Paper_V_Traditional_Final.tex
Realization Calculus
Paper_V_Traditional_Final.tex
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125
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
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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 \input{component-layout.tex}
218 \begin{document}
219 \KomorebiCover
220 \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.
221
222 \medskip
223 \textbf{Part II.} The complete extended development treats lattice coordinates, stable banks, singular packets, separated jump recovery, geometric error bounds, and multiscale localization.
224
225 \medskip
226 \textbf{Visual atlas.} Forty worked plates close the volume. The separate visual guide provides the same examples, and the Lean guide gives the precise formal contract.}
227
228 \section*{Reading this chapter}\addcontentsline{toc}{section}{Reading this chapter}
229 This chapter develops signal calculus in two complementary parts. Part I gives twenty results on exact coordinates, filtering, adaptive selection and packet error. Part II develops lattice coordinates, stable banks, singular packets and separated jump recovery, with E-prefixed section numbers. The two parts share constructions but use them under different hypotheses.
230
231 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.
232
233 The formal companion maps all twenty Part I results to 225 supporting Lean theorem targets across 23 proof modules. Part II has ordinary written proofs and is outside that twenty-result formal map. Exact examples and numerical studies have their own stated scopes. The formal companion and source records distinguish kernel verification, finite checks and numerical replay.
234
235 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.
236 \clearpage
237 \tableofcontents
238 \clearpage
239 \listoffigures
240 \listoftables
241 \clearpage
242 \pagenumbering{arabic}
243 \part*{Part I: The formal core}\addcontentsline{toc}{part}{Part I: The formal core}
244 \begin{readerroadmap}[title={Coordinates first, decisions second}]
245 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.
246 \end{readerroadmap}
247
248
249
250 \begin{abstract}
251 This chapter 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 accompanying formal development gives 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 chapter. Exact signal theorems, limited numerical estimators, classical constructions, and open recovery problems are kept distinct throughout.
252 \end{abstract}
253
254 \begin{interpretation}[title={Scope and claim boundary}]
255 The chapter 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 supplied two-class estimator is a universal segmentation method. The estimator is included as one reproducible finite example. For every finite packet refinement tree, the chapter 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.
256 \end{interpretation}
257
258 \clearpage
259 \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 modeling choices, or the two conjectures. The manuscript-to-declaration correspondence has been reviewed manually. Section~\ref{sec:formalization} records it result by result. Numerical replay has a separate record and does not enlarge the formal theorem scope.
260
261
262
263 \section{The signal realization of the seed calculus}\label{core:section:01}
264
265 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
266 \[
267 (Sx)[n]=x[n-1].
268 \]
269 Then
270 \[
271 S^n\delta_0=\delta_n.
272 \]
273 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.
274
275 \begin{theorem}[Impulse seed theorem]\label{result:01}
276 The closed shift-generated hull of $\delta_0$ is all of $\ell^2(\bbZ)$:
277 \[
278 \boxed{
279 \overline{\operatorname{span}\{S^n\delta_0:n\in\bbZ\}}^{\,\ell^2}
280 =\ell^2(\bbZ).
281 }
282 \]
283 Every finitely supported signal has the exact finite expression
284 \[
285 x=\sum_{n\in\bbZ}x[n]S^n\delta_0.
286 \]
287 \end{theorem}
288
289 \begin{proof}
290 The shift orbit is the standard orthonormal basis $\{\delta_n\}_{n\in\bbZ}$. For $x\in\ell^2$, the finite truncations
291 \[
292 x_F=\sum_{n\in F}x[n]\delta_n,\qquad F\subset\bbZ\text{ finite},
293 \]
294 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.
295 \end{proof}
296
297 This realization fixes the language used below:
298 \[
299 \begin{gathered}
300 \text{impulse seed}\longrightarrow\text{shift address}\longrightarrow\text{packet projection}\longrightarrow\text{filter transfer}\\
301 \longrightarrow\text{refinement}\longrightarrow\text{reconstruction}.
302 \end{gathered}
303 \]
304
305 \begin{remark}[Several completions]\label{core:record:02}
306 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 chapter use ordinary Euclidean norms. Theorem statements specify the ambient norm when it matters.
307 \end{remark}
308
309 \section{Residue projectors and exact polyphase coordinates}\label{core:section:02}
310
311 \begin{keyidea}[title={A packet has two useful representations}]
312 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.
313 \end{keyidea}
314
315 Fix an integer $N\ge1$. For $0\le r<N$, define
316 \[
317 (P_{N,r}x)[n]
318 =
319 \mathbf 1_{n\equiv r\, (\mathrm{mod}\,N)}x[n].
320 \]
321 The corresponding decimated channel is
322 \[
323 x_r[q]=x[Nq+r].
324 \]
325
326 \begin{theorem}[Polyphase projector theorem]\label{result:02}
327 The maps $P_{N,r}$ are bounded orthogonal projectors on $\ell^2$, with operator norm at most one, and satisfy
328 \[
329 P_{N,r}P_{N,s}=\delta_{rs}P_{N,r},
330 \qquad
331 \sum_{r=0}^{N-1}P_{N,r}=I.
332 \]
333 Consequently,
334 \[
335 \boxed{x=\sum_{r=0}^{N-1}P_{N,r}x}
336 \]
337 with no approximation.
338 \end{theorem}
339
340 \begin{proof}
341 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,
342 \[
343 \|P_{N,r}x\|_2^2=\sum_{n\equiv r\ (\mathrm{mod}\,N)}|x[n]|^2\le\|x\|_2^2,
344 \qquad
345 \langle P_{N,r}x,y\rangle=\langle x,P_{N,r}y\rangle.
346 \]
347 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.
348 \end{proof}
349
350 Define the analysis and synthesis maps
351 \[
352 \cA_Nx=(x_0,\ldots,x_{N-1}),
353 \qquad
354 \cS_N(x_0,\ldots,x_{N-1})[Nq+r]=x_r[q].
355 \]
356
357 \begin{corollary}[Analysis--synthesis identity]\label{result:03}
358 \[
359 \boxed{\cS_N\cA_N=I.}
360 \]
361 On the product of the channel spaces one also has $\cA_N\cS_N=I$. Equip this finite product with the Hilbert norm
362 \[
363 \|(x_0,\ldots,x_{N-1})\|_{\mathrm{ch}}^2
364 =\sum_{r=0}^{N-1}\|x_r\|_2^2.
365 \]
366 Then $\cA_N$ is a linear isometric equivalence and
367 \[
368 \|x\|_2^2=\sum_{r=0}^{N-1}\|x_r\|_2^2.
369 \]
370 \end{corollary}
371 \begin{proof}
372 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.
373 \end{proof}
374
375 \begin{figure}[htbp]
376 \centering
377 \includegraphics[width=0.94\textwidth]{01_polyphase_analysis.pdf}
378 \caption{Four-phase analysis of a finite signal. The channels are decimated coordinate packets. Interleaving them gives the original signal exactly.}
379 \label{fig:polyphase-analysis}
380 \end{figure}
381
382 \subsection{Nested refinement}
383
384 Suppose $N,M\ge1$ and $N\mid M$. Every coarse phase has a disjoint family of fine descendants:
385 \[
386 \operatorname{Child}_{M}(N,r)
387 =
388 \{s:0\le s<M,\ s\equiv r\pmod N\}.
389 \]
390
391 \begin{theorem}[Exact nested refinement]\label{result:04}
392 For $N,M\ge1$ and $N\mid M$,
393 \[
394 \boxed{
395 P_{N,r}
396 =
397 \sum_{\substack{0\le s<M\\s\equiv r\, (\mathrm{mod}\,N)}}P_{M,s}.
398 }
399 \]
400 \end{theorem}
401
402 \begin{proof}
403 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.
404 \end{proof}
405
406 This identity is the signal form of the refinement law introduced in Chapter 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.
407
408 \section{Linear time-invariant operators as transfer matrices}\label{core:section:03}
409
410 Let $h$ be a finite impulse response and
411 \[
412 y=h*x,
413 \qquad
414 y[n]=\sum_{q\in\bbZ}h[q]x[n-q].
415 \]
416 Resolve both input and output into $N$ phases. For output phase $k$ and input phase $s$, define the transfer kernel
417 \[
418 \boxed{
419 h_{ks}[m]=h[Nm+k-s].
420 }
421 \]
422 Out-of-range taps are understood to be zero.
423
424 \begin{theorem}[Exact FIR polyphase transfer law]\label{result:05}
425 The output channels satisfy
426 \[
427 \boxed{
428 y_k[n]
429 =
430 \sum_{s=0}^{N-1}\sum_{m\in\bbZ}
431 h_{ks}[m]x_s[n-m].
432 }
433 \]
434 For the numerical channel transform, take $z\in\bbC\setminus\{0\}$ and assume
435 \[
436 \sum_{n\in\bbZ}|x_s[n]z^{-n}|<\infty\quad(0\le s<N).
437 \]
438 Define $X_s(z)=\sum_nx_s[n]z^{-n}$. The output series then also converge absolutely, and
439 \[
440 \bm Y(z)=\bm H^{(N)}(z)\bm X(z),
441 \qquad
442 H^{(N)}_{ks}(z)=\sum_m h[Nm+k-s]z^{-m}.
443 \]
444 \end{theorem}
445
446 \begin{proof}
447 Evaluate $y$ at address $Nn+k$ and partition every input index by its residue $s$:
448 \[
449 \begin{aligned}
450 y[Nn+k]
451 &=\sum_qh[q]x[Nn+k-q]\\
452 &=\sum_{s=0}^{N-1}\sum_m
453 h[Nm+k-s]x[N(n-m)+s].
454 \end{aligned}
455 \]
456 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 $x$.
457
458 The last factor is $x_s[n-m]$, giving the time-domain formula. Under the additional transform hypothesis, each delayed channel has transform
459 \[
460 \sum_nx_s[n-m]z^{-n}
461 =z^{-m}\sum_jx_s[j]z^{-j}.
462 \]
463 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.
464 \end{proof}
465
466 \begin{figure}[htbp]
467 \centering
468 \includegraphics[width=0.80\textwidth]{02_filter_transfer_matrix.pdf}
469 \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.}
470 \label{fig:filter-matrix}
471 \end{figure}
472
473 \begin{remark}[Operator-block form]\label{core:record:07}
474 If $H$ denotes convolution by $h$, the same identity is
475 \[
476 H=\sum_{k,s}P_{N,k}HP_{N,s}.
477 \]
478 Thus the familiar polyphase matrix is exactly the transfer-block matrix of the seed calculus.
479 \end{remark}
480
481 \section{Analysis banks, synthesis banks, and perfect reconstruction}\label{core:section:04}
482
483 \begin{interpretation}[title={Read the delay in the coordinate system used}]
484 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.
485 \end{interpretation}
486 \begin{figure}[htbp]\centering
487 \includegraphics[width=.91\linewidth]{monograph_figures/delay_coordinates.pdf}
488 \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}
489 \end{figure}
490
491 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
492 \[
493 \widehat{\bm X}(z)=\bm R(z)\bm E(z)\bm X(z).
494 \]
495
496 \begin{theorem}[Perfect-reconstruction criterion]\label{result:06}
497 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:
498 \[
499 \boxed{
500 \bm R(z)\bm E(z)=z^{-d}I_N.
501 }
502 \]
503 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.
504 \end{theorem}
505
506 \begin{proof}
507 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 $c=1$.
508 \end{proof}
509
510 \paragraph{Original-sample delay.}
511 For an arbitrary original-sample delay $D\in\bbZ$, the output phase $k$ has
512 \[
513 y_k[n]=x_{r_k}[n+q_k],\qquad
514 r_k=(k-D)\bmod N,\qquad q_k=(k-D)\mathbin{\mathrm{div}}N.
515 \]
516 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
517 \[
518 \begin{pmatrix}0&z^{-1}\\1&0\end{pmatrix},
519 \]
520 whereas $z^{-1}I_2$ delays the original signal by two samples. The general address identity is valid for negative sample addresses as well.
521
522 \subsection{The Haar bank}
523
524 The Haar analysis matrix is
525 \[
526 \bm E_{\mathrm{H}}
527 =
528 \frac1{\sqrt2}
529 \begin{pmatrix}
530 1&1\\
531 1&-1
532 \end{pmatrix}.
533 \]
534 It is orthogonal, so $\bm R_{\mathrm{H}}=\bm E_{\mathrm{H}}^{\mathsf T}$ and
535 \[
536 \bm R_{\mathrm{H}}\bm E_{\mathrm{H}}=I_2.
537 \]
538 The channels are
539 \[
540 a[n]=\frac{x[2n]+x[2n+1]}{\sqrt2},
541 \qquad
542 d[n]=\frac{x[2n]-x[2n+1]}{\sqrt2}.
543 \]
544
545 \begin{corollary}[Haar energy identity]\label{result:07}
546 \[
547 \boxed{
548 \|x\|_2^2=\|a\|_2^2+\|d\|_2^2.
549 }
550 \]
551 \end{corollary}
552
553 \begin{proof}
554 For any $u,v\in\bbC$, the parallelogram identity gives
555 \[
556 \left|\frac{u+v}{\sqrt2}\right|^2+
557 \left|\frac{u-v}{\sqrt2}\right|^2=|u|^2+|v|^2.
558 \]
559 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.
560 \end{proof}
561
562 \begin{figure}[htbp]
563 \centering
564 \includegraphics[width=0.92\textwidth]{03_haar_edge_packets.pdf}
565 \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.}
566 \label{fig:haar-packets}
567 \end{figure}
568
569 \subsection{Predict--update lifting}
570
571 Split $x$ into even and odd channels $e,o$. Define
572 \[
573 d=o-e,
574 \qquad
575 s=e+\frac12d.
576 \]
577 The inverse is
578 \[
579 e=s-\frac12d,
580 \qquad
581 o=d+e.
582 \]
583
584 \begin{theorem}[Exact lifting reconstruction]\label{result:08}
585 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.
586 \end{theorem}
587
588 \begin{proof}
589 The forward matrix is
590 \[
591 \begin{pmatrix}s\\d\end{pmatrix}
592 =
593 \begin{pmatrix}1/2&1/2\\-1&1\end{pmatrix}
594 \begin{pmatrix}e\\o\end{pmatrix},
595 \]
596 whose determinant is one. Direct substitution gives the inverse formulas.
597 \end{proof}
598
599 \section{Wavelet packets as addressed refinements}\label{core:section:05}
600
601 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.
602
603 \begin{theorem}[Wavelet-packet reconstruction and energy]\label{result:09}
604 At every finite depth $\ell$,
605 \[
606 \boxed{
607 x=\sum_{\alpha\in\{0,1\}^{\ell}}W_{\ell,\alpha}^{*}W_{\ell,\alpha}x,
608 }
609 \]
610 and
611 \[
612 \boxed{
613 \|x\|_2^2
614 =
615 \sum_{\alpha\in\{0,1\}^{\ell}}
616 \|W_{\ell,\alpha}x\|_2^2.
617 }
618 \]
619 \end{theorem}
620
621 \begin{proof}
622 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 $\ell+1$.
623
624 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.
625 \end{proof}
626
627 \begin{figure}[htbp]
628 \centering
629 \includegraphics[width=0.92\textwidth]{04_wavelet_packet_energy.pdf}
630 \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.}
631 \label{fig:wavelet-energy}
632 \end{figure}
633
634 \begin{classical}[Wavelet boundary]\label{core:record:12}
635 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.
636 \end{classical}
637
638 \section{Known edge packets and spectral reconstruction}\label{core:section:06}
639
640 \begin{keyidea}[title={Retention changes the approximation target}]
641 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.
642 \end{keyidea}
643 \begin{figure}[htbp]\centering
644 \includegraphics[width=.95\linewidth]{monograph_figures/retained_packet_cancellation.pdf}
645 \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}
646 \end{figure}
647
648
649 The central distinction is between approximating a jump globally and retaining it as its own packet.
650
651 Let
652 \[
653 f=s+j
654 \]
655 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
656 \[
657 \widetilde f_m=Q_ms+j.
658 \]
659
660 \begin{theorem}[Exact known-edge theorem]\label{result:10}
661 For every norm for which the expressions are defined,
662 \[
663 \boxed{
664 f-\widetilde f_m=s-Q_ms,
665 \qquad
666 \|f-\widetilde f_m\|=\|s-Q_ms\|.
667 }
668 \]
669 No approximation of $j$ occurs.
670 \end{theorem}
671
672 \begin{proof}
673 Subtract:
674 \[
675 f-\widetilde f_m=(s+j)-(Q_ms+j)=s-Q_ms.
676 \]
677 Taking the norm gives the second identity.
678 \end{proof}
679
680 \begin{corollary}[Several exact edge packets]\label{result:11}
681 If
682 \[
683 f=s+\sum_{q=1}^{M}j_q
684 \]
685 and every $j_q$ is retained exactly, then
686 \[
687 \left\|f-\left(Q_ms+\sum_qj_q\right)\right\|
688 =\|s-Q_ms\|.
689 \]
690 Curved edges, corners, and different jump amplitudes do not alter the identity.
691 \end{corollary}
692 \begin{proof}
693 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.
694 \end{proof}
695
696 \begin{boundary}[What the theorem does and does not say]\label{core:record:15}
697 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.
698 \end{boundary}
699
700 \section{Mask, amplitude, and smooth-background error}\label{core:section:07}
701
702 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
703 \[
704 j=a\mathbf 1_{\Omega},
705 \]
706 and an estimated packet is
707 \[
708 \widehat j=\widehat a\mathbf 1_{\widehat\Omega}.
709 \]
710
711 \begin{theorem}[Packet perturbation bound]\label{result:12}
712 For $1\le p<\infty$, interpreted first as an inequality of extended $L^p$ seminorms,
713 \[
714 \boxed{
715 \|j-\widehat j\|_{L^p}
716 \le
717 |a-\widehat a|\,\mu(\Omega)^{1/p}
718 +
719 |\widehat a|\,\mu(\Omega\triangle\widehat\Omega)^{1/p}.
720 }
721 \]
722 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,
723 \[
724 \|j-\widehat j\|_{\infty}
725 \le |a-\widehat a|+|\widehat a|\,
726 \mathbf 1_{\Omega\triangle\widehat\Omega\ne\varnothing}.
727 \]
728 \end{theorem}
729
730 \begin{proof}
731 Write
732 \[
733 a\mathbf 1_{\Omega}-\widehat a\mathbf 1_{\widehat\Omega}
734 =(a-\widehat a)\mathbf 1_{\Omega}
735 +\widehat a(\mathbf 1_{\Omega}-\mathbf 1_{\widehat\Omega}).
736 \]
737 Apply the triangle inequality and use
738 \[
739 \|\mathbf 1_{\Omega}-\mathbf 1_{\widehat\Omega}\|_{L^p}
740 =\mu(\Omega\triangle\widehat\Omega)^{1/p}.
741 \]
742 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.
743 \end{proof}
744
745 If $s,\widehat s$ are measurable smooth-background models, the same triangle inequality gives, for finite $p$,
746 \[
747 \boxed{
748 \|f-(\widehat s+\widehat j)\|_p
749 \le
750 \|s-\widehat s\|_p
751 +|a-\widehat a|\mu(\Omega)^{1/p}
752 +|\widehat a|\mu(\Omega\triangle\widehat\Omega)^{1/p}.
753 }
754 \]
755 This is the correct separation of errors. Detection error is not hidden inside a generic reconstruction score.
756
757 \begin{conjecture}[Stable edge-packet recovery]\label{core:record:17}
758 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.
759 \end{conjecture}
760
761 \Needspace{12\baselineskip}
762 \section{One-dimensional numerical study}\label{core:section:08}
763
764 The one-dimensional signal contains a smooth non-bandlimited background and a positive rectangular packet. Three reconstructions are compared:
765
766 \begin{enumerate}[label=(\roman*)]
767 \item global Fourier truncation of the complete signal;
768 \item Fourier truncation of the known smooth packet plus the exact jump packet;
769 \item a two-class estimated packet followed by harmonic background filling, spectral smoothing, and jump-amplitude estimation.
770 \end{enumerate}
771
772 \begin{figure}[htbp]
773 \centering
774 \includegraphics[width=0.98\textwidth]{05_edge_reconstruction_1d.pdf}
775 \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.}
776 \label{fig:edge-1d}
777 \end{figure}
778
779 \begin{certificate}[One-dimensional errors]\label{core:record:18}
780 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
781 \[
782 \begin{array}{lrr}
783 \toprule
784 \text{method}&\RMSE&\text{maximum absolute error}\\
785 \midrule
786 \text{global Fourier truncation}&5.4023\times10^{-2}&3.4968\times10^{-1}\\
787 \text{known edge packet}&1.3692\times10^{-4}&1.4503\times10^{-3}\\
788 \text{estimated edge packet}&2.4029\times10^{-2}&5.9530\times10^{-2}\\
789 \bottomrule
790 \end{array}
791 \]
792 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.
793 \end{certificate}
794
795 \begin{figure}[htbp]
796 \centering
797 \includegraphics[width=0.84\textwidth]{06_edge_error_vs_resolution.pdf}
798 \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.}
799 \label{fig:edge-resolution}
800 \end{figure}
801
802 \Needspace{12\baselineskip}
803 \section{Two-dimensional edge packets}\label{core:section:09}
804
805 The first image consists of a smooth background and one sharp disk. The same three logical levels must remain separate:
806
807 \begin{enumerate}[label=(\alph*)]
808 \item the exact theorem using the known decomposition;
809 \item a limited estimator designed for a positive-contrast two-class packet;
810 \item the open problem of recovering arbitrary edge packets under noise and model ambiguity.
811 \end{enumerate}
812
813 \begin{figure}[htbp]
814 \centering
815 \includegraphics[width=0.99\textwidth]{07_known_edge_packet_2d.pdf}
816 \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.}
817 \label{fig:known-2d}
818 \end{figure}
819
820 \begin{certificate}[Two-dimensional errors]\label{core:record:19}
821 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
822 \[
823 \begin{array}{lrr}
824 \toprule
825 \text{method}&\text{full-image RMSE}&\text{boundary-band RMSE}\\
826 \midrule
827 \text{global Fourier truncation}&3.7643\times10^{-2}&1.8541\times10^{-1}\\
828 \text{known edge packet}&6.4752\times10^{-5}&1.2709\times10^{-5}\\
829 \text{estimated edge packet}&3.4043\times10^{-3}&8.8834\times10^{-3}\\
830 \bottomrule
831 \end{array}
832 \]
833 The mask intersection-over-union was one in this positive-contrast synthetic case. The amplitude was estimated rather than inserted from the generating model.
834 \end{certificate}
835
836 \begin{figure}[htbp]
837 \centering
838 \includegraphics[width=0.99\textwidth]{08_estimated_edge_packet_2d.pdf}
839 \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.}
840 \label{fig:estimated-2d}
841 \end{figure}
842
843 \subsection{Curves, corners, and multiple amplitudes}
844
845 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.
846
847 \begin{figure}[htbp]
848 \centering
849 \includegraphics[width=0.99\textwidth]{09_multiedge_stress_test.pdf}
850 \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.}
851 \label{fig:multiedge}
852 \end{figure}
853
854 \begin{certificate}[Multiedge errors]\label{core:record:20}
855 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.
856 \end{certificate}
857
858 \section{Sparse wavelet packets and global Fourier modes}\label{core:section:10}
859
860 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.
861
862 \begin{figure}[htbp]
863 \centering
864 \includegraphics[width=0.83\textwidth]{10_sparse_wavelet_fourier.pdf}
865 \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.}
866 \label{fig:sparse-comparison}
867 \end{figure}
868
869 \begin{table}[htbp]
870 \centering
871 \caption{Sparse approximation errors for the one-dimensional test signal.}
872 \begin{tabular}{rrr}
873 \toprule
874 \text{retained coefficients}&\text{Haar packet RMSE}&\text{Fourier RMSE}\\
875 \midrule
876 16&$1.7999\times10^{-1}$&$6.6522\times10^{-2}$\\
877 32&$3.8414\times10^{-2}$&$4.9153\times10^{-2}$\\
878 64&$1.6267\times10^{-2}$&$3.8056\times10^{-2}$\\
879 128&$1.7387\times10^{-3}$&$2.5631\times10^{-2}$\\
880 256&$1.5870\times10^{-4}$&$1.7424\times10^{-2}$\\
881 \bottomrule
882 \end{tabular}
883 \label{tab:sparse}
884 \end{table}
885
886 \begin{boundary}[Basis selection]\label{core:record:21}
887 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 seed calculus records both systems with the same analysis, transfer, refinement, and synthesis maps.
888 \end{boundary}
889
890 \section{Finite adaptive packet trees}\label{core:section:11}
891
892 \begin{readerroadmap}[title={Three levels of a finite tree decision}]
893 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.
894 \end{readerroadmap}
895
896 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.
897
898 \begin{definition}[Complete packet pruning]\label{core:record:22}
899 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
900 \[
901 P_v=\sum_{u\in\operatorname{ch}(v)}P_u,
902 \qquad
903 P_uP_w=0\quad(u\ne w\text{ in }\operatorname{ch}(v)).
904 \]
905 A set $\Lambda$ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets $\Lambda$ exactly once.
906 \end{definition}
907
908 \begin{theorem}[Exact synthesis from every complete pruning]\label{result:13}
909 For every complete pruning $\Lambda$,
910 \[
911 \boxed{\sum_{v\in\Lambda}P_v=I,}
912 \qquad
913 \boxed{x=\sum_{v\in\Lambda}P_vx.}
914 \]
915 Thus changing the selected tree changes the coordinates but does not introduce reconstruction error.
916 \end{theorem}
917
918 \begin{proof}
919 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.
920 \end{proof}
921
922 Let $y$ denote the observed data. For a complete pruning $\Lambda$, consider any finite criterion
923 \[
924 J_y(\Lambda)
925 =
926 R_y(\Lambda)
927 +\eta L_y(\Lambda)
928 +\tau K(\Lambda),
929 \]
930 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.
931
932 \begin{theorem}[Finite globally optimal stable selector]\label{result:14}
933 Let $\mathfrak P(\mathcal T)$ be the finite set of complete prunings of $\mathcal T$.
934
935 \begin{enumerate}[label=(\roman*)]
936 \item A global minimizer
937 \[
938 \Lambda_y^*\in\operatorname*{argmin}_{\Lambda\in\mathfrak P(\mathcal T)}J_y(\Lambda)
939 \]
940 always exists.
941 \item If the minimizer is unique and at least one competing pruning exists, its gap
942 \[
943 \Gamma_y
944 =
945 \min_{\Lambda\ne\Lambda_y^*}
946 \bigl(J_y(\Lambda)-J_y(\Lambda_y^*)\bigr)
947 \]
948 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.
949 \item In the case with competitors, if new data $y'$ satisfy
950 \[
951 \sup_{\Lambda\in\mathfrak P(\mathcal T)}
952 |J_{y'}(\Lambda)-J_y(\Lambda)|
953 <\frac{\Gamma_y}{2},
954 \]
955 then $\Lambda_{y'}^*=\Lambda_y^*$.
956 \item Every selected minimizer preserves the exact synthesis identity of the preceding theorem.
957 \end{enumerate}
958 \end{theorem}
959
960 \begin{proof}
961 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 $\Lambda\ne\Lambda_y^*$,
962 \[
963 \begin{aligned}
964 J_{y'}(\Lambda)-J_{y'}(\Lambda_y^*)
965 &\ge
966 J_y(\Lambda)-J_y(\Lambda_y^*)
967 -2\sup_{\Xi}|J_{y'}(\Xi)-J_y(\Xi)|\\
968 &>\Gamma_y-\Gamma_y=0.
969 \end{aligned}
970 \]
971 Hence the same pruning remains uniquely optimal. Exact synthesis depends only on completeness of the pruning, not on the cost used to select it.
972 \end{proof}
973
974 For tree-local costs, the optimum can be found without enumerating all prunings.
975
976 \begin{theorem}[Dynamic-programming best packet basis]\label{result:15}
977 Suppose
978 \[
979 J_y(\Lambda)=\sum_{v\in\Lambda}c_v(y).
980 \]
981 For a node $v$, define recursively
982 \[
983 V_y(v)
984 =
985 \begin{cases}
986 c_v(y),&v\text{ terminal},\\[2mm]
987 \displaystyle
988 \min\left\{c_v(y),\sum_{u\in\operatorname{ch}(v)}V_y(u)\right\},
989 &v\text{ internal}.
990 \end{cases}
991 \]
992 Stop at $v$ when $c_v(y)\le\sum_uV_y(u)$, including equality, and 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.
993 \end{theorem}
994
995 \begin{proof}
996 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.
997 \end{proof}
998
999 \begin{corollary}[Local perturbation certificate]\label{result:16}
1000 Assume
1001 \[
1002 |c_v(y')-c_v(y)|\le\varepsilon_v.
1003 \]
1004 Define
1005 \[
1006 E(v)
1007 =
1008 \begin{cases}
1009 \varepsilon_v,&v\text{ terminal},\\[1mm]
1010 \displaystyle
1011 \max\left\{\varepsilon_v,
1012 \sum_{u\in\operatorname{ch}(v)}E(u)\right\},&v\text{ internal}.
1013 \end{cases}
1014 \]
1015 For the complete prunings $\mathfrak P_v$ below $v$, the envelope is exactly
1016 \[
1017 E(v)=\max_{\Lambda\in\mathfrak P_v}\sum_{a\in\Lambda}\varepsilon_a.
1018 \]
1019 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,
1020 \[
1021 \boxed{\Gamma_y>2E(\varnothing)}
1022 \]
1023 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.
1024 \end{corollary}
1025
1026 \begin{proof}
1027 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.
1028 \end{proof}
1029
1030 \begin{figure}[htbp]
1031 \centering
1032 \includegraphics[width=0.97\textwidth]{17_adaptive_packet_tree.pdf}
1033 \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.}
1034 \label{fig:adaptive-tree}
1035 \end{figure}
1036
1037 \begin{figure}[htbp]
1038 \centering
1039 \includegraphics[width=0.97\textwidth]{18_adaptive_tree_stability.pdf}
1040 \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.}
1041 \label{fig:adaptive-stability}
1042 \end{figure}
1043
1044 \begin{certificate}[Adaptive-tree optimum]\label{core:record:27}
1045 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
1046 \[
1047 \{000,001,0100,0101,011,1\},
1048 \]
1049 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.
1050 \end{certificate}
1051
1052 \begin{boundary}[Finite versus asymptotic selection]\label{core:record:28}
1053 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.
1054 \end{boundary}
1055
1056
1057 \section{Nonlinear signal packets}\label{core:section:12}
1058
1059 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
1060 \[
1061 A(z)=\sum_{n\ge0}a_nz^n,
1062 \qquad
1063 A_i=P_{N,i}A.
1064 \]
1065 All phase indices range from $0$ to $N-1$. For another representation $B(z)$, multiplication gives a nonlinear or bilinear interaction.
1066
1067 \begin{theorem}[Cyclic product law]\label{result:17}
1068 \[
1069 \boxed{
1070 P_{N,k}(AB)
1071 =
1072 \sum_{i+j\equiv k\, (\mathrm{mod}\,N)}A_iB_j.
1073 }
1074 \]
1075 For $q\in\bbN$, the $q$th-power identity is
1076 \[
1077 P_{N,k}(A^q)
1078 =
1079 \sum_{i_1+\cdots+i_q\equiv k}
1080 A_{i_1}\cdots A_{i_q}.
1081 \]
1082 \end{theorem}
1083
1084 \begin{proof}
1085 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.
1086 \end{proof}
1087
1088 \begin{figure}[htbp]
1089 \centering
1090 \includegraphics[width=0.76\textwidth]{14_nonlinear_phase_coupling.pdf}
1091 \caption{Norms of pairwise product interactions for five residue phases. The label in each cell gives the output phase $(i+j)\bmod5$.}
1092 \label{fig:nonlinear-coupling}
1093 \end{figure}
1094
1095 For a finite nonlinear shift operator
1096 \[
1097 F(x)[n]=x[n]+\alpha x[n]x[n-1],
1098 \]
1099 the exact packet coordinates are simply
1100 \[
1101 F_r((x_s)_s)=P_{N,r}F\!\left(\sum_sP_{N,s}x\right).
1102 \]
1103 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.
1104
1105 \section{Smith coordinates as cyclic signal coordinates}\label{core:section:13}
1106
1107 Let $Z_0>0$ be a reference impedance and
1108 \[
1109 z=\frac{Z}{Z_0}=r+\ii x.
1110 \]
1111 The reflection coefficient is
1112 \[
1113 \boxed{
1114 \Gam=\frac{z-1}{z+1},
1115 \qquad
1116 z=\frac{1+\Gam}{1-\Gam}.
1117 }
1118 \]
1119
1120 \begin{classical}[Cayley geometry]\label{core:record:30}
1121 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.
1122 \end{classical}
1123
1124 Writing $\Gam=u+\ii v$, the grid equations are
1125 \[
1126 \left(u-\frac{r}{1+r}\right)^2+v^2=\frac1{(1+r)^2},
1127 \]
1128 and, for $x\ne0$,
1129 \[
1130 (u-1)^2+\left(v-\frac1x\right)^2=\frac1{x^2}.
1131 \]
1132
1133 Fix
1134 \[
1135 \omega_N=e^{2\pi\ii/N}
1136 \]
1137 and rotate the reflection coefficient by
1138 \[
1139 R_{N,k}(\Gam)=\omega_N^k\Gam.
1140 \]
1141
1142 \begin{theorem}[Polyphase M\"obius action]\label{result:18}
1143 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
1144 \[
1145 \boxed{
1146 T_{N,k}(z)
1147 =
1148 \frac{(1+\omega_N^k)z+(1-\omega_N^k)}
1149 {(1-\omega_N^k)z+(1+\omega_N^k)}.
1150 }
1151 \]
1152 As maps on $\mathbb P^1(\bbC)$ it satisfies
1153 \[
1154 T_{N,k}\circ T_{N,\ell}=T_{N,k+\ell\, (\mathrm{mod}\,N)}.
1155 \]
1156 \end{theorem}
1157
1158 \begin{proof}
1159 Let $C(z)=(z-1)/(z+1)$. Then
1160 \[
1161 T_{N,k}=C^{-1}\circ R_{N,k}\circ C.
1162 \]
1163 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
1164 \[
1165 A(w)=\frac12\begin{pmatrix}1+w&1-w\\1-w&1+w\end{pmatrix}.
1166 \]
1167 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.
1168 \end{proof}
1169
1170 \begin{corollary}[Impedance--admittance is the two-phase split]\label{result:19}
1171 For $N=2$ and finite $z\ne0$,
1172 \[
1173 \boxed{T_{2,1}(z)=z^{-1}.}
1174 \]
1175 On the full projective line it exchanges $0$ and $\infty$. Thus a half-turn of the reflection disk exchanges normalized impedance and admittance.
1176 \end{corollary}
1177 \begin{proof}
1178 $\omega_2=\exp(\pi\ii)=-1$, 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.
1179 \end{proof}
1180
1181 \begin{figure}[htbp]
1182 \centering
1183 \includegraphics[width=0.88\textwidth]{11_polyphase_smith_chart.pdf}
1184 \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.}
1185 \label{fig:smith-chart}
1186 \end{figure}
1187
1188 \subsection{Radial seeds and physical phase}
1189
1190 For $0\le\rho<1$ and angle $\theta$,
1191 \[
1192 \Gam=\rho e^{\ii\theta}
1193 \]
1194 gives
1195 \[
1196 z(\rho,\theta)
1197 =\frac{1+\rho e^{\ii\theta}}{1-\rho e^{\ii\theta}},
1198 \]
1199 with
1200 \[
1201 r(\rho,\theta)=\frac{1-\rho^2}{1-2\rho\cos\theta+\rho^2},
1202 \qquad
1203 x(\rho,\theta)=\frac{2\rho\sin\theta}{1-2\rho\cos\theta+\rho^2}.
1204 \]
1205 A fixed $\rho$ is a constant-mismatch circle. Refining $N$ increases angular resolution without changing $\rho$.
1206
1207 \begin{figure}[htbp]
1208 \centering
1209 \includegraphics[width=0.78\textwidth]{12_smith_nested_refinement.pdf}
1210 \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.}
1211 \label{fig:smith-refinement}
1212 \end{figure}
1213
1214 For a lossless transmission line,
1215 \[
1216 \Gam_{\rm in}=\Gam_L e^{-2\ii\beta\ell}.
1217 \]
1218 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.
1219
1220 \section{Dyadic--odd phase factorization and boundary weighting}\label{core:section:14}
1221
1222 Write
1223 \[
1224 N=2^sM,
1225 \qquad M\text{ odd}.
1226 \]
1227 The cyclic index splits by the Chinese remainder theorem:
1228 \[
1229 \bbZ/N\bbZ
1230 \cong
1231 \bbZ/2^s\bbZ\times\bbZ/M\bbZ.
1232 \]
1233
1234 \begin{proposition}[Dyadic--odd packet factorization]\label{result:20}
1235 After identifying the residue labels by the Chinese remainder map, the phase projector factors as
1236 \[
1237 \boxed{
1238 P_{N,j}=P_{2^s,j_2}P_{M,j_o}.
1239 }
1240 \]
1241 \end{proposition}
1242
1243 \begin{proof}
1244 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.
1245 \end{proof}
1246
1247 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.
1248
1249 Away from its pole $\Gam=1$, the inverse Cayley map has derivative
1250 \[
1251 \frac{dz}{d\Gam}=\frac{2}{(1-\Gam)^2},
1252 \]
1253 while normalized admittance has derivative proportional to $(1+\Gam)^{-2}$. Away from $\Gam=\pm1$, this motivates the symmetric sensitivity
1254 \[
1255 \boxed{
1256 w_{\rm Smith}(\Gam)
1257 =
1258 \frac1{|1-\Gam|^2}
1259 +
1260 \frac1{|1+\Gam|^2}.
1261 }
1262 \]
1263
1264 \begin{figure}[htbp]
1265 \centering
1266 \includegraphics[width=0.78\textwidth]{13_smith_sensitivity.pdf}
1267 \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.}
1268 \label{fig:smith-weight}
1269 \end{figure}
1270
1271 \begin{boundary}[Weighted refinement]\label{core:record:34}
1272 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 illustration displays the weight but does not prove optimality.
1273 \end{boundary}
1274
1275 \section{Boundary-localized error and claim control}\label{core:section:15}
1276
1277 The strongest visual distinction between global Fourier reconstruction and exact packet retention is seen in a band around the discontinuity.
1278
1279 \begin{figure}[htbp]
1280 \centering
1281 \includegraphics[width=0.94\textwidth]{15_boundary_leakage.pdf}
1282 \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.}
1283 \label{fig:boundary-error}
1284 \end{figure}
1285
1286 \begin{figure}[htbp]
1287 \centering
1288 \includegraphics[width=0.90\textwidth]{16_claim_status.pdf}
1289 \caption{Proof and claim boundary. Finite adaptive-tree selection is proved. The remaining open rows concern universal recovery and infinite-tree or asymptotic selection.}
1290 \label{fig:claim-status}
1291 \end{figure}
1292
1293 \begin{conjecture}[Edge-packet approximation class]\label{core:record:35}
1294 There should exist natural function spaces in which a signal is represented as
1295 \[
1296 f=s+\sum_q a_q\mathbf1_{\Omega_q},
1297 \]
1298 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.
1299 \end{conjecture}
1300
1301
1302 \section{Implementation and reproducibility}\label{core:section:16}
1303
1304 The companion package is organized by mathematical layer:
1305
1306 \begin{longtable}{L{0.27\textwidth}L{0.64\textwidth}}
1307 \caption{Numerical companion modules.}\label{tab:modules}\\
1308 \toprule
1309 \textbf{module}&\textbf{implemented content}\\
1310 \midrule
1311 \endfirsthead
1312 \toprule
1313 \textbf{module}&\textbf{implemented content}\\
1314 \midrule
1315 \endhead
1316 \texttt{polyphase.py}&residue projectors, analysis/synthesis, refinement, exact FIR transfer kernels, polyphase convolution\\
1317 \texttt{filterbanks.py}&Haar analysis/synthesis, complete Haar packet trees, energy accounting, predict--update lifting\\
1318 \texttt{adaptive.py}&complete-pruning enumeration, exact mixed-depth synthesis, dynamic-programming optimization, and stability certificates\\
1319 \texttt{edges.py}&one- and two-dimensional Fourier truncation, known-packet reconstruction, narrow two-class estimators, harmonic filling, error metrics\\
1320 \texttt{smith.py}&Cayley coordinates, cyclic M\"obius action, Smith circles, sensitivity weight, coefficient projectors\\
1321 \texttt{nonlinear.py}&coefficient packets, cyclic product convolution, nonlinear shifted packet reconstruction\\
1322 \texttt{experiments.py}&all machine-readable numerical certificates and tabular summaries\\
1323 \texttt{visuals.py}&the eighteen figure pairs used in the chapter\\
1324 \bottomrule
1325 \end{longtable}
1326
1327 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.
1328
1329 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.
1330
1331 \begin{certificate}[Finite numerical residuals]\label{core:record:36}
1332 The largest recorded residuals were
1333 \[
1334 \begin{array}{lr}
1335 \toprule
1336 \text{identity}&\text{residual}\\
1337 \midrule
1338 \text{polyphase reconstruction}&0\\
1339 \text{nested refinement}&0\\
1340 \text{polyphase convolution}&4.44\times10^{-16}\\
1341 \text{Haar reconstruction}&8.88\times10^{-16}\\
1342 \text{wavelet-packet reconstruction}&2.22\times10^{-15}\\
1343 \text{adaptive-tree optimum versus exhaustive search}&0\\
1344 \text{adaptive mixed-depth synthesis}&8.88\times10^{-16}\\
1345 \text{lifting reconstruction}&4.44\times10^{-16}\\
1346 \text{Smith rotation conjugacy}&3.34\times10^{-16}\\
1347 \text{Smith impedance--admittance duality}&0\\
1348 \text{nonlinear product-packet reconstruction}&1.33\times10^{-15}\\
1349 \text{nonlinear shifted-filter reconstruction}&0\\
1350 \bottomrule
1351 \end{array}
1352 \]
1353 The Haar and wavelet energy residuals were $5.68\times10^{-14}$ and $2.27\times10^{-13}$, respectively.
1354 \end{certificate}
1355
1356 \section{Conclusions}\label{core:section:17}
1357
1358 The signal realization makes the seed, address, packet, transfer, refinement, and synthesis operations concrete. The objects and laws can be read directly:
1359 \[
1360 \boxed{
1361 \begin{aligned}
1362 \text{seed}&=\delta_0,\\
1363 \text{address}&=S^n,\\
1364 \text{packet}&=P_{N,r}x,\\
1365 \text{transfer}&=P_{N,k}HP_{N,s},\\
1366 \text{refinement}&=P_{N,r}=\sum_{s\succ r}P_{M,s},\\
1367 \text{synthesis}&=\sum_rP_{N,r}x,\\
1368 \text{nonlinearity}&=P_{N,r}F\left(\sum_sP_{N,s}x\right).
1369 \end{aligned}
1370 }
1371 \]
1372 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.
1373
1374 Chapter VI turns from discrete signal transfer to semigroups, resolvents, conditioning, stochastic evolution, fractional powers, and PDE kernels.
1375
1376 \appendix
1377 \section{Algorithm statements}\label{core:section:18}
1378
1379 \subsection{Exact polyphase filtering}
1380
1381 Given $x$, $h$, and $N$:
1382
1383 \begin{enumerate}
1384 \item Form $x_s[n]=x[Nn+s]$ for $0\le s<N$.
1385 \item Form every transfer kernel $h_{ks}[m]=h[Nm+k-s]$.
1386 \item Compute
1387 \[
1388 y_k[n]=\sum_s\sum_mh_{ks}[m]x_s[n-m].
1389 \]
1390 \item Interleave the $y_k$ channels.
1391 \end{enumerate}
1392
1393 The algorithm is algebraically identical to direct convolution. Its purpose is to expose independent and interacting packets for block computation and refinement.
1394
1395 \subsection{Conditional known-edge reconstruction}
1396
1397 Given a decomposition $f=s+j$:
1398
1399 \begin{enumerate}
1400 \item Apply the selected smooth approximation $Q_m$ to $s$ only.
1401 \item Retain $j$ exactly.
1402 \item Return $Q_ms+j$.
1403 \end{enumerate}
1404
1405 The error certificate is $\|s-Q_ms\|$.
1406
1407 \subsection{Limited positive-contrast estimator}
1408
1409 The included finite estimator does the following:
1410
1411 \begin{enumerate}
1412 \item Split the observed values into two classes by an Otsu threshold.
1413 \item Keep coherent connected components and fill holes.
1414 \item Harmonically fill the selected packet to estimate the missing background.
1415 \item Smooth the filled background spectrally.
1416 \item Estimate one packet amplitude by a robust median residual.
1417 \item Recombine the estimated smooth and edge packets.
1418 \end{enumerate}
1419
1420 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.
1421
1422 \section{Numerical record}\label{core:section:19}
1423
1424 \begin{table}[H]
1425 \centering
1426 \caption{Main edge-study measurements.}
1427 \begin{tabular}{lrrr}
1428 \toprule
1429 \text{case}&\text{global RMSE}&\text{known-packet RMSE}&\text{estimated RMSE}\\
1430 \midrule
1431 1D rectangular packet&$5.4023\times10^{-2}$&$1.3692\times10^{-4}$&$2.4029\times10^{-2}$\\
1432 2D disk packet&$3.7643\times10^{-2}$&$6.4752\times10^{-5}$&$3.4043\times10^{-3}$\\
1433 2D multiedge packet&$3.3740\times10^{-2}$&\multicolumn{2}{c}{known composite packet only}\\
1434 \bottomrule
1435 \end{tabular}
1436 \end{table}
1437
1438 \begin{table}[H]
1439 \centering
1440 \caption{Main software artifacts.}
1441 \begin{tabular}{ll}
1442 \toprule
1443 \text{artifact}&\text{purpose}\\
1444 \midrule
1445 \texttt{results/numerical\_results.json}&complete machine-readable certificate record\\
1446 \texttt{tables/numerical\_summary.csv}&flat summary of errors and residuals\\
1447 \texttt{tables/sparse\_representation.csv}&wavelet/Fourier coefficient study\\
1448 \texttt{tests/}&automated finite identity and estimator tests\\
1449 \texttt{figures/}&PNG and vector-PDF figure pairs\\
1450 \texttt{VERIFY\_PAPER.py}&combined manuscript, numerical, and Lean checks\\
1451 \bottomrule
1452 \end{tabular}
1453 \end{table}
1454
1455 \section{Notation crosswalk}\label{core:section:20}
1456
1457 \begin{longtable}{L{0.20\textwidth}L{0.33\textwidth}L{0.37\textwidth}}
1458 \caption{Notation crosswalk.}\label{tab:notation}\\
1459 \toprule
1460 \textbf{common symbol}&\textbf{signal realization}&\textbf{interpretation}\\
1461 \midrule
1462 \endfirsthead
1463 \toprule
1464 \textbf{common symbol}&\textbf{signal realization}&\textbf{interpretation}\\
1465 \midrule
1466 \endhead
1467 $s$&$\delta_0$&primitive signal seed\\
1468 $A_n$&$S^n$&address operator\\
1469 $P_{N,r}$&residue mask&phase packet projector\\
1470 $T_{ks}$&$P_{N,k}HP_{N,s}$&input-to-output phase transfer\\
1471 $E,R$&analysis and synthesis matrices&coordinate conversion and reconstruction\\
1472 $W(\gamma)$&filter or path weight&refinement or implementation cost\\
1473 $\Delta(F)$&coarse/fine signal defect&aliasing, leakage, or estimator inconsistency\\
1474 $\operatorname{Hol}(\gamma)$&closed phase transfer&cycle consistency in a network\\
1475 $\Gam$&reflection coefficient&Smith-disk coordinate\\
1476 \bottomrule
1477 \end{longtable}
1478
1479 \clearpage
1480 \section{Formalization and reproducibility record}\label{core:section:21}\label{sec:formalization}
1481
1482 \subsection{What has been checked}
1483
1484 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 retained compiler record documents a successful build and axiom audit with warnings treated as errors. The formal companion identifies its source, toolchain, and reproduction procedure. 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.
1485
1486 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.
1487
1488 The mathematical text integrates the checked hypotheses and conclusions. Editorial changes do not increase the Lean theorem count. Eleven foundational formal-power-series proofs in \texttt{CyclicSeries} are adaptations of Chapter IV and are identified as such in the provenance record.
1489
1490 \subsection{Result-by-result correspondence}
1491
1492 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.
1493
1494 \begingroup\small
1495 \begin{longtable}{L{0.09\textwidth}L{0.28\textwidth}L{0.53\textwidth}}
1496 \caption{The twenty manuscript results and their Lean statement scopes.}\label{tab:lean-map}\\
1497 \toprule
1498 \textbf{Result}&\textbf{Principal modules}&\textbf{Checked mathematical content}\\
1499 \midrule\endfirsthead
1500 \toprule
1501 \textbf{Result}&\textbf{Principal modules}&\textbf{Checked mathematical content}\\
1502 \midrule\endhead
1503 \ref{result:01}&HilbertPackets&Complex $\ell^2$ impulse expansion, finite-support identity, unitary shifts, and closed shift-generated span.\\
1504 \ref{result:02}&HilbertPackets&Bounded self-adjoint residue projectors, pairwise annihilation, completeness, and orthogonal ranges.\\
1505 \ref{result:03}&IntegerPolyphase; HilbertPackets&Signed quotient/remainder coordinates, both inverse identities, and the joint Hilbert norm of the channels.\\
1506 \ref{result:04}&HilbertPackets&Literal bounded-projector refinement for positive divisible moduli.\\
1507 \ref{result:05}&FIRTransfer; ZTransform&Finite-support tap reindexing for all bilateral inputs; numerical channel transforms under weighted absolute summability.\\
1508 \ref{result:06}&FilterBank; FiniteTransforms&Finite Laurent/FIR bank composition, faithful action on sequences, gain-delay reconstruction criterion, and the $Nd$ original-sample delay.\\
1509 \ref{result:07}&HaarChannels; HaarHilbert&Literal even/odd input samples, normalized complex low/detail channels, and $\ell^2$ energy equality.\\
1510 \ref{result:08}&Lifting; FiniteTransforms&Predict/update inverse maps, determinant one, and both flat finite-signal inverse identities.\\
1511 \ref{result:09}&WaveletPackets&Every finite depth of literal Haar packet operators, energy equality, and reconstruction using actual Hilbert adjoints.\\
1512 \ref{result:10}&KnownEdge&Exact cancellation of one retained packet, and equality of approximation errors.\\
1513 \ref{result:11}&KnownEdge&Finite families of retained packets, with no geometry assumption on the packets.\\
1514 \ref{result:12}&MaskPerturbation&Measurable-mask symmetric-difference norm, extended finite-$p$ error bound, pointwise and essential supremum bounds, and smooth-background error.\\
1515 \ref{result:13}&CompletePrunings; PruningSums&Recursive cuts correspond to complete covering antichains; literal finite node sums preserve synthesis.\\
1516 \ref{result:14}&PruningSums&Finite arbitrary-objective minimization, attained positive gap with competitors, strict stability, and the singleton case.\\
1517 \ref{result:15}&FiniteAdaptive; PruningAlgorithm&Deterministic stop-on-ties selector, global optimum, arbitrary positive finite branching, and the stated unit-operation count.\\
1518 \ref{result:16}&PruningSums; PruningAlgorithm&Attained maximum error envelope, optimal-value perturbation, strict-gap selector stability, and probability transfer from a joint event.\\
1519 \ref{result:17}&CyclicSeries&Formal Cauchy-product residue convolution and the finite-tuple power identity, including the empty product at exponent zero.\\
1520 \ref{result:18}&SmithAction; SmithCyclic&Nonzero homogeneous pairs modulo scaling, Cayley conjugacy, the actual exponential phase, and cyclic composition on the projective line.\\
1521 \ref{result:19}&SmithCyclic&Two-phase coordinate swap, $0\leftrightarrow\infty$, and finite impedance inversion when $z\ne0$.\\
1522 \ref{result:20}&CRT&Signed integer residue-mask factorization for coprime moduli, including the dyadic--odd case.\\
1523 \bottomrule
1524 \end{longtable}
1525 \endgroup
1526
1527 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.
1528
1529 \subsection{Three bridges that matter for the statement}
1530
1531 \paragraph{Finite filters and numerical transforms.}
1532 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.
1533
1534 \paragraph{Recursive trees and set-valued prunings.}
1535 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.
1536
1537 \paragraph{Hilbert operators and finite experiments.}
1538 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.
1539
1540 \subsection{Recorded checks and numerical conventions}
1541
1542 \begin{center}
1543 \begin{tabular}{lr}
1544 \toprule
1545 \textbf{Check family}&\textbf{Accepted count}\\
1546 \midrule
1547 Original numerical companion tests&33\\
1548 Independent exact-model tests&20\\
1549 Verifier positive and negative controls&16\\
1550 Total tests in these three families&69\\
1551 Exact assertions inside the independent model suites&305,933\\
1552 Registered Lean theorem targets&225\\
1553 Lean proof modules&23\\
1554 Regenerated figure pairs&18\\
1555 \bottomrule
1556 \end{tabular}
1557 \end{center}
1558
1559 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.
1560
1561 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. The term ``harmonic filling'' describes that iteration, not a separately proved exact continuum Dirichlet solver.
1562
1563 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.
1564
1565 \subsection{Pinned environment and reproduction}
1566
1567 The Lean toolchain is \texttt{leanprover/lean4:v4.34.0-rc2}. The project pins mathlib to
1568 \begin{center}\small\texttt{aeb72f1e9981b6d35b426b16dca7759be3e7ba06}.\end{center}
1569 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.
1570
1571 From the preserved release root (\texttt{source/complete/original\_snapshot} in the final package), the combined verification entry point is:
1572 \begin{lstlisting}[language=bash,numbers=none]
1573 python3 VERIFY_PAPER.py
1574 \end{lstlisting}
1575 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
1576 \begin{lstlisting}[language=bash,numbers=none]
1577 cd companion
1578 python3 run_all.py
1579 python3 -m pytest -q tests
1580 \end{lstlisting}
1581 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.
1582
1583 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.
1584
1585 \subsection{What remains open}
1586
1587 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 formulas provide application context. Those physical and historical assertions are not additional Lean-certified theorem rows.
1588
1589 Historical cross-paper declaration counts are retained in the source records. The relevant formal denominator here is the twenty scoped Part I results. Supporting declarations, reused results and proofs in the broader written development have distinct roles.
1590
1591 \clearpage
1592 \part*{Part II: Extended mathematical development}\addcontentsline{toc}{part}{Part II: Extended mathematical development}
1593 \begin{center}\large Multiresolution Signal Calculus, Singular Packets, and Stable Reconstruction\end{center}
1594 \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.
1595 \setcounter{section}{0}\setcounter{figure}{0}\setcounter{table}{0}
1596 \renewcommand{\thesection}{E\arabic{section}}
1597 \renewcommand{\theHsection}{extension.\arabic{section}}
1598 \renewcommand{\thefigure}{E\arabic{figure}}
1599 \renewcommand{\theHfigure}{extension.\arabic{figure}}
1600 \renewcommand{\thetable}{E\arabic{table}}
1601 \renewcommand{\theHtable}{extension.\arabic{table}}
1602 \graphicspath{{extension_figures/}}
1603
1604 \begin{abstract}
1605 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.
1606 \end{abstract}
1607
1608
1609 \section{Introduction}\label{extension:section:01}\label{sec:introduction}
1610
1611 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 part is to place these constructions in one exact coordinate system.
1612
1613 The second aim is to separate representation from recovery. If a signal has already been written as
1614 \[
1615 f=s+j
1616 \]
1617 with a smooth part \(s\) and a singular packet \(j\), then
1618 \[
1619 \widetilde f=Qs+j
1620 \]
1621 satisfies
1622 \[
1623 f-\widetilde f=s-Qs.
1624 \]
1625 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 this part isolate all three terms.
1626
1627 The singular packets considered here range from indicator functions of regions to truncated powers
1628 \[
1629 \frac{(t-\tau)_+^m}{m!}.
1630 \]
1631 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.
1632
1633 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.
1634
1635 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 part 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.
1636
1637 This part 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.
1638
1639 Several conclusions are deliberately excluded. The chapter 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.
1640
1641 \subsection{Main results}
1642
1643 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
1644 \[
1645 x\longmapsto (x_r)_{r\in R},
1646 \qquad
1647 x_r(\ell)=x(r+\ell),
1648 \]
1649 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
1650 \[
1651 h_{rs}(m)=h(r-s+m).
1652 \]
1653 The matrix is therefore a coordinate representation of the original operator, not a reduced model.
1654
1655 The filter bank results distinguish exact reconstruction from quantitative near reconstruction. Exact recovery with delay \(d\) is the pointwise matrix identity
1656 \[
1657 R(z)E(z)=z^{-d}I.
1658 \]
1659 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.
1660
1661 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
1662 \[
1663 F(v)
1664 =
1665 \min\left\{
1666 c(v),
1667 \sum_{w\in\operatorname{child}(v)}F(w)
1668 \right\}
1669 \]
1670 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.
1671
1672 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
1673 \[
1674 \jump{f^{(m)}}_{\tau_q}
1675 \frac{(t-\tau_q)_+^m}{m!}
1676 \]
1677 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}.
1678
1679 The reconstruction formula with an estimated packet is
1680 \[
1681 f-\widehat f_m
1682 =
1683 (I-Q_m)s+(I-Q_m)(j-\widehat j).
1684 \]
1685 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.
1686
1687 The principal recovery theorem uses adjacent sample contrasts. Let
1688 \[
1689 b_h=Mh+2\varepsilon
1690 \]
1691 be the largest contrast that can be produced by a Lipschitz background and bounded noise in one grid cell. If
1692 \[
1693 a_{\min}>2b_h,
1694 \]
1695 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
1696 \[
1697 2\sigma\sqrt{\log(2n/\delta)}.
1698 \]
1699 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.
1700
1701 \subsection{Notation}
1702
1703 The following symbols recur throughout this part.
1704 \begin{center}
1705 \begin{tabular}{p{0.22\textwidth}p{0.66\textwidth}}
1706 \toprule
1707 symbol & meaning\\
1708 \midrule
1709 \(L\leq\ZZ^d\) & a finite index sublattice\\
1710 \(R\) & a set of representatives for \(\ZZ^d/L\)\\
1711 \(P_r\) & the coordinate projector onto the coset \(r+L\)\\
1712 \(\Acal_L,\Scal_L\) & lattice analysis and interleaving synthesis\\
1713 \(E,R\) & filter bank analysis and synthesis symbols\\
1714 \(Q_m\) & a bounded linear approximation operator acting on the smooth part\\
1715 \(j,\widehat j\) & the true and estimated singular packets\\
1716 \(H_\tau\) & the step packet \(\one_{[\tau,1]}\)\\
1717 \(\phi_{m,\tau}\) & the truncated power packet \((t-\tau)_+^m/m!\)\\
1718 \(h\) & the grid spacing in the recovery theorems\\
1719 \bottomrule
1720 \end{tabular}
1721 \end{center}
1722
1723 \section{Signal seeds and lattice addresses}\label{extension:section:02}\label{sec:signal_seeds}
1724
1725 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
1726 \[
1727 (T_ax)(n)=x(n-a).
1728 \]
1729 Let \(\delta_0\) be the impulse at the origin.
1730
1731 \begin{theorem}[Impulse generation]\label{thm:impulse_generation}
1732 For \(1\leq p<\infty\),
1733 \[
1734 \overline{\Span\{T_a\delta_0:a\in\ZZ^d\}}^{\ell^p}
1735 =
1736 \ell^p(\ZZ^d).
1737 \]
1738 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 \(c_0(\ZZ^d)\).
1739 \end{theorem}
1740
1741 \begin{proof}
1742 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 \(\ell^\infty\).
1743 \end{proof}
1744
1745 \begin{remark}\label{rem:infinity_boundary}
1746 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.
1747 \end{remark}
1748
1749 \begin{proposition}[Finite periodic signals]\label{prop:finite_periodic}
1750 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.
1751 \end{proposition}
1752
1753 \begin{proof}
1754 Translation sends the impulse at the identity to the impulse at each element of \(G\). These impulses form the standard basis of \(\CC^G\).
1755 \end{proof}
1756
1757 \section{Polyphase coordinates on finite index sublattices}\label{extension:section:03}\label{sec:polyphase}
1758
1759 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
1760 \[
1761 \ZZ^d=\bigsqcup_{r\in R}(r+L).
1762 \]
1763 For \(x:\ZZ^d\to\CC\), its channel at \(r\) is the sequence on \(L\)
1764 \[
1765 x_r(\ell)=x(r+\ell).
1766 \]
1767
1768 \begin{definition}[Lattice packet]\label{def:lattice_packet}
1769 For \(r\in R\), define the coordinate projector
1770 \[
1771 (P_rx)(n)=\one_{r+L}(n)x(n).
1772 \]
1773 The family \((P_r)_{r\in R}\) is called the packet system associated with \(L\) and \(R\).
1774 \end{definition}
1775
1776 For finite \(p\), equip \(\bigoplus_{r\in R}\ell^p(L)\) with
1777 \[
1778 \|(u_r)\|^p=\sum_{r\in R}\|u_r\|_p^p.
1779 \]
1780 For \(p=\infty\), use the maximum norm.
1781
1782 \begin{theorem}[Exact lattice polyphase decomposition]\label{thm:lattice_polyphase}
1783 For every \(1\leq p\leq\infty\), the analysis map
1784 \[
1785 \Acal_L:\ell^p(\ZZ^d)\longrightarrow\bigoplus_{r\in R}\ell^p(L),
1786 \qquad
1787 \Acal_Lx=(x_r)_{r\in R},
1788 \]
1789 is an isometric isomorphism. Its inverse is the interleaving map
1790 \[
1791 \Scal_L(u)(r+\ell)=u_r(\ell).
1792 \]
1793 The projectors satisfy
1794 \[
1795 P_rP_s=\delta_{rs}P_r,
1796 \qquad
1797 \sum_{r\in R}P_r=I.
1798 \]
1799 \end{theorem}
1800
1801 \begin{proof}
1802 Every address has a unique representation \(r+\ell\). For finite \(p\),
1803 \[
1804 \|x\|_p^p
1805 =
1806 \sum_{r\in R}\sum_{\ell\in L}|x(r+\ell)|^p
1807 =
1808 \sum_{r\in R}\|x_r\|_p^p.
1809 \]
1810 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.
1811 \end{proof}
1812
1813 \begin{corollary}[One dimensional residue coordinates]\label{cor:one_dimensional_polyphase}
1814 For \(L=N\ZZ\) and \(R=\{0,\ldots,N-1\}\),
1815 \[
1816 x_r(k)=x(Nk+r)
1817 \]
1818 and
1819 \[
1820 \ell^p(\ZZ)\cong\bigoplus_{r=0}^{N-1}\ell^p(\ZZ)
1821 \]
1822 isometrically.
1823 \end{corollary}
1824
1825 \begin{proof}
1826 Apply \cref{thm:lattice_polyphase} to the standard residue representatives.
1827 \end{proof}
1828
1829 \begin{proposition}[Translation covariance]\label{prop:translation_covariance}
1830 Fix \(a\in\ZZ^d\). For every \(r\in R\), write uniquely
1831 \[
1832 r-a=r_a+\lambda_{a,r},
1833 \qquad
1834 r_a\in R,
1835 \quad
1836 \lambda_{a,r}\in L.
1837 \]
1838 Then
1839 \[
1840 (\Acal_LT_ax)_r(\ell)
1841 =
1842 x_{r_a}(\ell+\lambda_{a,r}).
1843 \]
1844 Thus a fine translation permutes the lattice packets and applies an internal translation on \(L\).
1845 \end{proposition}
1846
1847 \begin{proof}
1848 Evaluate at \(r+\ell\):
1849 \[
1850 (T_ax)(r+\ell)
1851 =x(r+\ell-a)
1852 =x(r_a+\ell+\lambda_{a,r}).
1853 \]
1854 This is the stated channel formula.
1855 \end{proof}
1856
1857 \begin{figure}
1858 \centering
1859 \includegraphics[width=0.82\textwidth]{01_polyphase_coordinates.pdf}
1860 \caption{A signal and its four exact residue channels. Interleaving the channels returns the original samples without approximation.}
1861 \label{fig:polyphase_coordinates}
1862 \end{figure}
1863
1864 \subsection{Refinement of sublattices}
1865
1866 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
1867 \[
1868 \operatorname{Child}(r)
1869 =
1870 \{s\in R':s+L'\subset r+L\}.
1871 \]
1872
1873 \begin{theorem}[Exact lattice refinement]\label{thm:lattice_refinement}
1874 For every coarse packet,
1875 \[
1876 P_r^{L}
1877 =
1878 \sum_{s\in\operatorname{Child}(r)}P_s^{L'}.
1879 \]
1880 The fine packets in the sum are pairwise disjoint. Refinement through several nested sublattices is transitive.
1881 \end{theorem}
1882
1883 \begin{proof}
1884 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.
1885 \end{proof}
1886
1887 \begin{example}[Rectangular image phases]\label{ex:image_phases}
1888 For
1889 \[
1890 L=M\ZZ\times N\ZZ\subset\ZZ^2,
1891 \]
1892 one may take
1893 \[
1894 R=\{0,\ldots,M-1\}\times\{0,\ldots,N-1\}.
1895 \]
1896 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
1897 \[
1898 \frac{M'}M\frac{N'}N
1899 \]
1900 fine phases.
1901 \end{example}
1902
1903 \section{Convolution as a packet transfer matrix}\label{extension:section:04}\label{sec:convolution}
1904
1905 Let \(h\in\ell^1(\ZZ^d)\) and \(x\in\ell^p(\ZZ^d)\), with \(1\leq p\leq\infty\), and define
1906 \[
1907 (Hx)(n)
1908 =(h*x)(n)
1909 =\sum_{q\in\ZZ^d}h(q)x(n-q).
1910 \]
1911 Young's inequality gives
1912 \[
1913 \|Hx\|_p\leq\|h\|_1\|x\|_p
1914 \]
1915 for every \(1\leq p\leq\infty\).
1916
1917 \begin{definition}[Packet transfer kernel]\label{def:packet_transfer_kernel}
1918 For output residue \(r\in R\) and input residue \(s\in R\), define a kernel on \(L\) by
1919 \[
1920 h_{rs}(m)=h(r-s+m),
1921 \qquad
1922 m\in L.
1923 \]
1924 \end{definition}
1925
1926 \begin{theorem}[Exact convolution transfer law]\label{thm:convolution_transfer}
1927 Let \(y=h*x\). Then
1928 \[
1929 y_r(\ell)
1930 =
1931 \sum_{s\in R}\sum_{m\in L}h_{rs}(m)x_s(\ell-m).
1932 \]
1933 Equivalently,
1934 \[
1935 \Acal_LH\Acal_L^{-1}
1936 =
1937 (H_{rs})_{r,s\in R},
1938 \]
1939 where \(H_{rs}\) is convolution on \(L\) by \(h_{rs}\).
1940 \end{theorem}
1941
1942 \begin{proof}
1943 Evaluate the output at \(r+\ell\). Write every input address as \(s+\ell'\). The convolution offset is
1944 \[
1945 (r+\ell)-(s+\ell')
1946 =(r-s)+(\ell-\ell').
1947 \]
1948 Setting \(m=\ell-\ell'\) gives the formula.
1949 \end{proof}
1950
1951 \begin{proposition}[Transfer norm control]\label{prop:transfer_norm_control}
1952 For every output residue \(r\),
1953 \[
1954 \sum_{s\in R}\|h_{rs}\|_{\ell^1(L)}
1955 =
1956 \|h\|_{\ell^1(\ZZ^d)}.
1957 \]
1958 The packet transfer matrix acts on the polyphase direct sum with norm at most \(\|h\|_1\) on every \(\ell^p\) space.
1959 \end{proposition}
1960
1961 \begin{proof}
1962 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.
1963 \end{proof}
1964
1965 \begin{proposition}[Composition of transfer matrices]\label{prop:transfer_composition}
1966 If \(G\) and \(H\) are convolution operators with packet matrices \((G_{rs})\) and \((H_{rs})\), then the packet matrix of \(GH\) is
1967 \[
1968 (GH)_{rt}
1969 =
1970 \sum_{s\in R}G_{rs}H_{st}.
1971 \]
1972 In one-dimensional transform notation, the corresponding matrix symbols multiply pointwise.
1973 \end{proposition}
1974
1975 \begin{proof}
1976 Conjugation by \(\Acal_L\) preserves composition:
1977 \[
1978 \Acal_LGH\Acal_L^{-1}
1979 =
1980 (\Acal_LG\Acal_L^{-1})(\Acal_LH\Acal_L^{-1}).
1981 \]
1982 Reading the \((r,t)\) block gives the formula.
1983 \end{proof}
1984
1985 \subsection{The one-dimensional symbol}
1986
1987 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
1988 \[
1989 H_{ks}(z)=\sum_{m\in\ZZ}h(Nm+k-s)z^{-m}.
1990 \]
1991 For \(x\in\ell^1(\ZZ)\), the channel transforms converge absolutely on the unit circle and satisfy
1992 \[
1993 \bm Y(z)=\bm H_N(z)\bm X(z).
1994 \]
1995 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.
1996
1997 \begin{corollary}[Finite impulse response closure]\label{cor:fir_closure}
1998 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.
1999 \end{corollary}
2000
2001 \begin{proof}
2002 Finite support is preserved by taking residue subsequences. The composition statement is \cref{prop:transfer_composition}.
2003 \end{proof}
2004
2005 \begin{example}[A four phase finite impulse response operator]\label{ex:four_phase_fir}
2006 Let
2007 \[
2008 h(0)=1,
2009 \quad h(1)=2,
2010 \quad h(2)=3,
2011 \quad h(3)=4,
2012 \quad h(4)=5,
2013 \]
2014 with all other taps zero. For four residue channels, the transfer symbol is
2015 \[
2016 \bm H_4(z)
2017 =
2018 \begin{pmatrix}
2019 1+5z^{-1}&4z^{-1}&3z^{-1}&2z^{-1}\\
2020 2&1+5z^{-1}&4z^{-1}&3z^{-1}\\
2021 3&2&1+5z^{-1}&4z^{-1}\\
2022 4&3&2&1+5z^{-1}
2023 \end{pmatrix}.
2024 \]
2025 The delay term records the tap that crosses a coarse cell boundary. No approximation is introduced by the matrix representation.
2026 \end{example}
2027
2028 \begin{figure}
2029 \centering
2030 \includegraphics[width=0.62\textwidth]{02_transfer_blocks.pdf}
2031 \caption{Absolute tap weight in the transfer blocks of the four phase operator in \cref{ex:four_phase_fir}.}
2032 \label{fig:transfer_blocks}
2033 \end{figure}
2034
2035 \section{Filter banks, frame bounds, and lifting}\label{extension:section:05}\label{sec:filter_banks}
2036
2037 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.
2038
2039 \begin{theorem}[Perfect reconstruction criterion]\label{thm:perfect_reconstruction}
2040 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
2041 \[
2042 R(z)E(z)=z^{-d}I_N
2043 \]
2044 for almost every point on the unit circle. A nonzero scalar factor gives exact reconstruction up to gain and delay.
2045 \end{theorem}
2046
2047 \begin{proof}
2048 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.
2049 \end{proof}
2050
2051 \begin{theorem}[Quantitative imperfect reconstruction]\label{thm:imperfect_reconstruction}
2052 Define the delay compensated defect
2053 \[
2054 D(\e^{\ii\omega})
2055 =
2056 \e^{\ii d\omega}R(\e^{\ii\omega})E(\e^{\ii\omega})-I_N.
2057 \]
2058 If
2059 \[
2060 \varepsilon_{\mathrm{rec}}
2061 =
2062 \esssup_{\omega}\|D(\e^{\ii\omega})\|_{\mathrm{op}}<\infty,
2063 \]
2064 then the delay compensated reconstruction \(\widetilde x\) satisfies
2065 \[
2066 \|\widetilde x-x\|_2
2067 \leq
2068 \varepsilon_{\mathrm{rec}}\|x\|_2.
2069 \]
2070 The constant is sharp as an operator norm.
2071 \end{theorem}
2072
2073 \begin{proof}
2074 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.
2075 \end{proof}
2076
2077 \begin{example}[A reconstruction certificate]\label{ex:reconstruction_certificate}
2078 Suppose a two channel bank has delay compensated defect
2079 \[
2080 D(\e^{\ii\omega})
2081 =
2082 \begin{pmatrix}
2083 0.01\cos\omega&0\\
2084 0&0.02\sin\omega
2085 \end{pmatrix}.
2086 \]
2087 Then
2088 \[
2089 \esssup_\omega\|D(\e^{\ii\omega})\|_{\mathrm{op}}
2090 =0.02.
2091 \]
2092 Every input therefore has relative reconstruction error at most two percent in \(\ell^2\). The estimate is independent of the input length.
2093 \end{example}
2094
2095 \begin{theorem}[Stability under filter perturbation]\label{thm:filter_perturbation}
2096 Assume \(RE=z^{-d}I_N\). Let
2097 \[
2098 \widetilde E=E+\Delta E,
2099 \qquad
2100 \widetilde R=R+\Delta R.
2101 \]
2102 Set
2103 \[
2104 \begin{aligned}
2105 e_0&=\esssup\|E\|_{\mathrm{op}},
2106 & r_0&=\esssup\|R\|_{\mathrm{op}},\\
2107 \delta_E&=\esssup\|\Delta E\|_{\mathrm{op}},
2108 & \delta_R&=\esssup\|\Delta R\|_{\mathrm{op}}.
2109 \end{aligned}
2110 \]
2111 Then the delay compensated reconstruction defect obeys
2112 \[
2113 \varepsilon_{\mathrm{rec}}
2114 \leq
2115 r_0\delta_E+e_0\delta_R+\delta_R\delta_E.
2116 \]
2117 \end{theorem}
2118
2119 \begin{proof}
2120 Expand
2121 \[
2122 \widetilde R\widetilde E-RE
2123 =
2124 R\Delta E+\Delta R E+\Delta R\Delta E.
2125 \]
2126 Take pointwise operator norms, then the essential supremum. Multiplication by the delay factor has unit modulus.
2127 \end{proof}
2128
2129 \begin{theorem}[Frame bounds from the polyphase symbol]\label{thm:frame_bounds}
2130 Assume that
2131 \[
2132 A I_N
2133 \leq
2134 E(\e^{\ii\omega})^*E(\e^{\ii\omega})
2135 \leq
2136 B I_N
2137 \]
2138 for almost every \(\omega\), where \(0<A\leq B<\infty\). Then the analysis bank satisfies
2139 \[
2140 A\|x\|_2^2
2141 \leq
2142 \|\Acal_E x\|_2^2
2143 \leq
2144 B\|x\|_2^2.
2145 \]
2146 The pointwise matrix
2147 \[
2148 R_{\rm can}(\e^{\ii\omega})
2149 =
2150 (E^*E)^{-1}E^*
2151 \]
2152 is a bounded left inverse and gives the canonical dual synthesis bank.
2153 \end{theorem}
2154
2155 \begin{proof}
2156 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.
2157 \end{proof}
2158
2159 \begin{corollary}[Coefficient noise under canonical synthesis]\label{cor:canonical_noise}
2160 Under the hypotheses of \cref{thm:frame_bounds}, suppose the observed coefficient vector is
2161 \[
2162 c=\Acal_Ex+\nu.
2163 \]
2164 Canonical synthesis gives
2165 \[
2166 \widehat x=R_{\mathrm{can}}c
2167 \]
2168 and
2169 \[
2170 \|\widehat x-x\|_2
2171 \leq
2172 \frac1{\sqrt A}\|\nu\|_2.
2173 \]
2174 \end{corollary}
2175
2176 \begin{proof}
2177 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 \(R_{\mathrm{can}}\nu\).
2178 \end{proof}
2179
2180 \begin{theorem}[Perturbation of frame bounds]\label{thm:frame_perturbation}
2181 Under the hypotheses of \cref{thm:frame_bounds}, let \(\widetilde E=E+\Delta E\) and suppose
2182 \[
2183 \delta=\esssup_\omega\|\Delta E(\e^{\ii\omega})\|_{\mathrm{op}}<\sqrt A.
2184 \]
2185 Then \(\widetilde E\) is a frame analysis bank with bounds
2186 \[
2187 (\sqrt A-\delta)^2
2188 \quad\text{and}\quad
2189 (\sqrt B+\delta)^2.
2190 \]
2191 \end{theorem}
2192
2193 \begin{proof}
2194 For every vector \(v\),
2195 \[
2196 \|\widetilde Ev\|
2197 \geq
2198 \|Ev\|-\|\Delta Ev\|
2199 \geq
2200 (\sqrt A-\delta)\|v\|.
2201 \]
2202 The upper bound follows from
2203 \[
2204 \|\widetilde Ev\|
2205 \leq
2206 (\sqrt B+\delta)\|v\|.
2207 \]
2208 Square the inequalities and integrate in frequency.
2209 \end{proof}
2210
2211 \begin{remark}\label{rem:canonical_dual_support}
2212 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.
2213 \end{remark}
2214
2215 \begin{corollary}[Paraunitary bank]\label{cor:paraunitary}
2216 If \(E(z)^*E(z)=I_N\) on the unit circle, then analysis is an isometry and synthesis may be chosen as \(R(z)=E(z)^*\).
2217 \end{corollary}
2218
2219 \begin{proof}
2220 Use \cref{thm:frame_bounds} with \(A=B=1\), then multiply the matrices.
2221 \end{proof}
2222
2223 \begin{figure}
2224 \centering
2225 \includegraphics[width=0.72\textwidth]{03_frame_stability.pdf}
2226 \caption{Pointwise singular value bounds govern both the original frame and its perturbed analysis bank.}
2227 \label{fig:frame_stability}
2228 \end{figure}
2229
2230 \subsection{Haar analysis}
2231
2232 The Haar matrix is
2233 \[
2234 E_H
2235 =
2236 \frac1{\sqrt2}
2237 \begin{pmatrix}
2238 1&1\\
2239 1&-1
2240 \end{pmatrix}.
2241 \]
2242 For even and odd samples \(e(n)=x(2n)\) and \(o(n)=x(2n+1)\), the low and high coordinates are
2243 \[
2244 a(n)=\frac{e(n)+o(n)}{\sqrt2},
2245 \qquad
2246 d(n)=\frac{e(n)-o(n)}{\sqrt2}.
2247 \]
2248
2249 \begin{corollary}[Haar reconstruction and energy]\label{cor:haar_reconstruction}
2250 The Haar bank reconstructs exactly and
2251 \[
2252 \|x\|_2^2=\|a\|_2^2+\|d\|_2^2.
2253 \]
2254 \end{corollary}
2255
2256 \begin{proof}
2257 The matrix \(E_H\) is unitary.
2258 \end{proof}
2259
2260 \subsection{Lifting steps}
2261
2262 Let \(P(z)\) and \(Q(z)\) be Laurent polynomials. Define
2263 \[
2264 L_P(z)
2265 =
2266 \begin{pmatrix}
2267 1&0\\
2268 P(z)&1
2269 \end{pmatrix},
2270 \qquad
2271 U_Q(z)
2272 =
2273 \begin{pmatrix}
2274 1&Q(z)\\
2275 0&1
2276 \end{pmatrix}.
2277 \]
2278
2279 \begin{theorem}[Exact lifting update]\label{thm:lifting_update}
2280 The lifting matrices are invertible with
2281 \[
2282 L_P^{-1}=L_{-P},
2283 \qquad
2284 U_Q^{-1}=U_{-Q}.
2285 \]
2286 If \(R_0E_0=I\), then every finite product \(V\) of lifting matrices gives another exact bank by
2287 \[
2288 E=VE_0,
2289 \qquad
2290 R=R_0V^{-1}.
2291 \]
2292 Every lifting step has determinant one.
2293 \end{theorem}
2294
2295 \begin{proof}
2296 Direct multiplication proves the inverse formulas and the determinant statement. The reconstructed product is
2297 \[
2298 RE
2299 =R_0V^{-1}VE_0
2300 =R_0E_0
2301 =I.
2302 \]
2303 \end{proof}
2304
2305 \begin{example}[A two neighbor predict and update transform]\label{ex:predict_update}
2306 Split a sequence into even and odd samples. Define
2307 \[
2308 d(n)
2309 =o(n)-\frac{e(n)+e(n+1)}2
2310 \]
2311 and
2312 \[
2313 s(n)
2314 =e(n)+\frac{d(n-1)+d(n)}4.
2315 \]
2316 The inverse is obtained in reverse order:
2317 \[
2318 e(n)
2319 =s(n)-\frac{d(n-1)+d(n)}4,
2320 \]
2321 then
2322 \[
2323 o(n)
2324 =d(n)+\frac{e(n)+e(n+1)}2.
2325 \]
2326 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}.
2327 \end{example}
2328
2329 \section{Wavelet packet trees and admissible bases}\label{extension:section:06}\label{sec:packet_trees}
2330
2331 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.
2332
2333 \begin{definition}[Unitary packet tree]\label{def:unitary_packet_tree}
2334 A finite unitary packet tree consists of Hilbert spaces \(\Hcal_v\) at its vertices and, at every internal vertex \(v\), a unitary map
2335 \[
2336 U_v:\Hcal_v\longrightarrow\bigoplus_{w\in\operatorname{child}(v)}\Hcal_w.
2337 \]
2338 An admissible pruning is a set of vertices that meets every root to terminal path exactly once.
2339 \end{definition}
2340
2341 \begin{theorem}[Reconstruction on a pruned packet tree]\label{thm:pruned_packet_tree}
2342 Every admissible pruning \(\Lcal\) defines a unitary analysis map
2343 \[
2344 \Acal_{\Lcal}:\Hcal_{\rm root}
2345 \longrightarrow
2346 \bigoplus_{v\in\Lcal}\Hcal_v.
2347 \]
2348 Its inverse is obtained by applying the adjoints of the local splits from the leaves toward the root. In particular,
2349 \[
2350 \|x\|^2
2351 =
2352 \sum_{v\in\Lcal}\|x_v\|^2.
2353 \]
2354 \end{theorem}
2355
2356 \begin{proof}
2357 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.
2358 \end{proof}
2359
2360 \begin{corollary}[Coefficient perturbations in a unitary tree]\label{cor:unitary_tree_noise}
2361 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
2362 \[
2363 \|\widehat x-x\|=\|e\|.
2364 \]
2365 Thus coefficient quantization and coefficient noise are neither amplified nor attenuated by exact unitary synthesis.
2366 \end{corollary}
2367
2368 \begin{proof}
2369 The synthesis map is unitary by \cref{thm:pruned_packet_tree}. Hence
2370 \[
2371 \widehat x-x
2372 =\Acal_{\Lcal}^{-1}e
2373 \]
2374 has the same norm as \(e\).
2375 \end{proof}
2376
2377 \begin{proposition}[Bounds for nonunitary local splits]\label{prop:nonunitary_tree_bounds}
2378 Suppose every internal node has an analysis map \(U_v\) and constants \(0\leq A_v\leq B_v<\infty\) satisfying
2379 \[
2380 A_v\|x\|^2
2381 \leq
2382 \sum_{w\in\operatorname{child}(v)}\|(U_vx)_w\|^2
2383 \leq
2384 B_v\|x\|^2.
2385 \]
2386 Define recursively
2387 \[
2388 a(v)=
2389 \begin{cases}
2390 1,&v\text{ retained},\\
2391 A_v\min_w a(w),&v\text{ split},
2392 \end{cases}
2393 \]
2394 and
2395 \[
2396 b(v)=
2397 \begin{cases}
2398 1,&v\text{ retained},\\
2399 B_v\max_w b(w),&v\text{ split}.
2400 \end{cases}
2401 \]
2402 Then the leaf analysis of the pruned tree has bounds \(a({\rm root})\) and \(b({\rm root})\).
2403 \end{proposition}
2404
2405 \begin{proof}
2406 Assume the child subtree bounds have been proved. Then
2407 \[
2408 \sum_{w}\sum_{\ell\in\Lcal_w}\|x_\ell\|^2
2409 \geq
2410 \min_w a(w)
2411 \sum_w\|(U_vx)_w\|^2
2412 \geq
2413 A_v\min_w a(w)\|x\|^2.
2414 \]
2415 The upper bound follows in the same way with the maximum and \(B_v\). Induction completes the proof.
2416 \end{proof}
2417
2418 \begin{figure}
2419 \centering
2420 \includegraphics[width=0.72\textwidth]{04_packet_tree.pdf}
2421 \caption{A complete packet tree and one admissible pruning. Retained nodes may occur at different depths.}
2422 \label{fig:packet_tree}
2423 \end{figure}
2424
2425 \subsection{Best basis recursion}
2426
2427 \begin{interpretation}[title={The minimum is taken over feasible prunings}]
2428 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.
2429 \end{interpretation}
2430
2431 Assign a real cost \(c(v)\) to retaining each node. For an admissible pruning \(\Lcal\), let
2432 \[
2433 C(\Lcal)=\sum_{v\in\Lcal}c(v).
2434 \]
2435 The cost may measure entropy, coefficient count, an approximation penalty, or any other additive local objective.
2436
2437 \begin{theorem}[Exact additive best basis recursion]\label{thm:best_basis}
2438 For each vertex define
2439 \[
2440 F(v)
2441 =
2442 \min\left\{
2443 c(v),
2444 \sum_{w\in\operatorname{child}(v)}F(w)
2445 \right\},
2446 \]
2447 with \(F(v)=c(v)\) at terminal vertices. Then \(F(v)\) is the minimum cost among all admissible prunings 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.
2448 \end{theorem}
2449
2450 \begin{proof}
2451 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.
2452 \end{proof}
2453
2454 \begin{theorem}[Stability of the optimal cost]\label{thm:best_basis_cost_stability}
2455 Let \(c\) and \(\widetilde c\) be two node cost functions. For an admissible pruning \(\Lcal\), set
2456 \[
2457 \varepsilon(\Lcal)
2458 =
2459 \sum_{v\in\Lcal}|c(v)-\widetilde c(v)|
2460 \]
2461 and let
2462 \[
2463 \varepsilon_*
2464 =
2465 \max_{\Lcal}\varepsilon(\Lcal).
2466 \]
2467 If \(F_*\) and \(\widetilde F_*\) are the two optimal root costs, then
2468 \[
2469 |F_*-\widetilde F_*|
2470 \leq
2471 \varepsilon_*.
2472 \]
2473 \end{theorem}
2474
2475 \begin{proof}
2476 For every admissible pruning,
2477 \[
2478 |C(\Lcal)-\widetilde C(\Lcal)|
2479 \leq
2480 \varepsilon_*.
2481 \]
2482 Evaluate \(\widetilde C\) on a minimizer for \(C\) to obtain \(\widetilde F_*\leq F_*+\varepsilon_*\). Reverse the roles to obtain \(F_*\leq\widetilde F_*+\varepsilon_*\).
2483 \end{proof}
2484
2485 \begin{corollary}[Stability of the selected basis]\label{cor:best_basis_selection_stability}
2486 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
2487 \[
2488 2\varepsilon_*<g,
2489 \]
2490 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.
2491 \end{corollary}
2492
2493 \begin{proof}
2494 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.
2495 \end{proof}
2496
2497 \begin{figure}
2498 \centering
2499 \includegraphics[width=0.72\textwidth]{05_best_basis.pdf}
2500 \caption{An additive best basis recursion compares the cost of retaining a packet with the total cost of splitting it.}
2501 \label{fig:best_basis}
2502 \end{figure}
2503
2504 \section{Singular packets and exact reconstruction}\label{extension:section:07}\label{sec:singular_packets}
2505
2506 \begin{readerroadmap}[title={Trace data identify the singular part}]
2507 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.
2508 \end{readerroadmap}
2509
2510 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.
2511
2512 For \(m\geq0\) and \(0<\tau<1\), define
2513 \[
2514 \phi_{m,\tau}(t)
2515 =
2516 \frac{(t-\tau)_+^m}{m!}.
2517 \]
2518 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 \(\tau\).
2519
2520 \begin{definition}[Sobolev step packet class]\label{def:step_packet_class}
2521 Fix \(r\geq1\), \(1\leq p\leq\infty\), and an integer \(Q\geq0\). The class \(\Fcal_{r,p}(Q)\) consists of functions
2522 \[
2523 f=s+\sum_{q=1}^{Q}a_q\phi_{0,\tau_q}
2524 \]
2525 with \(s\in W^{r,p}(0,1)\) and
2526 \[
2527 0<\tau_1<\cdots<\tau_Q<1.
2528 \]
2529 No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.
2530 \end{definition}
2531
2532 \begin{theorem}[Higher order singular packet decomposition]\label{thm:higher_order_packet_decomposition}
2533 Let
2534 \[
2535 0=\tau_0<\tau_1<\cdots<\tau_Q<\tau_{Q+1}=1.
2536 \]
2537 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
2538 \[
2539 a_{q,m}
2540 =
2541 \jump{f^{(m)}}_{\tau_q}
2542 =
2543 f^{(m)}(\tau_q+)-f^{(m)}(\tau_q-)
2544 \]
2545 and
2546 \[
2547 j_f(t)
2548 =
2549 \sum_{q=1}^{Q}\sum_{m=0}^{r-1}a_{q,m}\phi_{m,\tau_q}(t).
2550 \]
2551 Then
2552 \[
2553 s_f=f-j_f
2554 \]
2555 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.
2556 \end{theorem}
2557
2558 \begin{proof}
2559 For \(0\leq \ell\leq m\), the classical derivative of \(\phi_{m,\tau}\) on the two open sides is
2560 \[
2561 \phi_{m,\tau}^{(\ell)}(t)
2562 =
2563 \frac{(t-\tau)_+^{m-\ell}}{(m-\ell)!}.
2564 \]
2565 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 \(r-1\).
2566
2567 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 \(f\).
2568 \end{proof}
2569
2570 \begin{corollary}[Uniqueness of the trace packet]\label{cor:trace_packet_uniqueness}
2571 Fix the breakpoints \(\tau_q\). If
2572 \[
2573 f=s+\sum_{q=1}^{Q}\sum_{m=0}^{r-1}b_{q,m}\phi_{m,\tau_q}
2574 \]
2575 with \(s\in W^{r,p}(0,1)\), then
2576 \[
2577 b_{q,m}=\jump{f^{(m)}}_{\tau_q}.
2578 \]
2579 The finite truncated power packet is therefore the unique packet supported at the declared breakpoints that removes all trace jumps through order \(r-1\).
2580 \end{corollary}
2581
2582 \begin{proof}
2583 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 \(b_{q,m}\).
2584 \end{proof}
2585
2586 \begin{remark}\label{rem:higher_order_scope}
2587 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}.
2588 \end{remark}
2589
2590 \begin{figure}
2591 \centering
2592 \includegraphics[width=0.72\textwidth]{06_higher_order_packets.pdf}
2593 \caption{Truncated power packets encode jumps in a function and in its successive derivatives.}
2594 \label{fig:higher_order_packets}
2595 \end{figure}
2596
2597 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.
2598
2599 \begin{theorem}[Exact retained packet identity]\label{thm:retained_packet_identity}
2600 Let \(f=s+j\) in \(X\). Define
2601 \[
2602 \widetilde f=Qs+j.
2603 \]
2604 Then
2605 \[
2606 \boxed{
2607 f-\widetilde f=(I-Q)s.
2608 }
2609 \]
2610 Consequently,
2611 \[
2612 \|f-\widetilde f\|=\|s-Qs\|.
2613 \]
2614 \end{theorem}
2615
2616 \begin{proof}
2617 Subtract:
2618 \[
2619 f-\widetilde f
2620 =s+j-Qs-j
2621 =(I-Q)s.
2622 \]
2623 \end{proof}
2624
2625 \begin{remark}\label{rem:retained_packet_boundary}
2626 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. Once a meaningful packet has been identified, its approximation error can be removed from the smooth approximation problem exactly.
2627 \end{remark}
2628
2629 \begin{corollary}[Sobolev rate after packet separation]\label{cor:sobolev_rate}
2630 Suppose \(Q_m\) satisfies
2631 \[
2632 \|s-Q_ms\|_{L^p}
2633 \leq
2634 C2^{-mr}\|s\|_{W^{r,p}}
2635 \]
2636 for every \(s\in W^{r,p}(0,1)\). If \(f=s+j\) and \(j\) is retained, then
2637 \[
2638 \|f-(Q_ms+j)\|_{L^p}
2639 \leq
2640 C2^{-mr}\|s\|_{W^{r,p}}.
2641 \]
2642 The same conclusion applies to the higher order decomposition in \cref{thm:higher_order_packet_decomposition}.
2643 \end{corollary}
2644
2645 \begin{proof}
2646 Use \cref{thm:retained_packet_identity} and the assumed approximation estimate.
2647 \end{proof}
2648
2649 \begin{theorem}[Reconstruction with an estimated packet]\label{thm:estimated_packet_reconstruction}
2650 Let \(Q_m:X\to X\) be linear, let \(f=s+j\), let \(\widehat j\) be an estimated packet, and define
2651 \[
2652 \widehat f_m
2653 =
2654 Q_m(f-\widehat j)+\widehat j.
2655 \]
2656 Then
2657 \[
2658 \boxed{
2659 f-\widehat f_m
2660 =
2661 (I-Q_m)s+(I-Q_m)(j-\widehat j).
2662 }
2663 \]
2664 If \(Q_m\) is bounded, then
2665 \[
2666 \|f-\widehat f_m\|
2667 \leq
2668 \|s-Q_ms\|
2669 +
2670 (1+\|Q_m\|)\|j-\widehat j\|.
2671 \]
2672 \end{theorem}
2673
2674 \begin{proof}
2675 Since \(f-\widehat j=s+(j-\widehat j)\),
2676 \[
2677 \widehat f_m
2678 =
2679 Q_ms+Q_m(j-\widehat j)+\widehat j.
2680 \]
2681 Subtract this expression from \(s+j\). The norm bound follows from
2682 \[
2683 \|(I-Q_m)u\|
2684 \leq
2685 (1+\|Q_m\|)\|u\|.
2686 \]
2687 \end{proof}
2688
2689 \begin{figure}
2690 \centering
2691 \includegraphics[width=0.84\textwidth]{07_known_packet_reconstruction.pdf}
2692 \caption{A global Fourier approximation must represent the jumps. A retained packet approximation applies the Fourier truncation only to the smooth component.}
2693 \label{fig:known_packet_reconstruction}
2694 \end{figure}
2695
2696 \section{Exact error formulas for estimated packets}\label{extension:section:08}\label{sec:packet_error}
2697
2698 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.
2699
2700 \begin{theorem}[Exact scalar region formula]\label{thm:scalar_region_formula}
2701 Let \(E,F\) be measurable subsets of a measure space, let \(a,b\in\CC\), and let \(1\leq p<\infty\). Then
2702 \[
2703 \begin{aligned}
2704 \|a\one_E-b\one_F\|_p^p
2705 ={}&
2706 |a-b|^p\mu(E\cap F)
2707 +|a|^p\mu(E\setminus F)\\
2708 &+|b|^p\mu(F\setminus E).
2709 \end{aligned}
2710 \]
2711 For equal amplitudes,
2712 \[
2713 \|a\one_E-a\one_F\|_p
2714 =
2715 |a|\mu(E\mathbin\triangle F)^{1/p}.
2716 \]
2717 \end{theorem}
2718
2719 \begin{proof}
2720 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.
2721 \end{proof}
2722
2723 \begin{corollary}[Location error for one step]\label{cor:step_location_error}
2724 Let \(0\leq\tau,\widehat\tau\leq1\) and \(H_\tau=\one_{[\tau,1]}\). For \(1\leq p<\infty\),
2725 \[
2726 \|aH_\tau-aH_{\widehat\tau}\|_p
2727 =
2728 |a||\tau-\widehat\tau|^{1/p}.
2729 \]
2730 In the supremum norm, the error equals \(|a|\) whenever \(\tau\neq\widehat\tau\).
2731 \end{corollary}
2732
2733 \begin{proof}
2734 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.
2735 \end{proof}
2736
2737 \begin{theorem}[Location stability for higher order packets]\label{thm:higher_order_location}
2738 Let \(m\geq1\), \(1\leq p<\infty\), and \(0\leq\tau,\widehat\tau\leq1\). Then
2739 \[
2740 \|\phi_{m,\tau}-\phi_{m,\widehat\tau}\|_{L^p(0,1)}
2741 \leq
2742 \frac{|\tau-\widehat\tau|}
2743 {(m-1)!((m-1)p+1)^{1/p}}.
2744 \]
2745 For \(p=\infty\),
2746 \[
2747 \|\phi_{m,\tau}-\phi_{m,\widehat\tau}\|_\infty
2748 \leq
2749 \frac{|\tau-\widehat\tau|}{(m-1)!}.
2750 \]
2751 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}.
2752 \end{theorem}
2753
2754 \begin{proof}
2755 Assume \(\tau<\widehat\tau\). Differentiation with respect to the location parameter gives
2756 \[
2757 \frac{\partial}{\partial u}\phi_{m,u}(t)
2758 =-\phi_{m-1,u}(t)
2759 \]
2760 for almost every \(t\). Hence
2761 \[
2762 \phi_{m,\tau}-\phi_{m,\widehat\tau}
2763 =
2764 \int_\tau^{\widehat\tau}\phi_{m-1,u}\dd u.
2765 \]
2766 Minkowski's inequality gives
2767 \[
2768 \|\phi_{m,\tau}-\phi_{m,\widehat\tau}\|_p
2769 \leq
2770 |\tau-\widehat\tau|
2771 \sup_u\|\phi_{m-1,u}\|_p.
2772 \]
2773 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.
2774 \end{proof}
2775
2776 \subsection{Several regions}
2777
2778 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.
2779
2780 \begin{theorem}[Overlap matrix formula]\label{thm:overlap_matrix}
2781 Define
2782 \[
2783 M_{ij}=\mu(E_i\cap F_j).
2784 \]
2785 For \(1\leq p<\infty\),
2786 \[
2787 \left\|
2788 \sum_{i=0}^{I}a_i\one_{E_i}
2789 -
2790 \sum_{j=0}^{J}b_j\one_{F_j}
2791 \right\|_p^p
2792 =
2793 \sum_{i=0}^{I}\sum_{j=0}^{J}|a_i-b_j|^pM_{ij}.
2794 \]
2795 The row sums of \(M\) are \(\mu(E_i)\), and the column sums are \(\mu(F_j)\).
2796 \end{theorem}
2797
2798 \begin{proof}
2799 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.
2800 \end{proof}
2801
2802 \begin{corollary}[Background and packet error]\label{cor:background_packet_error}
2803 Suppose the true and estimated packets use the same amplitude list \((a_i)_{i=0}^{I}\), but possibly different regions. Then
2804 \[
2805 \|j-\widehat j\|_p^p
2806 =
2807 \sum_{i,k}|a_i-a_k|^p\mu(E_i\cap F_k).
2808 \]
2809 Only off-diagonal overlaps contribute.
2810 \end{corollary}
2811
2812 \begin{proof}
2813 Apply \cref{thm:overlap_matrix} with \(b_k=a_k\). The diagonal terms vanish.
2814 \end{proof}
2815
2816 \begin{figure}
2817 \centering
2818 \includegraphics[width=0.60\textwidth]{08_overlap_matrix.pdf}
2819 \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.}
2820 \label{fig:overlap_matrix}
2821 \end{figure}
2822
2823 \subsection{Boundary tube control}
2824
2825 For a closed set \(\Gamma\), write
2826 \[
2827 N_\delta(\Gamma)
2828 =
2829 \{x:\dist(x,\Gamma)<\delta\}.
2830 \]
2831
2832 \begin{definition}[Linear tube bound]\label{def:linear_tube_bound}
2833 A boundary \(\Gamma\) has a linear tube bound if there are constants \(C_\Gamma\) and \(\delta_0>0\) such that
2834 \[
2835 \mu(N_\delta(\Gamma))
2836 \leq
2837 C_\Gamma\delta
2838 \]
2839 for \(0<\delta<\delta_0\).
2840 \end{definition}
2841
2842 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.
2843
2844 \begin{theorem}[Geometric packet bound]\label{thm:geometric_packet_bound}
2845 Let \(E\) be a region with boundary \(\Gamma\), and suppose
2846 \[
2847 E\mathbin\triangle F
2848 \subseteq
2849 N_\delta(\Gamma).
2850 \]
2851 If \(\Gamma\) has a linear tube bound and \(0<\delta<\delta_0\) lies in its stated range, then for \(1\leq p<\infty\),
2852 \[
2853 \|a\one_E-a\one_F\|_p
2854 \leq
2855 |a|(C_\Gamma\delta)^{1/p}.
2856 \]
2857 If amplitudes \(a\) and \(b\) differ, then
2858 \[
2859 \|a\one_E-b\one_F\|_p
2860 \leq
2861 |a-b|\mu(E\cap F)^{1/p}
2862 +
2863 (|a|+|b|)(C_\Gamma\delta)^{1/p}.
2864 \]
2865 \end{theorem}
2866
2867 \begin{proof}
2868 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.
2869 \end{proof}
2870
2871 \begin{example}[A perturbed disk]\label{ex:perturbed_disk}
2872 Let \(E\) and \(F\) be concentric disks of radii \(r\) and \(r+\delta\) in \(\RR^2\), with \(\delta>0\). Then
2873 \[
2874 \mu(E\mathbin\triangle F)
2875 =
2876 \pi\bigl((r+\delta)^2-r^2\bigr)
2877 =
2878 2\pi r\delta+\pi\delta^2.
2879 \]
2880 For equal amplitude \(a\), the exact \(L^2\) packet error is
2881 \[
2882 |a|\sqrt{2\pi r\delta+\pi\delta^2}.
2883 \]
2884 This gives the expected square root dependence on boundary displacement.
2885 \end{example}
2886
2887 \section{Stable recovery of separated jumps}\label{extension:section:09}\label{sec:jump_recovery}
2888
2889 \begin{keyidea}[title={The detector needs a visible interval of thresholds}]
2890 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.
2891 \end{keyidea}
2892 \begin{figure}[htbp]\centering
2893 \includegraphics[width=.93\linewidth]{monograph_figures/jump_contrast_gap.pdf}
2894 \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}
2895 \end{figure}
2896
2897
2898 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.
2899
2900 Let
2901 \[
2902 t_k=kh,
2903 \qquad
2904 k=0,\ldots,n,
2905 \qquad
2906 h=\frac1n.
2907 \]
2908 Assume
2909 \[
2910 f(t)
2911 =
2912 s(t)+\sum_{q=1}^{Q}a_qH_{\tau_q}(t),
2913 \qquad
2914 H_\tau(t)=\one_{[\tau,1]}(t),
2915 \]
2916 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
2917 \[
2918 y_k=f(t_k)+\eta_k.
2919 \]
2920 Define the adjacent contrast
2921 \[
2922 D_k=y_{k+1}-y_k.
2923 \]
2924
2925 \begin{lemma}[Contrast gap]\label{lem:contrast_gap}
2926 Assume \(|\eta_k|\leq\varepsilon\) and set
2927 \[
2928 b_h=Mh+2\varepsilon.
2929 \]
2930 If the cell \((t_k,t_{k+1})\) contains no jump, then
2931 \[
2932 |D_k|\leq b_h.
2933 \]
2934 If it contains one jump of amplitude \(a_q\), then
2935 \[
2936 |D_k-a_q|\leq b_h
2937 \]
2938 and therefore
2939 \[
2940 |D_k|\geq |a_q|-b_h.
2941 \]
2942 \end{lemma}
2943
2944 \begin{proof}
2945 In a cell without a jump,
2946 \[
2947 D_k=s(t_{k+1})-s(t_k)+\eta_{k+1}-\eta_k.
2948 \]
2949 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.
2950 \end{proof}
2951
2952 \begin{theorem}[Deterministic jump cell recovery]\label{thm:deterministic_jump_recovery}
2953 Let
2954 \[
2955 a_{\min}=\min_q|a_q|
2956 \]
2957 and assume
2958 \[
2959 a_{\min}>2b_h.
2960 \]
2961 Choose a threshold \(\theta\) satisfying
2962 \[
2963 b_h<\theta<a_{\min}-b_h.
2964 \]
2965 Then
2966 \[
2967 \widehat\Kcal
2968 =
2969 \{k:|D_k|>\theta\}
2970 \]
2971 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,
2972 \[
2973 |\widehat\tau_q-\tau_q|
2974 \leq
2975 \frac h2
2976 \]
2977 and
2978 \[
2979 |\widehat a_q-a_q|
2980 \leq
2981 b_h.
2982 \]
2983 \end{theorem}
2984
2985 \begin{proof}
2986 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}.
2987 \end{proof}
2988
2989 \begin{corollary}[Jump count and sign recovery]\label{cor:jump_count_sign}
2990 Under the hypotheses of \cref{thm:deterministic_jump_recovery}, the number of detected cells equals \(Q\), and
2991 \[
2992 \sgn(\widehat a_q)=\sgn(a_q)
2993 \]
2994 for every real valued jump.
2995 \end{corollary}
2996
2997 \begin{proof}
2998 Exact cell recovery gives the jump count. Since
2999 \[
3000 |\widehat a_q-a_q|\leq b_h<|a_q|,
3001 \]
3002 the amplitude estimate cannot cross zero.
3003 \end{proof}
3004
3005 \begin{corollary}[Packet reconstruction after detection]\label{cor:reconstruction_after_detection}
3006 Let
3007 \[
3008 j=\sum_{q=1}^{Q}a_qH_{\tau_q},
3009 \qquad
3010 \widehat j=\sum_{q=1}^{Q}\widehat a_qH_{\widehat\tau_q}.
3011 \]
3012 Under the hypotheses of \cref{thm:deterministic_jump_recovery}, for \(1\leq p<\infty\),
3013 \[
3014 \|j-\widehat j\|_p
3015 \leq
3016 Qb_h
3017 +
3018 \left(
3019 \frac h2\sum_{q=1}^{Q}|a_q|^p
3020 \right)^{1/p}.
3021 \]
3022 If \(Q_m\) is bounded and linear and
3023 \[
3024 \widehat f_m=Q_m(f-\widehat j)+\widehat j,
3025 \]
3026 then
3027 \[
3028 \begin{aligned}
3029 \|f-\widehat f_m\|_p
3030 \leq{}&
3031 \|s-Q_ms\|_p\\
3032 &+(1+\|Q_m\|)
3033 \left[
3034 Qb_h+
3035 \left(
3036 \frac h2\sum_q|a_q|^p
3037 \right)^{1/p}
3038 \right].
3039 \end{aligned}
3040 \]
3041 \end{corollary}
3042
3043 \begin{proof}
3044 Write
3045 \[
3046 j-\widehat j
3047 =
3048 \sum_q(a_q-\widehat a_q)H_{\widehat\tau_q}
3049 +
3050 \sum_q a_q(H_{\tau_q}-H_{\widehat\tau_q}).
3051 \]
3052 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}.
3053 \end{proof}
3054
3055 \begin{remark}[Why finite \(L^p\) matters]\label{rem:finite_p_recovery}
3056 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.
3057 \end{remark}
3058
3059 \begin{corollary}[Deterministic consistency]\label{cor:deterministic_consistency}
3060 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 \(L^p\).
3061 \end{corollary}
3062
3063 \begin{proof}
3064 The gap condition holds for sufficiently small \(h\). Then \(b_h\to0\), and the bound in \cref{cor:reconstruction_after_detection} tends to zero.
3065 \end{proof}
3066
3067 \begin{figure}
3068 \centering
3069 \includegraphics[width=0.83\textwidth]{09_jump_detection.pdf}
3070 \caption{Separated jumps create isolated adjacent sample contrasts above the smooth and noise background.}
3071 \label{fig:jump_detection}
3072 \end{figure}
3073
3074 \begin{example}[A numerical threshold certificate]\label{ex:jump_threshold_certificate}
3075 Take \(h=1/256\), \(M=0.5\), and \(\varepsilon=0.002\). Then
3076 \[
3077 b_h
3078 =\frac{0.5}{256}+0.004
3079 <0.00596.
3080 \]
3081 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
3082 \[
3083 \frac1{512}
3084 \]
3085 and every amplitude error is less than \(0.00596\).
3086 \end{example}
3087
3088 \subsection{Sub-Gaussian noise}
3089
3090 Assume the noise variables are independent, centered, and \(\sigma\) sub-Gaussian:
3091 \[
3092 \mathbb E\e^{\lambda\eta_k}
3093 \leq
3094 \e^{\sigma^2\lambda^2/2}.
3095 \]
3096
3097 \begin{theorem}[Probability recovery bound]\label{thm:probability_recovery}
3098 Fix \(0<\delta<1\) and define
3099 \[
3100 t_{n,\delta}
3101 =
3102 2\sigma
3103 \sqrt{\log\left(\frac{2n}{\delta}\right)}
3104 \]
3105 and
3106 \[
3107 b_{h,\delta}=Mh+t_{n,\delta}.
3108 \]
3109 With probability at least \(1-\delta\), every contrast satisfies the conclusions of \cref{lem:contrast_gap} with \(b_h\) replaced by \(b_{h,\delta}\).
3110
3111 If
3112 \[
3113 a_{\min}>2b_{h,\delta},
3114 \]
3115 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 \(b_{h,\delta}\).
3116 \end{theorem}
3117
3118 \begin{proof}
3119 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
3120 \[
3121 \mathbb P\bigl(|\eta_{k+1}-\eta_k|>t\bigr)
3122 \leq
3123 2\exp\left(-\frac{t^2}{4\sigma^2}\right).
3124 \]
3125 A union bound over the \(n\) cells gives
3126 \[
3127 \mathbb P\left(
3128 \max_k|\eta_{k+1}-\eta_k|>t_{n,\delta}
3129 \right)
3130 \leq
3131 \delta.
3132 \]
3133 On the complementary event, repeat the deterministic proof.
3134 \end{proof}
3135
3136 \begin{corollary}[Repeated measurements]\label{cor:repeated_measurements}
3137 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
3138 \[
3139 a_{\min}
3140 >
3141 2Mh+
3142 \frac{4\sigma}{\sqrt R}
3143 \sqrt{\log\left(\frac{2n}{\delta}\right)}.
3144 \]
3145 \end{corollary}
3146
3147 \begin{proof}
3148 The average of \(R\) independent \(\sigma\) sub-Gaussian variables is \(\sigma/\sqrt R\) sub-Gaussian. Substitute this parameter in \cref{thm:probability_recovery}.
3149 \end{proof}
3150
3151 \begin{remark}\label{rem:noise_scaling}
3152 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.
3153 \end{remark}
3154
3155 \begin{corollary}[Probability consistency]\label{cor:probability_consistency}
3156 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.
3157 \end{corollary}
3158
3159 \begin{proof}
3160 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.
3161 \end{proof}
3162
3163 \subsection{Signals with values in a normed space}
3164
3165 The recovery proof does not depend on an ordering of the signal values.
3166
3167 \begin{proposition}[Vector jump recovery]\label{prop:vector_jump_recovery}
3168 Let the values lie in a normed space \(V\). Assume
3169 \[
3170 \|s(t)-s(u)\|_V\leq M|t-u|,
3171 \qquad
3172 \|\eta_k\|_V\leq\varepsilon,
3173 \]
3174 and let the jump vectors satisfy \(\|a_q\|_V\geq a_{\min}\). Thresholding \(\|D_k\|_V\) gives the same recovery theorem under the condition
3175 \[
3176 a_{\min}>2(Mh+2\varepsilon).
3177 \]
3178 \end{proposition}
3179
3180 \begin{proof}
3181 Only the triangle inequality was used in \cref{lem:contrast_gap,thm:deterministic_jump_recovery}.
3182 \end{proof}
3183
3184 \section{Haar localization and annihilating differences}\label{extension:section:10}\label{sec:localization}
3185
3186 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.
3187
3188 \subsection{Haar coefficient separation}
3189
3190 Let
3191 \[
3192 I_{j,k}
3193 =
3194 [k2^{-j},(k+1)2^{-j})
3195 \]
3196 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.
3197
3198 \begin{theorem}[Smooth Haar coefficient bound]\label{thm:smooth_haar_bound}
3199 If \(s\) is Lipschitz with constant \(M\), then
3200 \[
3201 |\langle s,\psi_{j,k}\rangle|
3202 \leq
3203 \frac M4\,2^{-3j/2}.
3204 \]
3205 \end{theorem}
3206
3207 \begin{proof}
3208 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
3209 \[
3210 \langle s,\psi_{j,k}\rangle
3211 =
3212 L^{-1/2}
3213 \int_a^{a+L/2}
3214 \overline{s(t)-s(t+L/2)}\,\dd t.
3215 \]
3216 The integrand has magnitude at most \(ML/2\), and the integration interval has length \(L/2\). The result is \(ML^{3/2}/4\).
3217 \end{proof}
3218
3219 \begin{theorem}[Exact Haar coefficient of a step]\label{thm:haar_step_coefficient}
3220 Let \(\tau\in I_{j,k}\), let \(L=2^{-j}\), and write
3221 \[
3222 u=\frac{\tau-kL}{L}\in[0,1].
3223 \]
3224 Then
3225 \[
3226 \left|
3227 \left\langle
3228 aH_\tau,\psi_{j,k}
3229 \right\rangle
3230 \right|
3231 =
3232 |a|L^{1/2}\min\{u,1-u\}.
3233 \]
3234 Every Haar coefficient whose support does not cross \(\tau\) is zero.
3235 \end{theorem}
3236
3237 \begin{proof}
3238 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 \(|a|(1-u)L^{1/2}\).
3239 \end{proof}
3240
3241 \begin{corollary}[Scale separation]\label{cor:haar_scale_separation}
3242 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
3243 \[
3244 |\langle f,\psi_{j,k}\rangle|
3245 \geq
3246 \left(
3247 |a|\rho
3248 -\varepsilon
3249 -\frac M4L
3250 \right)L^{1/2}.
3251 \]
3252 A wavelet support without a jump satisfies
3253 \[
3254 |\langle f,\psi_{j,k}\rangle|
3255 \leq
3256 \left(
3257 \varepsilon+\frac M4L
3258 \right)L^{1/2}.
3259 \]
3260 Thus a threshold gap exists whenever
3261 \[
3262 |a|\rho
3263 >
3264 2\varepsilon+\frac M2L.
3265 \]
3266 \end{corollary}
3267
3268 \begin{proof}
3269 Combine \cref{thm:smooth_haar_bound,thm:haar_step_coefficient}. The noise coefficient is at most
3270 \[
3271 \|\eta\|_\infty\|\psi_{j,k}\|_1
3272 =
3273 \varepsilon L^{1/2}.
3274 \]
3275 The final inequality makes the lower bound on a jump support larger than the upper bound on a smooth support.
3276 \end{proof}
3277
3278 \begin{figure}
3279 \centering
3280 \includegraphics[width=0.68\textwidth]{10_haar_scale_separation.pdf}
3281 \caption{The smooth contribution decays like \(L^{3/2}\), while a separated jump contribution decays like \(L^{1/2}\).}
3282 \label{fig:haar_scale_separation}
3283 \end{figure}
3284
3285 \subsection{Two shifted grids}
3286
3287 A jump on a dyadic boundary can be invisible to a wavelet at that scale. One additional grid removes this exact alignment.
3288
3289 \begin{theorem}[Half cell shift guarantee]\label{thm:half_shift_guarantee}
3290 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.
3291 \end{theorem}
3292
3293 \begin{proof}
3294 Reduce \(\tau\) modulo \(L\) and write its representative as \(x\in[0,L)\). If
3295 \[
3296 L/4\leq x\leq 3L/4,
3297 \]
3298 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.
3299 \end{proof}
3300
3301 \begin{corollary}[Alignment free two grid Haar detection]\label{cor:two_grid_haar_detection}
3302 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
3303 \[
3304 \frac{|a|}{4}
3305 >
3306 2\varepsilon+\frac M2L.
3307 \]
3308 \end{corollary}
3309
3310 \begin{proof}
3311 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.
3312 \end{proof}
3313
3314 \begin{figure}
3315 \centering
3316 \includegraphics[width=0.78\textwidth]{11_shifted_grids.pdf}
3317 \caption{The boundaries of the two grids interlace. Every jump is well inside a cell in at least one grid.}
3318 \label{fig:shifted_grids}
3319 \end{figure}
3320
3321 \subsection{Higher differences}
3322
3323 Let
3324 \[
3325 (\Delta x)(n)=x(n+1)-x(n).
3326 \]
3327 Then
3328 \[
3329 \Delta^m x(n)
3330 =
3331 \sum_{r=0}^{m}
3332 (-1)^{m-r}\binom mr x(n+r).
3333 \]
3334
3335 \begin{theorem}[Polynomial annihilation and boundary support]\label{thm:polynomial_annihilation}
3336 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
3337 \[
3338 \Delta^{d+1}x(n)=0
3339 \]
3340 whenever the complete stencil
3341 \[
3342 \{n,n+1,\ldots,n+d+1\}
3343 \]
3344 lies inside one polynomial piece. Hence \(\Delta^{d+1}x\) is supported in a band of at most \(d+1\) addresses before each breakpoint.
3345 \end{theorem}
3346
3347 \begin{proof}
3348 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.
3349 \end{proof}
3350
3351 \begin{proposition}[Noise amplification by differences]\label{prop:difference_noise}
3352 If \(\|\eta\|_\infty\leq\varepsilon\), then
3353 \[
3354 \|\Delta^m\eta\|_\infty
3355 \leq
3356 2^m\varepsilon.
3357 \]
3358 Therefore no false response above \(2^{d+1}\varepsilon\) can occur on a stencil lying inside one polynomial piece.
3359 \end{proposition}
3360
3361 \begin{proof}
3362 The absolute values of the binomial coefficients in \(\Delta^m\) sum to \(2^m\).
3363 \end{proof}
3364
3365 \begin{corollary}[Robust breakpoint bands]\label{cor:robust_breakpoint_bands}
3366 Assume every true breakpoint produces at least one response of magnitude \(\rho\) in its \((d+1)\)st difference band. If
3367 \[
3368 \rho>2^{d+2}\varepsilon,
3369 \]
3370 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.
3371 \end{corollary}
3372
3373 \begin{proof}
3374 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.
3375 \end{proof}
3376
3377 \section{Packet separation and the Gibbs phenomenon}\label{extension:section:11}\label{sec:gibbs}
3378
3379 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}.
3380
3381 \begin{standardfact}[Classical square-wave Gibbs behavior]\label{sf:gibbs_behavior}
3382 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
3383 \[
3384 Q_n(x)=\frac{2}{\pi}\sum_{k=0}^{n-1}\frac{\sin((2k+1)x)}{2k+1}.
3385 \]
3386 At \(x_n=\pi/(2n)\),
3387 \[
3388 x_n\longrightarrow0,\qquad Q_n(x_n)\longrightarrow\frac{\operatorname{Si}(\pi)}{\pi}
3389 =\frac12+G,\qquad G\approx0.089489872236,
3390 \]
3391 where \(\operatorname{Si}(z)=\int_0^z\sin(t)/t\,dt\) and \(G=\operatorname{Si}(\pi)/\pi-1/2>0\).
3392 Thus a nonzero overshoot persists along points approaching the jump. Scaling by a real amplitude \(a\) gives an overshoot or undershoot of magnitude \(|a|G\).
3393
3394 For the limit, put \(u_k=(k+\tfrac12)/n\). The identity
3395 \[
3396 Q_n(x_n)=\frac{1}{\pi n}\sum_{k=0}^{n-1}\frac{\sin(\pi u_k)}{u_k}
3397 \longrightarrow\frac1\pi\int_0^1\frac{\sin(\pi u)}u\,du
3398 =\frac{\operatorname{Si}(\pi)}{\pi}
3399 \]
3400 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}.
3401 \end{standardfact}
3402
3403 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.
3404
3405 \begin{theorem}[Uniform convergence with exact jump packets]\label{thm:uniform_packet_convergence}
3406 Let \(f=s+j\) on a compact domain, where \(j\) is any bounded singular packet. Suppose \(Q_ms\to s\) uniformly. Then
3407 \[
3408 \widetilde f_m=Q_ms+j
3409 \]
3410 converges uniformly to \(f\), even when \(f\) is discontinuous:
3411 \[
3412 \|f-\widetilde f_m\|_\infty
3413 =
3414 \|s-Q_ms\|_\infty
3415 \longrightarrow0.
3416 \]
3417 There is no error proportional to the jump size because the jump is shared exactly by \(f\) and \(\widetilde f_m\).
3418 \end{theorem}
3419
3420 \begin{proof}
3421 Apply \cref{thm:retained_packet_identity} in the supremum norm.
3422 \end{proof}
3423
3424 \Needspace{20\baselineskip}
3425 \begin{theorem}[Estimated packet Gibbs bound]\label{thm:estimated_packet_gibbs}
3426 Let \(Q_m\) be linear and uniformly bounded on \(L^p\), with
3427 \[
3428 \sup_m\|Q_m\|\leq C_Q.
3429 \]
3430 For \(f=s+j\) and an estimated packet \(\widehat j_m\), define
3431 \[
3432 \widehat f_m
3433 =Q_m(f-\widehat j_m)+\widehat j_m.
3434 \]
3435 Then
3436 \[
3437 \|f-\widehat f_m\|_p
3438 \leq
3439 \|s-Q_ms\|_p
3440 +(1+C_Q)\|j-\widehat j_m\|_p.
3441 \]
3442 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.
3443 \end{theorem}
3444
3445 \begin{proof}
3446 This is \cref{thm:estimated_packet_reconstruction} with the uniform operator bound.
3447 \end{proof}
3448
3449 \begin{corollary}[Boundary band error]\label{cor:boundary_band_error}
3450 Let \(\Gamma\) be the singular boundary and assume
3451 \[
3452 \mu(N_\delta(\Gamma))\leq C_\Gamma\delta
3453 \]
3454 for small \(\delta\). For finite \(p\),
3455 \[
3456 \|f-\widetilde f_m\|_{L^p(N_\delta(\Gamma))}
3457 \leq
3458 (C_\Gamma\delta)^{1/p}
3459 \|s-Q_ms\|_\infty.
3460 \]
3461 \end{corollary}
3462
3463 \begin{proof}
3464 The error equals \(s-Q_ms\). Bound it by its supremum and multiply by the measure of the boundary tube.
3465 \end{proof}
3466
3467 \begin{example}[Square wave Fourier tail]\label{ex:square_wave_tail}
3468 Let
3469 \[
3470 j(t)=\one_{[-a,a]}(t)
3471 \]
3472 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,
3473 \[
3474 Q_ms+j
3475 \]
3476 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.
3477 \end{example}
3478
3479 \begin{figure}
3480 \centering
3481 \includegraphics[width=0.82\textwidth]{12_gibbs_comparison.pdf}
3482 \caption{Global Fourier truncation represents the jump and exhibits an oscillatory boundary layer. Packet separation retains the jump and approximates only the smooth component.}
3483 \label{fig:gibbs_comparison}
3484 \end{figure}
3485
3486 \FloatBarrier
3487 \section{Nonlinear interactions in residue coordinates}\label{extension:section:12}\label{sec:nonlinear}
3488
3489 Linear transfer matrices do not directly describe nonlinear products. Residue coordinates still obey exact conservation laws because integer addresses add modulo the packet modulus.
3490
3491 \begin{theorem}[Residue law for Cauchy products]\label{thm:cauchy_residue_law}
3492 Let
3493 \[
3494 F(z)=\sum_{n\geq0}f_nz^n,
3495 \qquad
3496 G(z)=\sum_{n\geq0}g_nz^n
3497 \]
3498 be formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo \(N\):
3499 \[
3500 F=\sum_{r=0}^{N-1}F_r,
3501 \qquad
3502 G=\sum_{s=0}^{N-1}G_s.
3503 \]
3504 If \(H=FG\), then its residue packet \(H_t\) is
3505 \[
3506 \boxed{
3507 H_t
3508 =
3509 \sum_{r+s\equiv t\!\!\pmod N}F_rG_s.
3510 }
3511 \]
3512 \end{theorem}
3513
3514 \begin{proof}
3515 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.
3516 \end{proof}
3517
3518 \begin{theorem}[Shifted pointwise monomial law]\label{thm:shifted_monomial_law}
3519 Let
3520 \[
3521 y(n)=\prod_{j=1}^{m}x(n-a_j)
3522 \]
3523 and write
3524 \[
3525 a_j=Nc_j+b_j,
3526 \qquad
3527 0\leq b_j<N.
3528 \]
3529 For output residue \(r\), define
3530 \[
3531 r_j\equiv r-b_j\pmod N,
3532 \qquad
3533 0\leq r_j<N,
3534 \]
3535 and let
3536 \[
3537 \kappa_{r,b_j}
3538 =
3539 \begin{cases}
3540 1,&r<b_j,\\
3541 0,&r\geq b_j.
3542 \end{cases}
3543 \]
3544 Then the output channel is
3545 \[
3546 \boxed{
3547 y_r(q)
3548 =
3549 \prod_{j=1}^{m}
3550 x_{r_j}
3551 \bigl(q-c_j-\kappa_{r,b_j}\bigr).
3552 }
3553 \]
3554 \end{theorem}
3555
3556 \begin{proof}
3557 For \(n=Nq+r\),
3558 \[
3559 n-a_j
3560 =N(q-c_j)+r-b_j.
3561 \]
3562 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.
3563 \end{proof}
3564
3565 \begin{corollary}[Polynomial shift filters]\label{cor:polynomial_shift_filters}
3566 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.
3567 \end{corollary}
3568
3569 \begin{proof}
3570 Apply \cref{thm:shifted_monomial_law} to each monomial and add the results.
3571 \end{proof}
3572
3573 \begin{example}[Quadratic nearest neighbor interaction]\label{ex:quadratic_neighbor}
3574 For
3575 \[
3576 y(n)=x(n)x(n-1)
3577 \]
3578 and modulus \(N\),
3579 \[
3580 y_r(q)
3581 =
3582 \begin{cases}
3583 x_r(q)x_{r-1}(q),&r\geq1,\\
3584 x_0(q)x_{N-1}(q-1),&r=0.
3585 \end{cases}
3586 \]
3587 The carry at residue zero is the nonlinear analogue of a transfer crossing a coarse cell boundary.
3588 \end{example}
3589
3590 \section{Finite numerical illustrations}\label{extension:section:13}\label{sec:numerical}
3591
3592 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.
3593
3594 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.
3595
3596 \begin{figure}
3597 \centering
3598 \includegraphics[width=0.70\textwidth]{13_sparse_comparison.pdf}
3599 \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.}
3600 \label{fig:sparse_comparison}
3601 \end{figure}
3602
3603 \begin{table}
3604 \centering
3605 \caption{Finite sparse approximation record for one test signal.}
3606 \label{tab:sparse_record}
3607 \begin{tabular}{rrr}
3608 \toprule
3609 retained & Haar packet RMSE & Fourier RMSE\\
3610 \midrule
3611 16 & \(1.7999\times10^{-1}\) & \(6.6522\times10^{-2}\)\\
3612 32 & \(3.8414\times10^{-2}\) & \(4.9153\times10^{-2}\)\\
3613 64 & \(1.6267\times10^{-2}\) & \(3.8056\times10^{-2}\)\\
3614 128 & \(1.7387\times10^{-3}\) & \(2.5631\times10^{-2}\)\\
3615 256 & \(1.5870\times10^{-4}\) & \(1.7424\times10^{-2}\)\\
3616 \bottomrule
3617 \end{tabular}
3618 \end{table}
3619
3620 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.
3621
3622 \begin{table}
3623 \centering
3624 \caption{Finite edge reconstruction record from the original computational study.}
3625 \label{tab:edge_record}
3626 \small
3627 \begin{tabular}{lccc}
3628 \toprule
3629 case & global & exact packet & estimated packet\\
3630 \midrule
3631 one-dimensional step & \(5.4023\times10^{-2}\) & \(1.3692\times10^{-4}\) & \(2.4029\times10^{-2}\)\\
3632 two-dimensional disk & \(3.7643\times10^{-2}\) & \(6.4752\times10^{-5}\) & \(3.4043\times10^{-3}\)\\
3633 multiple regions & \(3.3740\times10^{-2}\) & \(5.2118\times10^{-7}\) & not evaluated\\
3634 \bottomrule
3635 \end{tabular}
3636 \end{table}
3637
3638 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.
3639
3640 \section{Cyclic Cayley coordinates}\label{extension:section:14}\label{sec:cayley}
3641
3642 \begin{interpretation}[title={Keep the rotation and its chart separate}]
3643 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.
3644 \end{interpretation}
3645
3646 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}.
3647
3648 Let
3649 \[
3650 \mathbb H
3651 =
3652 \{z\in\CC:\Re z>0\}
3653 \]
3654 and define
3655 \[
3656 C(z)=\frac{z-1}{z+1}.
3657 \]
3658 It maps \(\mathbb H\) conformally onto the unit disk \(\mathbb D\), with inverse
3659 \[
3660 C^{-1}(w)=\frac{1+w}{1-w}.
3661 \]
3662 For \(\theta\in\RR\), let
3663 \[
3664 R_\theta(w)=\e^{\ii\theta}w.
3665 \]
3666
3667 \begin{theorem}[Cyclic M\"obius action]\label{thm:cyclic_mobius}
3668 The pullback
3669 \[
3670 \Phi_\theta
3671 =C^{-1}R_\theta C
3672 \]
3673 is the fractional linear map
3674 \[
3675 \boxed{
3676 \Phi_\theta(z)
3677 =
3678 \frac{(1+\e^{\ii\theta})z+(1-\e^{\ii\theta})}
3679 {(1-\e^{\ii\theta})z+(1+\e^{\ii\theta})}.
3680 }
3681 \]
3682 It preserves \(\mathbb H\) and satisfies
3683 \[
3684 \Phi_\theta\Phi_\varphi
3685 =
3686 \Phi_{\theta+\varphi}.
3687 \]
3688 For an integer \(N\geq1\) and \(\theta=2\pi/N\), the map has order \(N\).
3689 \end{theorem}
3690
3691 \begin{proof}
3692 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 \(N\).
3693 \end{proof}
3694
3695 \begin{corollary}[Impedance and admittance]\label{cor:impedance_admittance}
3696 For \(N=2\), \(\theta=\pi\) and
3697 \[
3698 \Phi_\pi(z)=\frac1z.
3699 \]
3700 Thus the two phase action is normalized impedance and admittance inversion.
3701 \end{corollary}
3702
3703 \begin{proof}
3704 Insert \(\e^{\ii\pi}=-1\) in the formula of \cref{thm:cyclic_mobius}.
3705 \end{proof}
3706
3707 \begin{proposition}[Radial and phase coordinates]\label{prop:radial_phase}
3708 Write
3709 \[
3710 C(z)=\rho\e^{\ii\varphi},
3711 \qquad
3712 0\leq\rho<1.
3713 \]
3714 Then
3715 \[
3716 C(\Phi_\theta(z))
3717 =
3718 \rho\e^{\ii(\varphi+\theta)}.
3719 \]
3720 The radial mismatch \(\rho\) is invariant and the phase coordinate advances by \(\theta\).
3721 \end{proposition}
3722
3723 \begin{proof}
3724 This is the defining conjugacy \(C\Phi_\theta=R_\theta C\).
3725 \end{proof}
3726
3727 \begin{proposition}[Coprime phase factorization]\label{prop:phase_factorization}
3728 Let \(M\) and \(N\) be coprime. The cyclic phase labels modulo \(MN\) are equivalent to pairs of labels modulo \(M\) and modulo \(N\):
3729 \[
3730 \ZZ/(MN)\ZZ
3731 \cong
3732 \ZZ/M\ZZ\times\ZZ/N\ZZ.
3733 \]
3734 Under this identification, the rotation by \(2\pi/(MN)\) is determined by its two projected cyclic actions.
3735 \end{proposition}
3736
3737 \begin{proof}
3738 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.
3739 \end{proof}
3740
3741 \begin{figure}
3742 \centering
3743 \includegraphics[width=0.61\textwidth]{14_smith_coordinates.pdf}
3744 \caption{The cyclic action is an ordinary phase rotation in the reflection disk and a fractional linear action in the impedance half plane.}
3745 \label{fig:smith_coordinates}
3746 \end{figure}
3747
3748 \section{Limits and further directions}\label{extension:section:15}\label{sec:further_questions}
3749
3750 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.
3751
3752 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}.
3753
3754 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.
3755
3756 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.
3757
3758 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.
3759
3760 \section{Conclusion}\label{extension:section:16}\label{sec:conclusion}
3761
3762 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.
3763
3764 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.
3765
3766 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.
3767
3768 The resulting chain is
3769 \[
3770 \begin{gathered}
3771 \text{lattice coordinates}
3772 \longrightarrow
3773 \text{transfer blocks}
3774 \longrightarrow
3775 \text{stable filter banks}
3776 \longrightarrow
3777 \text{packet trees}
3778 \\
3779 \longrightarrow
3780 \text{singular packet model}
3781 \longrightarrow
3782 \text{local recovery}
3783 \longrightarrow
3784 \text{smooth approximation rate}.
3785 \end{gathered}
3786 \]
3787 Every arrow is exact or quantitatively controlled under the hypotheses stated in the corresponding result.
3788
3789
3790 \clearpage
3791 \part*{Part III: Forty worked visual plates}\addcontentsline{toc}{part}{Part III: Forty worked visual plates}
3792 \begin{readerroadmap}[title={A second route through the mathematics}]
3793 These forty plates develop the finite models used in the separate visual guide. Plates V.1--V.40 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.
3794 \end{readerroadmap}
3795 \clearpage
3796 \begingroup
3797 \setlength{\parskip}{3pt}
3798 \setlength{\abovedisplayskip}{6pt}\setlength{\belowdisplayskip}{6pt}
3799 \setstretch{1.01}
3800 \fontsize{10}{12}\selectfont
3801 \input{atlas-pages.tex}
3802 \endgroup
3803 \clearpage
3804
3805 \bibliographystyle{alpha}
3806 \bibliography{references}
3807
3808
3809 \end{document}
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