Signal Reconstruction

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11 pdfauthor={Komorebi P.},pdftitle={Signal Reconstruction: Polyphase Sampling, Unknown Edges, and Self-Calibration}}
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15 \fancyhead[L]{\small\textsc{Signal Reconstruction}}
16 \fancyhead[R]{\small Komorebi P.}\fancyfoot[C]{\thepage}
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18 \titleformat{\subsection}{\large\bfseries\color{Navy}}{\thesubsection}{0.6em}{}
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21 \theoremstyle{remark}\newtheorem{remark}[theorem]{Remark}
22 
23 % Operation Komorebi: shared publication profile.
24 \usepackage{komorebi-publication}
25 \renewcommand{\KomorebiTitle}{Signal Reconstruction}
26 \renewcommand{\KomorebiSubtitle}{Polyphase sampling, unknown edges, and self-calibration}
27 \renewcommand{\KomorebiSeries}{Research library}
28 \renewcommand{\KomorebiKind}{Traditional paper}
29 \renewcommand{\KomorebiEdition}{Standardized publication edition / 8 October 2026}
30 \renewcommand{\KomorebiShortTitle}{Signal Reconstruction}
31 \renewcommand{\KomorebiTitleSize}{38}
32 \renewcommand{\KomorebiTitleLeading}{43}
33 \renewcommand{\KomorebiProfile}{classical}
34 \renewcommand{\KomorebiOrientation}{portrait}
35 \KomorebiSetup
36 \begin{document}
37 \KomorebiCover
38 \KomorebiOpeningNote{\textbf{Reading set.} Traditional paper, visual guide, Lean guide, and complete
39 source bundle. The bundle includes the \texttt{.md} files, editable manuscript
40 sources, Python implementations, exact figure generators, related Lean
41 sources, and verification records.\par\medskip
42 \textbf{Scope.} Exact finite-rate edge recovery and native synthesis;
43 conditional noisy and smooth-tail recovery; finite geometric dictionaries;
44 and explicitly separated adaptive image approximation.}
45 \hypersetup{pageanchor=false}
46 
47 \hypersetup{pageanchor=true}\pagenumbering{arabic}
48 \begin{abstract}
49 Finite signal reconstruction depends on a declared model and an injective
50 observation. This paper develops a packet-based reconstruction chain from
51 complete projector coordinates and transfer kernels to exact off-grid jump
52 recovery, polyphase alias calibration, and native singularity synthesis.
53 Consecutive differentiated Fourier moments determine an unknown finite jump
54 count through Hankel rank, followed by an annihilating polynomial and
55 Vandermonde recovery. Matched adjacent offsets recover the physical phase
56 lost by decimation. Noisy order, root, and branch conclusions retain explicit
57 conditioning and margin gates, with conditional asymptotic rates.
58 A corrected periodic mean-zero jump estimate controls edge synthesis in
59 $L^1$. Classical Gibbs ringing is derived and localized to the globally
60 truncated edge block. Finite dictionaries and adaptive image experiments are
61 kept distinct from universal continuous-edge recovery. The current numerical
62 replay includes successful controlled checks and reproduced identification
63 failures. Related Lean infrastructure is mapped without claiming a formal
64 certificate for the new recovery chain.
65 \end{abstract}
66 \tableofcontents\newpage
67 \section{The reconstruction problem}
68 
69 A measured signal is an object seen through a declared observation map. A
70 reconstruction theorem therefore has two obligations: identify the object
71 within a stated class and synthesize it in a representation suited to that
72 class. These obligations differ. A Fourier series can determine a jump signal
73 uniquely while its finite partial sums still ring near the jump. Conversely,
74 a native edge representation can reproduce a known discontinuity exactly
75 while supplying no method to discover an unknown edge.
76 
77 We treat both obligations. First we organize signals and operators by complete
78 packet systems. We then derive known-edge cancellation, recovery in finite
79 geometric dictionaries, exact off-grid Prony recovery, and self-calibration of
80 polyphase aliases from adjacent offsets. Quantitative conclusions retain their
81 rank, conditioning, separation, and branch hypotheses. The final synthesis
82 error is split into the smooth approximation error and the recovered edge error.
83 
84 The default periodic domain is $\mathbb T=\mathbb R/(2\pi\mathbb Z)$, with
85 \[
86  \widehat f(n)=\frac1{2\pi}\int_0^{2\pi}f(x)e^{-inx}\,dx.
87 \]
88 Signal identities inferred from Fourier coefficients are identities of
89 $L^1$ classes. For a piecewise-constant edge we select the right-continuous
90 representative. This convention fixes the values at jumps, which Fourier
91 integrals alone cannot determine. When an $L^1$ norm occurs it is the
92 unnormalized integral over a period. On the unit circle we use chord distance
93 $|z-w|$ and circular angular distance $d_{\mathbb T}(\tau,\sigma)$.
94 
95 \subsection{What belongs to this edition}
96 The inherited unknown-edge and self-calibration manuscripts are consolidated
97 here with full proofs and corrected contracts. The Gibbs case studies explain
98 the operator choice. The complex signal/image experiments probe model
99 mismatch. The seed calculus supplies the finite packet language. The infinite
100 packet program supplies a separate adaptive approximation experiment.
101 The raw sources remain unchanged in the bundle. The claim concordance records
102 which source each published result develops and where a printed contract is
103 narrower or corrected.
104 
105 Prony systems, finite-rate sampling, polyphase decomposition, Fourier Gibbs
106 asymptotics, sparse secant injectivity, and shifted exponential coefficients
107 have established histories. In particular the shifted-coefficient phase
108 relation used below occurs in VEXPA \cite{vexpa}. Our presentation makes those
109 ingredients explicit within one packet analysis, recovery, validation, and
110 synthesis chain. It does not assert that these ingredients were first
111 discovered here.
112 
113 
114 \section{Packets, transfer blocks, and polyphase coordinates}
115 
116 A packet is a component specified by a projector and its image. Complete
117 packets keep every distinction in the declared object space. A transfer block
118 records how an operation sends one packet to another. These definitions are
119 structural: they do not themselves promise that measurements recover the
120 packets. Recovery is established by the later injectivity and conditioning
121 contracts.
122 
123 
124 \par{\small\textbf{Domain and hypotheses.} Vector spaces over a field. Let $P_1,\ldots,P_m\in\operatorname{End}(X)$ satisfy $P_aP_b=\delta_{ab}P_a$ and $\sum_aP_a=I_X$. Set $X_a=P_aX$.}\par
125 \begin{theorem}[SG-01\quad Complete packet analysis and synthesis]\label{SG-01}
126 
127 The maps
128 \[
129  \mathcal A x=(P_ax)_a,\qquad
130  \mathcal S(x_a)_a=\sum_ax_a
131 \]
132 are inverse isomorphisms between $X$ and $\bigoplus_aX_a$. In particular,
133 $x=\sum_aP_ax$ uniquely.
134 
135 \end{theorem}
136 \begin{proof}
137 
138 Completeness gives $\mathcal S\mathcal A x=x$. If $x_a\in X_a$, then
139 $P_ax_a=x_a$ and $P_bx_a=0$ for $b\ne a$. Thus the $b$th component of
140 $\mathcal A\mathcal S(x_a)_a$ is $P_b\sum_ax_a=x_b$.
141 Both inverse identities follow. For uniqueness, apply $P_b$ to any proposed
142 sum of components. This proof uses pairwise annihilation, not orthogonality;
143 the packet projections can be oblique.
144 
145 \end{proof}
146 
147 \par{\small\color{Teal}\textit{Source lineage:} \textit{seed}, thm:object-reconstruction.}\par
148 
149 \par{\small\textbf{Domain and hypotheses.} The packet space of SG-01. Let $S,T\in\operatorname{End}(X)$. Set $T_{ba}=P_bTP_a$, viewed as a map $X_a\to X_b$.}\par
150 \begin{theorem}[SG-02\quad Transfer algebra and packet paths]\label{SG-02}
151 
152 \[
153  T=\sum_{a,b}P_bTP_a,\qquad
154  (ST)_{ca}=\sum_bS_{cb}T_{ba}.
155 \]
156 The block representation is an algebra isomorphism to
157 $\operatorname{End}(\bigoplus_aX_a)$, with multiplication by contraction of
158 intermediate packets. For every integer $k\ge1$, iterating the product gives
159 \[
160  P_{a_k}T^kP_{a_0}
161  =\sum_{a_1,\ldots,a_{k-1}}
162  (P_{a_k}TP_{a_{k-1}})\cdots(P_{a_1}TP_{a_0}).
163 \]
164 
165 \end{theorem}
166 \begin{proof}
167 
168 Insert $I=\sum_aP_a$ on both sides of $T$ to obtain reconstruction. Insert
169 that identity between $S$ and $T$ to obtain the product formula. Equivalently,
170 the block map is $T\mapsto\mathcal A T\mathcal S$; SG-01 shows that its
171 inverse is $B\mapsto\mathcal S B\mathcal A$ and that it preserves multiplication.
172 The path formula follows by inserting the same identity between every
173 successive pair of operators. These are finite sums, so no convergence
174 hypothesis is hidden in the insertion.
175 
176 \end{proof}
177 
178 \par{\small\color{Teal}\textit{Source lineage:} \textit{seed}, thm:operator-reconstruction; thm:block-product; thm:path-power.}\par
179 
180 \par{\small\textbf{Domain and hypotheses.} Finite complete packet systems. Let $(Q_\rho)$ refine $(P_a)$, with a partition of fine labels into $F_a$ and $P_a=\sum_{\rho\in F_a}Q_\rho$.}\par
181 \begin{theorem}[SG-03\quad Refinement preserves represented operators]\label{SG-03}
182 
183 \[
184  P_bTP_a=\sum_{\sigma\in F_b}\sum_{\rho\in F_a}Q_\sigma TQ_\rho.
185 \]
186 The coordinate changes $J=\mathcal A_Q\mathcal S_P$ and
187 $C=\mathcal A_P\mathcal S_Q$ satisfy $CJ=I$ and $JC=I$ on their packet
188 coordinate spaces. Refinement is exact before any child packet is discarded.
189 
190 \end{theorem}
191 \begin{proof}
192 
193 Expand the two finite projector sums to obtain the block formula. Apply
194 SG-01 twice to compute
195 $CJ=\mathcal A_P\mathcal S_Q\mathcal A_Q\mathcal S_P
196 =\mathcal A_P\mathcal S_P=I$, and similarly $JC=I$.
197 The identities concern the direct sums of packet images, not an unrestricted
198 product with duplicated arbitrary ambient components.
199 
200 \end{proof}
201 
202 \par{\small\color{Teal}\textit{Source lineage:} \textit{seed}, prop:block-refinement; thm:coordinate-refinement.}\par
203 
204 \par{\small\textbf{Domain and hypotheses.} Vector spaces with an invertible transform. Let $V:X\to Y$ be an isomorphism and $(E_a)$ a finite complete packet family on $Y$.}\par
205 \begin{theorem}[SG-04\quad Pulled-back transform packets]\label{SG-04}
206 
207 The operators $P_a=V^{-1}E_aV$ are a complete packet family on $X$.
208 For a unitary $V$ and orthogonal coordinate selectors $E_a$ in a Hilbert space,
209 \[
210  \|x\|^2=\sum_a\|P_ax\|^2.
211 \]
212 
213 \end{theorem}
214 \begin{proof}
215 
216 Compute $P_aP_b=V^{-1}E_aE_bV=\delta_{ab}P_a$ and
217 $\sum_aP_a=V^{-1}IV=I$. In the unitary case the pulled-back projections are
218 orthogonal: $P_a^*=P_a$. Their images are mutually orthogonal, and SG-01
219 writes $x$ as their finite orthogonal sum. The Pythagorean identity gives the
220 energy formula. Without unitarity and orthogonality the algebraic identities
221 remain valid while that norm identity need not hold.
222 
223 \end{proof}
224 
225 \par{\small\color{Teal}\textit{Source lineage:} \textit{seed}, thm:transform-atlas; Area II Fourier analysis.}\par
226 
227 \par{\small\textbf{Domain and hypotheses.} Absolutely summable sequences on $\mathbb Z$. Fix an integer $N\ge1$. Let $h,x\in\ell^1(\mathbb Z)$ and $y=h*x$. Define $x_a[q]=x[Nq+a]$ and $y_b[q]=y[Nq+b]$, for $0\le a,b<N$.}\par
228 \begin{theorem}[SG-05\quad Polyphase analysis and convolution kernels]\label{SG-05}
229 
230 Interleaving the $N$ channels recovers $x$ exactly. With
231 \[
232  h_{ba}[k]=h[Nk+b-a],
233 \]
234 the output channels satisfy
235 \[
236  y_b[q]=\sum_{a=0}^{N-1}\sum_{t\in\mathbb Z}h_{ba}[q-t]x_a[t].
237 \]
238 Thus convolution becomes an $N\times N$ transfer-kernel system.
239 
240 \end{theorem}
241 \begin{proof}
242 
243 Every integer has a unique Euclidean decomposition $Nt+a$, so the analysis
244 and interleaving maps are inverse. In the convolution sum for $y[Nq+b]$, split
245 the input index $m$ as $Nt+a$. Then the kernel index becomes
246 $Nq+b-m=N(q-t)+b-a$, which is precisely $h_{ba}[q-t]$.
247 Absolute summability permits the regrouping: the total absolute double sum
248 is bounded by the corresponding convolution of $|h|$ and $|x|$.
249 The formula also holds for finitely supported sequences directly. It is the
250 sequence version of SG-02, with the residue projectors selecting addresses.
251 
252 \end{proof}
253 
254 \par{\small\color{Teal}\textit{Source lineage:} \textit{seed}, Area II exact polyphase analysis and synthesis.}\par
255 
256 \par{\small\textbf{Domain and hypotheses.} Unit-modulus exponential nodes. Let $z\in\mathbb C$, $|z|=1$, and suppose $u=z^N$ and $w_j=z^{r_j}$ are known, with $N\ge1$ and integer offsets $r_j$. Let $d=\gcd(N,r_1,\ldots,r_k)$.}\par
257 \begin{theorem}[SG-06\quad The arithmetic of residual alias ambiguity]\label{SG-06}
258 
259 Exactly $d$ unit-modulus nodes satisfy the same data. Hence the physical node
260 is unique precisely when $d=1$. If integers $c,b_1,\ldots,b_k$ obey
261 $Nc+\sum_jr_jb_j=d$, then
262 \[
263  u^c\prod_jw_j^{b_j}=z^d.
264 \]
265 For $d=1$ this is an explicit exact de-aliasing formula.
266 
267 \end{theorem}
268 \begin{proof}
269 
270 If $z'$ gives the same data, the quotient $v=z'/z$ satisfies
271 $v^N=v^{r_1}=\cdots=v^{r_k}=1$. Bezout's identity gives $v^d=1$.
272 Conversely $d$ divides every supplied exponent, so every $d$th root of unity
273 has all those powers equal to one. Therefore the fiber consists exactly of
274 $z$ multiplied by the $d$ distinct $d$th roots of unity. Applying the Bezout
275 identity directly to the measured powers gives the displayed formula.
276 Negative integer powers are legitimate because the data are nonzero.
277 
278 \end{proof}
279 
280 \par{\small\color{Teal}\textit{Source lineage:} \textit{self}, Why decimation helps and why it aliases; Offset coefficients and internal phase recovery.}\par
281 
282 
283 The offset data in SG-06 must be actual powers of the same node. In the
284 multi-edge problem they are obtained only after matching coefficient vectors
285 to common aliased nodes. A gcd computation does not bypass that matching or
286 undo a collision of two edges in a single decimated channel. Adjacent offsets
287 supply the exponent $1$, so they remove the scalar branch ambiguity once the
288 individual coefficient progressions have been resolved.
289 
290 
291 \section{Identifiability and its unavoidable boundaries}
292 
293 
294 \par{\small\textbf{Domain and hypotheses.} An arbitrary measurement model and linear packet classes. Let $M:X\to Y$ be any map and let an edge assignment $J:X\to\mathcal J$ be specified. Separately, let $\mathcal S,\mathcal J$ be linear subspaces of a common vector space.}\par
295 \begin{theorem}[SG-07\quad Measurement and decomposition obstructions]\label{SG-07}
296 
297 If $Mf_0=Mf_1$ but $J(f_0)\ne J(f_1)$, no map $R:Y\to\mathcal J$ recovers
298 the correct edge for both. If $\mathcal S\cap\mathcal J\ne\{0\}$, the
299 sum decomposition $f=s+j$ is not unique. In particular, finitely many point
300 samples on a periodic interval cannot recover every arbitrary interval packet.
301 
302 \end{theorem}
303 \begin{proof}
304 
305 A recovery map has one value at the common datum, so it cannot equal the two
306 different required edges. For the decomposition obstruction take nonzero
307 $h\in\mathcal S\cap\mathcal J$: the pairs $(s,j)$ and $(s+h,j-h)$ have the
308 same sum. Finally, the complement of a finite sample set contains a nonempty
309 open interval. A nonzero indicator supported inside that interval vanishes at
310 every sample point. Compare it with the zero signal to obtain the first
311 obstruction concretely.
312 
313 \end{proof}
314 
315 \par{\small\color{Teal}\textit{Source lineage:} \textit{unknown}, The measurement obstruction; The decomposition obstruction.}\par
316 
317 \par{\small\textbf{Domain and hypotheses.} Two-jump periodic packets of fixed known amplitudes. Fix a finite set $F\subset\mathbb Z$ of derivative-moment indices. For $0<\delta<\pi/4$, compare locations $(0,\delta)$ and $(\pi,\pi+\delta)$, both with amplitudes $(1,-1)$.}\par
318 \begin{theorem}[SG-08\quad No uniform location stability without separation]\label{SG-08}
319 
320 Their unordered location sets have permutation-matched bottleneck distance at
321 least $\pi-\delta$, while their measured derivative moments satisfy
322 \[
323  |m_n-\widetilde m_n|
324  =|(1-e^{-in\pi})(1-e^{-in\delta})|
325  \le2|n|\delta\qquad(n\in F).
326 \]
327 There is no Lipschitz inverse for locations with one constant on all distinct
328 two-jump configurations, even at this fixed order and with known amplitudes.
329 
330 \end{theorem}
331 \begin{proof}
332 
333 Every circular distance from a point of $\{0,\delta\}$ to a point of
334 $\{\pi,\pi+\delta\}$ is at least $\pi-\delta$. This bounds every matching.
335 The moments of the first packet are $1-e^{-in\delta}$; translating by $\pi$
336 multiplies them by $e^{-in\pi}$. Subtraction gives the formula.
337 The elementary inequality $|1-e^{-it}|\le|t|$ proves the bound. Every fixed
338 finite norm of the measurement difference tends to zero as $\delta\to0$,
339 whereas the location distance stays bounded away from zero. A common finite
340 Lipschitz constant is therefore impossible. This argument compares two
341 admissible fixed-order packets; it does not require assigning location
342 parameters to the zero-edge signal.
343 
344 \end{proof}
345 
346 \par{\small\color{Teal}\textit{Source lineage:} \textit{unknown}, A collapsing pair of jumps; Collision is the unavoidable hard boundary.}\par
347 
348 \par{\small\textbf{Publication correction.} The original collision argument compared an edge parameter vector with the zero packet. The published proof compares two fixed-order configurations and specifies a permutation-invariant location metric.}\par
349 
350 \section{Known edges and the source of Gibbs ringing}
351 
352 The simplest reconstruction problem already separates discovery from
353 synthesis. Suppose a singular packet is known, but its amplitude and the
354 smooth background are not. A high-frequency functional can identify the
355 amplitude exactly on a bandlimited background. The recovered singular packet
356 then has its own synthesis route.
357 
358 
359 \par{\small\textbf{Domain and hypotheses.} Periodic integrable signals with one known edge shape. Let $\mathcal S_K$ be the trigonometric polynomials of degree at most $K$. Let $m\in L^1(\mathbb T)$ and a finite nonempty set $F\subset\{n:|n|>K\}$ satisfy $D_F=\sum_{n\in F}|\widehat m(n)|^2>0$.}\par
360 \begin{theorem}[SG-09\quad Out-of-band matched identification]\label{SG-09}
361 
362 On $X=\mathcal S_K+\operatorname{span}\{m\}$, the functional
363 \[
364  \ell_F(g)=D_F^{-1}\sum_{n\in F}\overline{\widehat m(n)}\widehat g(n)
365 \]
366 satisfies $\ell_F(s+am)=a$. The decomposition is direct. If the measured
367 coefficients have errors $\eta_n$, then
368 \[
369  |\widetilde a-a|\le D_F^{-1/2}\Bigl(\sum_{n\in F}|\eta_n|^2\Bigr)^{1/2}.
370 \]
371 
372 \end{theorem}
373 \begin{proof}
374 
375 The coefficients of $s\in\mathcal S_K$ vanish on $F$. Substituting $s+am$
376 into the functional leaves $aD_F/D_F=a$. If $m$ belonged to $\mathcal S_K$,
377 all its coefficients on $F$ would vanish, contradicting $D_F>0$; hence the
378 sum is direct. The amplitude error is the inner product of the coefficient
379 error with the known coefficient vector divided by its squared norm.
380 Cauchy--Schwarz gives the displayed bound. The frequencies must actually see
381 the edge: an all-zero identifying set makes the formula undefined.
382 
383 \end{proof}
384 
385 \par{\small\color{Teal}\textit{Source lineage:} \textit{singularity}, An edge functional from out-of-band modes.}\par
386 
387 \par{\small\textbf{Domain and hypotheses.} The model space of SG-09. Let $P_Jg=\ell_F(g)m$, $P_S=I-P_J$, and let $S_M$ be Fourier truncation of degree $M\ge K$.}\par
388 \begin{theorem}[SG-10\quad The singularity-exact block operator]\label{SG-10}
389 
390 \[
391  P_J^2=P_J,\quad P_S^2=P_S,\quad P_SP_J=P_JP_S=0,
392  \quad P_S+P_J=I_X,
393 \]
394 and the resolved reconstruction obeys
395 \[
396  \mathcal R_M=S_MP_S+P_J=I_X.
397 \]
398 
399 \end{theorem}
400 \begin{proof}
401 
402 For $g=s+am$, SG-09 gives $P_Jg=am$ and $P_Sg=s$.
403 These coordinate formulas immediately imply idempotence and pairwise
404 annihilation. Since $s$ is bandlimited and $M\ge K$, $S_Ms=s$.
405 Thus $\mathcal R_Mg=s+am=g$. The equality is in the declared signal class;
406 for a selected right-continuous packet it also reproduces that representative.
407 This is a known-packet theorem. An unknown interval still needs the discovery
408 step proved later.
409 
410 \end{proof}
411 
412 \par{\small\color{Teal}\textit{Source lineage:} \textit{singularity}, The singularity projector; The singularity-exact reconstruction theorem; \textit{gibbs}, The first exact packet decomposition.}\par
413 
414 \par{\small\textbf{Domain and hypotheses.} Periodic signals with a supplied exact edge packet. Let $f=s+j$ with $|\widehat s(n)|\le Ce^{-\rho|n|}$, where $C\ge0$ and $\rho>0$. Use the canonical uniformly convergent Fourier-series representative of $s$. Set $\mathcal R_Mf=S_Ms+j$.}\par
415 \begin{theorem}[SG-11\quad Resolved error for an analytic background]\label{SG-11}
416 
417 The reconstruction error is exactly $f-\mathcal R_Mf=s-S_Ms$, and
418 \[
419  \|f-\mathcal R_Mf\|_\infty
420  \le\frac{2Ce^{-\rho(M+1)}}{1-e^{-\rho}}.
421 \]
422 The bound is uniform across the jumps of $j$.
423 
424 \end{theorem}
425 \begin{proof}
426 
427 Cancel $j$ from the defining identity. Absolute summability of the Fourier
428 coefficients gives an everywhere uniformly convergent series for $s$.
429 The omitted positive and negative frequencies are bounded by
430 $2C\sum_{n>M}e^{-\rho n}$, which is the displayed geometric series.
431 No limiting trigonometric approximation of $j$ occurs in this estimate.
432 
433 \end{proof}
434 
435 \par{\small\color{Teal}\textit{Source lineage:} \textit{gibbs}, Analytic smooth background.}\par
436 
437 \par{\small\textbf{Domain and hypotheses.} The interval packet $j=\mathbf1_{[-a,a)}$ on $\mathbb T$. Assume $0<a<\pi$, let $\lambda_M=M+1/2$, and use the right edge coordinate $x_M(z)=a-z/\lambda_M$.}\par
438 \begin{theorem}[SG-12\quad The scaled interval-edge Gibbs profile]\label{SG-12}
439 
440 Uniformly for $z$ in compact real intervals,
441 \[
442  S_Mj(x_M(z))\longrightarrow\frac12+\frac1\pi\operatorname{Si}(z),
443  \qquad \operatorname{Si}(z)=\int_0^z\frac{\sin t}{t}\,dt.
444 \]
445 The first positive maximum of the limiting profile is $1+G$, where
446 \[
447  G=\frac{\operatorname{Si}(\pi)}\pi-\frac12
448    =0.089489872236\ldots.
449 \]
450 
451 \end{theorem}
452 \begin{proof}
453 
454 Direct integration gives $\widehat j(0)=a/\pi$ and
455 $\widehat j(n)=\sin(na)/(\pi n)$ for $n\ne0$. The Dirichlet kernel
456 $D_M(t)=\sin(\lambda_Mt)/\sin(t/2)$ yields
457 \[
458  S_Mj(x)=\frac1{2\pi}\int_{-a}^{a}D_M(x-t)\,dt,
459  \qquad (S_Mj)'(x)=\frac{D_M(x+a)-D_M(x-a)}{2\pi}.
460 \]
461 At $x=a$, the coefficient formula gives
462 \[
463  S_Mj(a)=\frac a\pi+\frac1\pi\sum_{n=1}^M\frac{\sin(2na)}n
464  \longrightarrow\frac12.
465 \]
466 For completeness, the sine-series limit follows by summation by parts from
467 bounded partial sums of $e^{2ina}$ and by taking radial limits in
468 $\sum_{n\ge1}r^ne^{2ina}/n=-\log(1-re^{2ia})$; its imaginary part tends to
469 $(\pi-2a)/2$ for $0<2a<2\pi$.
470 Now differentiate the scaled function:
471 \[
472  \frac d{dz}S_Mj(x_M(z))=
473  \frac{D_M(-z/\lambda_M)-D_M(2a-z/\lambda_M)}{2\pi\lambda_M}.
474 \]
475 The first term converges uniformly on compact $z$ sets to
476 $\sin z/(\pi z)$, with its continuous value at zero. The second tends
477 uniformly to zero because $\sin(a-z/(2\lambda_M))$ stays bounded away from
478 zero there. Integrate from $0$ to $z$ and use the base-point limit.
479 The derivative of the limiting profile is $\sin z/(\pi z)$, positive on
480 $(0,\pi)$ and negative on $(\pi,2\pi)$, so its first positive maximum is at
481 $z=\pi$. Substitution gives $G$. The constant is relative to a unit jump;
482 scaling a packet scales its overshoot by the jump amplitude.
483 
484 \end{proof}
485 
486 \par{\small\color{Teal}\textit{Source lineage:} \textit{gibbs}, The scaled Gibbs limit; \textit{singularity}, Detailed proof of the local scaling limit.}\par
487 
488 \par{\small\textbf{Domain and hypotheses.} Periodic bounded real signals. Let $\sigma_M=(M+1)^{-1}\sum_{k=0}^MS_k$ be the Fejer mean. Suppose $0\le j\le1$ almost everywhere.}\par
489 \begin{theorem}[SG-13\quad Positive Fourier averaging preserves range]\label{SG-13}
490 
491 Then $0\le\sigma_Mj\le1$ everywhere. For an interval packet, this suppresses
492 overshoot while retaining a continuous transition rather than an exact jump.
493 
494 \end{theorem}
495 \begin{proof}
496 
497 The averaging kernel is
498 \[
499  F_M(t)=\frac1{M+1}\left|\sum_{k=0}^Me^{ikt}\right|^2\ge0,
500  \qquad\frac1{2\pi}\int_0^{2\pi}F_M(t)\,dt=1.
501 \]
502 Its convolution with $j$ is therefore a weighted mean of numbers in $[0,1]$.
503 The resulting finite trigonometric polynomial is continuous. A nonconstant
504 interval indicator is discontinuous, so the mean cannot equal its selected
505 representative at all points or have its exact one-sided jump.
506 
507 \end{proof}
508 
509 \par{\small\color{Teal}\textit{Source lineage:} \textit{gibbs}, Fejer averaging.}\par
510 
511 \section{Finite dictionaries and residualized geometric discovery}
512 
513 In a finite dictionary model the unknown is a coefficient vector. The smooth
514 background is eliminated before testing the edge dictionary. This removal is
515 exact only for a declared background subspace. A residualized dictionary can
516 lose distinctions that the original dictionary had, so its kernel is the
517 correct object to inspect.
518 
519 
520 \par{\small\textbf{Domain and hypotheses.} Finite real or complex Euclidean spaces. Let $\mathcal S\subseteq\mathbb F^m$ be a subspace, $P$ its orthogonal projector, $R=I-P$, $D\in\mathbb F^{m\times d}$, and $A=RD$. Let $s\ge0$ be an integer.}\par
521 \begin{theorem}[SG-14\quad Residualized sparse injectivity]\label{SG-14}
522 
523 Every vector $\alpha$ with $\|\alpha\|_0\le s$ is uniquely determined by
524 $A\alpha$ if and only if
525 \[
526  \ker A\cap\{h:\|h\|_0\le\min(2s,d)\}=\{0\}.
527 \]
528 Equivalently every set of at most $\min(2s,d)$ columns is independent.
529 With the convention $\operatorname{spark}(A)=d+1$ when all $d$ columns are
530 independent, the equivalent condition is
531 $\operatorname{spark}(A)>\min(2s,d)$.
532 For $s=0$ uniqueness is automatic.
533 
534 \end{theorem}
535 \begin{proof}
536 
537 The difference of two $s$-sparse vectors has support of size at most $2s$
538 and at most $d$. Thus the kernel condition implies uniqueness.
539 Conversely, if a nonzero kernel vector $h$ has support size at most $2s$,
540 partition that support into two sets of at most $s$ indices. Let $\alpha$
541 equal $h$ on the first set and zero elsewhere; let $\beta$ equal $-h$ on
542 the second set. Then $h=\alpha-\beta$, and $A\alpha=A\beta$ contradicts
543 uniqueness. Column dependence is exactly the existence of a nonzero kernel
544 vector supported on the selected columns. The spark convention then gives
545 the stated capped inequality, including a fully independent matrix when
546 $2s>d$. When $s=0$, the only admissible vector is zero and the kernel test
547 concerns only zero. If $y=s_0+D\alpha$ with $s_0\in\mathcal S$, then
548 $Ry=A\alpha$, so the same theorem applies after background elimination.
549 
550 \end{proof}
551 
552 \par{\small\color{Teal}\textit{Source lineage:} \textit{unknown}, The residualized secant theorem; \textit{complex}, Two-dimensional finite dictionary model.}\par
553 
554 \par{\small\textbf{Publication correction.} The uncapped inequality spark(A)>2s is replaced by the kernel-secant condition, or its capped spark equivalent. This also covers full-column-rank dictionaries with 2s greater than the number of columns.}\par
555 
556 \par{\small\textbf{Domain and hypotheses.} The Euclidean model of SG-14. Assume $s\ge1$, $d\ge1$, and the injectivity condition of SG-14. Let $y=s_0+D\alpha_0+\eta$, $s_0\in\mathcal S$, $\|\alpha_0\|_0\le s$, and $\|R\eta\|_2\le\epsilon$. Let $\widehat\alpha$ be $s$-sparse with $\|A\widehat\alpha-Ry\|_2\le\epsilon$.}\par
557 \begin{theorem}[SG-15\quad Sparse noise stability and support thresholding]\label{SG-15}
558 
559 Define
560 \[
561  \gamma=\min_{\substack{\|h\|_2=1\\\|h\|_0\le\min(2s,d)}}\|Ah\|_2>0.
562 \]
563 Then
564 \[
565  \|\widehat\alpha-\alpha_0\|_2\le2\epsilon/\gamma,
566  \qquad\|D(\widehat\alpha-\alpha_0)\|_2
567  \le2\|D\|_2\epsilon/\gamma.
568 \]
569 If every nonzero true coefficient has magnitude greater than $4\epsilon/\gamma$,
570 thresholding at $2\epsilon/\gamma$ with a strict greater-than rule recovers
571 the true support.
572 
573 \end{theorem}
574 \begin{proof}
575 
576 The admissible unit vectors form a finite union of compact coordinate
577 spheres. The continuous function $\|Ah\|_2$ attains its minimum. A zero
578 minimum would contradict SG-14, hence $\gamma>0$.
579 For $h=\widehat\alpha-\alpha_0$, subtract the two residuals to obtain
580 \[
581  \|Ah\|_2\le\|A\widehat\alpha-Ry\|_2+\|R\eta\|_2\le2\epsilon.
582 \]
583 Its support is admissible, so homogeneity gives
584 $\gamma\|h\|_2\le\|Ah\|_2$. Apply $D$ for the packet bound.
585 Every coordinate error is at most $2\epsilon/\gamma$. A true zero therefore
586 never passes the strict threshold, while a true coefficient larger than twice
587 that threshold remains strictly above it. For $s=0$ both vectors vanish and
588 no restricted constant is required.
589 
590 \end{proof}
591 
592 \par{\small\color{Teal}\textit{Source lineage:} \textit{unknown}, A deterministic stability theorem.}\par
593 
594 \par{\small\textbf{Domain and hypotheses.} Finite real dictionaries with positive amplitudes. Let $B$ have full column rank, $P$ be the orthogonal projector onto its range, $R=I-P$, and let $\mathcal Q$ be a finite mask library with $Rq\ne0$ for every $q$. Suppose $y=B\theta_*+a_*q_*$, with $a_*>0$ and $q_*\in\mathcal Q$.}\par
595 \begin{theorem}[SG-16\quad Profiled one-packet discovery and its finite gap]\label{SG-16}
596 
597 For an unrestricted scalar amplitude the least-squares profile is
598 \[
599  \widehat a(q)=\frac{\langle Rq,Ry\rangle}{\|Rq\|_2^2},
600  \qquad J(q)=\|Ry-\widehat a(q)Rq\|_2^2.
601 \]
602 The true mask has $\widehat a(q_*)=a_*$ and $J(q_*)=0$.
603 It is the unique zero-residual positive-amplitude mask exactly when no other
604 residualized mask lies on its positive ray. Define the closed rays
605 $C_q=\{aRq:a\ge0\}$. If
606 \[
607  \Gamma=\inf_{q\ne q_*}\operatorname{dist}(Ry,C_q)>0,
608 \]
609 with the convention $\inf\varnothing=+\infty$, then every perturbed residualized datum $Ry+e$ with $\|e\|_2<\Gamma/2$
610 selects $q_*$ uniquely by nearest-ray distance.
611 
612 \end{theorem}
613 \begin{proof}
614 
615 Differentiate the quadratic function $\|Ry-aRq\|_2^2$ with respect to $a$.
616 Its unique stationary point is the profile formula and is the global
617 minimum. At the true mask $Ry=a_*Rq_*$, giving the stated values.
618 A wrong mask gives an admissible zero residual precisely when
619 $Ry=aRq$ for some $a>0$. This is the positive-ray condition.
620 For the noise statement, the distance to any nonempty closed set is
621 1-Lipschitz by the triangle inequality. The true ray has perturbed distance
622 at most $\|e\|_2$, and every wrong ray has distance at least
623 $\Gamma-\|e\|_2>\|e\|_2$. Thus the true ray wins strictly.
624 A squared objective gap must be square-rooted before it is used in this
625 norm-distance condition. For positive constraints the optimum on a closed
626 ray uses $\max(0,\widehat a(q))$, rather than an unconditionally signed
627 amplitude.
628 
629 \end{proof}
630 
631 \par{\small\color{Teal}\textit{Source lineage:} \textit{gibbs}, Profile least squares over interval packets; The finite identifiability gap; \textit{singularity}, Noise stability of finite packet discovery.}\par
632 
633 \par{\small\textbf{Domain and hypotheses.} Finite binary masks. Let $q,\widetilde q\in\{0,1\}^m$ and $a,\widetilde a\in\mathbb R$.}\par
634 \begin{theorem}[SG-17\quad Mask error is a support geometry calculation]\label{SG-17}
635 
636 \begin{align*}
637  \|aq-\widetilde a\widetilde q\|_2^2
638  &=|a-\widetilde a|^2|\operatorname{supp}q\cap\operatorname{supp}\widetilde q|\\
639  &\quad+|a|^2|\operatorname{supp}q\setminus\operatorname{supp}\widetilde q|
640  +|\widetilde a|^2|\operatorname{supp}\widetilde q\setminus\operatorname{supp}q|.
641 \end{align*}
642 For $a=\widetilde a=1$ the squared norm is the cardinality of the symmetric
643 support difference.
644 
645 \end{theorem}
646 \begin{proof}
647 
648 Partition the coordinates into common support, first-only support,
649 second-only support, and neither. The difference equals respectively
650 $a-\widetilde a$, $a$, $-\widetilde a$, and zero. Sum the squared values.
651 The same pointwise partition gives a continuous-measure formula when support
652 cardinalities are replaced by support measures.
653 
654 \end{proof}
655 
656 \par{\small\color{Teal}\textit{Source lineage:} \textit{gibbs}, Packet perturbation identity.}\par
657 
658 \section{Exact off-grid recovery from finitely many Fourier coefficients}
659 
660 Let $\mathcal U_{K,L}$ consist of real periodic signals $f=s+j$, where
661 $s\in\mathcal S_K$ and $j$ is right-continuous, piecewise constant, mean zero,
662 and has at most $L$ distinct active jumps. An active jump has nonzero amplitude.
663 For an actual jump count $r$, write
664 \[
665  Dj=\sum_{\ell=1}^r a_\ell\delta_{\tau_\ell},\quad
666  z_\ell=e^{-i\tau_\ell},\quad\sum_\ell a_\ell=0.
667 \]
668 The sum-zero condition is required by periodicity. A single nonzero active
669 jump cannot occur, so the nonconstant real class starts at $r=2$. The case
670 $r=0$ is allowed and will be handled explicitly. The smooth-edge
671 representation is unique with this normalization: a piecewise-constant
672 trigonometric polynomial is constant, and mean zero fixes that constant.
673 
674 
675 \par{\small\textbf{Domain and hypotheses.} The class $\mathcal U_{K,L}$. Use the Fourier convention in Section 1. Let $n>K$ and $z_\ell=e^{-i\tau_\ell}$.}\par
676 \begin{theorem}[SG-18\quad Differentiation isolates an exponential moment packet]\label{SG-18}
677 
678 \[
679  Dj=\sum_\ell a_\ell\delta_{\tau_\ell},\qquad
680  m_n:=2\pi in\widehat f(n)=\sum_{\ell=1}^ra_\ell z_\ell^n.
681 \]
682 For a non-bandlimited integrable background the same formula holds with
683 an additive defect $e_n=2\pi in\widehat s(n)$.
684 
685 \end{theorem}
686 \begin{proof}
687 
688 For a smooth periodic test function $\varphi$, integrate by parts over each
689 constant segment of $j$. Boundary terms telescope at each jump to
690 $a_\ell\varphi(\tau_\ell)$, which is exactly the displayed distribution.
691 Evaluate the distribution on $e^{-inx}$:
692 \[
693  \langle Dj,e^{-inx}\rangle
694  =-\int j(x)(-in)e^{-inx}\,dx
695  =2\pi in\widehat j(n).
696 \]
697 The Dirac sum gives $\sum_\ell a_\ell e^{-in\tau_\ell}$.
698 For $n>K$, the smooth polynomial contributes zero. Without bandlimiting it
699 contributes the stated defect. At $n=0$ the derivative moment is zero, in
700 agreement with $\sum a_\ell=0$; it cannot determine the signal mean.
701 
702 \end{proof}
703 
704 \par{\small\color{Teal}\textit{Source lineage:} \textit{unknown}, The derivative packet; Removing the unknown smooth background; \textit{self}, Signal class and Fourier convention.}\par
705 
706 \par{\small\textbf{Domain and hypotheses.} Finite exponential moments over $\mathbb C$. Let $m_{n_0+k}=\sum_{\ell=1}^r c_\ell z_\ell^k$, with distinct nonzero nodes and nonzero coefficients. Let $H_{h,w}=(m_{n_0+i+j})_{0\le i<h,\ 0\le j<w}$, with $h,w\ge r$.}\par
707 \begin{theorem}[SG-19\quad Hankel rank detects the actual order]\label{SG-19}
708 
709 Writing $V_h=(z_\ell^i)_{i,\ell}$ and $V_w=(z_\ell^j)_{j,\ell}$,
710 \[
711  H_{h,w}=V_h\operatorname{diag}(c_\ell)V_w^{\mathsf T},
712  \qquad\operatorname{rank}H_{h,w}=r.
713 \]
714 For $r=0$ the Hankel matrix is zero. The transpose is algebraic and is not
715 a conjugate transpose.
716 
717 \end{theorem}
718 \begin{proof}
719 
720 Expand a matrix entry to obtain the factorization. The leading $r\times r$
721 Vandermonde submatrix has determinant
722 $\prod_{\ell<k}(z_k-z_\ell)$, which is nonzero. Thus both rectangular
723 Vandermonde factors have full column rank, and their transposes have full
724 row rank. The diagonal factor is invertible. Its product with $V_w^T$ is
725 surjective onto $\mathbb C^r$, and subsequent multiplication by the injective
726 $V_h$ has image dimension $r$. For an empty exponential sum every entry is
727 zero and the rank is zero.
728 
729 \end{proof}
730 
731 \par{\small\color{Teal}\textit{Source lineage:} \textit{unknown}, The Hankel packet and its rank; \textit{self}, Unknown jump count from a Hankel gap.}\par
732 
733 \par{\small\textbf{Domain and hypotheses.} The nonzero-order moment model of SG-19. Assume $r\ge1$ and $2r$ consecutive moments $m_{n_0},\ldots,m_{n_0+2r-1}$ are known.}\par
734 \begin{theorem}[SG-20\quad Exact Prony annihilation and location recovery]\label{SG-20}
735 
736 The unique monic annihilator
737 \[
738  Q(z)=\prod_{\ell=1}^r(z-z_\ell)=\sum_{k=0}^{r-1}q_kz^k+z^r
739 \]
740 is recovered from
741 \[
742  H_{r,r}q=-v,\qquad v=(m_{n_0+r+i})_{i=0}^{r-1}.
743 \]
744 Its roots are exactly the nodes. Once the nodes are known, $r$ consecutive
745 moments determine the coefficients uniquely by a Vandermonde solve.
746 For unit-modulus nodes the corresponding locations are
747 $\tau_\ell=-\arg z_\ell\pmod{2\pi}$.
748 
749 \end{theorem}
750 \begin{proof}
751 
752 For every available recurrence index,
753 \[
754  \sum_{k=0}^rq_km_{n+k}
755  =\sum_\ell c_\ell z_\ell^{n-n_0}Q(z_\ell)=0,
756  \qquad q_r=1.
757 \]
758 SG-19 makes $H_{r,r}$ invertible. Thus the system has one solution and the
759 true polynomial supplies it. Factoring the resulting polynomial recovers the
760 distinct nodes up to a permutation. The amplitude matrix for consecutive
761 moments is a square Vandermonde multiplied by an invertible diagonal node
762 power matrix, so it too is invertible. The argument convention identifies
763 locations modulo a period, as required. Polynomial root extraction is a
764 finite algebraic step; the theorem does not claim that arbitrary real
765 locations have rational closed forms.
766 
767 \end{proof}
768 
769 \par{\small\color{Teal}\textit{Source lineage:} \textit{unknown}, The annihilating polynomial; Recovering jump amplitudes and the edge packet.}\par
770 
771 \par{\small\textbf{Domain and hypotheses.} A period of length $T>0$. Let distinct locations $\tau_\ell\in\mathbb R/(T\mathbb Z)$ and real amplitudes $a_\ell$ satisfy $\sum_\ell a_\ell=0$. Define the mean-zero periodic sawtooth $G_T(x)=1/2-x/T$ for $0<x<T$, with the right-continuous endpoint value.}\par
772 \begin{theorem}[SG-21\quad Canonical periodic integration of jump data]\label{SG-21}
773 
774 \[
775  j(x)=\sum_\ell a_\ell G_T(x-\tau_\ell)
776 \]
777 is the unique mean-zero right-continuous piecewise-constant packet with those
778 jumps. Its distributional derivative is $\sum_\ell a_\ell\delta_{\tau_\ell}$.
779 
780 \end{theorem}
781 \begin{proof}
782 
783 The sawtooth derivative is $DG_T=\delta_0-T^{-1}dx$: its slope is $-1/T$
784 away from the endpoint and its endpoint jump is one. Summing translated
785 copies gives the prescribed Dirac masses and a constant-density term
786 $-T^{-1}\sum a_\ell$, which vanishes. Consequently $j$ is constant between
787 the jump points. Each translated sawtooth has mean zero, so $j$ does too.
788 Any other packet with the same jumps has distributional derivative difference
789 zero. Since both are piecewise constant, their difference is the same
790 constant on every segment and across every jump. The mean-zero condition
791 sets it to zero almost everywhere, and right continuity fixes the endpoint
792 values. This explicit formula also supplies the perturbation estimate later.
793 
794 \end{proof}
795 
796 \par{\small\color{Teal}\textit{Source lineage:} \textit{unknown}, Periodic integration.}\par
797 
798 \par{\small\textbf{Domain and hypotheses.} The real class $\mathcal U_{K,L}$. Let $K\ge0$ and $L\ge1$ be known integers. Assume exact Fourier coefficients for $-K\le n\le K$ and $K+1\le n\le K+2L$.}\par
799 \begin{theorem}[SG-22\quad Finite exact unknown-order reconstruction]\label{SG-22}
800 
801 Those data uniquely determine the actual active order $r$, every jump
802 location and amplitude up to permutation, the canonical edge packet $j$,
803 the smooth coefficients of $s$, and the complete signal $f$ as an $L^1$ class
804 and as its selected right-continuous representative. The high-frequency
805 budget is $2L$ complex coefficients. The low budget is $2K+1$ coefficients,
806 with conjugate symmetry redundancy for a real signal.
807 
808 \end{theorem}
809 \begin{proof}
810 
811 Use SG-18 to form $2L$ exact high derivative moments. With $n_0=K+1$,
812 form $H_{L,L}$ using the first $2L-1$ of them. SG-19 says its rank is the
813 actual $r$. If $r=0$, $j=0$ and the low coefficients directly give $s$.
814 Otherwise the first $2r$ moments are available. SG-20 recovers the nodes and
815 amplitudes. SG-21 constructs the unique normalized edge.
816 For each $|n|\le K$, subtract its coefficient from $\widehat f(n)$ to obtain
817 $\widehat s(n)$; all other smooth coefficients are zero by the model.
818 Their finite trigonometric sum gives $s$. Every step is uniquely determined,
819 so another signal in the class with those data must have the same edge and
820 background. Finally add $s+j$. Values at jump points are selected by the
821 stated convention, rather than inferred from measure-zero Fourier data.
822 
823 \end{proof}
824 
825 \par{\small\color{Teal}\textit{Source lineage:} \textit{unknown}, The universal off-grid recovery theorem.}\par
826 
827 \par{\small\textbf{Publication correction.} Exactness from Fourier data is stated for the L1 class and its declared right-continuous representative. Arbitrarily assigned point values at discontinuities are not recoverable from Fourier integrals.}\par
828 
829 
830 The measurement bound is a sufficient budget for this construction, not a
831 minimal sample theorem. Exact order is an algebraic rank. A floating-point
832 rank threshold requires a quantitative gap, addressed below. Distinct nodes
833 need not have a prescribed minimum separation for exact uniqueness, but
834 stability constants deteriorate as nodes approach one another.
835 
836 
837 \section{Perturbation contracts for the finite recovery steps}
838 
839 
840 \par{\small\textbf{Domain and hypotheses.} A rectangular Hankel matrix of derivative moments. Let measured Fourier coefficients have errors $\eta_n$ and moment errors $\epsilon_n=2\pi in\eta_n$. Let $E=(\epsilon_{n_0+i+j})$ have $h\ge1$ rows and $w\ge1$ columns.}\par
841 \begin{theorem}[SG-23\quad Hankel perturbation from Fourier noise]\label{SG-23}
842 
843 \[
844  \|E\|_2\le\|E\|_F
845  \le\sqrt{hw}\max_{0\le i<h,\ 0\le j<w}|\epsilon_{n_0+i+j}|.
846 \]
847 In particular, an error bound $|\eta_n|\le\eta$ implies
848 \[
849  \|E\|_2\le2\pi\sqrt{hw}\eta\max_{i,j}|n_0+i+j|.
850 \]
851 
852 \end{theorem}
853 \begin{proof}
854 
855 Differentiate the noisy Fourier coefficients to obtain the exact moment
856 error identity. The spectral norm is at most the Frobenius norm. Its square
857 is the sum of $hw$ squared entry magnitudes, each bounded by the squared
858 maximum. Substitute $|\epsilon_n|=2\pi|n||\eta_n|$.
859 Repeated moments occur at multiple matrix entries; the Frobenius bound
860 correctly includes those repetitions. Smooth background tails can be included
861 in $\epsilon_n$ by the same calculation.
862 
863 \end{proof}
864 
865 \par{\small\color{Teal}\textit{Source lineage:} \textit{unknown}, Fourier noise becomes derivative-moment noise; Hankel perturbation.}\par
866 
867 \par{\small\textbf{Domain and hypotheses.} The invertible square Prony system of SG-20. Write $Hq=-v$, and suppose $\widetilde H=H+E$, $\widetilde v=v+e$. Assume $\|H^{-1}\|_2\|E\|_2<1$.}\par
868 \begin{theorem}[SG-24\quad Annihilator coefficient perturbation]\label{SG-24}
869 
870 The perturbed system is invertible, and its solution satisfies
871 \[
872  \|\widetilde q-q\|_2
873  \le\frac{\|H^{-1}\|_2}{1-\|H^{-1}\|_2\|E\|_2}
874        (\|e\|_2+\|E\|_2\|q\|_2).
875 \]
876 
877 \end{theorem}
878 \begin{proof}
879 
880 Factor $\widetilde H=H(I+H^{-1}E)$. The Neumann series converges in operator
881 norm under the supplied inequality, so
882 $\|\widetilde H^{-1}\|_2\le
883 \|H^{-1}\|_2/(1-\|H^{-1}\|_2\|E\|_2)$.
884 Subtract the equations to obtain
885 $\widetilde H(\widetilde q-q)=-e-Eq$. Apply the inverse bound and the
886 triangle inequality. The estimate assumes that the same true order is used;
887 it does not itself choose the order from noisy data.
888 
889 \end{proof}
890 
891 \par{\small\color{Teal}\textit{Source lineage:} \textit{unknown}, Hankel perturbation; Root localization by Rouche theorem.}\par
892 
893 \par{\small\textbf{Domain and hypotheses.} Monic complex polynomials of the same degree. Let $r\ge1$ and $Q(z)=\prod_{\ell=1}^r(z-z_\ell)$ have distinct roots and $\widetilde Q=Q+E_Q$. Choose $0<\rho<\tfrac12\min_{\ell\ne k}|z_\ell-z_k|$ when $r\ge2$, or any $\rho>0$ when $r=1$.}\par
894 \begin{theorem}[SG-25\quad Conditional simple-root disk localization]\label{SG-25}
895 
896 If for every $\ell$
897 \[
898  \sup_{|z-z_\ell|=\rho}|E_Q(z)|
899  <\min_{|z-z_\ell|=\rho}|Q(z)|,
900 \]
901 then $\widetilde Q$ has exactly one root, counted with multiplicity, in each
902 disk $|z-z_\ell|<\rho$. All its roots are thereby matched with error less
903 than $\rho$. A sufficient explicit bound on each circle is
904 \[
905  \sum_{k=0}^{r-1}|\widetilde q_k-q_k|(|z_\ell|+\rho)^k
906  <\rho\prod_{j\ne\ell}(|z_\ell-z_j|-\rho).
907 \]
908 
909 \end{theorem}
910 \begin{proof}
911 
912 Each disk contains exactly one root of $Q$. The strict boundary inequality
913 is the hypothesis of Rouche's theorem, which preserves the number of zeros
914 inside. There are $r$ disjoint disks, each containing one zero of the degree
915 $r$ perturbed polynomial, so they account for every zero.
916 For the explicit version, apply the triangle inequality to the coefficient
917 error polynomial and use $|z|\le|z_\ell|+\rho$. On the circle,
918 $|z-z_\ell|=\rho$ and
919 $|z-z_j|\ge|z_\ell-z_j|-\rho$, giving the lower product bound.
920 For unit true nodes and a recovered root inside a disk with $\rho<1$, radial
921 normalization changes its chord error to at most $2\rho$ by the triangle
922 inequality. Its angular location error is then at most $2\arcsin\rho$.
923 The disk radius is a complex-root error, not directly an angular error.
924 
925 \end{proof}
926 
927 \par{\small\color{Teal}\textit{Source lineage:} \textit{unknown}, Root localization by Rouche theorem.}\par
928 
929 \par{\small\textbf{Publication correction.} This is a conditional stability theorem involving true root centers and true coefficient errors. It is not a certificate computable from noisy data alone, and a complex disk radius is not asserted to equal the angular location error.}\par
930 
931 \par{\small\textbf{Domain and hypotheses.} A square or overdetermined Vandermonde amplitude fit. Let $V$ have full column rank, $m=Va$, $\widetilde V=V+\Delta V$, and $\widetilde m=m+e$. Assume $\sigma_{\min}(V)>\|\Delta V\|_2$. Let $\widehat a$ minimize $\|\widetilde Vb-\widetilde m\|_2$.}\par
932 \begin{theorem}[SG-26\quad Amplitude stability with perturbed nodes]\label{SG-26}
933 
934 \[
935  \|\widehat a-a\|_2
936  \le\frac{\|e\|_2+\|\Delta V\|_2\|a\|_2}
937  {\sigma_{\min}(V)-\|\Delta V\|_2}.
938 \]
939 For unit nodes in rows with nonnegative powers $n$, matched chord error
940 $|\widehat z_\ell-z_\ell|\le\delta$ gives the entry bound
941 $|\widehat z_\ell^n-z_\ell^n|\le n\delta$.
942 
943 \end{theorem}
944 \begin{proof}
945 
946 The minimum singular value of $\widetilde V$ is at least
947 $\sigma_{\min}(V)-\|\Delta V\|_2>0$. Thus its least-squares inverse exists.
948 Since $\widetilde V a=m+\Delta V a$,
949 \[
950  \widehat a-a=\widetilde V^\dagger(e-\Delta V a).
951 \]
952 Apply $\|\widetilde V^\dagger\|_2=1/\sigma_{\min}(\widetilde V)$.
953 The power bound follows by factoring
954 $\widehat z^n-z^n=(\widehat z-z)\sum_{k=0}^{n-1}\widehat z^{n-1-k}z^k$;
955 each summand has modulus one. The statement assumes aligned nodes and unit
956 normalization. Arbitrary unnormalized roots require extra modulus factors.
957 
958 \end{proof}
959 
960 \par{\small\color{Teal}\textit{Source lineage:} \textit{unknown}, Amplitude stability.}\par
961 
962 \section{Self-calibration of polyphase aliases}
963 
964 Decimation reorganizes the differentiated moments, and it also maps each
965 physical node $z$ to $z^N$. This can improve a selected geometry or create a
966 collision. The physical phase is retained in the offset coefficients. The
967 central recovery step uses those coefficients rather than choosing a principal
968 root by convention. The shifted-exponential ingredient is classical
969 \cite{vexpa}; the contracts below specify what is sufficient for its use in
970 the complete unknown-order pipeline.
971 
972 
973 \par{\small\textbf{Domain and hypotheses.} Derivative moments with an additive defect. Fix integers $N\ge2$, $q_0\ge1$, and an offset $r\ge0$. Suppose $m_n=\sum_\ell a_\ell z_\ell^n+e_n$, with $|z_\ell|=1$.}\par
974 \begin{theorem}[SG-27\quad Shared-node offset channels]\label{SG-27}
975 
976 \[
977  m_{(q_0+q)N+r}=\sum_\ell b_{\ell,r}u_\ell^q+e_{(q_0+q)N+r},
978  \quad u_\ell=z_\ell^N,\quad b_{\ell,r}=a_\ell u_\ell^{q_0}z_\ell^r.
979 \]
980 For each matched edge $b_{\ell,r+1}=z_\ell b_{\ell,r}$.
981 If $|\widehat s(n)|\le C_s(1+|n|)^{-p-2}$ with $p\ge0$, then for
982 nonnegative $q,r$
983 \[
984  |e_{(q_0+q)N+r}|\le2\pi C_s(q_0N)^{-p-1}.
985 \]
986 
987 \end{theorem}
988 \begin{proof}
989 
990 Split the exponent as $(q_0+q)N+r=q_0N+r+qN$ to obtain the channel
991 identity. Incrementing the offset multiplies only $z_\ell^r$ by $z_\ell$.
992 For the tail bound, use
993 $|e_n|=2\pi|n||\widehat s(n)|\le2\pi C_s(1+|n|)^{-p-1}$ and
994 $n\ge q_0N>0$. The channel identity is valid even when aliased nodes collide,
995 but a Prony solve on that channel then observes combined coefficients and
996 cannot separate those nodes using the simple distinct-node theorem.
997 
998 \end{proof}
999 
1000 \par{\small\color{Teal}\textit{Source lineage:} \textit{self}, Why decimation helps and why it aliases; \textit{complex}, One-dimensional finite-rate packet model.}\par
1001 
1002 \par{\small\textbf{Domain and hypotheses.} Finite rectangular complex matrices. Let $H=H_0+E$, with $H_0$ of rank $r\ge1$, $k=\min(h,w)\ge r$, and $\epsilon=\|E\|_2<\sigma_r(H_0)/2$. Supply a threshold $t$ with $\epsilon<t<\sigma_r(H_0)-\epsilon$.}\par
1003 \begin{theorem}[SG-28\quad Order recovery with a supplied singular threshold]\label{SG-28}
1004 
1005 Exactly $r$ singular values of $H$ exceed $t$. If $r=0$, a supplied
1006 threshold $t>\|E\|_2$ selects zero. If $r=k$, all $k$ singular values exceed
1007 the threshold and no nonexistent $\sigma_{k+1}$ is used.
1008 
1009 \end{theorem}
1010 \begin{proof}
1011 
1012 The singular-value perturbation inequality \cite{stewart} gives
1013 $|\sigma_j(H)-\sigma_j(H_0)|\le\|E\|_2$ for $1\le j\le k$.
1014 It holds equally over the complex field: in the min--max characterization
1015 \[
1016  \sigma_j(A)=\min_{\dim V=w-j+1}\max_{x\in V,\,\|x\|=1}\|Ax\|.
1017 \]
1018 the triangle inequality bounds each restricted maximum for $A+E$ by that
1019 for $A$ plus $\|E\|_2$. Minimize and then interchange the two matrices.
1020 For $j\le r$ this yields $\sigma_j(H)\ge\sigma_r(H_0)-\epsilon>t$.
1021 For $j>r$, when such indices exist, it gives $\sigma_j(H)\le\epsilon<t$.
1022 For rank zero the same upper bound applies to every singular value. For full
1023 rectangular rank the lower bound already accounts for every singular value.
1024 The theorem assumes a threshold satisfying the inequalities; it does not
1025 prove that every heuristic gap selector finds it.
1026 
1027 \end{proof}
1028 
1029 \par{\small\color{Teal}\textit{Source lineage:} \textit{self}, Unknown jump count from a Hankel gap.}\par
1030 
1031 \subsection{Why an empirical rank rule is a separate algorithm}
1032 
1033 The native implementation estimates order from singular-value ratios and
1034 validation heuristics. A largest adjacent ratio is not implied to select the
1035 true order by SG-28. An exact witness makes the distinction concrete. Take
1036 four-by-four Hankel matrices of moments beginning at $n_0=4$ from the nodes
1037 $(1,i,-1)$ and real amplitudes $(100,-99,-1)$. The four-point Fourier columns
1038 are orthogonal, so the true singular values are $(400,396,4,0)$. Add the
1039 moment perturbation $0.1(-i)^n$. It contributes a fourth orthogonal column and
1040 produces singular values $(400,396,4,0.4)$ with perturbation norm $0.4$.
1041 The hypothesis $0.4<4/2$ holds. A threshold in $(0.4,3.6)$ selects the true
1042 order three, while the largest adjacent ratio $396/4=99$ selects order two.
1043 Thus a valid quantitative threshold theorem and a successful practical
1044 heuristic remain different claims.
1045 
1046 
1047 \par{\small\textbf{Domain and hypotheses.} One matched unit-circle coefficient progression. Let $R\ge1$ be an integer counting adjacent products, and let $c_r=bz^r$ for $0\le r\le R$, with $b\ne0$ and $|z|=1$. Define $S(c)=\sum_{r=0}^{R-1}\overline{c_r}c_{r+1}$.}\par
1048 \begin{theorem}[SG-29\quad Exact phase from adjacent offset products]\label{SG-29}
1049 
1050 \[
1051  S(c)=R|b|^2z\ne0,\qquad\Phi(c):=S(c)/|S(c)|=z.
1052 \]
1053 Two channels already give $z=c_1/c_0$ exactly. More channels provide repeated
1054 adjacent relations, without an unconditional deterministic averaging gain.
1055 
1056 \end{theorem}
1057 \begin{proof}
1058 
1059 Each summand is
1060 $\overline{bz^r}\,bz^{r+1}=|b|^2z$ because $|z|=1$.
1061 There are $R$ summands and $R\ge1$, so normalization returns $z$.
1062 For two channels direct division cancels $b$. No statistical independence is
1063 assumed, so one cannot improve a worst-case noise order merely by increasing
1064 $R$; the next bound shows the cancellation of the number of products.
1065 
1066 \end{proof}
1067 
1068 \par{\small\color{Teal}\textit{Source lineage:} \textit{self}, Offset coefficients and internal phase recovery.}\par
1069 
1070 \par{\small\textbf{Publication correction.} R denotes the number of adjacent products, so R+1 offset channels are used and R is at least one. An empty sum has no phase.}\par
1071 
1072 \par{\small\textbf{Domain and hypotheses.} A noisy progression as in SG-29. Let $c_r=bz^r+e_r$ for $0\le r\le R$, with $R\ge1$, $b\ne0$, $|z|=1$, and $|e_r|\le\epsilon$. Put $A=R|b|^2$ and $\Delta=R(2|b|\epsilon+\epsilon^2)$. Assume $\Delta<A$.}\par
1073 \begin{theorem}[SG-30\quad Deterministic offset-phase perturbation]\label{SG-30}
1074 
1075 The phase sum is nonzero and
1076 \[
1077  |\Phi(c)-z|\le\frac{2\Delta}{A}
1078  =\frac{4\epsilon}{|b|}+2\left(\frac\epsilon{|b|}\right)^2.
1079 \]
1080 
1081 \end{theorem}
1082 \begin{proof}
1083 
1084 Expand each product:
1085 \[
1086  \overline{c_r}c_{r+1}=|b|^2z
1087  +\overline{bz^r}e_{r+1}+\overline{e_r}bz^{r+1}
1088  +\overline{e_r}e_{r+1}.
1089 \]
1090 The three error terms have total magnitude at most
1091 $2|b|\epsilon+\epsilon^2$. Summing gives $S(c)=Az+E$ with $|E|\le\Delta$.
1092 Therefore $|S(c)|\ge A-\Delta>0$.
1093 For any nonzero $v$ and unit $z$,
1094 \[
1095  \left|\frac v{|v|}-z\right|
1096  \le\left|\frac v{|v|}-\frac vA\right|+\left|\frac vA-z\right|
1097  =\frac{\bigl|A-|v|\bigr|}{A}+\frac{|v-Az|}{A}
1098  \le\frac{2|v-Az|}{A}.
1099 \]
1100 Apply this inequality to $v=S(c)$ and substitute the definitions.
1101 The number of products cancels, as appropriate for coherent adversarial errors.
1102 
1103 \end{proof}
1104 
1105 \par{\small\color{Teal}\textit{Source lineage:} \textit{self}, Deterministic phase perturbation; Detailed algebra for the multi-offset estimator.}\par
1106 
1107 \par{\small\textbf{Domain and hypotheses.} Finite complex offset data. Let $R\ge1$, $c_0\ne0$, and $|\zeta|=1$. Define $\nu(c,\zeta)=\bigl(\sum_{r=0}^{R-1}|c_{r+1}-\zeta c_r|^2\bigr)^{1/2}/\bigl(\sum_{r=0}^{R-1}|c_r|^2\bigr)^{1/2}$.}\par
1108 \begin{theorem}[SG-31\quad A zero offset residual tests a progression]\label{SG-31}
1109 
1110 Then $\nu(c,\zeta)=0$ if and only if $c_r=c_0\zeta^r$ for
1111 $0\le r\le R$.
1112 
1113 \end{theorem}
1114 \begin{proof}
1115 
1116 The denominator is positive because it includes $|c_0|^2$.
1117 A sum of nonnegative squared magnitudes is zero precisely when each recurrence
1118 $c_{r+1}=\zeta c_r$ holds. Induction gives the progression. The reverse
1119 implication follows by substitution. This is a consistency check for the
1120 fitted offsets. Another physical model or structured background could produce
1121 the same progression; zero residual alone does not certify the true edge
1122 model or a true-location error bound. Unused offset channels provide a
1123 stronger practical check because they are not part of the fit, but no
1124 universal probability guarantee is asserted here.
1125 
1126 \end{proof}
1127 
1128 \par{\small\color{Teal}\textit{Source lineage:} \textit{self}, A posteriori phase validation.}\par
1129 
1130 \par{\small\textbf{Domain and hypotheses.} Nth roots of a unit-modulus estimate. Let $N\ge2$, $|z|=1$, $u=z^N$, and $\overline u=\widehat u/|\widehat u|$ with $\widehat u\ne0$. Let $\rho$ be a root of $\overline u$ continuously associated with $z$, and $\widetilde z$ a coarse phase estimate.}\par
1131 \begin{theorem}[SG-32\quad Unique branch selection under a strict margin]\label{SG-32}
1132 
1133 If
1134 \[
1135  |\widetilde z-z|+|\rho-z|<\sin(\pi/N),
1136 \]
1137 then $\rho$ is the unique root of $\overline u$ nearest to $\widetilde z$.
1138 It equals the branch an oracle chooses by the true-node neighborhood.
1139 
1140 \end{theorem}
1141 \begin{proof}
1142 
1143 The $N$ candidates lie on the unit circle at angles spaced by $2\pi/N$.
1144 The smallest distance between distinct candidates is $2\sin(\pi/N)$.
1145 The triangle inequality gives
1146 $|\widetilde z-\rho|\le|\widetilde z-z|+|z-\rho|<\sin(\pi/N)$.
1147 For any other candidate $\rho'$,
1148 $|\widetilde z-\rho'|\ge|\rho'-\rho|-|\rho-\widetilde z|
1149 >\sin(\pi/N)$. Thus the nearest candidate is unique. The same spacing
1150 argument places every wrong root farther from the true node than the
1151 associated root under the given margin. The oracle and data-based choices
1152 therefore coincide. This is conditional on the margin; the true-node terms
1153 are not available directly to the algorithm.
1154 
1155 \end{proof}
1156 
1157 \par{\small\color{Teal}\textit{Source lineage:} \textit{self}, Branch lifting from aliased nodes.}\par
1158 
1159 \par{\small\textbf{Publication correction.} The candidate-spacing and branch-margin statement explicitly assumes N at least two.}\par
1160 
1161 \par{\small\textbf{Domain and hypotheses.} Unit-modulus nodes and their associated root branches. Let $N\ge1$, $z^N=u$, $|z|=|u|=|\overline u|=1$, and $|\overline u-u|\le\eta<2$. Let $\rho$ be the associated root along the shortest angular perturbation.}\par
1162 \begin{theorem}[SG-33\quad Associated roots divide the angular error]\label{SG-33}
1163 
1164 \[
1165  |\rho-z|\le2\sin\left(\frac1N\arcsin\frac\eta2\right)
1166  \le\frac2N\arcsin\frac\eta2.
1167 \]
1168 For a nonzero raw estimate, $|\widehat u-u|\le\epsilon$ implies
1169 $|\widehat u/|\widehat u|-u|\le2\epsilon$. Thus when the raw error tends to
1170 zero, correct lifting gains a factor $1/N$ in chord and angular accuracy.
1171 
1172 \end{theorem}
1173 \begin{proof}
1174 
1175 If $\theta\in(-\pi,\pi)$ is the shortest angular difference, the unit-circle
1176 chord identity gives $|\overline u-u|=2\sin(|\theta|/2)$.
1177 Hence $|\theta|\le2\arcsin(\eta/2)$. On the associated branch the root
1178 angle changes by $\theta/N$. Convert to chord distance and use $\sin t\le t$.
1179 For normalization, add and subtract $\widehat u$:
1180 \[
1181  |\widehat u/|\widehat u|-u|
1182  \le|1-|\widehat u||+|\widehat u-u|
1183  \le2|\widehat u-u|.
1184 \]
1185 The shortest-angle ambiguity at an antipodal estimate is excluded by
1186 $\eta<2$. The gain applies to an associated branch, so a branch-selection
1187 argument such as SG-32 is needed to transfer it to an estimator.
1188 
1189 \end{proof}
1190 
1191 \par{\small\color{Teal}\textit{Source lineage:} \textit{self}, One-order root gain; Detailed algebra for the branch lift.}\par
1192 
1193 \par{\small\textbf{Domain and hypotheses.} Bandlimited background with a bounded number of jumps. Let $f\in\mathcal U_{K,L_{\max}}$, with $L_{\max}\ge1$. Supply $N\ge2$, $q_0\ge1$ with $q_0N>K$, and assume the active aliased nodes $u_\ell=z_\ell^N$ are distinct.}\par
1194 \begin{theorem}[SG-34\quad An explicit exact self-calibration budget]\label{SG-34}
1195 
1196 The Fourier coefficients at the high indices
1197 \[
1198  \mathcal I_0=\{(q_0+q)N:0\le q\le2L_{\max}\},\qquad
1199  \mathcal I_1=\{(q_0+q)N+1:0\le q<L_{\max}\}
1200 \]
1201 determine the unknown edge order, every physical node, and every jump amplitude
1202 exactly. The high-frequency budget is $3L_{\max}+1$. Low modes
1203 $-K\le n\le K$ then give the complete signal. No supplied jump neighborhood
1204 is required.
1205 
1206 \end{theorem}
1207 \begin{proof}
1208 
1209 All high indices exceed $K$, so the derivative moments have no smooth defect.
1210 The base channel is an exponential sum in the distinct $u_\ell$ with
1211 coefficients $b_{\ell,0}=a_\ell u_\ell^{q_0}\ne0$.
1212 Its $2L_{\max}+1$ moments form an $(L_{\max}+1)$ square Hankel matrix;
1213 SG-19 gives its rank $r$. If $r=0$, the edge is zero and the low modes finish
1214 the recovery. Otherwise SG-20 recovers the $u_\ell$ and then the base
1215 coefficients. The adjacent channel supplies at least $r$ moments of
1216 $\sum_\ell b_{\ell,0}z_\ell u_\ell^q$. Using the same recovered nodes, its
1217 Vandermonde system recovers $b_{\ell,1}=b_{\ell,0}z_\ell$.
1218 Division gives $z_\ell=b_{\ell,1}/b_{\ell,0}$, and
1219 $a_\ell=b_{\ell,0}u_\ell^{-q_0}$.
1220 SG-21 constructs $j$, and subtraction from the low modes constructs $s$.
1221 The two index sets are disjoint because $N\ge2$ and their residues are zero
1222 and one. Their sizes add to $3L_{\max}+1$. This is an upper bound for the
1223 construction and excludes the low reconstruction coefficients.
1224 
1225 \end{proof}
1226 
1227 \par{\small\color{Teal}\textit{Source lineage:} \textit{self}, Self-calibrating reconstruction theorem; Explicit finite innovation budget.}\par
1228 
1229 \par{\small\textbf{Publication correction.} The budget is conditional on one supplied noncolliding decimation. A finite search over candidate decimations is not guaranteed to contain such a decimation without an additional existence hypothesis.}\par
1230 
1231 \par{\small\textbf{Domain and hypotheses.} A selected sequence of decimation scales. Let $p\ge0$, fix $q_0$ and the finite number of channel samples and offsets, and take a sequence $N\to\infty$. Assume distinct aliased nodes, uniform rank-threshold gaps/local inverse bounds for the aligned Prony and coefficient fits, $|b_{\ell,0}|\ge b_{\min}>0$, uniformly bounded amplitudes, and the strict branch margin of SG-32. Suppose the moment defect is $O(N^{-p-1})$.}\par
1232 \begin{theorem}[SG-35\quad Conditional asymptotic rates after calibrated lifting]\label{SG-35}
1233 
1234 The threshold-selected order is eventually correct. The aligned lifted nodes
1235 and locations satisfy
1236 \[
1237  |\widehat z_\ell-z_\ell|=O(N^{-p-2}),\qquad
1238  d_{\mathbb T}(\widehat\tau_\ell,\tau_\ell)=O(N^{-p-2}).
1239 \]
1240 The recovered amplitudes satisfy
1241 $|\widehat a_\ell-a_\ell|=O(N^{-p-1})$.
1242 
1243 \end{theorem}
1244 \begin{proof}
1245 
1246 Uniform local inverse bounds transfer the moment error to aliased node and
1247 matched coefficient errors of order $N^{-p-1}$. The supplied rank gap and
1248 threshold select the correct order by SG-28. Nonvanishing base coefficients
1249 allow SG-30 to give the same order for the coarse phase error.
1250 The strict branch margin selects the associated root by SG-32. Normalize the
1251 aliased estimate and use SG-33: its root error is $N^{-1}$ times an error of
1252 order $N^{-p-1}$, giving $N^{-p-2}$. On the unit circle,
1253 $d_{\mathbb T}=2\arcsin(|\widehat z-z|/2)$, which has the same order for
1254 small errors. For the amplitude use
1255 $\widehat a=\widehat b_0\,\overline u^{-q_0}$ and $a=b_0u^{-q_0}$.
1256 Since $q_0$ is fixed and both powers have unit modulus, SG-26's telescoping
1257 power estimate bounds the power error by $q_0|\overline u-u|$.
1258 Together with the coefficient error this gives the stated amplitude order.
1259 The smooth-tail assumption in SG-27 supplies the moment defect.
1260 The conclusion is uniform only on the selected scales and parameter families
1261 for which the inverse, rank, and branch bounds are uniform.
1262 
1263 \end{proof}
1264 
1265 \par{\small\color{Teal}\textit{Source lineage:} \textit{self}, Rate-preserving polyphase lifting; Perturbed self-calibration for a non-bandlimited background.}\par
1266 
1267 \par{\small\textbf{Publication correction.} For p=0 the coarse phase error and root spacing are both of order 1/N, so the branch margin is retained as an explicit hypothesis. Physical separation alone does not imply uniform aliased separation for every N; rates are asserted only along admissible selected scales.}\par
1268 
1269 
1270 The native pipeline does not receive the true order or jump neighborhoods.
1271 It tests several decimations, fits common nodes, uses offsets for branch
1272 selection, and performs validation and optional nonlinear refinement.
1273 Those algorithmic choices are supported by the displayed conditional
1274 contracts and finite experiments. The rate theorem does not assert that a
1275 largest-gap heuristic, every decimation, or every non-bandlimited signal
1276 satisfies its hypotheses.
1277 
1278 
1279 \section{Native edge synthesis and a corrected perturbation estimate}
1280 
1281 
1282 \par{\small\textbf{Domain and hypotheses.} A normed signal space. Let $f=s+j$, let $\widehat j$ belong to the same space, and let $A_M$ be a bounded linear operator with norm at most $C_M$. Define $\widehat f_M=A_M(f-\widehat j)+\widehat j$.}\par
1283 \begin{theorem}[SG-36\quad Resolved synthesis splits the two error sources]\label{SG-36}
1284 
1285 \[
1286  f-\widehat f_M=(s-A_Ms)+(I-A_M)(j-\widehat j),
1287 \]
1288 and therefore
1289 \[
1290  \|f-\widehat f_M\|\le\|s-A_Ms\|+(1+C_M)\|j-\widehat j\|.
1291 \]
1292 If $\widehat j=j$, the complete error equals the smooth approximation error.
1293 
1294 \end{theorem}
1295 \begin{proof}
1296 
1297 Substitute $f=s+j$ into the definition and collect the two terms. Linearity
1298 of $A_M$ is used in that collection. The triangle inequality and
1299 $\|I-A_M\|\le1+\|A_M\|$ give the estimate. With the exact edge, its error
1300 term vanishes as an object identity. The theorem holds for the declared norm;
1301 a misplaced discontinuity need not be small in supremum norm even when its
1302 location error is small. In $L^1$, the next theorem quantifies that error.
1303 A nonlinear clipping operator is not an instance of this linear identity.
1304 
1305 \end{proof}
1306 
1307 \par{\small\color{Teal}\textit{Source lineage:} \textit{self}, A stable synthesis theorem; \textit{complex}, Resolved synthesis.}\par
1308 
1309 \par{\small\textbf{Domain and hypotheses.} Mean-zero periodic packets of period $T>0$. Let $j,\widehat j$ have matched real jump data $(a_\ell,\tau_\ell)$ and $(\widehat a_\ell,\widehat\tau_\ell)$, with both amplitude sums zero. Use their canonical representatives and let $d_\ell=d_T(\tau_\ell,\widehat\tau_\ell)\le T/2$.}\par
1310 \begin{theorem}[SG-37\quad Sharp additive L1 control for normalized jump packets]\label{SG-37}
1311 
1312 \[
1313  \|j-\widehat j\|_1
1314  \le\frac T4\sum_\ell|a_\ell-\widehat a_\ell|
1315  +2\sum_\ell|\widehat a_\ell|d_\ell(1-d_\ell/T).
1316 \]
1317 In particular the simpler location term $2\sum_\ell|\widehat a_\ell|d_\ell$
1318 is valid. The printed location coefficient one in the inherited
1319 self-calibration manuscript is not valid for mean-zero normalization.
1320 
1321 \end{theorem}
1322 \begin{proof}
1323 
1324 SG-21 writes both packets as translated sawtooth sums. Subtract in the form
1325 \[
1326  j-\widehat j=
1327  \sum_\ell(a_\ell-\widehat a_\ell)G_T(\cdot-\tau_\ell)
1328  +\sum_\ell\widehat a_\ell
1329  [G_T(\cdot-\tau_\ell)-G_T(\cdot-\widehat\tau_\ell)].
1330 \]
1331 Direct integration gives $\|G_T\|_1=T/4$.
1332 For a shift by $d\in[0,T/2]$, on an interval of length $T-d$ the difference
1333 has magnitude $d/T$, and on an interval of length $d$ it has magnitude
1334 $1-d/T$. Hence its $L^1$ norm is
1335 $(T-d)d/T+d(1-d/T)=2d(1-d/T)$.
1336 Apply the triangle inequality term by term. Periodic translation invariance
1337 makes the calculation depend only on circular shift distance.
1338 To refute coefficient one, take
1339 \[
1340  j=\mathbf1_{[0,T/2)}-\tfrac12,\qquad
1341  \widehat j=\mathbf1_{[T/4,T/2)}-\tfrac14.
1342 \]
1343 Both have jumps $(1,-1)$, with one jump shifted by $T/4$. Their norm
1344 difference is $3T/8$, while the inherited bound gives $T/4$.
1345 The sharper location term is attained in this example. The amplitude
1346 coefficient $T/4$ is also attained by changing opposite antipodal jump
1347 amplitudes: the difference packet has constant magnitude half the amplitude
1348 change on the two half-periods.
1349 
1350 \end{proof}
1351 
1352 \par{\small\color{Teal}\textit{Source lineage:} \textit{self}, A stable synthesis theorem.}\par
1353 
1354 \par{\small\textbf{Publication correction.} The old L1 estimate is refuted by two mean-zero interval packets. The published bound includes the normalization-induced constant shift and uses the exact translated-sawtooth norm.}\par
1355 
1356 \par{\small\textbf{Domain and hypotheses.} The rate family of SG-35 with $T=2\pi$. Assume a fixed finite jump count, bounded jump amplitudes, the hypotheses of SG-35, and a bounded linear smooth reconstruction $A_M$ acting on $L^1(\mathbb T)$. Require $\widehat j$ to be the canonical packet of real matched estimated amplitudes whose sum is zero. Let $b_M=\|s-A_Ms\|_1$ and $C_M=\|A_M\|_{L^1\to L^1}$.}\par
1357 \begin{theorem}[SG-38\quad A conditional global L1 reconstruction rate]\label{SG-38}
1358 
1359 \[
1360  \|j-\widehat j\|_1=O(N^{-p-1}),\qquad
1361  \|f-\widehat f_M\|_1\le b_M+O((1+C_M)N^{-p-1}).
1362 \]
1363 For Fourier truncation, $C_M=O(1+\log(M+1))$. If also
1364 $|\widehat s(n)|\le C_s(1+|n|)^{-p-2}$, its smooth remainder satisfies
1365 $b_M=O(M^{-p-1})$. Taking $M$ comparable to $N$ yields
1366 $O((1+\log N)N^{-p-1})$.
1367 
1368 \end{theorem}
1369 \begin{proof}
1370 
1371 SG-35 gives an amplitude error of order $N^{-p-1}$ and a location error of
1372 order $N^{-p-2}$. If the fitted amplitudes are complex or do not sum to zero,
1373 take their real parts and subtract their common arithmetic mean. Since the
1374 true real amplitudes sum to zero, this changes each amplitude error by at
1375 most another maximum amplitude error, preserving its order. Apply SG-21 to
1376 the resulting real zero-sum data to define $\widehat j$. Insert these errors into SG-37, with finitely many bounded
1377 amplitudes, to obtain the edge norm bound. SG-36 then gives the general
1378 synthesis estimate.
1379 For Fourier truncation, its convolution norm is bounded by the normalized
1380 $L^1$ norm of $D_M$. On $[-\pi,\pi]$ use
1381 $|D_M(t)|\le\min(2M+1,\pi/|t|)$, with the removable value at zero.
1382 Split the integral at $1/(M+1)$ to obtain the logarithmic bound.
1383 The smooth Fourier tail has supremum bound
1384 $2C_s\sum_{n>M}(1+n)^{-p-2}=O(M^{-p-1})$ by an integral comparison;
1385 multiplication by $2\pi$ gives the $L^1$ estimate.
1386 This conclusion accounts for the norm growth of partial sums instead of
1387 assuming a uniformly bounded Fourier truncation operator on $L^1$.
1388 
1389 \end{proof}
1390 
1391 \par{\small\color{Teal}\textit{Source lineage:} \textit{self}, A stable synthesis theorem; Limitations and next theorem.}\par
1392 
1393 \section{Shared edges, jump jets, and geometric image models}
1394 
1395 
1396 \par{\small\textbf{Domain and hypotheses.} A finite collection of exponential moment channels. Let $m_n^{(c)}=\sum_{\ell=1}^ra_{c\ell}z_\ell^n$, with distinct common unit-modulus nodes. At each node at least one channel amplitude is nonzero. Form the stacked linear recurrence system for a monic polynomial of degree $r$ and assume its coefficient matrix has column rank $r$.}\par
1397 \begin{theorem}[SG-39\quad Shared-location multichannel recovery]\label{SG-39}
1398 
1399 The stacked system uniquely recovers the common annihilator
1400 $Q(z)=\prod_\ell(z-z_\ell)$. Its roots recover the common locations.
1401 With $r$ consecutive moments per channel, a channelwise Vandermonde solve
1402 recovers every amplitude, including amplitudes that vanish in that channel.
1403 
1404 \end{theorem}
1405 \begin{proof}
1406 
1407 The polynomial $Q$ annihilates every exponential term and therefore every
1408 channel, as in SG-20. Full column rank of the stacked recurrence matrix means
1409 that a consistent coefficient vector is unique; the true $Q$ provides one.
1410 Its distinct roots give the nodes. Once known, the same nonsingular
1411 Vandermonde system solves each channel independently. No individual channel
1412 has to see all nodes, but the stated stacked rank is essential. Merely
1413 collecting channels does not prove that rank condition automatically.
1414 
1415 \end{proof}
1416 
1417 \par{\small\color{Teal}\textit{Source lineage:} \textit{unknown}, Several channels with shared edge locations.}\par
1418 
1419 \par{\small\textbf{Domain and hypotheses.} Polynomial-exponential sequences. Let $m_n=\sum_{\ell=1}^rz_\ell^nP_\ell(n)$ with distinct nonzero nodes and $\deg P_\ell\le d$. Let $E$ be the forward shift. For an offset ratio assume $N\ge1$ and the leading coefficient of $P_\ell$ at degree $d$ is nonzero.}\par
1420 \begin{theorem}[SG-40\quad Confluent annihilation and the leading offset phase]\label{SG-40}
1421 
1422 \[
1423  \prod_{\ell=1}^r(E-z_\ell)^{d+1}m=0.
1424 \]
1425 For offset $v\ge0$, writing a decimated channel as
1426 \[
1427  m_{(q_0+q)N+v}=\sum_\ell u_\ell^q
1428                    \sum_{h=0}^d\gamma_{\ell,v,h}q^h,
1429  \qquad u_\ell=z_\ell^N,
1430 \]
1431 the highest-degree coefficients satisfy
1432 $\gamma_{\ell,1,d}/\gamma_{\ell,0,d}=z_\ell$.
1433 This ratio applies to separated, matched confluent terms with nonzero leading
1434 coefficient. The displayed annihilator is not asserted to have minimal order
1435 when leading coefficients vanish.
1436 
1437 \end{theorem}
1438 \begin{proof}
1439 
1440 For one term, $(E-z)(z^nP(n))=z^{n+1}(P(n+1)-P(n))$.
1441 A finite difference lowers a nonconstant polynomial's degree by one, and
1442 annihilates a constant. After $d+1$ differences the term vanishes. The shift
1443 factors commute, so the product annihilates every term in the sum.
1444 If $P_\ell(n)$ has leading coefficient $c_\ell\ne0$ at degree $d$, substitution
1445 of $(q_0+q)N+v$ has leading $q$ coefficient $c_\ell N^d$.
1446 The remaining exponential prefactor is $z_\ell^{q_0N+v}$; hence
1447 $\gamma_{\ell,v,d}=c_\ell N^dz_\ell^{q_0N+v}$ and the ratio is $z_\ell$.
1448 Repeated roots require confluent recovery and stability arguments; the
1449 simple-root disk theorem SG-25 cannot be used to claim one perturbed root
1450 per physical node or a Lipschitz individual-root bound at a multiple root.
1451 
1452 \end{proof}
1453 
1454 \par{\small\color{Teal}\textit{Source lineage:} \textit{unknown}, Piecewise-polynomial packets; \textit{self}, Higher jump jets and confluent Prony systems.}\par
1455 
1456 \par{\small\textbf{Publication correction.} Annihilation is proved without a minimal-order assertion. Phase division requires a nonzero highest-degree coefficient, and simple-root perturbation claims are not transferred to repeated roots.}\par
1457 
1458 \par{\small\textbf{Domain and hypotheses.} A finite-dimensional background-edge model. Use $A=RD$ from SG-14 and assume $s\ge1$, $d\ge1$, and its restricted constant $\gamma>0$. Suppose $y=s_0+D\alpha_0+e+\eta$, with $s_0\in\mathcal S$, $\|\alpha_0\|_0\le s$, and $\|R(e+\eta)\|_2\le\epsilon$. Let $\widehat\alpha$ be $s$-sparse with $\|A\widehat\alpha-Ry\|_2\le\epsilon$.}\par
1459 \begin{theorem}[SG-41\quad Model mismatch enters the same sparse stability bound]\label{SG-41}
1460 
1461 \[
1462  \|\widehat\alpha-\alpha_0\|_2\le2\epsilon/\gamma.
1463 \]
1464 If an actual edge $j$ obeys $\|j-D\alpha_0\|_2\le\delta$, then
1465 \[
1466  \|j-D\widehat\alpha\|_2\le\delta+2\|D\|_2\epsilon/\gamma.
1467 \]
1468 The theorem applies to finite image dictionaries under these printed
1469 sparsity, residual, and restricted-injectivity hypotheses.
1470 
1471 \end{theorem}
1472 \begin{proof}
1473 
1474 The nuisance $e+\eta$ has exactly the role of the residualized noise in
1475 SG-15, so its proof gives the coefficient bound. Add and subtract
1476 $D\alpha_0$ in the edge difference and apply the operator norm bound to the
1477 coefficient error. A best approximation can be substituted when it exists;
1478 otherwise any admissible approximate minimizer gives the same statement.
1479 The theorem does not show that a greedy finite dictionary selector outputs
1480 an $s$-sparse vector with the specified tolerance. Nor does it establish
1481 restricted injectivity for every geometric dictionary; both remain
1482 hypotheses to verify for a concrete model.
1483 
1484 \end{proof}
1485 
1486 \par{\small\color{Teal}\textit{Source lineage:} \textit{complex}, Two-dimensional finite dictionary model; A theorem for model mismatch.}\par
1487 
1488 \subsection{What the image experiments test}
1489 
1490 The complex benchmark uses a low-frequency DCT background and 318 geometric
1491 atoms: rectangles, disks, ellipses, and horizontal and vertical bars.
1492 The named exact-model images and forty archived random dictionary cases select
1493 the intended supports under their deterministic settings. Those are finite
1494 results. SG-14 and SG-41 explain the contract a universal statement about
1495 such a library would require; the benchmark does not establish all its
1496 restricted singular values or make a greedy selector universally exact.
1497 
1498 The seven-point star, text over a gradient, and nested rings lie outside the
1499 fixed dictionary. Their comparisons are retained as model-mismatch tests.
1500 Four natural images are exploratory examples, not instances of a finite-rate
1501 exact edge theorem. In the natural-background hybrid the selector returns
1502 five atoms for three true overlays. Background structure outside the chosen
1503 DCT space is being explained as extra geometry. This exposes a background
1504 model failure rather than just an edge-library failure.
1505 
1506 
1507 \section{Countable packet grammars and finite image approximation}
1508 
1509 A countable grammar can describe an unlimited ideal family while a program
1510 uses a finite prefix or a finite store. The infinite packet code in this
1511 bundle performs adaptive approximation of an available pixel image. Its
1512 constructor receives the full image, and residual seeds are generated from
1513 that image minus the current reconstruction. This is a different observation
1514 model from the finitely measured Fourier recovery theorem.
1515 The exact statements below specify what survives in the ideal mathematical
1516 model and what changes under the native ridge and clipping operations.
1517 
1518 
1519 \par{\small\textbf{Domain and hypotheses.} A Hilbert space with finite selected packet spans. Let $V_1\subseteq V_2\subseteq\cdots$ be finite-dimensional subspaces. Let $P_k$ be the orthogonal projector onto $V_k$.}\par
1520 \begin{theorem}[SG-42\quad Ideal nested-span refitting decreases residual]\label{SG-42}
1521 
1522 \[
1523  \|f-P_{k+1}f\|\le\|f-P_kf\|,
1524  \qquad f-P_kf\perp V_k.
1525 \]
1526 If $\bigcup_kV_k$ is dense, then $P_kf\to f$ in norm.
1527 
1528 \end{theorem}
1529 \begin{proof}
1530 
1531 Orthogonal projection minimizes distance to its subspace. Since $P_kf$
1532 belongs to the larger subspace $V_{k+1}$, its residual is a feasible upper
1533 bound on the next minimum. Orthogonality follows from the defining projection
1534 identity, or by varying the quadratic norm along every vector in $V_k$.
1535 For convergence, fix $\epsilon>0$ and choose $v\in V_K$ within $\epsilon$
1536 of $f$ by density. For all $k\ge K$, minimum distance is at most
1537 $\|f-v\|<\epsilon$. Thus the residual converges to zero.
1538 This theorem assumes dense actual selected spans. A countably dense candidate
1539 pool alone does not show that a particular greedy selection sequence becomes
1540 dense.
1541 
1542 \end{proof}
1543 
1544 \par{\small\color{Teal}\textit{Source lineage:} \textit{infinite}, InfinitePacketReconstructor.\_refit; InfinitePacketGenerator.}\par
1545 
1546 \par{\small\textbf{Domain and hypotheses.} Finite-dimensional real packet designs. Let $M\in\mathbb R^{m\times d}$, $y\in\mathbb R^m$, and $\lambda>0$. Let $c$ minimize $\|y-Mc\|_2^2+\lambda\|c\|_2^2$ and set $r=y-Mc$.}\par
1547 \begin{theorem}[SG-43\quad Ridge refitting has a different stationarity law]\label{SG-43}
1548 
1549 The minimizer is unique and
1550 \[
1551  (M^{\mathsf T}M+\lambda I)c=M^{\mathsf T}y,
1552  \qquad M^{\mathsf T}r=\lambda c.
1553 \]
1554 Adding candidate columns cannot increase the minimized ridge objective,
1555 but exact orthogonality $M^{\mathsf T}r=0$ is not the ridge stationarity law.
1556 Even a complete pixel basis need not give zero residual when $\lambda>0$.
1557 
1558 \end{theorem}
1559 \begin{proof}
1560 
1561 The Hessian is twice $M^TM+\lambda I$, which is positive definite because
1562 $v^T(M^TM+\lambda I)v=\|Mv\|^2+\lambda\|v\|^2>0$ for nonzero $v$.
1563 Setting the gradient to zero gives the unique minimizer and rearrangement gives
1564 $M^Tr=\lambda c$. When columns are added, extending the previous coefficient
1565 vector by zero is a feasible point with exactly the previous objective, so the
1566 new optimum is no larger. This concerns the ridge objective, not an automatic
1567 monotonicity theorem for the unpenalized residual alone.
1568 For the full pixel basis $M=I$, $c=y/(1+\lambda)$ and
1569 $r=\lambda y/(1+\lambda)$, which is nonzero for $y\ne0$.
1570 The native default $\lambda=10^{-9}$ makes this residual small, not
1571 mathematically zero. Numerical tolerances must remain numerical tolerances.
1572 
1573 \end{proof}
1574 
1575 \par{\small\color{Teal}\textit{Source lineage:} \textit{infinite}, InfinitePacketReconstructor.\_refit; default ridge=1e-9.}\par
1576 
1577 \par{\small\textbf{Publication correction.} Claims of exact unregularized refitting are separated from the implemented positive-ridge solve. The native normal-equation law includes the ridge term.}\par
1578 
1579 \par{\small\textbf{Domain and hypotheses.} Real pixel arrays with Euclidean norms. Define $C(x)$ coordinatewise by $C(t)=\min(1,\max(0,t))$. Suppose the truth $f$ has every coordinate in $[0,1]$.}\par
1580 \begin{theorem}[SG-44\quad Clipping is nonexpansive toward a bounded truth]\label{SG-44}
1581 
1582 \[
1583  \|C(x)-C(y)\|_2\le\|x-y\|_2,\qquad
1584  \|C(x)-f\|_2\le\|x-f\|_2.
1585 \]
1586 Clipping is nonlinear; it is not the bounded linear synthesis operator in
1587 SG-36. Its truth-error guarantee depends on the stated box constraint.
1588 
1589 \end{theorem}
1590 \begin{proof}
1591 
1592 For real scalars the clipping function is nondecreasing with slope between
1593 zero and one on each of its three affine pieces. A direct case split at zero
1594 and one gives $|C(x)-C(y)|\le|x-y|$ for every pair.
1595 Square and sum the scalar inequalities. Since $C(f)=f$, substitute $y=f$ to
1596 obtain the second estimate. Nonlinearity is explicit: $C(1+1)=1$ whereas
1597 $C(1)+C(1)=2$. If the truth lies outside the box, clipping can increase its
1598 error; for instance $x=f=2$ becomes $C(x)=1$.
1599 
1600 \end{proof}
1601 
1602 \par{\small\color{Teal}\textit{Source lineage:} \textit{infinite}, bounded\_synthesis.}\par
1603 
1604 \par{\small\textbf{Domain and hypotheses.} Any normed signal space. Let an approximation $q$ to an available object $f$ be supplied, and retain the complete residual $r=f-q$.}\par
1605 \begin{theorem}[SG-45\quad An exact residual archive preserves the object]\label{SG-45}
1606 
1607 The synthesis $q+r=f$ is exact. This identity supplies no sparsity, compression,
1608 or recovery-from-fewer-measurements conclusion. In finite pixel space the
1609 full pixel basis likewise spans every image in the ideal unregularized model.
1610 
1611 \end{theorem}
1612 \begin{proof}
1613 
1614 The first statement is the defining subtraction identity. Retaining $r$
1615 can carry every component omitted from $q$, so its data cost is not bounded by
1616 a small number of packet coefficients. For an image $f\in\mathbb R^m$, the
1617 standard pixel vectors satisfy $f=\sum_{j=1}^mf_je_j$ and form a complete
1618 finite basis. Those coefficients are all the pixels themselves.
1619 A program using the truth residual thus certifies faithful storage of its
1620 available input, rather than uniqueness under a smaller measurement map.
1621 
1622 \end{proof}
1623 
1624 \par{\small\color{Teal}\textit{Source lineage:} \textit{infinite}, exact\_archive\_residual; benchmark\_case.}\par
1625 
1626 \par{\small\textbf{Domain and hypotheses.} Normed spaces and bounded linear maps. Let $D$ be a dense linear subspace of $X$, and let $A,B:X\to Y$ be bounded linear operators agreeing on $D$.}\par
1627 \begin{theorem}[SG-46\quad Bounded identities extend from a dense core]\label{SG-46}
1628 
1629 Then $A=B$ on $X$. Consequently a finite packet identity established on a
1630 dense core extends to a completion when both represented operators have
1631 bounded extensions. An infinite projector sum additionally requires a
1632 specified convergent synthesis; bounded finite identities alone do not
1633 supply that convergence.
1634 
1635 \end{theorem}
1636 \begin{proof}
1637 
1638 For $x\in X$ choose $x_n\in D$ with $x_n\to x$.
1639 Continuity gives $Ax=\lim_nAx_n=\lim_nBx_n=Bx$.
1640 This proves uniqueness of bounded extensions. For an infinite packet sum one
1641 would need partial sums to converge to the declared operator, in a stated
1642 operator or strong topology. Neither density nor the finite identity by
1643 itself proves such a limit. In SG-42 density of the actual selected spans
1644 provides a specific strong convergence theorem; a capped dictionary store or
1645 an arbitrary greedy prefix does not inherit it automatically.
1646 
1647 \end{proof}
1648 
1649 \par{\small\color{Teal}\textit{Source lineage:} \textit{seed}, thm:dense-extension; cor:packet-completion.}\par
1650 
1651 
1652 The native infinite-packet generator uses finite stores, including mask caps
1653 of 120 or 180 in reported settings. Its discrete masks, quadtree refinements,
1654 and residual-based seeds are a finite adaptive realization, not an
1655 implemented unbounded continuous atlas. The code's clipping stage should be
1656 interpreted through SG-44 and its ridge refit through SG-43. Case-level
1657 wall time is meaningful in the existing benchmark record; per-method runtime
1658 fields that remain zero because of method-key bookkeeping do not supply
1659 valid method timing comparisons.
1660 
1661 
1662 \section{A closed four-jump calculation}
1663 
1664 Take a bandlimited background
1665 \[
1666  s(x)=\frac14+\frac25\cos(2x)-\frac13\sin(3x)+\frac15\cos(5x)
1667 \]
1668 and a mean-zero edge with locations and jumps
1669 \[
1670  (\tau_\ell)=(0,\pi/2,\pi,3\pi/2),\qquad
1671  (a_\ell)=(1,-2,-1,2).
1672 \]
1673 The physical nodes are $(1,-i,-1,i)$. Their derivative moments are
1674 \[
1675  m_n=1-2(-i)^n-(-1)^n+2i^n,
1676 \]
1677 so in residues modulo four they are
1678 \[
1679  m_{4q}=0,\quad m_{4q+1}=2+4i,\quad
1680  m_{4q+2}=0,\quad m_{4q+3}=2-4i.
1681 \]
1682 Since the smooth degree is five, the first clean consecutive moment is
1683 $n_0=6$. With an order bound four, coefficients at $n=6,\ldots,13$ determine
1684 the edge. The Hankel factorization has four distinct nodes and four nonzero
1685 amplitudes, hence rank four. Its recurrence is
1686 $m_{n+4}-m_n=0$, with monic polynomial $Q(z)=z^4-1$.
1687 Recovering those roots and solving the Vandermonde system returns the displayed
1688 jumps. SG-21 then constructs the edge, and coefficients at $-5,\ldots,5$
1689 return the smooth background.
1690 
1691 The same object demonstrates internal phase recovery. Choose $N=3$ and
1692 $q_0=2$. The nodes are permuted by cubing, so the aliased nodes remain distinct.
1693 The sufficient self-calibration sets for $L_{\max}=4$ are
1694 \[
1695  \mathcal I_0=\{6,9,12,15,18,21,24,27,30\},\qquad
1696  \mathcal I_1=\{7,10,13,16\}.
1697 \]
1698 For each matched term the base coefficient is
1699 $b_{\ell,0}=a_\ell z_\ell^6$, and the adjacent coefficient is
1700 $b_{\ell,1}=b_{\ell,0}z_\ell$. Their ratio is the original physical node,
1701 not the principal cube root. The total high budget is thirteen, agreeing with
1702 $3L_{\max}+1$. The un-decimated construction uses eight high coefficients;
1703 the polyphase construction trades additional channel information for its
1704 chosen decimation geometry and branch recovery. Neither budget is claimed
1705 minimal. Both reconstruct the same canonical object exactly.
1706 
1707 
1708 \section{Evidence, implementations, and unresolved scope}
1709 
1710 \subsection{Current finite checks}
1711 The current release replay contains ten bounded suites with 288 recorded
1712 checks. After excluding the seven repeated checks in the SPD variant of the
1713 same infinite-packet tests, the count is 281 distinct recorded checks under
1714 the stated counting convention. A named check can compare a whole vector or
1715 family; these numbers are neither theorem counts nor claims of independent
1716 proofs. The manifests include source hashes, fixed seeds or deterministic
1717 inputs, tolerances, resource limits, and scientific result records.
1718 
1719 \begin{center}\small
1720 \begin{tabular}{lrr}
1721 \toprule Suite & Recorded checks & Counted distinct checks\\\midrule
1722 Exact invariants & 134 & 134\\
1723 Unknown edge & 24 & 24\\
1724 Self-calibration & 31 & 31\\
1725 Singularity dictionary & 26 & 26\\
1726 Gibbs case & 8 & 8\\
1727 Complex benchmark & 23 & 23\\
1728 Infinite packet & 7 & 7\\
1729 Infinite packet SPD & 7 & 0\\
1730 Rank-aware execution copy & 16 & 16\\
1731 Rank-ratio boundary & 12 & 12\\\midrule
1732 Total & 288 & 281\\\bottomrule
1733 \end{tabular}\normalsize\end{center}
1734 
1735 The controlled finite-recovery checks pass in their printed ranges and
1736 tolerances. The replay also reproduces two recovery failures, and its status
1737 explicitly includes them. In the localized-burst signal the true order six
1738 is inferred as nine. Location and jump comparison errors are nonfinite
1739 because there is no same-cardinality match, while reconstructed waveform
1740 RMSE remains finite and improved. Improved waveform error therefore does not
1741 establish correct edge identification. The rank-ratio witness in Section 7
1742 returns order two rather than three under the native heuristic, although the
1743 supplied-threshold rule returns three.
1744 
1745 An initial replay check mistakenly required every complex stress example to
1746 have correct order and finite parameter errors. It failed on the burst case.
1747 That failed record is retained. The follow-up uses the actual scientific
1748 scope: controlled examples pass their recovery criteria and stress failures
1749 are reported as failures. The scientific claim was narrowed; the burst
1750 outcome was not changed or hidden.
1751 
1752 The self-calibration replay includes one hundred deterministic separated
1753 random cases with correct order in the tested settings. The archived wider
1754 benchmark figures and natural-image comparisons remain historical finite
1755 evidence unless their particular experiment is covered by a fresh manifest.
1756 A bounded current replay is not a universal probability statement and does
1757 not formalize the Python implementation.
1758 
1759 \subsection{Executable boundaries and a separate repaired copy}
1760 The original unknown-edge implementation's generic background projector uses
1761 an unpivoted reduced QR construction. Its usual full-column-rank design is
1762 appropriate, but dependent background columns need numerical-rank handling.
1763 The original restricted-singular-value helper also needs an explicit zero
1764 case for supports wider than the number of residual rows. A separate
1765 rank-aware execution copy supplies those two local repairs and has sixteen
1766 finite regression controls. The immutable source payload is preserved;
1767 the repair provenance records the changed copy and its hashes.
1768 
1769 The zero-edge signal is a valid trivial branch of SG-22 and SG-34. The
1770 nonconstant native edge-model interface does not thereby acquire a zero-edge
1771 wrapper. Exact model assertions and accepted input domains of a program are
1772 listed separately. Floating-point root, rank, and amplitude tolerances are
1773 finite checks, not exact kernel certificates.
1774 
1775 \subsection{Formal coverage}
1776 The included related Paper V checkpoint contains 225 theorem declarations
1777 in twenty-three theorem-bearing modules. It supports packet, mask, polyphase,
1778 finite transfer, and known-edge reconstruction infrastructure under its own
1779 printed types. It is an earlier intact checkpoint, and its inherited logs
1780 belong to that checkpoint. It does not certify the July unknown-edge and
1781 self-calibration claims of this paper. The Lean guide maps each SG result to
1782 related infrastructure or an explicit uncovered formalization obligation.
1783 No Signal-specific Lean proof or fresh Lean compilation is claimed in this
1784 edition. The manuscript proofs and the finite numerical evidence remain
1785 separate from that formal coverage.
1786 
1787 \subsection{What the paper closes}
1788 For a bandlimited periodic background plus finitely many distinct off-grid
1789 jumps, finitely many exact Fourier coefficients recover the complete canonical
1790 signal. Under a supplied noncolliding decimation, adjacent offsets internally
1791 recover physical phase with a stated finite budget. Under uniform local
1792 conditioning and a strict branch margin, smooth Fourier tails yield the
1793 printed conditional location and amplitude rates. Native edge synthesis
1794 removes the edge's global-truncation error, and SG-37 supplies a valid
1795 mean-zero periodic $L^1$ perturbation estimate.
1796 
1797 \subsection{What remains open here}
1798 The paper does not provide model-free unknown-edge recovery, a universal
1799 noise bound without conditioning, an automatic proof that an empirical order
1800 heuristic meets the rank-threshold gate, or a data-only root enclosure from
1801 arbitrary noisy Fourier coefficients. It does not guarantee that every finite
1802 decimation search contains a noncolliding candidate. General piecewise-smooth
1803 backgrounds can imitate extra edges. Continuous arbitrary two-dimensional
1804 boundaries need a geometry-adaptive identifiability and approximation theorem.
1805 A sparse convergent adaptive infinite grammar needs a selection theorem,
1806 not just a dense candidate description or a truth-residual archive.
1807 
1808 These boundaries are mathematical distinctions exposed by the proofs and
1809 negative controls. They guide the next work: robust unknown-order selection,
1810 validated root enclosures, model-mismatch control, and adaptive geometry with
1811 a measured-data observation model.
1812 
1813 
1814 
1815 \appendix
1816 \section{Reading the source and claim concordance}
1817 Every SG result has a stable identifier, a domain, printed hypotheses,
1818 a statement and proof, dependency identifiers, and hash-pinned source
1819 lineage in \texttt{Claim\_Concordance.json}. Source-section labels identify
1820 the inherited mathematical material; a lineage anchor is not an assertion
1821 that every sentence of the new proof appeared verbatim in that source.
1822 The CSV is a compact index of the same results. Reconstructed proofs and
1823 publication corrections are marked explicitly. All SG results are
1824 manuscript results rather than Signal-specific Lean-certified declarations.
1825 
1826 The bundle includes the full \texttt{.md} manuscript files, editable
1827 \LaTeX{} sources, Python implementations and figure generators, original
1828 source archives, related Lean sources, fresh and inherited verification
1829 records, and the two companion guides. The raw inherited manuscripts are
1830 retained alongside the corrected edition so that changes remain reviewable.
1831 
1832 \section{Principal proof dependencies}
1833 SG-01--SG-06 supply exact finite coordinates, transfer multiplication,
1834 polyphase kernels, and alias arithmetic. SG-07--SG-08 state the
1835 identifiability boundaries. Known-edge reconstruction SG-09--SG-11 precedes
1836 the Gibbs profile SG-12 and positive averaging SG-13. Finite dictionary
1837 recovery follows the kernel-secant path SG-14 to SG-15; SG-16 gives the
1838 one-packet profiled search and its independent finite gap.
1839 
1840 The off-grid chain is differentiation SG-18, Hankel rank SG-19,
1841 annihilation SG-20, canonical integration SG-21, and complete recovery
1842 SG-22. SG-23--SG-26 quantify its perturbed linear and root steps.
1843 Self-calibration uses SG-27's offset identity and SG-28's supplied threshold,
1844 then SG-29--SG-33 for phase and branch control. SG-34 is the exact finite
1845 budget; SG-35 is the conditional asymptotic theorem. SG-36--SG-38 return
1846 those parameters to a normed signal with the corrected mean-zero edge bound.
1847 SG-39--SG-41 extend only the declared shared-edge, confluent, and dictionary
1848 contracts. SG-42--SG-46 keep ideal projection, ridge, clipping, residual
1849 storage, and completion as separate operations.
1850 
1851 \clearpage
1852 \begin{thebibliography}{9}
1853 \bibitem{stewart} G. W. Stewart.
1854 Perturbation Theory for the Singular Value Decomposition.
1855 UMIACS-TR-90-124 / CS-TR-2539, September 1990, Section 2, Theorem 1.
1856 \url{https://drum.lib.umd.edu/bitstreams/a38cc6db-9bb7-4f3b-ae43-55b5827bb9d4/download}.
1857 \bibitem{vexpa} M. Briani, A. Cuyt, F. Knaepkens, and W.-s. Lee.
1858 VEXPA: Validated EXPonential Analysis through regular sub-sampling.
1859 \emph{Signal Processing} 177 (2020), 107722.
1860 \url{https://doi.org/10.1016/j.sigpro.2020.107722}.
1861 \bibitem{batenkov} D. Batenkov.
1862 Complete algebraic reconstruction of piecewise-smooth functions from Fourier data.
1863 \emph{Mathematics of Computation} 84 (2015), 2329--2350.
1864 \url{https://doi.org/10.1090/S0025-5718-2015-02948-2}.
1865 \bibitem{sparse} D. L. Donoho and M. Elad.
1866 Optimally sparse representation in general (nonorthogonal) dictionaries
1867 via $\ell^1$ minimization. \emph{PNAS} 100 (2003), 2197--2202.
1868 \url{https://doi.org/10.1073/pnas.0437847100}.
1869 \bibitem{rouche} J. Lebl. \emph{Guide to Cultivating Complex Analysis:
1870 Working the Complex Field}, version 1.9, July 11, 2026,
1871 Section 5.4.2. \url{https://www.jirka.org/ca/ca.pdf}.
1872 \bibitem{fri} M. Vetterli, P. Marziliano, and T. Blu.
1873 Sampling Signals with Finite Rate of Innovation.
1874 \emph{IEEE Transactions on Signal Processing} 50 (2002), 1417--1428.
1875 \url{https://doi.org/10.1109/TSP.2002.1003065}.
1876 \end{thebibliography}
1877 The finite-rate paper is cited for historical attribution. Exact sampling
1878 hypotheses from it are not imported: the consecutive-moment theorem is
1879 proved directly in this manuscript. The literature review records the
1880 checked versions and the limits of the source search. No global novelty
1881 claim follows from that search.
1882 
1883 \end{document}

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