidth]{05_best_basis.pdf}
2499 \caption{An additive best basis recursion compares the cost of retaining a packet with the total cost of splitting it.}
2500 \label{fig:best_basis}
2501 \end{figure}
2502 
2503 \section{Singular packets and exact reconstruction}\label{extension:section:07}\label{sec:singular_packets}
2504 
2505 \begin{readerroadmap}[title={Trace data identify the singular part}]
2506 A value jump needs a step packet. A jump in a derivative needs a truncated power of the corresponding order. Subtracting these finite packets matches the one sided traces and leaves the regular remainder covered by the theorem's Sobolev assumptions. The packet construction and the approximation operator play separate roles.
2507 \end{readerroadmap}
2508 
2509 A packet decomposition becomes useful for approximation when it separates a component that should be retained from a component that should be compressed. The simplest singular packets are steps. Higher order packets record jumps in derivatives.
2510 
2511 For \(m\geq0\) and \(0<\tau<1\), define
2512 \[
2513 \phi_{m,\tau}(t)
2514 =
2515 \frac{(t-\tau)_+^m}{m!}.
2516 \]
2517 Thus \(\phi_{0,\tau}=\one_{[\tau,1]}\). On each side of \(\tau\), the packet is a polynomial. Its derivative of order \(m\) has a unit jump at \(\tau\).
2518 
Statement: Sobolev step packet class2519 \begin{definition}[Sobolev step packet class]\label{def:step_packet_class}
2520 Fix \(r\geq1\), \(1\leq p\leq\infty\), and an integer \(Q\geq0\). The class \(\Fcal_{r,p}(Q)\) consists of functions
2521 \[
2522 f=s+\sum_{q=1}^{Q}a_q\phi_{0,\tau_q}
2523 \]
2524 with \(s\in W^{r,p}(0,1)\) and
2525 \[
2526 0<\tau_1<\cdots<\tau_Q<1.
2527 \]
2528 No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.
2529 \end{definition}
2530 
Statement: Higher order singular packet decomposition · Go to proof · Read statement and proof side by side2531 \begin{theorem}[Higher order singular packet decomposition]\label{thm:higher_order_packet_decomposition}
2532 Let
2533 \[
2534 0=\tau_0<\tau_1<\cdots<\tau_Q<\tau_{Q+1}=1.
2535 \]
2536 Suppose the restriction of \(f\) to every interval \((\tau_q,\tau_{q+1})\) belongs to \(W^{r,p}\), and choose the one dimensional Sobolev representatives whose derivatives through order \(r-1\) have one sided traces at the endpoints. Define
2537 \[
2538 a_{q,m}
2539 =
2540 \jump{f^{(m)}}_{\tau_q}
2541 =
2542 f^{(m)}(\tau_q+)-f^{(m)}(\tau_q-)
2543 \]
2544 and
2545 \[
2546 j_f(t)
2547 =
2548 \sum_{q=1}^{Q}\sum_{m=0}^{r-1}a_{q,m}\phi_{m,\tau_q}(t).
2549 \]
2550 Then
2551 \[
2552 s_f=f-j_f
2553 \]
2554 has matching one sided derivatives through order \(r-1\) at every breakpoint and belongs to \(W^{r,p}(0,1)\). Its weak derivative of order \(r\) equals the piecewise classical derivative \(f^{(r)}\) almost everywhere.
2555 \end{theorem}
2556 
Proof block: Higher order singular packet decomposition · Go to statement · Read statement and proof side by side2557 \begin{proof}
2558 For \(0\leq \ell\leq m\), the classical derivative of \(\phi_{m,\tau}\) on the two open sides is
2559 \[
2560 \phi_{m,\tau}^{(\ell)}(t)
2561 =
2562 \frac{(t-\tau)_+^{m-\ell}}{(m-\ell)!}.
2563 \]
2564 This derivative has zero jump when \(\ell<m\) and a unit jump when \(\ell=m\). Derivatives of order larger than \(m\) vanish on both open sides. It follows that subtracting the term \(a_{q,m}\phi_{m,\tau_q}\) removes exactly the jump in the derivative of order \(m\) without changing the other classical trace jumps. Hence \(s_f\) has matching traces through order \(r-1\).
2565 
2566 On each partition interval, \(j_f\) is a polynomial of degree at most \(r-1\), so its weak derivative of order \(r\) is zero there. A piecewise \(W^{r,p}\) function whose traces through order \(r-1\) match across every breakpoint belongs to \(W^{r,p}(0,1)\). Repeated integration by parts on the partition intervals shows that all boundary terms cancel. Thus the weak derivative of order \(r\) is the piecewise weak derivative of \(f\).
2567 \end{proof}
2568 
Statement: Uniqueness of the trace packet · Go to proof · Read statement and proof side by side2569 \begin{corollary}[Uniqueness of the trace packet]\label{cor:trace_packet_uniqueness}
2570 Fix the breakpoints \(\tau_q\). If
2571 \[
2572 f=s+\sum_{q=1}^{Q}\sum_{m=0}^{r-1}b_{q,m}\phi_{m,\tau_q}
2573 \]
2574 with \(s\in W^{r,p}(0,1)\), then
2575 \[
2576 b_{q,m}=\jump{f^{(m)}}_{\tau_q}.
2577 \]
2578 The finite truncated power packet is therefore the unique packet supported at the declared breakpoints that removes all trace jumps through order \(r-1\).
2579 \end{corollary}
2580 
Proof block: Uniqueness of the trace packet · Go to statement · Read statement and proof side by side2581 \begin{proof}
2582 Every representative in \(W^{r,p}(0,1)\) has derivatives through order \(r-1\) that admit absolutely continuous representatives on compact subintervals, with matching one sided traces at each interior point. Taking the jump of the derivative of order \(m\) therefore removes the contribution of \(s\). The packet \(\phi_{m,\tau_q}\) has unit jump in its derivative of order \(m\), while all other declared packets have zero jump at that derivative order and breakpoint. Hence the remaining jump is exactly \(b_{q,m}\).
2583 \end{proof}
2584 
2585 \begin{remark}\label{rem:higher_order_scope}
2586 The theorem isolates only finitely many trace jumps. More general free discontinuity models may contain jump sets, Cantor parts, or singular derivatives that are not represented by a finite truncated power sum. The natural ambient theory is related to bounded variation and free discontinuity spaces \cite{AmbrosioFuscoPallara2000}.
2587 \end{remark}
2588 
2589 \begin{figure}
2590 \centering
2591 \includegraphics[width=0.72\textwidth]{06_higher_order_packets.pdf}
2592 \caption{Truncated power packets encode jumps in a function and in its successive derivatives.}
2593 \label{fig:higher_order_packets}
2594 \end{figure}
2595 
2596 Let \(X\) be a normed linear space and let \(Q:X\to X\) be bounded and linear. The next identity is elementary, but it is the central separation principle.
2597 
Statement: Exact retained packet identity · Go to proof · Read statement and proof side by side2598 \begin{theorem}[Exact retained packet identity]\label{thm:retained_packet_identity}
2599 Let \(f=s+j\) in \(X\). Define
2600 \[
2601 \widetilde f=Qs+j.
2602 \]
2603 Then
2604 \[
2605 \boxed{
2606 f-\widetilde f=(I-Q)s.
2607 }
2608 \]
2609 Consequently,
2610 \[
2611 \|f-\widetilde f\|=\|s-Qs\|.
2612 \]
2613 \end{theorem}
2614 
Proof block: Exact retained packet identity · Go to statement · Read statement and proof side by side2615 \begin{proof}
2616 Subtract:
2617 \[
2618 f-\widetilde f
2619 =s+j-Qs-j
2620 =(I-Q)s.
2621 \]
2622 \end{proof}
2623 
2624 \begin{remark}\label{rem:retained_packet_boundary}
2625 The theorem assumes that the packet \(j\) is known. It does not supply a detector. It also does not say that every representation of \(f\) as \(s+j\) is useful. The point is that once a meaningful packet has been identified, its approximation error can be removed from the smooth approximation problem exactly.
2626 \end{remark}
2627 
Statement: Sobolev rate after packet separation · Go to proof · Read statement and proof side by side2628 \begin{corollary}[Sobolev rate after packet separation]\label{cor:sobolev_rate}
2629 Suppose \(Q_m\) satisfies
2630 \[
2631 \|s-Q_ms\|_{L^p}
2632 \leq
2633 C2^{-mr}\|s\|_{W^{r,p}}
2634 \]
2635 for every \(s\in W^{r,p}(0,1)\). If \(f=s+j\) and \(j\) is retained, then
2636 \[
2637 \|f-(Q_ms+j)\|_{L^p}
2638 \leq
2639 C2^{-mr}\|s\|_{W^{r,p}}.
2640 \]
2641 The same conclusion applies to the higher order decomposition in \cref{thm:higher_order_packet_decomposition}.
2642 \end{corollary}
2643 
Proof block: Sobolev rate after packet separation · Go to statement · Read statement and proof side by side2644 \begin{proof}
2645 Use \cref{thm:retained_packet_identity} and the assumed approximation estimate.
2646 \end{proof}
2647 
Statement: Reconstruction with an estimated packet · Go to proof · Read statement and proof side by side2648 \begin{theorem}[Reconstruction with an estimated packet]\label{thm:estimated_packet_reconstruction}
2649 Let \(Q_m:X\to X\) be linear, let \(f=s+j\), let \(\widehat j\) be an estimated packet, and define
2650 \[
2651 \widehat f_m
2652 =
2653 Q_m(f-\widehat j)+\widehat j.
2654 \]
2655 Then
2656 \[
2657 \boxed{
2658 f-\widehat f_m
2659 =
2660 (I-Q_m)s+(I-Q_m)(j-\widehat j).
2661 }
2662 \]
2663 If \(Q_m\) is bounded, then
2664 \[
2665 \|f-\widehat f_m\|
2666 \leq
2667 \|s-Q_ms\|
2668 +
2669 (1+\|Q_m\|)\|j-\widehat j\|.
2670 \]
2671 \end{theorem}
2672 
Proof block: Reconstruction with an estimated packet · Go to statement · Read statement and proof side by side2673 \begin{proof}
2674 Since \(f-\widehat j=s+(j-\widehat j)\),
2675 \[
2676 \widehat f_m
2677 =
2678 Q_ms+Q_m(j-\widehat j)+\widehat j.
2679 \]
2680 Subtract this expression from \(s+j\). The norm bound follows from
2681 \[
2682 \|(I-Q_m)u\|
2683 \leq
2684 (1+\|Q_m\|)\|u\|.
2685 \]
2686 \end{proof}
2687 
2688 \begin{figure}
2689 \centering
2690 \includegraphics[width=0.84\textwidth]{07_known_packet_reconstruction.pdf}
2691 \caption{A global Fourier approximation must represent the jumps. A retained packet approximation applies the Fourier truncation only to the smooth component.}
2692 \label{fig:known_packet_reconstruction}
2693 \end{figure}
2694 
2695 \section{Exact error formulas for estimated packets}\label{extension:section:08}\label{sec:packet_error}
2696 
2697 The error of an estimated packet is often described by a segmentation score. For approximation, an exact norm formula is more useful because it separates value errors from geometric disagreement.
2698 
Statement: Exact scalar region formula · Go to proof · Read statement and proof side by side2699 \begin{theorem}[Exact scalar region formula]\label{thm:scalar_region_formula}
2700 Let \(E,F\) be measurable subsets of a measure space, let \(a,b\in\CC\), and let \(1\leq p<\infty\). Then
2701 \[
2702 \begin{aligned}
2703 \|a\one_E-b\one_F\|_p^p
2704 ={}&
2705 |a-b|^p\mu(E\cap F)
2706 +|a|^p\mu(E\setminus F)\\
2707 &+|b|^p\mu(F\setminus E).
2708 \end{aligned}
2709 \]
2710 For equal amplitudes,
2711 \[
2712 \|a\one_E-a\one_F\|_p
2713 =
2714 |a|\mu(E\mathbin\triangle F)^{1/p}.
2715 \]
2716 \end{theorem}
2717 
Proof block: Exact scalar region formula · Go to statement · Read statement and proof side by side2718 \begin{proof}
2719 The sets \(E\cap F\), \(E\setminus F\), and \(F\setminus E\) are disjoint. The difference takes the values \(a-b\), \(a\), and \(-b\) on these three sets and is zero elsewhere. Integrating its \(p\)th power gives the formula.
2720 \end{proof}
2721 
Statement: Location error for one step · Go to proof · Read statement and proof side by side2722 \begin{corollary}[Location error for one step]\label{cor:step_location_error}
2723 Let \(0\leq\tau,\widehat\tau\leq1\) and \(H_\tau=\one_{[\tau,1]}\). For \(1\leq p<\infty\),
2724 \[
2725 \|aH_\tau-aH_{\widehat\tau}\|_p
2726 =
2727 |a||\tau-\widehat\tau|^{1/p}.
2728 \]
2729 In the supremum norm, the error equals \(|a|\) whenever \(\tau\neq\widehat\tau\).
2730 \end{corollary}
2731 
Proof block: Location error for one step · Go to statement · Read statement and proof side by side2732 \begin{proof}
2733 The symmetric difference of the two intervals has length \(|\tau-\widehat\tau|\). The supremum statement follows because the two steps differ by \(a\) on a nonempty interval.
2734 \end{proof}
2735 
Statement: Location stability for higher order packets · Go to proof · Read statement and proof side by side2736 \begin{theorem}[Location stability for higher order packets]\label{thm:higher_order_location}
2737 Let \(m\geq1\), \(1\leq p<\infty\), and \(0\leq\tau,\widehat\tau\leq1\). Then
2738 \[
2739 \|\phi_{m,\tau}-\phi_{m,\widehat\tau}\|_{L^p(0,1)}
2740 \leq
2741 \frac{|\tau-\widehat\tau|}
2742 {(m-1)!((m-1)p+1)^{1/p}}.
2743 \]
2744 For \(p=\infty\),
2745 \[
2746 \|\phi_{m,\tau}-\phi_{m,\widehat\tau}\|_\infty
2747 \leq
2748 \frac{|\tau-\widehat\tau|}{(m-1)!}.
2749 \]
2750 Thus location error is Lipschitz for every derivative jump packet of positive order, while a value jump has only the finite \(L^p\) modulus in \cref{cor:step_location_error}.
2751 \end{theorem}
2752 
Proof block: Location stability for higher order packets · Go to statement · Read statement and proof side by side2753 \begin{proof}
2754 Assume \(\tau<\widehat\tau\). Differentiation with respect to the location parameter gives
2755 \[
2756 \frac{\partial}{\partial u}\phi_{m,u}(t)
2757 =-\phi_{m-1,u}(t)
2758 \]
2759 for almost every \(t\). Hence
2760 \[
2761 \phi_{m,\tau}-\phi_{m,\widehat\tau}
2762 =
2763 \int_\tau^{\widehat\tau}\phi_{m-1,u}\dd u.
2764 \]
2765 Minkowski's inequality gives
2766 \[
2767 \|\phi_{m,\tau}-\phi_{m,\widehat\tau}\|_p
2768 \leq
2769 |\tau-\widehat\tau|
2770 \sup_u\|\phi_{m-1,u}\|_p.
2771 \]
2772 The supremum is bounded by the norm of \(t^{m-1}/(m-1)!\) on \([0,1]\), which is the stated constant. The supremum norm argument is identical.
2773 \end{proof}
2774 
2775 \subsection{Several regions}
2776 
2777 Let \((E_i)_{i=0}^{I}\) and \((F_j)_{j=0}^{J}\) be measurable partitions of the same domain, up to null sets. Let \(a_i\) and \(b_j\) be the corresponding values. The index zero may be used for a background region.
2778 
Statement: Overlap matrix formula · Go to proof · Read statement and proof side by side2779 \begin{theorem}[Overlap matrix formula]\label{thm:overlap_matrix}
2780 Define
2781 \[
2782 M_{ij}=\mu(E_i\cap F_j).
2783 \]
2784 For \(1\leq p<\infty\),
2785 \[
2786 \left\|
2787 \sum_{i=0}^{I}a_i\one_{E_i}
2788 -
2789 \sum_{j=0}^{J}b_j\one_{F_j}
2790 \right\|_p^p
2791 =
2792 \sum_{i=0}^{I}\sum_{j=0}^{J}|a_i-b_j|^pM_{ij}.
2793 \]
2794 The row sums of \(M\) are \(\mu(E_i)\), and the column sums are \(\mu(F_j)\).
2795 \end{theorem}
2796 
Proof block: Overlap matrix formula · Go to statement · Read statement and proof side by side2797 \begin{proof}
2798 The intersections \(E_i\cap F_j\) form a measurable partition. On such an intersection, the two piecewise constant functions have values \(a_i\) and \(b_j\). Integrate the constant error over every intersection. The marginal identities follow by summing the partition pieces in a fixed row or column.
2799 \end{proof}
2800 
Statement: Background and packet error · Go to proof · Read statement and proof side by side2801 \begin{corollary}[Background and packet error]\label{cor:background_packet_error}
2802 Suppose the true and estimated packets use the same amplitude list \((a_i)_{i=0}^{I}\), but possibly different regions. Then
2803 \[
2804 \|j-\widehat j\|_p^p
2805 =
2806 \sum_{i,k}|a_i-a_k|^p\mu(E_i\cap F_k).
2807 \]
2808 Only off diagonal overlaps contribute.
2809 \end{corollary}
2810 
Proof block: Background and packet error · Go to statement · Read statement and proof side by side2811 \begin{proof}
2812 Apply \cref{thm:overlap_matrix} with \(b_k=a_k\). The diagonal terms vanish.
2813 \end{proof}
2814 
2815 \begin{figure}
2816 \centering
2817 \includegraphics[width=0.60\textwidth]{08_overlap_matrix.pdf}
2818 \caption{An overlap matrix records both correct regions and every type of misclassification. The exact packet norm weights each entry by the corresponding value difference.}
2819 \label{fig:overlap_matrix}
2820 \end{figure}
2821 
2822 \subsection{Boundary tube control}
2823 
2824 For a closed set \(\Gamma\), write
2825 \[
2826 N_\delta(\Gamma)
2827 =
2828 \{x:\dist(x,\Gamma)<\delta\}.
2829 \]
2830 
Statement: Linear tube bound2831 \begin{definition}[Linear tube bound]\label{def:linear_tube_bound}
2832 A boundary \(\Gamma\) has a linear tube bound if there are constants \(C_\Gamma\) and \(\delta_0>0\) such that
2833 \[
2834 \mu(N_\delta(\Gamma))
2835 \leq
2836 C_\Gamma\delta
2837 \]
2838 for \(0<\delta<\delta_0\).
2839 \end{definition}
2840 
2841 Bounded Lipschitz domains and many finite perimeter sets have estimates of this form, with constants controlled by geometric size. The next theorem only requires the displayed bound.
2842 
Statement: Geometric packet bound · Go to proof · Read statement and proof side by side2843 \begin{theorem}[Geometric packet bound]\label{thm:geometric_packet_bound}
2844 Let \(E\) be a region with boundary \(\Gamma\), and suppose
2845 \[
2846 E\mathbin\triangle F
2847 \subseteq
2848 N_\delta(\Gamma).
2849 \]
2850 If \(\Gamma\) has a linear tube bound and \(0<\delta<\delta_0\) lies in its stated range, then for \(1\leq p<\infty\),
2851 \[
2852 \|a\one_E-a\one_F\|_p
2853 \leq
2854 |a|(C_\Gamma\delta)^{1/p}.
2855 \]
2856 If amplitudes \(a\) and \(b\) differ, then
2857 \[
2858 \|a\one_E-b\one_F\|_p
2859 \leq
2860 |a-b|\mu(E\cap F)^{1/p}
2861 +
2862 (|a|+|b|)(C_\Gamma\delta)^{1/p}.
2863 \]
2864 \end{theorem}
2865 
Proof block: Geometric packet bound · Go to statement · Read statement and proof side by side2866 \begin{proof}
2867 The equal amplitude case follows from \cref{thm:scalar_region_formula} and the tube inclusion. For unequal amplitudes, split the error into its value mismatch on \(E\cap F\) and the two disagreement pieces. Apply the triangle inequality and the same tube estimate.
2868 \end{proof}
2869 
2870 \begin{example}[A perturbed disk]\label{ex:perturbed_disk}
2871 Let \(E\) and \(F\) be concentric disks of radii \(r\) and \(r+\delta\) in \(\RR^2\), with \(\delta>0\). Then
2872 \[
2873 \mu(E\mathbin\triangle F)
2874 =
2875 \pi\bigl((r+\delta)^2-r^2\bigr)
2876 =
2877 2\pi r\delta+\pi\delta^2.
2878 \]
2879 For equal amplitude \(a\), the exact \(L^2\) packet error is
2880 \[
2881 |a|\sqrt{2\pi r\delta+\pi\delta^2}.
2882 \]
2883 This gives the expected square root dependence on boundary displacement.
2884 \end{example}
2885 
2886 \section{Stable recovery of separated jumps}\label{extension:section:09}\label{sec:jump_recovery}
2887 
2888 \begin{keyidea}[title={The detector needs a visible interval of thresholds}]
2889 A smooth cell and bounded sample noise give one upper contrast bound. A jump cell gives a lower contrast bound after the same uncertainty is subtracted. Recovery is possible when the lower jump bound exceeds the upper smooth bound. The gap is a condition on the observation model, rather than an automatic consequence of packet coordinates.
2890 \end{keyidea}
2891 \begin{figure}[htbp]\centering
2892 \includegraphics[width=.93\linewidth]{monograph_figures/jump_contrast_gap.pdf}
2893 \caption{A sufficient threshold interval from exact rational contrast bounds. With the displayed model parameters, smooth and noise contrasts are at most $1/10$ and jump contrasts are at least $9/10$. Every threshold strictly between those values separates the cells under the sampling assumptions stated below. The figure does not assert an optimal threshold for other observation models.}\label{monograph:fig:contrast}
2894 \end{figure}
2895 
2896 
2897 The previous sections describe the error after a packet has been estimated. We now give a direct recovery theorem for one dimensional separated jumps. The detector is intentionally simple because the aim is to expose the exact contrast gap.
2898 
2899 Let
2900 \[
2901 t_k=kh,
2902 \qquad
2903 k=0,\ldots,n,
2904 \qquad
2905 h=\frac1n.
2906 \]
2907 Assume
2908 \[
2909 f(t)
2910 =
2911 s(t)+\sum_{q=1}^{Q}a_qH_{\tau_q}(t),
2912 \qquad
2913 H_\tau(t)=\one_{[\tau,1]}(t),
2914 \]
2915 where \(s\) is Lipschitz with constant \(M\), and \(Q\geq1\) is finite with nonzero jump amplitudes. The zero-jump case requires no minimum-amplitude condition. Suppose no jump lies at a sample point and no grid cell contains more than one jump. Observe
2916 \[
2917 y_k=f(t_k)+\eta_k.
2918 \]
2919 Define the adjacent contrast
2920 \[
2921 D_k=y_{k+1}-y_k.
2922 \]
2923 
Statement: Contrast gap · Go to proof · Read statement and proof side by side2924 \begin{lemma}[Contrast gap]\label{lem:contrast_gap}
2925 Assume \(|\eta_k|\leq\varepsilon\) and set
2926 \[
2927 b_h=Mh+2\varepsilon.
2928 \]
2929 If the cell \((t_k,t_{k+1})\) contains no jump, then
2930 \[
2931 |D_k|\leq b_h.
2932 \]
2933 If it contains one jump of amplitude \(a_q\), then
2934 \[
2935 |D_k-a_q|\leq b_h
2936 \]
2937 and therefore
2938 \[
2939 |D_k|\geq |a_q|-b_h.
2940 \]
2941 \end{lemma}
2942 
Proof block: Contrast gap · Go to statement · Read statement and proof side by side2943 \begin{proof}
2944 In a cell without a jump,
2945 \[
2946 D_k=s(t_{k+1})-s(t_k)+\eta_{k+1}-\eta_k.
2947 \]
2948 The Lipschitz and noise bounds give \(|D_k|\leq Mh+2\varepsilon\). In a cell with one jump, the same formula has the additional term \(a_q\), which proves the second estimate.
2949 \end{proof}
2950 
Statement: Deterministic jump cell recovery · Go to proof · Read statement and proof side by side2951 \begin{theorem}[Deterministic jump cell recovery]\label{thm:deterministic_jump_recovery}
2952 Let
2953 \[
2954 a_{\min}=\min_q|a_q|
2955 \]
2956 and assume
2957 \[
2958 a_{\min}>2b_h.
2959 \]
2960 Choose a threshold \(\theta\) satisfying
2961 \[
2962 b_h<\theta<a_{\min}-b_h.
2963 \]
2964 Then
2965 \[
2966 \widehat\Kcal
2967 =
2968 \{k:|D_k|>\theta\}
2969 \]
2970 is exactly the set of grid cells containing jumps. If \(\widehat\tau_q\) is the midpoint of the detected cell and \(\widehat a_q=D_k\), then after matching by order,
2971 \[
2972 |\widehat\tau_q-\tau_q|
2973 \leq
2974 \frac h2
2975 \]
2976 and
2977 \[
2978 |\widehat a_q-a_q|
2979 \leq
2980 b_h.
2981 \]
2982 \end{theorem}
2983 
Proof block: Deterministic jump cell recovery · Go to statement · Read statement and proof side by side2984 \begin{proof}
2985 By \cref{lem:contrast_gap}, every cell without a jump has contrast at most \(b_h<\theta\). Every cell with a jump has contrast at least \(a_{\min}-b_h>\theta\). Hence the detected set is exact. The midpoint of a cell is within \(h/2\) of every point in that cell. The amplitude estimate is the second inequality in \cref{lem:contrast_gap}.
2986 \end{proof}
2987 
Statement: Jump count and sign recovery · Go to proof · Read statement and proof side by side2988 \begin{corollary}[Jump count and sign recovery]\label{cor:jump_count_sign}
2989 Under the hypotheses of \cref{thm:deterministic_jump_recovery}, the number of detected cells equals \(Q\), and
2990 \[
2991 \sgn(\widehat a_q)=\sgn(a_q)
2992 \]
2993 for every real valued jump.
2994 \end{corollary}
2995 
Proof block: Jump count and sign recovery · Go to statement · Read statement and proof side by side2996 \begin{proof}
2997 Exact cell recovery gives the jump count. Since
2998 \[
2999 |\widehat a_q-a_q|\leq b_h<|a_q|,
3000 \]
3001 the amplitude estimate cannot cross zero.
3002 \end{proof}
3003 
Statement: Packet reconstruction after detection · Go to proof · Read statement and proof side by side3004 \begin{corollary}[Packet reconstruction after detection]\label{cor:reconstruction_after_detection}
3005 Let
3006 \[
3007 j=\sum_{q=1}^{Q}a_qH_{\tau_q},
3008 \qquad
3009 \widehat j=\sum_{q=1}^{Q}\widehat a_qH_{\widehat\tau_q}.
3010 \]
3011 Under the hypotheses of \cref{thm:deterministic_jump_recovery}, for \(1\leq p<\infty\),
3012 \[
3013 \|j-\widehat j\|_p
3014 \leq
3015 Qb_h
3016 +
3017 \left(
3018 \frac h2\sum_{q=1}^{Q}|a_q|^p
3019 \right)^{1/p}.
3020 \]
3021 If \(Q_m\) is bounded and linear and
3022 \[
3023 \widehat f_m=Q_m(f-\widehat j)+\widehat j,
3024 \]
3025 then
3026 \[
3027 \begin{aligned}
3028 \|f-\widehat f_m\|_p
3029 \leq{}&
3030 \|s-Q_ms\|_p\\
3031 &+(1+\|Q_m\|)
3032 \left[
3033 Qb_h+
3034 \left(
3035 \frac h2\sum_q|a_q|^p
3036 \right)^{1/p}
3037 \right].
3038 \end{aligned}
3039 \]
3040 \end{corollary}
3041 
Proof block: Packet reconstruction after detection · Go to statement · Read statement and proof side by side3042 \begin{proof}
3043 Write
3044 \[
3045 j-\widehat j
3046 =
3047 \sum_q(a_q-\widehat a_q)H_{\widehat\tau_q}
3048 +
3049 \sum_q a_q(H_{\tau_q}-H_{\widehat\tau_q}).
3050 \]
3051 The first sum has norm at most \(\sum_q|a_q-\widehat a_q|\leq Qb_h\). The location error interval for the \(q\)th jump is contained in its detected grid cell and has length at most \(h/2\). Distinct detected cells are disjoint, so the location errors have disjoint supports. Their \(p\)th powers therefore add exactly, which gives the displayed location term. The reconstruction estimate follows from \cref{thm:estimated_packet_reconstruction}.
3052 \end{proof}
3053 
3054 \begin{remark}[Why finite \(L^p\) matters]\label{rem:finite_p_recovery}
3055 A shifted step does not converge in the supremum norm until the location is exact. The finite \(L^p\) error does converge because the disagreement is confined to an interval whose length tends to zero.
3056 \end{remark}
3057 
Statement: Deterministic consistency · Go to proof · Read statement and proof side by side3058 \begin{corollary}[Deterministic consistency]\label{cor:deterministic_consistency}
3059 Assume \(h\to0\) along grids satisfying the preceding sampling assumptions, \(\varepsilon_h\to0\), the number of jumps is fixed, and the nonzero jump amplitudes and positive minimum separation remain fixed. Then the detected packet converges to the true packet in every finite \(L^p\) norm. If \(Q_m\) has a vanishing smooth approximation error and uniformly bounded norm, the complete reconstruction also converges in \(L^p\).
3060 \end{corollary}
3061 
Proof block: Deterministic consistency · Go to statement · Read statement and proof side by side3062 \begin{proof}
3063 The gap condition holds for sufficiently small \(h\). Then \(b_h\to0\), and the bound in \cref{cor:reconstruction_after_detection} tends to zero.
3064 \end{proof}
3065 
3066 \begin{figure}
3067 \centering
3068 \includegraphics[width=0.83\textwidth]{09_jump_detection.pdf}
3069 \caption{Separated jumps create isolated adjacent sample contrasts above the smooth and noise background.}
3070 \label{fig:jump_detection}
3071 \end{figure}
3072 
3073 \begin{example}[A numerical threshold certificate]\label{ex:jump_threshold_certificate}
3074 Take \(h=1/256\), \(M=0.5\), and \(\varepsilon=0.002\). Then
3075 \[
3076 b_h
3077 =\frac{0.5}{256}+0.004
3078 <0.00596.
3079 \]
3080 If every jump has magnitude at least \(0.08\), then \(0.08>2b_h\). Any threshold between \(0.00596\) and \(0.07404\) recovers the exact jump cells. The location error is at most
3081 \[
3082 \frac1{512}
3083 \]
3084 and every amplitude error is less than \(0.00596\).
3085 \end{example}
3086 
3087 \subsection{Sub Gaussian noise}
3088 
3089 Assume the noise variables are independent, centered, and \(\sigma\) sub Gaussian:
3090 \[
3091 \mathbb E\e^{\lambda\eta_k}
3092 \leq
3093 \e^{\sigma^2\lambda^2/2}.
3094 \]
3095 
Statement: Probability recovery bound · Go to proof · Read statement and proof side by side3096 \begin{theorem}[Probability recovery bound]\label{thm:probability_recovery}
3097 Fix \(0<\delta<1\) and define
3098 \[
3099 t_{n,\delta}
3100 =
3101 2\sigma
3102 \sqrt{\log\left(\frac{2n}{\delta}\right)}
3103 \]
3104 and
3105 \[
3106 b_{h,\delta}=Mh+t_{n,\delta}.
3107 \]
3108 With probability at least \(1-\delta\), every contrast satisfies the conclusions of \cref{lem:contrast_gap} with \(b_h\) replaced by \(b_{h,\delta}\).
3109 
3110 If
3111 \[
3112 a_{\min}>2b_{h,\delta},
3113 \]
3114 then thresholding between \(b_{h,\delta}\) and \(a_{\min}-b_{h,\delta}\) recovers every jump cell with probability at least \(1-\delta\). The same location and amplitude bounds hold with \(b_{h,\delta}\).
3115 \end{theorem}
3116 
Proof block: Probability recovery bound · Go to statement · Read statement and proof side by side3117 \begin{proof}
3118 If \(\sigma=0\), every centered zero-sub-Gaussian noise variable vanishes almost surely, and the noise-free deterministic conclusion applies. For \(\sigma>0\), the difference \(\eta_{k+1}-\eta_k\) is sub Gaussian with variance proxy \(2\sigma^2\), so
3119 \[
3120 \mathbb P\bigl(|\eta_{k+1}-\eta_k|>t\bigr)
3121 \leq
3122 2\exp\left(-\frac{t^2}{4\sigma^2}\right).
3123 \]
3124 A union bound over the \(n\) cells gives
3125 \[
3126 \mathbb P\left(
3127 \max_k|\eta_{k+1}-\eta_k|>t_{n,\delta}
3128 \right)
3129 \leq
3130 \delta.
3131 \]
3132 On the complementary event, repeat the deterministic proof.
3133 \end{proof}
3134 
Statement: Repeated measurements · Go to proof · Read statement and proof side by side3135 \begin{corollary}[Repeated measurements]\label{cor:repeated_measurements}
3136 Suppose \(R\) observations are averaged at every sample location, with all noise variables independent across sites and repetitions and each centered and \(\sigma\) sub Gaussian. Then \cref{thm:probability_recovery} holds with \(\sigma\) replaced by \(\sigma/\sqrt R\). In particular, a sufficient recovery condition is
3137 \[
3138 a_{\min}
3139 >
3140 2Mh+
3141 \frac{4\sigma}{\sqrt R}
3142 \sqrt{\log\left(\frac{2n}{\delta}\right)}.
3143 \]
3144 \end{corollary}
3145 
Proof block: Repeated measurements · Go to statement · Read statement and proof side by side3146 \begin{proof}
3147 The average of \(R\) independent \(\sigma\) sub Gaussian variables is \(\sigma/\sqrt R\) sub Gaussian. Substitute this parameter in \cref{thm:probability_recovery}.
3148 \end{proof}
3149 
3150 \begin{remark}\label{rem:noise_scaling}
3151 The displayed uniform noise certificate scales as \(\sigma\sqrt{\log n}\). At fixed positive \(\sigma\), it does not imply consistency of this detector. Decreasing the noise scale or averaging repeated measurements gives sufficient regimes; other averaging schemes require their own analysis.
3152 \end{remark}
3153 
Statement: Probability consistency · Go to proof · Read statement and proof side by side3154 \begin{corollary}[Probability consistency]\label{cor:probability_consistency}
3155 Let \(\sigma_n\sqrt{\log n}\to0\) and \(h=1/n\). For every fixed \(0<\delta<1\), along grids satisfying the preceding sampling assumptions, all jump cells are recovered with probability at least \(1-\delta\) for sufficiently large \(n\). For the fixed finite jump model above, along grids satisfying the preceding sampling assumptions, the packet error converges to zero in probability in every finite \(L^p\) norm.
3156 \end{corollary}
3157 
Proof block: Probability consistency · Go to statement · Read statement and proof side by side3158 \begin{proof}
3159 For each fixed \(\delta>0\), \(b_{h,\delta}\to0\). Apply \cref{thm:probability_recovery,cor:reconstruction_after_detection}. For any error tolerance \(\epsilon>0\), the high-probability bound is eventually smaller than \(\epsilon\), so the limit superior of the probability that the packet error exceeds \(\epsilon\) is at most \(\delta\). Since \(\delta\) is arbitrary, convergence is in probability.
3160 \end{proof}
3161 
3162 \subsection{Signals with values in a normed space}
3163 
3164 The recovery proof does not depend on an ordering of the signal values.
3165 
Statement: Vector jump recovery · Go to proof · Read statement and proof side by side3166 \begin{proposition}[Vector jump recovery]\label{prop:vector_jump_recovery}
3167 Let the values lie in a normed space \(V\). Assume
3168 \[
3169 \|s(t)-s(u)\|_V\leq M|t-u|,
3170 \qquad
3171 \|\eta_k\|_V\leq\varepsilon,
3172 \]
3173 and let the jump vectors satisfy \(\|a_q\|_V\geq a_{\min}\). Thresholding \(\|D_k\|_V\) gives the same recovery theorem under the condition
3174 \[
3175 a_{\min}>2(Mh+2\varepsilon).
3176 \]
3177 \end{proposition}
3178 
Proof block: Vector jump recovery · Go to statement · Read statement and proof side by side3179 \begin{proof}
3180 Only the triangle inequality was used in \cref{lem:contrast_gap,thm:deterministic_jump_recovery}.
3181 \end{proof}
3182 
3183 \section{Haar localization and annihilating differences}\label{extension:section:10}\label{sec:localization}
3184 
3185 The adjacent difference detector is local in the sample domain. Wavelet coefficients and higher differences give related multiscale packets. In the Haar subsections the signal is defined on \(\RR\). Here the step convention is \(H_\tau(t)=\one_{[\tau,\infty)}(t)\), whose restriction to \((0,1)\) agrees with the preceding interval step. A chosen extension beyond the observation interval must use this half-line step convention, and the smooth extension must satisfy the stated Lipschitz hypothesis on every displayed wavelet support.
3186 
3187 \subsection{Haar coefficient separation}
3188 
3189 Let
3190 \[
3191 I_{j,k}
3192 =
3193 [k2^{-j},(k+1)2^{-j})
3194 \]
3195 and let \(\psi_{j,k}\) be the normalized Haar wavelet supported on \(I_{j,k}\), positive on the left half and negative on the right half.
3196 
Statement: Smooth Haar coefficient bound · Go to proof · Read statement and proof side by side3197 \begin{theorem}[Smooth Haar coefficient bound]\label{thm:smooth_haar_bound}
3198 If \(s\) is Lipschitz with constant \(M\), then
3199 \[
3200 |\langle s,\psi_{j,k}\rangle|
3201 \leq
3202 \frac M4\,2^{-3j/2}.
3203 \]
3204 \end{theorem}
3205 
Proof block: Smooth Haar coefficient bound · Go to statement · Read statement and proof side by side3206 \begin{proof}
3207 Let \(L=2^{-j}\) and let \(a\) be the left endpoint of the support. Pair every point \(t\in[a,a+L/2]\) with \(t+L/2\). Then
3208 \[
3209 \langle s,\psi_{j,k}\rangle
3210 =
3211 L^{-1/2}
3212 \int_a^{a+L/2}
3213 \overline{s(t)-s(t+L/2)}\,\dd t.
3214 \]
3215 The integrand has magnitude at most \(ML/2\), and the integration interval has length \(L/2\). The result is \(ML^{3/2}/4\).
3216 \end{proof}
3217 
Statement: Exact Haar coefficient of a step · Go to proof · Read statement and proof side by side3218 \begin{theorem}[Exact Haar coefficient of a step]\label{thm:haar_step_coefficient}
3219 Let \(\tau\in I_{j,k}\), let \(L=2^{-j}\), and write
3220 \[
3221 u=\frac{\tau-kL}{L}\in[0,1].
3222 \]
3223 Then
3224 \[
3225 \left|
3226 \left\langle
3227  aH_\tau,\psi_{j,k}
3228 \right\rangle
3229 \right|
3230 =
3231 |a|L^{1/2}\min\{u,1-u\}.
3232 \]
3233 Every Haar coefficient whose support does not cross \(\tau\) is zero.
3234 \end{theorem}
3235 
Proof block: Exact Haar coefficient of a step · Go to statement · Read statement and proof side by side3236 \begin{proof}
3237 Away from the jump, the step is constant on the support and the Haar wavelet has zero mean. When the jump lies in the support, integrate the positive and negative halves directly. If \(u\leq1/2\), the magnitude is \(|a|uL^{1/2}\). If \(u\geq1/2\), it is \(|a|(1-u)L^{1/2}\).
3238 \end{proof}
3239 
Statement: Scale separation · Go to proof · Read statement and proof side by side3240 \begin{corollary}[Scale separation]\label{cor:haar_scale_separation}
3241 Suppose \(f=s+aH_\tau+\eta\), where \(s\) is \(M\) Lipschitz and \(\|\eta\|_\infty\leq\varepsilon\). If the jump lies at relative distance at least \(\rho\) from both ends of \(I_{j,k}\), then
3242 \[
3243 |\langle f,\psi_{j,k}\rangle|
3244 \geq
3245 \left(
3246 |a|\rho
3247 -\varepsilon
3248 -\frac M4L
3249 \right)L^{1/2}.
3250 \]
3251 A wavelet support without a jump satisfies
3252 \[
3253 |\langle f,\psi_{j,k}\rangle|
3254 \leq
3255 \left(
3256 \varepsilon+\frac M4L
3257 \right)L^{1/2}.
3258 \]
3259 Thus a threshold gap exists whenever
3260 \[
3261 |a|\rho
3262 >
3263 2\varepsilon+\frac M2L.
3264 \]
3265 \end{corollary}
3266 
Proof block: Scale separation · Go to statement · Read statement and proof side by side3267 \begin{proof}
3268 Combine \cref{thm:smooth_haar_bound,thm:haar_step_coefficient}. The noise coefficient is at most
3269 \[
3270 \|\eta\|_\infty\|\psi_{j,k}\|_1
3271 =
3272 \varepsilon L^{1/2}.
3273 \]
3274 The final inequality makes the lower bound on a jump support larger than the upper bound on a smooth support.
3275 \end{proof}
3276 
3277 \begin{figure}
3278 \centering
3279 \includegraphics[width=0.68\textwidth]{10_haar_scale_separation.pdf}
3280 \caption{The smooth contribution decays like \(L^{3/2}\), while a separated jump contribution decays like \(L^{1/2}\).}
3281 \label{fig:haar_scale_separation}
3282 \end{figure}
3283 
3284 \subsection{Two shifted grids}
3285 
3286 A jump on a dyadic boundary can be invisible to a wavelet at that scale. One additional grid removes this exact alignment.
3287 
Statement: Half cell shift guarantee · Go to proof · Read statement and proof side by side3288 \begin{theorem}[Half cell shift guarantee]\label{thm:half_shift_guarantee}
3289 Fix a cell length \(L>0\). Consider the grid with boundaries \(kL\) and the grid with boundaries \((k+1/2)L\). Every point \(\tau\in\RR\) lies in a cell of at least one grid at distance at least \(L/4\) from both cell boundaries.
3290 \end{theorem}
3291 
Proof block: Half cell shift guarantee · Go to statement · Read statement and proof side by side3292 \begin{proof}
3293 Reduce \(\tau\) modulo \(L\) and write its representative as \(x\in[0,L)\). If
3294 \[
3295 L/4\leq x\leq 3L/4,
3296 \]
3297 then the distance from \(x\) to both boundaries of its cell in the original grid is at least \(L/4\). If \(x<L/4\) or \(x>3L/4\), then \(x\) lies in the cell of the shifted grid bounded by \(-L/2\) and \(L/2\), or in its translate, at distance at least \(L/4\) from both shifted boundaries. One of the two grids therefore has the required property.
3298 \end{proof}
3299 
Statement: Alignment free two grid Haar detection · Go to proof · Read statement and proof side by side3300 \begin{corollary}[Alignment free two grid Haar detection]\label{cor:two_grid_haar_detection}
3301 Assume the support length \(L\) is smaller than the minimum jump separation. Analyze the signal with Haar wavelets on the original grid and on the half shifted grid. Every jump appears in a support with \(\rho\geq1/4\) in one of the two systems. Therefore a threshold gap exists independently of alignment whenever
3302 \[
3303 \frac{|a|}{4}
3304 >
3305 2\varepsilon+\frac M2L.
3306 \]
3307 \end{corollary}
3308 
Proof block: Alignment free two grid Haar detection · Go to statement · Read statement and proof side by side3309 \begin{proof}
3310 Apply \cref{thm:half_shift_guarantee} and then \cref{cor:haar_scale_separation} with \(\rho=1/4\). The separation assumption prevents a selected support from containing two jumps.
3311 \end{proof}
3312 
3313 \begin{figure}
3314 \centering
3315 \includegraphics[width=0.78\textwidth]{11_shifted_grids.pdf}
3316 \caption{The boundaries of the two grids interlace. Every jump is well inside a cell in at least one grid.}
3317 \label{fig:shifted_grids}
3318 \end{figure}
3319 
3320 \subsection{Higher differences}
3321 
3322 Let
3323 \[
3324 (\Delta x)(n)=x(n+1)-x(n).
3325 \]
3326 Then
3327 \[
3328 \Delta^m x(n)
3329 =
3330 \sum_{r=0}^{m}
3331 (-1)^{m-r}\binom mr x(n+r).
3332 \]
3333 
Statement: Polynomial annihilation and boundary support · Go to proof · Read statement and proof side by side3334 \begin{theorem}[Polynomial annihilation and boundary support]\label{thm:polynomial_annihilation}
3335 Let \(x\) be a sequence that agrees with a polynomial of degree at most \(d\) on each interval of a finite partition of \(\ZZ\). Then
3336 \[
3337 \Delta^{d+1}x(n)=0
3338 \]
3339 whenever the complete stencil
3340 \[
3341 \{n,n+1,\ldots,n+d+1\}
3342 \]
3343 lies inside one polynomial piece. Hence \(\Delta^{d+1}x\) is supported in a band of at most \(d+1\) addresses before each breakpoint.
3344 \end{theorem}
3345 
Proof block: Polynomial annihilation and boundary support · Go to statement · Read statement and proof side by side3346 \begin{proof}
3347 The \((d+1)\)st forward difference of a polynomial of degree at most \(d\) is zero. The only stencils for which this argument does not apply are those crossing a breakpoint.
3348 \end{proof}
3349 
Statement: Noise amplification by differences · Go to proof · Read statement and proof side by side3350 \begin{proposition}[Noise amplification by differences]\label{prop:difference_noise}
3351 If \(\|\eta\|_\infty\leq\varepsilon\), then
3352 \[
3353 \|\Delta^m\eta\|_\infty
3354 \leq
3355 2^m\varepsilon.
3356 \]
3357 Therefore no false response above \(2^{d+1}\varepsilon\) can occur on a stencil lying inside one polynomial piece.
3358 \end{proposition}
3359 
Proof block: Noise amplification by differences · Go to statement · Read statement and proof side by side3360 \begin{proof}
3361 The absolute values of the binomial coefficients in \(\Delta^m\) sum to \(2^m\).
3362 \end{proof}
3363 
Statement: Robust breakpoint bands · Go to proof · Read statement and proof side by side3364 \begin{corollary}[Robust breakpoint bands]\label{cor:robust_breakpoint_bands}
3365 Assume every true breakpoint produces at least one response of magnitude \(\rho\) in its \((d+1)\)st difference band. If
3366 \[
3367 \rho>2^{d+2}\varepsilon,
3368 \]
3369 then thresholding at any level between \(2^{d+1}\varepsilon\) and \(\rho-2^{d+1}\varepsilon\) detects a nonempty cluster near every breakpoint and creates no cluster away from the breakpoint bands.
3370 \end{corollary}
3371 
Proof block: Robust breakpoint bands · Go to statement · Read statement and proof side by side3372 \begin{proof}
3373 Use \cref{thm:polynomial_annihilation,prop:difference_noise}. Noise cannot cross the threshold away from a breakpoint. At a response of magnitude \(\rho\), the perturbed magnitude remains at least \(\rho-2^{d+1}\varepsilon\), which is above the threshold.
3374 \end{proof}
3375 
3376 \section{Packet separation and the Gibbs phenomenon}\label{extension:section:11}\label{sec:gibbs}
3377 
3378 A Fourier partial sum is global. When the target has a jump, the partial sum must approximate that jump using continuous trigonometric polynomials. This creates the classical Gibbs oscillation. Standard accounts and recovery methods appear in \cite{Zygmund2002,GottliebShu1997}.
3379 
Statement: Classical square-wave Gibbs behavior3380 \begin{standardfact}[Classical square-wave Gibbs behavior]\label{sf:gibbs_behavior}
3381 Let \(q(x)=\tfrac12\operatorname{sign}(\sin x)\), whose jump at zero has size one. For \(n\geq1\), its partial sum through the first \(n\) positive odd frequencies is
3382 \[
3383 Q_n(x)=\frac{2}{\pi}\sum_{k=0}^{n-1}\frac{\sin((2k+1)x)}{2k+1}.
3384 \]
3385 At \(x_n=\pi/(2n)\),
3386 \[
3387 x_n\longrightarrow0,\qquad Q_n(x_n)\longrightarrow\frac{\operatorname{Si}(\pi)}{\pi}
3388 =\frac12+G,\qquad G\approx0.089489872236,
3389 \]
3390 where \(\operatorname{Si}(z)=\int_0^z\sin(t)/t\,dt\) and \(G=\operatorname{Si}(\pi)/\pi-1/2>0\).
3391 Thus a nonzero overshoot persists along points approaching the jump. Scaling by a real amplitude \(a\) gives an overshoot or undershoot of magnitude \(|a|G\).
3392 
3393 For the limit, put \(u_k=(k+\tfrac12)/n\). The identity
3394 \[
3395 Q_n(x_n)=\frac{1}{\pi n}\sum_{k=0}^{n-1}\frac{\sin(\pi u_k)}{u_k}
3396 \longrightarrow\frac1\pi\int_0^1\frac{\sin(\pi u)}u\,du
3397 =\frac{\operatorname{Si}(\pi)}{\pi}
3398 \]
3399 is a midpoint Riemann sum, since the integrand has continuous value \(\pi\) at zero. This is the normalized classical example in \cite[Section~6.16(i), equations~6.16.1--6.16.4]{DLMFGibbs}.
3400 \end{standardfact}
3401 
3402 The square-wave example supplies a classical comparison. The following retained-packet identities do not require a general overshoot theorem. They change the approximation problem because the discontinuity is no longer represented by the Fourier partial sum.
3403 
Statement: Uniform convergence with exact jump packets · Go to proof · Read statement and proof side by side3404 \begin{theorem}[Uniform convergence with exact jump packets]\label{thm:uniform_packet_convergence}
3405 Let \(f=s+j\) on a compact domain, where \(j\) is any bounded singular packet. Suppose \(Q_ms\to s\) uniformly. Then
3406 \[
3407 \widetilde f_m=Q_ms+j
3408 \]
3409 converges uniformly to \(f\), even when \(f\) is discontinuous:
3410 \[
3411 \|f-\widetilde f_m\|_\infty
3412 =
3413 \|s-Q_ms\|_\infty
3414 \longrightarrow0.
3415 \]
3416 There is no error proportional to the jump size because the jump is shared exactly by \(f\) and \(\widetilde f_m\).
3417 \end{theorem}
3418 
Proof block: Uniform convergence with exact jump packets · Go to statement · Read statement and proof side by side3419 \begin{proof}
3420 Apply \cref{thm:retained_packet_identity} in the supremum norm.
3421 \end{proof}
3422 
3423 \Needspace{20\baselineskip}
Statement: Estimated packet Gibbs bound · Go to proof · Read statement and proof side by side3424 \begin{theorem}[Estimated packet Gibbs bound]\label{thm:estimated_packet_gibbs}
3425 Let \(Q_m\) be linear and uniformly bounded on \(L^p\), with
3426 \[
3427 \sup_m\|Q_m\|\leq C_Q.
3428 \]
3429 For \(f=s+j\) and an estimated packet \(\widehat j_m\), define
3430 \[
3431 \widehat f_m
3432 =Q_m(f-\widehat j_m)+\widehat j_m.
3433 \]
3434 Then
3435 \[
3436 \|f-\widehat f_m\|_p
3437 \leq
3438 \|s-Q_ms\|_p
3439 +(1+C_Q)\|j-\widehat j_m\|_p.
3440 \]
3441 Consequently, smooth approximation and packet recovery may be improved independently. No claim of Gibbs removal is justified unless both terms are controlled in the norm under consideration.
3442 \end{theorem}
3443 
Proof block: Estimated packet Gibbs bound · Go to statement · Read statement and proof side by side3444 \begin{proof}
3445 This is \cref{thm:estimated_packet_reconstruction} with the uniform operator bound.
3446 \end{proof}
3447 
Statement: Boundary band error · Go to proof · Read statement and proof side by side3448 \begin{corollary}[Boundary band error]\label{cor:boundary_band_error}
3449 Let \(\Gamma\) be the singular boundary and assume
3450 \[
3451 \mu(N_\delta(\Gamma))\leq C_\Gamma\delta
3452 \]
3453 for small \(\delta\). For finite \(p\),
3454 \[
3455 \|f-\widetilde f_m\|_{L^p(N_\delta(\Gamma))}
3456 \leq
3457 (C_\Gamma\delta)^{1/p}
3458 \|s-Q_ms\|_\infty.
3459 \]
3460 \end{corollary}
3461 
Proof block: Boundary band error · Go to statement · Read statement and proof side by side3462 \begin{proof}
3463 The error equals \(s-Q_ms\). Bound it by its supremum and multiply by the measure of the boundary tube.
3464 \end{proof}
3465 
3466 \begin{example}[Square wave Fourier tail]\label{ex:square_wave_tail}
3467 Let
3468 \[
3469 j(t)=\one_{[-a,a]}(t)
3470 \]
3471 on the circle and let \(s\) be smooth. If \(Q_m\) is a Fourier partial sum, then \(Q_m(s+j)\) has the ordinary oscillatory boundary layer. In contrast,
3472 \[
3473 Q_ms+j
3474 \]
3475 has error exactly \(s-Q_ms\). The visual comparison in \cref{fig:gibbs_comparison} illustrates this change in approximation target. It is not a new Fourier summability theorem.
3476 \end{example}
3477 
3478 \begin{figure}
3479 \centering
3480 \includegraphics[width=0.82\textwidth]{12_gibbs_comparison.pdf}
3481 \caption{Global Fourier truncation represents the jump and exhibits an oscillatory boundary layer. Packet separation retains the jump and approximates only the smooth component.}
3482 \label{fig:gibbs_comparison}
3483 \end{figure}
3484 
3485 \FloatBarrier
3486 \section{Nonlinear interactions in residue coordinates}\label{extension:section:12}\label{sec:nonlinear}
3487 
3488 Linear transfer matrices do not directly describe nonlinear products. Residue coordinates still obey exact conservation laws because integer addresses add modulo the packet modulus.
3489 
Statement: Residue law for Cauchy products · Go to proof · Read statement and proof side by side3490 \begin{theorem}[Residue law for Cauchy products]\label{thm:cauchy_residue_law}
3491 Let
3492 \[
3493 F(z)=\sum_{n\geq0}f_nz^n,
3494 \qquad
3495 G(z)=\sum_{n\geq0}g_nz^n
3496 \]
3497 be formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo \(N\):
3498 \[
3499 F=\sum_{r=0}^{N-1}F_r,
3500 \qquad
3501 G=\sum_{s=0}^{N-1}G_s.
3502 \]
3503 If \(H=FG\), then its residue packet \(H_t\) is
3504 \[
3505 \boxed{
3506 H_t
3507 =
3508 \sum_{r+s\equiv t\!\!\pmod N}F_rG_s.
3509 }
3510 \]
3511 \end{theorem}
3512 
Proof block: Residue law for Cauchy products · Go to statement · Read statement and proof side by side3513 \begin{proof}
3514 Every monomial in \(F_rG_s\) has degree congruent to \(r+s\) modulo \(N\). Conversely, every coefficient of degree congruent to \(t\) is formed from pairs whose residues sum to \(t\). Grouping the Cauchy product by this residue gives the formula.
3515 \end{proof}
3516 
Statement: Shifted pointwise monomial law · Go to proof · Read statement and proof side by side3517 \begin{theorem}[Shifted pointwise monomial law]\label{thm:shifted_monomial_law}
3518 Let
3519 \[
3520 y(n)=\prod_{j=1}^{m}x(n-a_j)
3521 \]
3522 and write
3523 \[
3524 a_j=Nc_j+b_j,
3525 \qquad
3526 0\leq b_j<N.
3527 \]
3528 For output residue \(r\), define
3529 \[
3530 r_j\equiv r-b_j\pmod N,
3531 \qquad
3532 0\leq r_j<N,
3533 \]
3534 and let
3535 \[
3536 \kappa_{r,b_j}
3537 =
3538 \begin{cases}
3539 1,&r<b_j,\\
3540 0,&r\geq b_j.
3541 \end{cases}
3542 \]
3543 Then the output channel is
3544 \[
3545 \boxed{
3546 y_r(q)
3547 =
3548 \prod_{j=1}^{m}
3549 x_{r_j}
3550 \bigl(q-c_j-\kappa_{r,b_j}\bigr).
3551 }
3552 \]
3553 \end{theorem}
3554 
Proof block: Shifted pointwise monomial law · Go to statement · Read statement and proof side by side3555 \begin{proof}
3556 For \(n=Nq+r\),
3557 \[
3558 n-a_j
3559 =N(q-c_j)+r-b_j.
3560 \]
3561 If \(r\geq b_j\), this address has residue \(r-b_j\) and coarse index \(q-c_j\). If \(r<b_j\), add \(N\) to the residue and subtract one from the coarse index. Multiplying the resulting channel values gives the formula.
3562 \end{proof}
3563 
Statement: Polynomial shift filters · Go to proof · Read statement and proof side by side3564 \begin{corollary}[Polynomial shift filters]\label{cor:polynomial_shift_filters}
3565 Every finite polynomial in shifted samples has an exact finite expression in residue coordinates. Each monomial selects a deterministic list of input residues and coarse shifts, and the complete output is their finite sum.
3566 \end{corollary}
3567 
Proof block: Polynomial shift filters · Go to statement · Read statement and proof side by side3568 \begin{proof}
3569 Apply \cref{thm:shifted_monomial_law} to each monomial and add the results.
3570 \end{proof}
3571 
3572 \begin{example}[Quadratic nearest neighbor interaction]\label{ex:quadratic_neighbor}
3573 For
3574 \[
3575 y(n)=x(n)x(n-1)
3576 \]
3577 and modulus \(N\),
3578 \[
3579 y_r(q)
3580 =
3581 \begin{cases}
3582 x_r(q)x_{r-1}(q),&r\geq1,\\
3583 x_0(q)x_{N-1}(q-1),&r=0.
3584 \end{cases}
3585 \]
3586 The carry at residue zero is the nonlinear analogue of a transfer crossing a coarse cell boundary.
3587 \end{example}
3588 
3589 \section{Finite numerical illustrations}\label{extension:section:13}\label{sec:numerical}
3590 
3591 The figures and tables in this section are finite checks and examples. They are not substitutes for the preceding proofs. Their purpose is to show the scale and shape of the terms appearing in the exact estimates.
3592 
3593 A fixed depth Haar packet transform and a Fourier basis were applied to the same piecewise smooth test signal. The largest coefficients were retained in each representation.
3594 
3595 \begin{figure}
3596 \centering
3597 \includegraphics[width=0.70\textwidth]{13_sparse_comparison.pdf}
3598 \caption{One finite sparse approximation comparison. Fourier coordinates are better at the smallest budget, while the fixed Haar packet basis becomes better after localized edge coordinates are resolved.}
3599 \label{fig:sparse_comparison}
3600 \end{figure}
3601 
3602 \begin{table}
3603 \centering
3604 \caption{Finite sparse approximation record for one test signal.}
3605 \label{tab:sparse_record}
3606 \begin{tabular}{rrr}
3607 \toprule
3608 retained & Haar packet RMSE & Fourier RMSE\\
3609 \midrule
3610 16 & \(1.7999\times10^{-1}\) & \(6.6522\times10^{-2}\)\\
3611 32 & \(3.8414\times10^{-2}\) & \(4.9153\times10^{-2}\)\\
3612 64 & \(1.6267\times10^{-2}\) & \(3.8056\times10^{-2}\)\\
3613 128 & \(1.7387\times10^{-3}\) & \(2.5631\times10^{-2}\)\\
3614 256 & \(1.5870\times10^{-4}\) & \(1.7424\times10^{-2}\)\\
3615 \bottomrule
3616 \end{tabular}
3617 \end{table}
3618 
3619 The table does not prove that Haar packets dominate Fourier coordinates for every signal. Smooth periodic oscillations may favor Fourier coordinates. Curved edges in two dimensions may favor directional systems. The geometric approximation results in \cite{Donoho1999,CandesDonoho2004} give much deeper statements for specific image classes.
3620 
3621 \begin{table}
3622 \centering
3623 \caption{Finite edge reconstruction record from the original computational study.}
3624 \label{tab:edge_record}
3625 \small
3626 \begin{tabular}{lccc}
3627 \toprule
3628 case & global & exact packet & estimated packet\\
3629 \midrule
3630 one dimensional step & \(5.4023\times10^{-2}\) & \(1.3692\times10^{-4}\) & \(2.4029\times10^{-2}\)\\
3631 two dimensional disk & \(3.7643\times10^{-2}\) & \(6.4752\times10^{-5}\) & \(3.4043\times10^{-3}\)\\
3632 multiple regions & \(3.3740\times10^{-2}\) & \(5.2118\times10^{-7}\) & not evaluated\\
3633 \bottomrule
3634 \end{tabular}
3635 \end{table}
3636 
3637 The estimated packet rows used a narrow positive contrast synthetic model. They are not evidence for a general segmentation theorem. The exact packet rows verify \cref{thm:retained_packet_identity} numerically after the packet is supplied.
3638 
3639 \section{Cyclic Cayley coordinates}\label{extension:section:14}\label{sec:cayley}
3640 
3641 \begin{interpretation}[title={Keep the rotation and its chart separate}]
3642 For the phase factorization below, $M$ and $N$ are positive phase orders. The reflection coordinate rotates in the unit disk. The Cayley map expresses that same motion as a fractional linear impedance action. A pole of the chart does not mean that the projective action fails. This realization is independent of the separated jump detector.
3643 \end{interpretation}
3644 
3645 This section records a secondary realization from the original signal manuscript. It is logically independent of the recovery theorems. The classical Smith chart uses the Cayley transform between normalized impedance and a reflection coordinate \cite{Smith1969}.
3646 
3647 Let
3648 \[
3649 \mathbb H
3650 =
3651 \{z\in\CC:\Re z>0\}
3652 \]
3653 and define
3654 \[
3655 C(z)=\frac{z-1}{z+1}.
3656 \]
3657 It maps \(\mathbb H\) conformally onto the unit disk \(\mathbb D\), with inverse
3658 \[
3659 C^{-1}(w)=\frac{1+w}{1-w}.
3660 \]
3661 For \(\theta\in\RR\), let
3662 \[
3663 R_\theta(w)=\e^{\ii\theta}w.
3664 \]
3665 
Statement: Cyclic Mobius action · Go to proof · Read statement and proof side by side3666 \begin{theorem}[Cyclic Mobius action]\label{thm:cyclic_mobius}
3667 The pullback
3668 \[
3669 \Phi_\theta
3670 =C^{-1}R_\theta C
3671 \]
3672 is the fractional linear map
3673 \[
3674 \boxed{
3675 \Phi_\theta(z)
3676 =
3677 \frac{(1+\e^{\ii\theta})z+(1-\e^{\ii\theta})}
3678 {(1-\e^{\ii\theta})z+(1+\e^{\ii\theta})}.
3679 }
3680 \]
3681 It preserves \(\mathbb H\) and satisfies
3682 \[
3683 \Phi_\theta\Phi_\varphi
3684 =
3685 \Phi_{\theta+\varphi}.
3686 \]
3687 For an integer \(N\geq1\) and \(\theta=2\pi/N\), the map has order \(N\).
3688 \end{theorem}
3689 
Proof block: Cyclic Mobius action · Go to statement · Read statement and proof side by side3690 \begin{proof}
3691 Substitute \(\e^{\ii\theta}(z-1)/(z+1)\) into \(C^{-1}\) and simplify. Preservation of \(\mathbb H\) follows because \(C\) maps it to \(\mathbb D\), rotation preserves \(\mathbb D\), and \(C^{-1}\) maps back. The group law follows by conjugating the rotation group law. When \(\theta=2\pi/N\), the \(N\)th power is the identity. Conversely, if its \(k\)th power is the identity, conjugation by \(C\) and evaluation at \(w=1/2\) give \(\e^{2\pi\ii k/N}=1\), so \(N\) divides \(k\). Hence the order is exactly \(N\).
3692 \end{proof}
3693 
Statement: Impedance and admittance · Go to proof · Read statement and proof side by side3694 \begin{corollary}[Impedance and admittance]\label{cor:impedance_admittance}
3695 For \(N=2\), \(\theta=\pi\) and
3696 \[
3697 \Phi_\pi(z)=\frac1z.
3698 \]
3699 Thus the two phase action is normalized impedance and admittance inversion.
3700 \end{corollary}
3701 
Proof block: Impedance and admittance · Go to statement · Read statement and proof side by side3702 \begin{proof}
3703 Insert \(\e^{\ii\pi}=-1\) in the formula of \cref{thm:cyclic_mobius}.
3704 \end{proof}
3705 
Statement: Radial and phase coordinates · Go to proof · Read statement and proof side by side3706 \begin{proposition}[Radial and phase coordinates]\label{prop:radial_phase}
3707 Write
3708 \[
3709 C(z)=\rho\e^{\ii\varphi},
3710 \qquad
3711 0\leq\rho<1.
3712 \]
3713 Then
3714 \[
3715 C(\Phi_\theta(z))
3716 =
3717 \rho\e^{\ii(\varphi+\theta)}.
3718 \]
3719 The radial mismatch \(\rho\) is invariant and the phase coordinate advances by \(\theta\).
3720 \end{proposition}
3721 
Proof block: Radial and phase coordinates · Go to statement · Read statement and proof side by side3722 \begin{proof}
3723 This is the defining conjugacy \(C\Phi_\theta=R_\theta C\).
3724 \end{proof}
3725 
Statement: Coprime phase factorization · Go to proof · Read statement and proof side by side3726 \begin{proposition}[Coprime phase factorization]\label{prop:phase_factorization}
3727 Let \(M\) and \(N\) be coprime. The cyclic phase labels modulo \(MN\) are equivalent to pairs of labels modulo \(M\) and modulo \(N\):
3728 \[
3729 \ZZ/(MN)\ZZ
3730 \cong
3731 \ZZ/M\ZZ\times\ZZ/N\ZZ.
3732 \]
3733 Under this identification, the rotation by \(2\pi/(MN)\) is determined by its two projected cyclic actions.
3734 \end{proposition}
3735 
Proof block: Coprime phase factorization · Go to statement · Read statement and proof side by side3736 \begin{proof}
3737 This is the Chinese remainder theorem applied to the additive phase label. A label modulo \(MN\) is uniquely determined by its two residues, and addition of labels is preserved.
3738 \end{proof}
3739 
3740 \begin{figure}
3741 \centering
3742 \includegraphics[width=0.61\textwidth]{14_smith_coordinates.pdf}
3743 \caption{The cyclic action is an ordinary phase rotation in the reflection disk and a fractional linear action in the impedance half plane.}
3744 \label{fig:smith_coordinates}
3745 \end{figure}
3746 
3747 \section{Limits and further directions}\label{extension:section:15}\label{sec:further_questions}
3748 
3749 The one dimensional recovery theorem identifies an explicit admissible threshold interval for an adjacent contrast detector. More general recovery requires a broader observation model. A multiscale average can reduce fixed sample noise, but its bias depends on the smoothness class and the separation of nearby jumps. A useful next theorem should select the averaging scale from the data while retaining a finite probability certificate.
3750 
3751 Higher dimensional recovery requires geometry. The overlap and tube formulas in \cref{sec:packet_error} become quantitative once a boundary estimate is known. What remains is a stable method that recovers topology, orientation, curvature, and region values under a declared noise model. Wedgelets and curvelets show why directional geometry is essential for broad image classes \cite{Donoho1999,CandesDonoho2004}.
3752 
3753 The best basis recursion is exact for additive objectives. Cross packet leakage, shared computation, and global hardware constraints introduce couplings between subtrees. Those problems require a larger compiler or an approximation theorem. The cost stability results in \cref{thm:best_basis_cost_stability,cor:best_basis_selection_stability} identify one condition under which the additive decision survives perturbation.
3754 
3755 The higher order singular packet theorem removes finitely many trace jumps. A more complete approximation theory should connect packet norms to geometric measure quantities in \(\BV\), piecewise Sobolev spaces, and free discontinuity models. It should also determine when a redundant packet dictionary can identify the derivative order of a singularity rather than merely its location.
3756 
3757 Finally, the two shifted Haar grids remove one exact alignment failure, but they are redundant. A complete redundant theory should provide stable synthesis, noise aware coefficient aggregation, and a best basis rule that does not count the same singularity several times.
3758 
3759 \section{Conclusion}\label{extension:section:16}\label{sec:conclusion}
3760 
3761 Finite index sublattices give exact coordinates for lattice signals. In those coordinates, convolution is a matrix of transfer kernels. Perfect reconstruction is a matrix inverse identity. Approximate reconstruction is controlled by a multiplier norm. Frame bounds are pointwise singular value bounds. Lifting steps are exact triangular coordinate changes. Every admissible packet tree gives a complete coordinate system, and every additive tree objective has an exact dynamic recursion.
3762 
3763 The singular packet theory separates representation from recovery. A finite family of truncated powers records jumps in a function and its derivatives, leaving a globally Sobolev remainder under explicit trace assumptions. If the packet is known and retained, the error is exactly the smooth approximation error. If the packet is estimated, exact region overlap and location formulas quantify the remaining defect, while the reconstruction operator contributes a transparent stability factor.
3764 
3765 For separated one dimensional jumps, a deterministic contrast gap gives exact cell recovery, midpoint location accuracy, and amplitude control. A sub Gaussian version follows from one union bound, and repeated measurements reduce the effective noise scale. Haar coefficients provide a second separation law because smooth coefficients decay one full power of scale faster than jump coefficients. Two shifted grids remove boundary alignment at a fixed scale. Higher differences annihilate piecewise polynomial interiors and confine their response to breakpoint bands.
3766 
3767 The resulting chain is
3768 \[
3769 \begin{gathered}
3770 \text{lattice coordinates}
3771 \longrightarrow
3772 \text{transfer blocks}
3773 \longrightarrow
3774 \text{stable filter banks}
3775 \longrightarrow
3776 \text{packet trees}
3777 \\
3778 \longrightarrow
3779 \text{singular packet model}
3780 \longrightarrow
3781 \text{local recovery}
3782 \longrightarrow
3783 \text{smooth approximation rate}.
3784 \end{gathered}
3785 \]
3786 Every arrow is exact or quantitatively controlled under the hypotheses stated in the corresponding result.
3787 
3788 
3789 \clearpage
3790 \part*{Part III: Forty worked visual plates}\addcontentsline{toc}{part}{Part III: Forty worked visual plates}
3791 \begin{readerroadmap}[title={A second route through the mathematics}]
3792 These forty plates develop the finite models used in the separate visual guide. The identifiers V1--V40 stay fixed across both documents. They lead from exact signal coordinates through filter transfers, adaptive decisions, packet errors, stable banks, and separated jump recovery. Every page gives the relevant manuscript locator. Finite arithmetic and diagrams explain the stated hypotheses and do not extend the formal contract.
3793 \end{readerroadmap}
3794 \clearpage
3795 \begingroup
3796 \setlength{\parskip}{3pt}
3797 \setlength{\abovedisplayskip}{6pt}\setlength{\belowdisplayskip}{6pt}
3798 \setstretch{1.01}
3799 \fontsize{10}{12}\selectfont
3800 \input{atlas-pages.tex}
3801 \endgroup
3802 \clearpage
3803 
3804 \bibliographystyle{alpha}
3805 \bibliography{references}
3806 
3807 
3808 \end{document}