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Matched adjacent offsets recover the physical phase\nlost by decimation. Noisy order, root, and branch conclusions retain explicit\nconditioning and margin gates, with conditional asymptotic rates.\nA corrected periodic mean-zero jump estimate controls edge synthesis in\n$L^1$. Classical Gibbs ringing is derived and localized to the globally\ntruncated edge block. Finite dictionaries and adaptive image experiments are\nkept distinct from universal continuous-edge recovery. The current numerical\nreplay includes successful controlled checks and reproduced identification\nfailures. Related Lean infrastructure is mapped without claiming a formal\ncertificate for the new recovery chain.\n\\end{abstract}\n\\tableofcontents\\newpage\n\\section{The reconstruction problem}","The default periodic domain is $\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)$, with\n\\[\n \\widehat f(n)=\\frac1{2\\pi}\\int_0^{2\\pi}f(x)e^{-inx}\\,dx.\n\\]\nSignal identities inferred from Fourier coefficients are identities of\n$L^1$ classes. For a piecewise-constant edge we select the right-continuous\nrepresentative. This convention fixes the values at jumps, which Fourier\nintegrals alone cannot determine. When an $L^1$ norm occurs it is the\nunnormalized integral over a period. On the unit circle we use chord distance\n$|z-w|$ and circular angular distance $d_{\\mathbb T}(\\tau,\\sigma)$.","e default periodic domain is $\\mathbb T=\\mathbb R/(2\\pi\\mathbb Z)$, with\n\\[\n \\widehat f(n)=\\frac1{2\\pi}\\int_0^{2\\pi}f(x)e^{-inx}\\,dx.\n\\]\nSignal identities inferred from Fourier coefficients are identities of\n$L^1$ classes. For a piecewise-constant edge we select the right-continuous\nrepresentative. This convention fixes the values at jumps, which Fourier\nintegrals alone cannot determine. When an $L^1$ norm occurs it is the\nunnormalized integral over a period. On the unit circle we use chord distance\n$|z-w|$ and circular angular distance $d_{\\mathbb T}(\\tau,\\sigma)$.","t_0^{2\\pi}f(x)e^{-inx}\\,dx.\n\\]\nSignal identities inferred from Fourier coefficients are identities of\n$L^1$ classes. For a piecewise-constant edge we select the right-continuous\nrepresentative. This convention fixes the values at jumps, which Fourier\nintegrals alone cannot determine. When an $L^1$ norm occurs it is the\nunnormalized integral over a period. On the unit circle we use chord distance\n$|z-w|$ and circular angular distance $d_{\\mathbb T}(\\tau,\\sigma)$.","identities inferred from Fourier coefficients are identities of\n$L^1$ classes. For a piecewise-constant edge we select the right-continuous\nrepresentative. This convention fixes the values at jumps, which Fourier\nintegrals alone cannot determine. When an $L^1$ norm occurs it is the\nunnormalized integral over a period. On the unit circle we use chord distance\n$|z-w|$ and circular angular distance $d_{\\mathbb T}(\\tau,\\sigma)$.","\\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\n\\begin{theorem}[SG-01\\quad Complete packet analysis and synthesis]\\label{SG-01}","The maps\n\\[\n \\mathcal A x=(P_ax)_a,\\qquad\n \\mathcal S(x_a)_a=\\sum_ax_a\n\\]\nare inverse isomorphisms between $X$ and $\\bigoplus_aX_a$. In particular,\n$x=\\sum_aP_ax$ uniquely.","Completeness gives $\\mathcal S\\mathcal A x=x$. If $x_a\\in X_a$, then\n$P_ax_a=x_a$ and $P_bx_a=0$ for $b\\ne a$. Thus the $b$th component of\n$\\mathcal A\\mathcal S(x_a)_a$ is $P_b\\sum_ax_a=x_b$.\nBoth inverse identities follow. For uniqueness, apply $P_b$ to any proposed\nsum of components. This proof uses pairwise annihilation, not orthogonality;\nthe packet projections can be oblique.","\\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\n\\begin{theorem}[SG-02\\quad Transfer algebra and packet paths]\\label{SG-02}","\\[\n T=\\sum_{a,b}P_bTP_a,\\qquad\n (ST)_{ca}=\\sum_bS_{cb}T_{ba}.\n\\]\nThe block representation is an algebra isomorphism to\n$\\operatorname{End}(\\bigoplus_aX_a)$, with multiplication by contraction of\nintermediate packets. For every integer $k\\ge1$, iterating the product gives\n\\[\n P_{a_k}T^kP_{a_0}\n =\\sum_{a_1,\\ldots,a_{k-1}}\n (P_{a_k}TP_{a_{k-1}})\\cdots(P_{a_1}TP_{a_0}).\n\\]","Insert $I=\\sum_aP_a$ on both sides of $T$ to obtain reconstruction. Insert\nthat identity between $S$ and $T$ to obtain the product formula. Equivalently,\nthe block map is $T\\mapsto\\mathcal A T\\mathcal S$; SG-01 shows that its\ninverse is $B\\mapsto\\mathcal S B\\mathcal A$ and that it preserves multiplication.\nThe path formula follows by inserting the same identity between every\nsuccessive pair of operators. These are finite sums, so no convergence\nhypothesis is hidden in the insertion.","\\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\n\\begin{theorem}[SG-03\\quad Refinement preserves represented operators]\\label{SG-03}","\\[\n P_bTP_a=\\sum_{\\sigma\\in F_b}\\sum_{\\rho\\in F_a}Q_\\sigma TQ_\\rho.\n\\]\nThe coordinate changes $J=\\mathcal A_Q\\mathcal S_P$ and\n$C=\\mathcal A_P\\mathcal S_Q$ satisfy $CJ=I$ and $JC=I$ on their packet\ncoordinate spaces. Refinement is exact before any child packet is discarded.","Expand the two finite projector sums to obtain the block formula. Apply\nSG-01 twice to compute\n$CJ=\\mathcal A_P\\mathcal S_Q\\mathcal A_Q\\mathcal S_P\n=\\mathcal A_P\\mathcal S_P=I$, and similarly $JC=I$.\nThe identities concern the direct sums of packet images, not an unrestricted\nproduct with duplicated arbitrary ambient components.","\\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\n\\begin{theorem}[SG-04\\quad Pulled-back transform packets]\\label{SG-04}","The operators $P_a=V^{-1}E_aV$ are a complete packet family on $X$.\nFor a unitary $V$ and orthogonal coordinate selectors $E_a$ in a Hilbert space,\n\\[\n \\|x\\|^2=\\sum_a\\|P_ax\\|^2.\n\\]","Compute $P_aP_b=V^{-1}E_aE_bV=\\delta_{ab}P_a$ and\n$\\sum_aP_a=V^{-1}IV=I$. In the unitary case the pulled-back projections are\northogonal: $P_a^*=P_a$. Their images are mutually orthogonal, and SG-01\nwrites $x$ as their finite orthogonal sum. The Pythagorean identity gives the\nenergy formula. Without unitarity and orthogonality the algebraic identities\nremain valid while that norm identity need not hold.","\\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\n\\begin{theorem}[SG-05\\quad Polyphase analysis and convolution kernels]\\label{SG-05}","Interleaving the $N$ channels recovers $x$ exactly. With\n\\[\n h_{ba}[k]=h[Nk+b-a],\n\\]\nthe output channels satisfy\n\\[\n y_b[q]=\\sum_{a=0}^{N-1}\\sum_{t\\in\\mathbb Z}h_{ba}[q-t]x_a[t].\n\\]\nThus convolution becomes an $N\\times N$ transfer-kernel system.","Every integer has a unique Euclidean decomposition $Nt+a$, so the analysis\nand interleaving maps are inverse. In the convolution sum for $y[Nq+b]$, split\nthe input index $m$ as $Nt+a$. Then the kernel index becomes\n$Nq+b-m=N(q-t)+b-a$, which is precisely $h_{ba}[q-t]$.\nAbsolute summability permits the regrouping: the total absolute double sum\nis bounded by the corresponding convolution of $|h|$ and $|x|$.\nThe formula also holds for finitely supported sequences directly. It is the\nsequence version of SG-02, with the residue projectors selecting addresses.","ry integer has a unique Euclidean decomposition $Nt+a$, so the analysis\nand interleaving maps are inverse. In the convolution sum for $y[Nq+b]$, split\nthe input index $m$ as $Nt+a$. Then the kernel index becomes\n$Nq+b-m=N(q-t)+b-a$, which is precisely $h_{ba}[q-t]$.\nAbsolute summability permits the regrouping: the total absolute double sum\nis bounded by the corresponding convolution of $|h|$ and $|x|$.\nThe formula also holds for finitely supported sequences directly. It is the\nsequence version of SG-02, with the residue projectors selecting addresses.","\\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\n\\begin{theorem}[SG-06\\quad The arithmetic of residual alias ambiguity]\\label{SG-06}","Exactly $d$ unit-modulus nodes satisfy the same data. Hence the physical node\nis unique precisely when $d=1$. If integers $c,b_1,\\ldots,b_k$ obey\n$Nc+\\sum_jr_jb_j=d$, then\n\\[\n u^c\\prod_jw_j^{b_j}=z^d.\n\\]\nFor $d=1$ this is an explicit exact de-aliasing formula.","If $z'$ gives the same data, the quotient $v=z'/z$ satisfies\n$v^N=v^{r_1}=\\cdots=v^{r_k}=1$. Bezout's identity gives $v^d=1$.\nConversely $d$ divides every supplied exponent, so every $d$th root of unity\nhas all those powers equal to one. Therefore the fiber consists exactly of\n$z$ multiplied by the $d$ distinct $d$th roots of unity. Applying the Bezout\nidentity directly to the measured powers gives the displayed formula.\nNegative integer powers are legitimate because the data are nonzero.","The offset data in SG-06 must be actual powers of the same node. In the\nmulti-edge problem they are obtained only after matching coefficient vectors\nto common aliased nodes. A gcd computation does not bypass that matching or\nundo a collision of two edges in a single decimated channel. Adjacent offsets\nsupply the exponent $1$, so they remove the scalar branch ambiguity once the\nindividual coefficient progressions have been resolved.","\\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\n\\begin{theorem}[SG-07\\quad Measurement and decomposition obstructions]\\label{SG-07}","If $Mf_0=Mf_1$ but $J(f_0)\\ne J(f_1)$, no map $R:Y\\to\\mathcal J$ recovers\nthe correct edge for both. If $\\mathcal S\\cap\\mathcal J\\ne\\{0\\}$, the\nsum decomposition $f=s+j$ is not unique. In particular, finitely many point\nsamples on a periodic interval cannot recover every arbitrary interval packet.","A recovery map has one value at the common datum, so it cannot equal the two\ndifferent required edges. For the decomposition obstruction take nonzero\n$h\\in\\mathcal S\\cap\\mathcal J$: the pairs $(s,j)$ and $(s+h,j-h)$ have the\nsame sum. Finally, the complement of a finite sample set contains a nonempty\nopen interval. A nonzero indicator supported inside that interval vanishes at\nevery sample point. Compare it with the zero signal to obtain the first\nobstruction concretely.","\\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\n\\begin{theorem}[SG-08\\quad No uniform location stability without separation]\\label{SG-08}","Their unordered location sets have permutation-matched bottleneck distance at\nleast $\\pi-\\delta$, while their measured derivative moments satisfy\n\\[\n |m_n-\\widetilde m_n|\n =|(1-e^{-in\\pi})(1-e^{-in\\delta})|\n \\le2|n|\\delta\\qquad(n\\in F).\n\\]\nThere is no Lipschitz inverse for locations with one constant on all distinct\ntwo-jump configurations, even at this fixed order and with known amplitudes.","Every circular distance from a point of $\\{0,\\delta\\}$ to a point of\n$\\{\\pi,\\pi+\\delta\\}$ is at least $\\pi-\\delta$. This bounds every matching.\nThe moments of the first packet are $1-e^{-in\\delta}$; translating by $\\pi$\nmultiplies them by $e^{-in\\pi}$. Subtraction gives the formula.\nThe elementary inequality $|1-e^{-it}|\\le|t|$ proves the bound. Every fixed\nfinite norm of the measurement difference tends to zero as $\\delta\\to0$,\nwhereas the location distance stays bounded away from zero. A common finite\nLipschitz constant is therefore impossible. This argument compares two\nadmissible fixed-order packets; it does not require assigning location\nparameters to the zero-edge signal.","nce from a point of $\\{0,\\delta\\}$ to a point of\n$\\{\\pi,\\pi+\\delta\\}$ is at least $\\pi-\\delta$. This bounds every matching.\nThe moments of the first packet are $1-e^{-in\\delta}$; translating by $\\pi$\nmultiplies them by $e^{-in\\pi}$. Subtraction gives the formula.\nThe elementary inequality $|1-e^{-it}|\\le|t|$ proves the bound. Every fixed\nfinite norm of the measurement difference tends to zero as $\\delta\\to0$,\nwhereas the location distance stays bounded away from zero. A common finite\nLipschitz constant is therefore impossible. This argument compares two\nadmissible fixed-order packets; it does not require assigning location\nparameters to the zero-edge signal.","\\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\n\\begin{theorem}[SG-09\\quad Out-of-band matched identification]\\label{SG-09}","On $X=\\mathcal S_K+\\operatorname{span}\\{m\\}$, the functional\n\\[\n \\ell_F(g)=D_F^{-1}\\sum_{n\\in F}\\overline{\\widehat m(n)}\\widehat g(n)\n\\]\nsatisfies $\\ell_F(s+am)=a$. The decomposition is direct. If the measured\ncoefficients have errors $\\eta_n$, then\n\\[\n |\\widetilde a-a|\\le D_F^{-1/2}\\Bigl(\\sum_{n\\in F}|\\eta_n|^2\\Bigr)^{1/2}.\n\\]","The coefficients of $s\\in\\mathcal S_K$ vanish on $F$. Substituting $s+am$\ninto the functional leaves $aD_F/D_F=a$. If $m$ belonged to $\\mathcal S_K$,\nall its coefficients on $F$ would vanish, contradicting $D_F>0$; hence the\nsum is direct. The amplitude error is the inner product of the coefficient\nerror with the known coefficient vector divided by its squared norm.\nCauchy--Schwarz gives the displayed bound. The frequencies must actually see\nthe edge: an all-zero identifying set makes the formula undefined.","\\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\n\\begin{theorem}[SG-10\\quad The singularity-exact block operator]\\label{SG-10}","\\[\n P_J^2=P_J,\\quad P_S^2=P_S,\\quad P_SP_J=P_JP_S=0,\n \\quad P_S+P_J=I_X,\n\\]\nand the resolved reconstruction obeys\n\\[\n \\mathcal R_M=S_MP_S+P_J=I_X.\n\\]","For $g=s+am$, SG-09 gives $P_Jg=am$ and $P_Sg=s$.\nThese coordinate formulas immediately imply idempotence and pairwise\nannihilation. Since $s$ is bandlimited and $M\\ge K$, $S_Ms=s$.\nThus $\\mathcal R_Mg=s+am=g$. The equality is in the declared signal class;\nfor a selected right-continuous packet it also reproduces that representative.\nThis is a known-packet theorem. An unknown interval still needs the discovery\nstep proved later.","\\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\n\\begin{theorem}[SG-11\\quad Resolved error for an analytic background]\\label{SG-11}","The reconstruction error is exactly $f-\\mathcal R_Mf=s-S_Ms$, and\n\\[\n \\|f-\\mathcal R_Mf\\|_\\infty\n \\le\\frac{2Ce^{-\\rho(M+1)}}{1-e^{-\\rho}}.\n\\]\nThe bound is uniform across the jumps of $j$.","Cancel $j$ from the defining identity. Absolute summability of the Fourier\ncoefficients gives an everywhere uniformly convergent series for $s$.\nThe omitted positive and negative frequencies are bounded by\n$2C\\sum_{n>M}e^{-\\rho n}$, which is the displayed geometric series.\nNo limiting trigonometric approximation of $j$ occurs in this estimate.","\\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\n\\begin{theorem}[SG-12\\quad The scaled interval-edge Gibbs profile]\\label{SG-12}","Uniformly for $z$ in compact real intervals,\n\\[\n S_Mj(x_M(z))\\longrightarrow\\frac12+\\frac1\\pi\\operatorname{Si}(z),\n \\qquad \\operatorname{Si}(z)=\\int_0^z\\frac{\\sin t}{t}\\,dt.\n\\]\nThe first positive maximum of the limiting profile is $1+G$, where\n\\[\n G=\\frac{\\operatorname{Si}(\\pi)}\\pi-\\frac12\n   =0.089489872236\\ldots.\n\\]","Direct integration gives $\\widehat j(0)=a/\\pi$ and\n$\\widehat j(n)=\\sin(na)/(\\pi n)$ for $n\\ne0$. The Dirichlet kernel\n$D_M(t)=\\sin(\\lambda_Mt)/\\sin(t/2)$ yields\n\\[\n S_Mj(x)=\\frac1{2\\pi}\\int_{-a}^{a}D_M(x-t)\\,dt,\n \\qquad (S_Mj)'(x)=\\frac{D_M(x+a)-D_M(x-a)}{2\\pi}.\n\\]\nAt $x=a$, the coefficient formula gives\n\\[\n S_Mj(a)=\\frac a\\pi+\\frac1\\pi\\sum_{n=1}^M\\frac{\\sin(2na)}n\n \\longrightarrow\\frac12.\n\\]\nFor completeness, the sine-series limit follows by summation by parts from\nbounded partial sums of $e^{2ina}$ and by taking radial limits in\n$\\sum_{n\\ge1}r^ne^{2ina}/n=-\\log(1-re^{2ia})$; its imaginary part tends to\n$(\\pi-2a)/2$ for $0<2a<2\\pi$.\nNow differentiate the scaled function:\n\\[\n \\frac d{dz}S_Mj(x_M(z))=\n \\frac{D_M(-z/\\lambda_M)-D_M(2a-z/\\","Direct integration gives $\\widehat j(0)=a/\\pi$ and\n$\\widehat j(n)=\\sin(na)/(\\pi n)$ for $n\\ne0$. The Dirichlet kernel\n$D_M(t)=\\sin(\\lambda_Mt)/\\sin(t/2)$ yields\n\\[\n S_Mj(x)=\\frac1{2\\pi}\\int_{-a}^{a}D_M(x-t)\\,dt,\n \\qquad (S_Mj)'(x)=\\frac{D_M(x+a)-D_M(x-a)}{2\\pi}.\n\\]\nAt $x=a$, the coefficient formula gives\n\\[\n S_Mj(a)=\\frac a\\pi+\\frac1\\pi\\sum_{n=1}^M\\frac{\\sin(2na)}n\n \\longrightarrow\\frac12.\n\\]\nFor completeness, the sine-series limit follows by summation by parts from\nbounded partial sums of $e^{2ina}$ and by taking radial limits in\n$\\sum_{n\\ge1}r^ne^{2ina}/n=-\\log(1-re^{2ia})$; its imaginary part tends to\n$(\\pi-2a)/2$ for $0<2a<2\\pi$.\nNow differentiate the scaled function:\n\\[\n \\frac d{dz}S_Mj(x_M(z))=\n \\frac{D_M(-z/\\lambda_M)-D_M(2a-z/\\lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe fir","Direct integration gives $\\widehat j(0)=a/\\pi$ and\n$\\widehat j(n)=\\sin(na)/(\\pi n)$ for $n\\ne0$. The Dirichlet kernel\n$D_M(t)=\\sin(\\lambda_Mt)/\\sin(t/2)$ yields\n\\[\n S_Mj(x)=\\frac1{2\\pi}\\int_{-a}^{a}D_M(x-t)\\,dt,\n \\qquad (S_Mj)'(x)=\\frac{D_M(x+a)-D_M(x-a)}{2\\pi}.\n\\]\nAt $x=a$, the coefficient formula gives\n\\[\n S_Mj(a)=\\frac a\\pi+\\frac1\\pi\\sum_{n=1}^M\\frac{\\sin(2na)}n\n \\longrightarrow\\frac12.\n\\]\nFor completeness, the sine-series limit follows by summation by parts from\nbounded partial sums of $e^{2ina}$ and by taking radial limits in\n$\\sum_{n\\ge1}r^ne^{2ina}/n=-\\log(1-re^{2ia})$; its imaginary part tends to\n$(\\pi-2a)/2$ for $0<2a<2\\pi$.\nNow differentiate the scaled function:\n\\[\n \\frac d{dz}S_Mj(x_M(z))=\n \\frac{D_M(-z/\\lambda_M)-D_M(2a-z/\\lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe first term conv","Direct integration gives $\\widehat j(0)=a/\\pi$ and\n$\\widehat j(n)=\\sin(na)/(\\pi n)$ for $n\\ne0$. The Dirichlet kernel\n$D_M(t)=\\sin(\\lambda_Mt)/\\sin(t/2)$ yields\n\\[\n S_Mj(x)=\\frac1{2\\pi}\\int_{-a}^{a}D_M(x-t)\\,dt,\n \\qquad (S_Mj)'(x)=\\frac{D_M(x+a)-D_M(x-a)}{2\\pi}.\n\\]\nAt $x=a$, the coefficient formula gives\n\\[\n S_Mj(a)=\\frac a\\pi+\\frac1\\pi\\sum_{n=1}^M\\frac{\\sin(2na)}n\n \\longrightarrow\\frac12.\n\\]\nFor completeness, the sine-series limit follows by summation by parts from\nbounded partial sums of $e^{2ina}$ and by taking radial limits in\n$\\sum_{n\\ge1}r^ne^{2ina}/n=-\\log(1-re^{2ia})$; its imaginary part tends to\n$(\\pi-2a)/2$ for $0<2a<2\\pi$.\nNow differentiate the scaled function:\n\\[\n \\frac d{dz}S_Mj(x_M(z))=\n \\frac{D_M(-z/\\lambda_M)-D_M(2a-z/\\lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe first term converges uniformly on compact $z$ sets to\n$\\sin z/(\\pi z)$, w","Direct integration gives $\\widehat j(0)=a/\\pi$ and\n$\\widehat j(n)=\\sin(na)/(\\pi n)$ for $n\\ne0$. The Dirichlet kernel\n$D_M(t)=\\sin(\\lambda_Mt)/\\sin(t/2)$ yields\n\\[\n S_Mj(x)=\\frac1{2\\pi}\\int_{-a}^{a}D_M(x-t)\\,dt,\n \\qquad (S_Mj)'(x)=\\frac{D_M(x+a)-D_M(x-a)}{2\\pi}.\n\\]\nAt $x=a$, the coefficient formula gives\n\\[\n S_Mj(a)=\\frac a\\pi+\\frac1\\pi\\sum_{n=1}^M\\frac{\\sin(2na)}n\n \\longrightarrow\\frac12.\n\\]\nFor completeness, the sine-series limit follows by summation by parts from\nbounded partial sums of $e^{2ina}$ and by taking radial limits in\n$\\sum_{n\\ge1}r^ne^{2ina}/n=-\\log(1-re^{2ia})$; its imaginary part tends to\n$(\\pi-2a)/2$ for $0<2a<2\\pi$.\nNow differentiate the scaled function:\n\\[\n \\frac d{dz}S_Mj(x_M(z))=\n \\frac{D_M(-z/\\lambda_M)-D_M(2a-z/\\lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe first term converges uniformly on compact $z$ sets to\n$\\sin z/(\\pi z)$, with its continuous value at zero. The second tends\nuniformly to zero because $\\sin(a-z/(2\\lambda_M))$ stays bou","Direct integration gives $\\widehat j(0)=a/\\pi$ and\n$\\widehat j(n)=\\sin(na)/(\\pi n)$ for $n\\ne0$. The Dirichlet kernel\n$D_M(t)=\\sin(\\lambda_Mt)/\\sin(t/2)$ yields\n\\[\n S_Mj(x)=\\frac1{2\\pi}\\int_{-a}^{a}D_M(x-t)\\,dt,\n \\qquad (S_Mj)'(x)=\\frac{D_M(x+a)-D_M(x-a)}{2\\pi}.\n\\]\nAt $x=a$, the coefficient formula gives\n\\[\n S_Mj(a)=\\frac a\\pi+\\frac1\\pi\\sum_{n=1}^M\\frac{\\sin(2na)}n\n \\longrightarrow\\frac12.\n\\]\nFor completeness, the sine-series limit follows by summation by parts from\nbounded partial sums of $e^{2ina}$ and by taking radial limits in\n$\\sum_{n\\ge1}r^ne^{2ina}/n=-\\log(1-re^{2ia})$; its imaginary part tends to\n$(\\pi-2a)/2$ for $0<2a<2\\pi$.\nNow differentiate the scaled function:\n\\[\n \\frac d{dz}S_Mj(x_M(z))=\n \\frac{D_M(-z/\\lambda_M)-D_M(2a-z/\\lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe first term converges uniformly on compact $z$ sets to\n$\\sin z/(\\pi z)$, with its continuous value at zero. The second tends\nuniformly to zero because $\\sin(a-z/(2\\lambda_M))$ stays bounded away","Direct integration gives $\\widehat j(0)=a/\\pi$ and\n$\\widehat j(n)=\\sin(na)/(\\pi n)$ for $n\\ne0$. The Dirichlet kernel\n$D_M(t)=\\sin(\\lambda_Mt)/\\sin(t/2)$ yields\n\\[\n S_Mj(x)=\\frac1{2\\pi}\\int_{-a}^{a}D_M(x-t)\\,dt,\n \\qquad (S_Mj)'(x)=\\frac{D_M(x+a)-D_M(x-a)}{2\\pi}.\n\\]\nAt $x=a$, the coefficient formula gives\n\\[\n S_Mj(a)=\\frac a\\pi+\\frac1\\pi\\sum_{n=1}^M\\frac{\\sin(2na)}n\n \\longrightarrow\\frac12.\n\\]\nFor completeness, the sine-series limit follows by summation by parts from\nbounded partial sums of $e^{2ina}$ and by taking radial limits in\n$\\sum_{n\\ge1}r^ne^{2ina}/n=-\\log(1-re^{2ia})$; its imaginary part tends to\n$(\\pi-2a)/2$ for $0<2a<2\\pi$.\nNow differentiate the scaled function:\n\\[\n \\frac d{dz}S_Mj(x_M(z))=\n \\frac{D_M(-z/\\lambda_M)-D_M(2a-z/\\lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe first term converges uniformly on compact $z$ sets to\n$\\sin z/(\\pi z)$, with its continuous value at zero. The second tends\nuniformly to zero because $\\sin(a-z/(2\\lambda_M))$ stays bounded away from\nzero there. Integrate from $0$ to $z$ and use the base-point limit.\nThe derivative of the limiting profile is $\\sin","The Dirichlet kernel\n$D_M(t)=\\sin(\\lambda_Mt)/\\sin(t/2)$ yields\n\\[\n S_Mj(x)=\\frac1{2\\pi}\\int_{-a}^{a}D_M(x-t)\\,dt,\n \\qquad (S_Mj)'(x)=\\frac{D_M(x+a)-D_M(x-a)}{2\\pi}.\n\\]\nAt $x=a$, the coefficient formula gives\n\\[\n S_Mj(a)=\\frac a\\pi+\\frac1\\pi\\sum_{n=1}^M\\frac{\\sin(2na)}n\n \\longrightarrow\\frac12.\n\\]\nFor completeness, the sine-series limit follows by summation by parts from\nbounded partial sums of $e^{2ina}$ and by taking radial limits in\n$\\sum_{n\\ge1}r^ne^{2ina}/n=-\\log(1-re^{2ia})$; its imaginary part tends to\n$(\\pi-2a)/2$ for $0<2a<2\\pi$.\nNow differentiate the scaled function:\n\\[\n \\frac d{dz}S_Mj(x_M(z))=\n \\frac{D_M(-z/\\lambda_M)-D_M(2a-z/\\lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe first term converges uniformly on compact $z$ sets to\n$\\sin z/(\\pi z)$, with its continuous value at zero. The second tends\nuniformly to zero because $\\sin(a-z/(2\\lambda_M))$ stays bounded away from\nzero there. Integrate from $0$ to $z$ and use the base-point limit.\nThe derivative of the limiting profile is $\\sin z/(\\pi z)$, positive on\n$(0,\\pi)$ and negative on $(\\pi,2\\pi)$, so its first positive maximum is at\n$z=…","_Mt)/\\sin(t/2)$ yields\n\\[\n S_Mj(x)=\\frac1{2\\pi}\\int_{-a}^{a}D_M(x-t)\\,dt,\n \\qquad (S_Mj)'(x)=\\frac{D_M(x+a)-D_M(x-a)}{2\\pi}.\n\\]\nAt $x=a$, the coefficient formula gives\n\\[\n S_Mj(a)=\\frac a\\pi+\\frac1\\pi\\sum_{n=1}^M\\frac{\\sin(2na)}n\n \\longrightarrow\\frac12.\n\\]\nFor completeness, the sine-series limit follows by summation by parts from\nbounded partial sums of $e^{2ina}$ and by taking radial limits in\n$\\sum_{n\\ge1}r^ne^{2ina}/n=-\\log(1-re^{2ia})$; its imaginary part tends to\n$(\\pi-2a)/2$ for $0<2a<2\\pi$.\nNow differentiate the scaled function:\n\\[\n \\frac d{dz}S_Mj(x_M(z))=\n \\frac{D_M(-z/\\lambda_M)-D_M(2a-z/\\lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe first term converges uniformly on compact $z$ sets to\n$\\sin z/(\\pi z)$, with its continuous value at zero. The second tends\nuniformly to zero because $\\sin(a-z/(2\\lambda_M))$ stays bounded away from\nzero there. Integrate from $0$ to $z$ and use the base-point limit.\nThe derivative of the limiting profile is $\\sin z/(\\pi z)$, positive on\n$(0,\\pi)$ and negative on $(\\pi,2\\pi)$, so its first positive maximum is at\n$z=\\pi$. Substitution gives $G$. The constan…","\\qquad (S_Mj)'(x)=\\frac{D_M(x+a)-D_M(x-a)}{2\\pi}.\n\\]\nAt $x=a$, the coefficient formula gives\n\\[\n S_Mj(a)=\\frac a\\pi+\\frac1\\pi\\sum_{n=1}^M\\frac{\\sin(2na)}n\n \\longrightarrow\\frac12.\n\\]\nFor completeness, the sine-series limit follows by summation by parts from\nbounded partial sums of $e^{2ina}$ and by taking radial limits in\n$\\sum_{n\\ge1}r^ne^{2ina}/n=-\\log(1-re^{2ia})$; its imaginary part tends to\n$(\\pi-2a)/2$ for $0<2a<2\\pi$.\nNow differentiate the scaled function:\n\\[\n \\frac d{dz}S_Mj(x_M(z))=\n \\frac{D_M(-z/\\lambda_M)-D_M(2a-z/\\lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe first term converges uniformly on compact $z$ sets to\n$\\sin z/(\\pi z)$, with its continuous value at zero. The second tends\nuniformly to zero because $\\sin(a-z/(2\\lambda_M))$ stays bounded away from\nzero there. Integrate from $0$ to $z$ and use the base-point limit.\nThe derivative of the limiting profile is $\\sin z/(\\pi z)$, positive on\n$(0,\\pi)$ and negative on $(\\pi,2\\pi)$, so its first positive maximum is at\n$z=\\pi$. Substitution gives $G$. The constant is relative to a unit jump;\nscaling a packet scales its overshoot by the …","=\\frac{D_M(x+a)-D_M(x-a)}{2\\pi}.\n\\]\nAt $x=a$, the coefficient formula gives\n\\[\n S_Mj(a)=\\frac a\\pi+\\frac1\\pi\\sum_{n=1}^M\\frac{\\sin(2na)}n\n \\longrightarrow\\frac12.\n\\]\nFor completeness, the sine-series limit follows by summation by parts from\nbounded partial sums of $e^{2ina}$ and by taking radial limits in\n$\\sum_{n\\ge1}r^ne^{2ina}/n=-\\log(1-re^{2ia})$; its imaginary part tends to\n$(\\pi-2a)/2$ for $0<2a<2\\pi$.\nNow differentiate the scaled function:\n\\[\n \\frac d{dz}S_Mj(x_M(z))=\n \\frac{D_M(-z/\\lambda_M)-D_M(2a-z/\\lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe first term converges uniformly on compact $z$ sets to\n$\\sin z/(\\pi z)$, with its continuous value at zero. The second tends\nuniformly to zero because $\\sin(a-z/(2\\lambda_M))$ stays bounded away from\nzero there. Integrate from $0$ to $z$ and use the base-point limit.\nThe derivative of the limiting profile is $\\sin z/(\\pi z)$, positive on\n$(0,\\pi)$ and negative on $(\\pi,2\\pi)$, so its first positive maximum is at\n$z=\\pi$. Substitution gives $G$. The constant is relative to a unit jump;\nscaling a packet scales its overshoot by the jump amplitude.","fficient formula gives\n\\[\n S_Mj(a)=\\frac a\\pi+\\frac1\\pi\\sum_{n=1}^M\\frac{\\sin(2na)}n\n \\longrightarrow\\frac12.\n\\]\nFor completeness, the sine-series limit follows by summation by parts from\nbounded partial sums of $e^{2ina}$ and by taking radial limits in\n$\\sum_{n\\ge1}r^ne^{2ina}/n=-\\log(1-re^{2ia})$; its imaginary part tends to\n$(\\pi-2a)/2$ for $0<2a<2\\pi$.\nNow differentiate the scaled function:\n\\[\n \\frac d{dz}S_Mj(x_M(z))=\n \\frac{D_M(-z/\\lambda_M)-D_M(2a-z/\\lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe first term converges uniformly on compact $z$ sets to\n$\\sin z/(\\pi z)$, with its continuous value at zero. The second tends\nuniformly to zero because $\\sin(a-z/(2\\lambda_M))$ stays bounded away from\nzero there. Integrate from $0$ to $z$ and use the base-point limit.\nThe derivative of the limiting profile is $\\sin z/(\\pi z)$, positive on\n$(0,\\pi)$ and negative on $(\\pi,2\\pi)$, so its first positive maximum is at\n$z=\\pi$. Substitution gives $G$. The constant is relative to a unit jump;\nscaling a packet scales its overshoot by the jump amplitude.","-series limit follows by summation by parts from\nbounded partial sums of $e^{2ina}$ and by taking radial limits in\n$\\sum_{n\\ge1}r^ne^{2ina}/n=-\\log(1-re^{2ia})$; its imaginary part tends to\n$(\\pi-2a)/2$ for $0<2a<2\\pi$.\nNow differentiate the scaled function:\n\\[\n \\frac d{dz}S_Mj(x_M(z))=\n \\frac{D_M(-z/\\lambda_M)-D_M(2a-z/\\lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe first term converges uniformly on compact $z$ sets to\n$\\sin z/(\\pi z)$, with its continuous value at zero. The second tends\nuniformly to zero because $\\sin(a-z/(2\\lambda_M))$ stays bounded away from\nzero there. Integrate from $0$ to $z$ and use the base-point limit.\nThe derivative of the limiting profile is $\\sin z/(\\pi z)$, positive on\n$(0,\\pi)$ and negative on $(\\pi,2\\pi)$, so its first positive maximum is at\n$z=\\pi$. Substitution gives $G$. The constant is relative to a unit jump;\nscaling a packet scales its overshoot by the jump amplitude.","t follows by summation by parts from\nbounded partial sums of $e^{2ina}$ and by taking radial limits in\n$\\sum_{n\\ge1}r^ne^{2ina}/n=-\\log(1-re^{2ia})$; its imaginary part tends to\n$(\\pi-2a)/2$ for $0<2a<2\\pi$.\nNow differentiate the scaled function:\n\\[\n \\frac d{dz}S_Mj(x_M(z))=\n \\frac{D_M(-z/\\lambda_M)-D_M(2a-z/\\lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe first term converges uniformly on compact $z$ sets to\n$\\sin z/(\\pi z)$, with its continuous value at zero. The second tends\nuniformly to zero because $\\sin(a-z/(2\\lambda_M))$ stays bounded away from\nzero there. Integrate from $0$ to $z$ and use the base-point limit.\nThe derivative of the limiting profile is $\\sin z/(\\pi z)$, positive on\n$(0,\\pi)$ and negative on $(\\pi,2\\pi)$, so its first positive maximum is at\n$z=\\pi$. Substitution gives $G$. The constant is relative to a unit jump;\nscaling a packet scales its overshoot by the jump amplitude.","its in\n$\\sum_{n\\ge1}r^ne^{2ina}/n=-\\log(1-re^{2ia})$; its imaginary part tends to\n$(\\pi-2a)/2$ for $0<2a<2\\pi$.\nNow differentiate the scaled function:\n\\[\n \\frac d{dz}S_Mj(x_M(z))=\n \\frac{D_M(-z/\\lambda_M)-D_M(2a-z/\\lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe first term converges uniformly on compact $z$ sets to\n$\\sin z/(\\pi z)$, with its continuous value at zero. The second tends\nuniformly to zero because $\\sin(a-z/(2\\lambda_M))$ stays bounded away from\nzero there. Integrate from $0$ to $z$ and use the base-point limit.\nThe derivative of the limiting profile is $\\sin z/(\\pi z)$, positive on\n$(0,\\pi)$ and negative on $(\\pi,2\\pi)$, so its first positive maximum is at\n$z=\\pi$. Substitution gives $G$. The constant is relative to a unit jump;\nscaling a packet scales its overshoot by the jump amplitude.","ds to\n$(\\pi-2a)/2$ for $0<2a<2\\pi$.\nNow differentiate the scaled function:\n\\[\n \\frac d{dz}S_Mj(x_M(z))=\n \\frac{D_M(-z/\\lambda_M)-D_M(2a-z/\\lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe first term converges uniformly on compact $z$ sets to\n$\\sin z/(\\pi z)$, with its continuous value at zero. The second tends\nuniformly to zero because $\\sin(a-z/(2\\lambda_M))$ stays bounded away from\nzero there. Integrate from $0$ to $z$ and use the base-point limit.\nThe derivative of the limiting profile is $\\sin z/(\\pi z)$, positive on\n$(0,\\pi)$ and negative on $(\\pi,2\\pi)$, so its first positive maximum is at\n$z=\\pi$. Substitution gives $G$. The constant is relative to a unit jump;\nscaling a packet scales its overshoot by the jump amplitude.","(\\pi-2a)/2$ for $0<2a<2\\pi$.\nNow differentiate the scaled function:\n\\[\n \\frac d{dz}S_Mj(x_M(z))=\n \\frac{D_M(-z/\\lambda_M)-D_M(2a-z/\\lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe first term converges uniformly on compact $z$ sets to\n$\\sin z/(\\pi z)$, with its continuous value at zero. The second tends\nuniformly to zero because $\\sin(a-z/(2\\lambda_M))$ stays bounded away from\nzero there. Integrate from $0$ to $z$ and use the base-point limit.\nThe derivative of the limiting profile is $\\sin z/(\\pi z)$, positive on\n$(0,\\pi)$ and negative on $(\\pi,2\\pi)$, so its first positive maximum is at\n$z=\\pi$. Substitution gives $G$. The constant is relative to a unit jump;\nscaling a packet scales its overshoot by the jump amplitude.","c d{dz}S_Mj(x_M(z))=\n \\frac{D_M(-z/\\lambda_M)-D_M(2a-z/\\lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe first term converges uniformly on compact $z$ sets to\n$\\sin z/(\\pi z)$, with its continuous value at zero. The second tends\nuniformly to zero because $\\sin(a-z/(2\\lambda_M))$ stays bounded away from\nzero there. Integrate from $0$ to $z$ and use the base-point limit.\nThe derivative of the limiting profile is $\\sin z/(\\pi z)$, positive on\n$(0,\\pi)$ and negative on $(\\pi,2\\pi)$, so its first positive maximum is at\n$z=\\pi$. Substitution gives $G$. The constant is relative to a unit jump;\nscaling a packet scales its overshoot by the jump amplitude.","M(-z/\\lambda_M)-D_M(2a-z/\\lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe first term converges uniformly on compact $z$ sets to\n$\\sin z/(\\pi z)$, with its continuous value at zero. The second tends\nuniformly to zero because $\\sin(a-z/(2\\lambda_M))$ stays bounded away from\nzero there. Integrate from $0$ to $z$ and use the base-point limit.\nThe derivative of the limiting profile is $\\sin z/(\\pi z)$, positive on\n$(0,\\pi)$ and negative on $(\\pi,2\\pi)$, so its first positive maximum is at\n$z=\\pi$. Substitution gives $G$. The constant is relative to a unit jump;\nscaling a packet scales its overshoot by the jump amplitude.","lambda_M)}{2\\pi\\lambda_M}.\n\\]\nThe first term converges uniformly on compact $z$ sets to\n$\\sin z/(\\pi z)$, with its continuous value at zero. The second tends\nuniformly to zero because $\\sin(a-z/(2\\lambda_M))$ stays bounded away from\nzero there. Integrate from $0$ to $z$ and use the base-point limit.\nThe derivative of the limiting profile is $\\sin z/(\\pi z)$, positive on\n$(0,\\pi)$ and negative on $(\\pi,2\\pi)$, so its first positive maximum is at\n$z=\\pi$. Substitution gives $G$. The constant is relative to a unit jump;\nscaling a packet scales its overshoot by the jump amplitude.","rges uniformly on compact $z$ sets to\n$\\sin z/(\\pi z)$, with its continuous value at zero. The second tends\nuniformly to zero because $\\sin(a-z/(2\\lambda_M))$ stays bounded away from\nzero there. Integrate from $0$ to $z$ and use the base-point limit.\nThe derivative of the limiting profile is $\\sin z/(\\pi z)$, positive on\n$(0,\\pi)$ and negative on $(\\pi,2\\pi)$, so its first positive maximum is at\n$z=\\pi$. Substitution gives $G$. The constant is relative to a unit jump;\nscaling a packet scales its overshoot by the jump amplitude.","$ sets to\n$\\sin z/(\\pi z)$, with its continuous value at zero. The second tends\nuniformly to zero because $\\sin(a-z/(2\\lambda_M))$ stays bounded away from\nzero there. Integrate from $0$ to $z$ and use the base-point limit.\nThe derivative of the limiting profile is $\\sin z/(\\pi z)$, positive on\n$(0,\\pi)$ and negative on $(\\pi,2\\pi)$, so its first positive maximum is at\n$z=\\pi$. Substitution gives $G$. The constant is relative to a unit jump;\nscaling a packet scales its overshoot by the jump amplitude.","\\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\n\\begin{theorem}[SG-13\\quad Positive Fourier averaging preserves range]\\label{SG-13}","Then $0\\le\\sigma_Mj\\le1$ everywhere. For an interval packet, this suppresses\novershoot while retaining a continuous transition rather than an exact jump.","The averaging kernel is\n\\[\n F_M(t)=\\frac1{M+1}\\left|\\sum_{k=0}^Me^{ikt}\\right|^2\\ge0,\n \\qquad\\frac1{2\\pi}\\int_0^{2\\pi}F_M(t)\\,dt=1.\n\\]\nIts convolution with $j$ is therefore a weighted mean of numbers in $[0,1]$.\nThe resulting finite trigonometric polynomial is continuous. A nonconstant\ninterval indicator is discontinuous, so the mean cannot equal its selected\nrepresentative at all points or have its exact one-sided jump.","\\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\n\\begin{theorem}[SG-14\\quad Residualized sparse injectivity]\\label{SG-14}","Every vector $\\alpha$ with $\\|\\alpha\\|_0\\le s$ is uniquely determined by\n$A\\alpha$ if and only if\n\\[\n \\ker A\\cap\\{h:\\|h\\|_0\\le\\min(2s,d)\\}=\\{0\\}.\n\\]\nEquivalently every set of at most $\\min(2s,d)$ columns is independent.\nWith the convention $\\operatorname{spark}(A)=d+1$ when all $d$ columns are\nindependent, the equivalent condition is\n$\\operatorname{spark}(A)>\\min(2s,d)$.\nFor $s=0$ uniqueness is automatic.","The difference of two $s$-sparse vectors has support of size at most $2s$\nand at most $d$. Thus the kernel condition implies uniqueness.\nConversely, if a nonzero kernel vector $h$ has support size at most $2s$,\npartition that support into two sets of at most $s$ indices. Let $\\alpha$\nequal $h$ on the first set and zero elsewhere; let $\\beta$ equal $-h$ on\nthe second set. Then $h=\\alpha-\\beta$, and $A\\alpha=A\\beta$ contradicts\nuniqueness. Column dependence is exactly the existence of a nonzero kernel\nvector supported on the selected columns. The spark convention then gives\nthe stated capped inequality, including a fully independent matrix when\n$2s>d$. When $s=0$, the only admissible vector is zero and the kernel tes","The difference of two $s$-sparse vectors has support of size at most $2s$\nand at most $d$. Thus the kernel condition implies uniqueness.\nConversely, if a nonzero kernel vector $h$ has support size at most $2s$,\npartition that support into two sets of at most $s$ indices. Let $\\alpha$\nequal $h$ on the first set and zero elsewhere; let $\\beta$ equal $-h$ on\nthe second set. Then $h=\\alpha-\\beta$, and $A\\alpha=A\\beta$ contradicts\nuniqueness. Column dependence is exactly the existence of a nonzero kernel\nvector supported on the selected columns. The spark convention then gives\nthe stated capped inequality, including a fully independent matrix when\n$2s>d$. When $s=0$, the only admissible vector is zero and the kernel test\nconcerns only zero. If $y=s_0+D\\alpha$ with $s","The difference of two $s$-sparse vectors has support of size at most $2s$\nand at most $d$. Thus the kernel condition implies uniqueness.\nConversely, if a nonzero kernel vector $h$ has support size at most $2s$,\npartition that support into two sets of at most $s$ indices. Let $\\alpha$\nequal $h$ on the first set and zero elsewhere; let $\\beta$ equal $-h$ on\nthe second set. Then $h=\\alpha-\\beta$, and $A\\alpha=A\\beta$ contradicts\nuniqueness. Column dependence is exactly the existence of a nonzero kernel\nvector supported on the selected columns. The spark convention then gives\nthe stated capped inequality, including a fully independent matrix when\n$2s>d$. When $s=0$, the only admissible vector is zero and the kernel test\nconcerns only zero. If $y=s_0+D\\alpha$ with $s_0\\in\\mathcal S$","The difference of two $s$-sparse vectors has support of size at most $2s$\nand at most $d$. Thus the kernel condition implies uniqueness.\nConversely, if a nonzero kernel vector $h$ has support size at most $2s$,\npartition that support into two sets of at most $s$ indices. Let $\\alpha$\nequal $h$ on the first set and zero elsewhere; let $\\beta$ equal $-h$ on\nthe second set. Then $h=\\alpha-\\beta$, and $A\\alpha=A\\beta$ contradicts\nuniqueness. Column dependence is exactly the existence of a nonzero kernel\nvector supported on the selected columns. The spark convention then gives\nthe stated capped inequality, including a fully independent matrix when\n$2s>d$. When $s=0$, the only admissible vector is zero and the kernel test\nconcerns only zero. If $y=s_0+D\\alpha$ with $s_0\\in\\mathcal S$, then\n$Ry=A\\alpha$, so the same theorem applies after background elimination.","e difference of two $s$-sparse vectors has support of size at most $2s$\nand at most $d$. Thus the kernel condition implies uniqueness.\nConversely, if a nonzero kernel vector $h$ has support size at most $2s$,\npartition that support into two sets of at most $s$ indices. Let $\\alpha$\nequal $h$ on the first set and zero elsewhere; let $\\beta$ equal $-h$ on\nthe second set. Then $h=\\alpha-\\beta$, and $A\\alpha=A\\beta$ contradicts\nuniqueness. Column dependence is exactly the existence of a nonzero kernel\nvector supported on the selected columns. The spark convention then gives\nthe stated capped inequality, including a fully independent matrix when\n$2s>d$. When $s=0$, the only admissible vector is zero and the kernel test\nconcerns only zero. If $y=s_0+D\\alpha$ with $s_0\\in\\mathcal S$, then\n$Ry=A\\alpha$, so the same theorem applies after background elimination.","t most $s$ indices. Let $\\alpha$\nequal $h$ on the first set and zero elsewhere; let $\\beta$ equal $-h$ on\nthe second set. Then $h=\\alpha-\\beta$, and $A\\alpha=A\\beta$ contradicts\nuniqueness. Column dependence is exactly the existence of a nonzero kernel\nvector supported on the selected columns. The spark convention then gives\nthe stated capped inequality, including a fully independent matrix when\n$2s>d$. When $s=0$, the only admissible vector is zero and the kernel test\nconcerns only zero. If $y=s_0+D\\alpha$ with $s_0\\in\\mathcal S$, then\n$Ry=A\\alpha$, so the same theorem applies after background elimination.","dices. Let $\\alpha$\nequal $h$ on the first set and zero elsewhere; let $\\beta$ equal $-h$ on\nthe second set. Then $h=\\alpha-\\beta$, and $A\\alpha=A\\beta$ contradicts\nuniqueness. Column dependence is exactly the existence of a nonzero kernel\nvector supported on the selected columns. The spark convention then gives\nthe stated capped inequality, including a fully independent matrix when\n$2s>d$. When $s=0$, the only admissible vector is zero and the kernel test\nconcerns only zero. If $y=s_0+D\\alpha$ with $s_0\\in\\mathcal S$, then\n$Ry=A\\alpha$, so the same theorem applies after background elimination.","$-h$ on\nthe second set. Then $h=\\alpha-\\beta$, and $A\\alpha=A\\beta$ contradicts\nuniqueness. Column dependence is exactly the existence of a nonzero kernel\nvector supported on the selected columns. The spark convention then gives\nthe stated capped inequality, including a fully independent matrix when\n$2s>d$. When $s=0$, the only admissible vector is zero and the kernel test\nconcerns only zero. If $y=s_0+D\\alpha$ with $s_0\\in\\mathcal S$, then\n$Ry=A\\alpha$, so the same theorem applies after background elimination.","t. Then $h=\\alpha-\\beta$, and $A\\alpha=A\\beta$ contradicts\nuniqueness. Column dependence is exactly the existence of a nonzero kernel\nvector supported on the selected columns. The spark convention then gives\nthe stated capped inequality, including a fully independent matrix when\n$2s>d$. When $s=0$, the only admissible vector is zero and the kernel test\nconcerns only zero. If $y=s_0+D\\alpha$ with $s_0\\in\\mathcal S$, then\n$Ry=A\\alpha$, so the same theorem applies after background elimination.","and $A\\alpha=A\\beta$ contradicts\nuniqueness. Column dependence is exactly the existence of a nonzero kernel\nvector supported on the selected columns. The spark convention then gives\nthe stated capped inequality, including a fully independent matrix when\n$2s>d$. When $s=0$, the only admissible vector is zero and the kernel test\nconcerns only zero. If $y=s_0+D\\alpha$ with $s_0\\in\\mathcal S$, then\n$Ry=A\\alpha$, so the same theorem applies after background elimination.","\\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\n\\begin{theorem}[SG-15\\quad Sparse noise stability and support thresholding]\\label{SG-15}","Define\n\\[\n \\gamma=\\min_{\\substack{\\|h\\|_2=1\\\\\\|h\\|_0\\le\\min(2s,d)}}\\|Ah\\|_2>0.\n\\]\nThen\n\\[\n \\|\\widehat\\alpha-\\alpha_0\\|_2\\le2\\epsilon/\\gamma,\n \\qquad\\|D(\\widehat\\alpha-\\alpha_0)\\|_2\n \\le2\\|D\\|_2\\epsilon/\\gamma.\n\\]\nIf every nonzero true coefficient has magnitude greater than $4\\epsilon/\\gamma$,\nthresholding at $2\\epsilon/\\gamma$ with a strict greater-than rule recovers\nthe true support.","|h\\|_0\\le\\min(2s,d)}}\\|Ah\\|_2>0.\n\\]\nThen\n\\[\n \\|\\widehat\\alpha-\\alpha_0\\|_2\\le2\\epsilon/\\gamma,\n \\qquad\\|D(\\widehat\\alpha-\\alpha_0)\\|_2\n \\le2\\|D\\|_2\\epsilon/\\gamma.\n\\]\nIf every nonzero true coefficient has magnitude greater than $4\\epsilon/\\gamma$,\nthresholding at $2\\epsilon/\\gamma$ with a strict greater-than rule recovers\nthe true support.","The admissible unit vectors form a finite union of compact coordinate\nspheres. The continuous function $\\|Ah\\|_2$ attains its minimum. A zero\nminimum would contradict SG-14, hence $\\gamma>0$.\nFor $h=\\widehat\\alpha-\\alpha_0$, subtract the two residuals to obtain\n\\[\n \\|Ah\\|_2\\le\\|A\\widehat\\alpha-Ry\\|_2+\\|R\\eta\\|_2\\le2\\epsilon.\n\\]\nIts support is admissible, so homogeneity gives\n$\\gamma\\|h\\|_2\\le\\|Ah\\|_2$. Apply $D$ for the packet bound.\nEvery coordinate error is at most $2\\epsilon/\\gamma$. A true zero therefore\nnever passes the strict threshold, while a true coefficient larger than twice\nthat threshold remains strictly above it. For $s=0$ both vectors vanish and\nno restricted constant is required.","e unit vectors form a finite union of compact coordinate\nspheres. The continuous function $\\|Ah\\|_2$ attains its minimum. A zero\nminimum would contradict SG-14, hence $\\gamma>0$.\nFor $h=\\widehat\\alpha-\\alpha_0$, subtract the two residuals to obtain\n\\[\n \\|Ah\\|_2\\le\\|A\\widehat\\alpha-Ry\\|_2+\\|R\\eta\\|_2\\le2\\epsilon.\n\\]\nIts support is admissible, so homogeneity gives\n$\\gamma\\|h\\|_2\\le\\|Ah\\|_2$. Apply $D$ for the packet bound.\nEvery coordinate error is at most $2\\epsilon/\\gamma$. A true zero therefore\nnever passes the strict threshold, while a true coefficient larger than twice\nthat threshold remains strictly above it. For $s=0$ both vectors vanish and\nno restricted constant is required.","eres. The continuous function $\\|Ah\\|_2$ attains its minimum. A zero\nminimum would contradict SG-14, hence $\\gamma>0$.\nFor $h=\\widehat\\alpha-\\alpha_0$, subtract the two residuals to obtain\n\\[\n \\|Ah\\|_2\\le\\|A\\widehat\\alpha-Ry\\|_2+\\|R\\eta\\|_2\\le2\\epsilon.\n\\]\nIts support is admissible, so homogeneity gives\n$\\gamma\\|h\\|_2\\le\\|Ah\\|_2$. Apply $D$ for the packet bound.\nEvery coordinate error is at most $2\\epsilon/\\gamma$. A true zero therefore\nnever passes the strict threshold, while a true coefficient larger than twice\nthat threshold remains strictly above it. For $s=0$ both vectors vanish and\nno restricted constant is required.","wo residuals to obtain\n\\[\n \\|Ah\\|_2\\le\\|A\\widehat\\alpha-Ry\\|_2+\\|R\\eta\\|_2\\le2\\epsilon.\n\\]\nIts support is admissible, so homogeneity gives\n$\\gamma\\|h\\|_2\\le\\|Ah\\|_2$. Apply $D$ for the packet bound.\nEvery coordinate error is at most $2\\epsilon/\\gamma$. A true zero therefore\nnever passes the strict threshold, while a true coefficient larger than twice\nthat threshold remains strictly above it. For $s=0$ both vectors vanish and\nno restricted constant is required.","\\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\n\\begin{theorem}[SG-16\\quad Profiled one-packet discovery and its finite gap]\\label{SG-16}","For an unrestricted scalar amplitude the least-squares profile is\n\\[\n \\widehat a(q)=\\frac{\\langle Rq,Ry\\rangle}{\\|Rq\\|_2^2},\n \\qquad J(q)=\\|Ry-\\widehat a(q)Rq\\|_2^2.\n\\]\nThe true mask has $\\widehat a(q_*)=a_*$ and $J(q_*)=0$.\nIt is the unique zero-residual positive-amplitude mask exactly when no other\nresidualized mask lies on its positive ray. Define the closed rays\n$C_q=\\{aRq:a\\ge0\\}$. If\n\\[\n \\Gamma=\\inf_{q\\ne q_*}\\operatorname{dist}(Ry,C_q)>0,\n\\]\nwith the convention $\\inf\\varnothing=+\\infty$, then every perturbed residualized datum $Ry+e$ with $\\|e\\|_2<\\Gamma/2$\nselects $q_*$ uniquely by nearest-ray distance.","ehat a(q)=\\frac{\\langle Rq,Ry\\rangle}{\\|Rq\\|_2^2},\n \\qquad J(q)=\\|Ry-\\widehat a(q)Rq\\|_2^2.\n\\]\nThe true mask has $\\widehat a(q_*)=a_*$ and $J(q_*)=0$.\nIt is the unique zero-residual positive-amplitude mask exactly when no other\nresidualized mask lies on its positive ray. Define the closed rays\n$C_q=\\{aRq:a\\ge0\\}$. If\n\\[\n \\Gamma=\\inf_{q\\ne q_*}\\operatorname{dist}(Ry,C_q)>0,\n\\]\nwith the convention $\\inf\\varnothing=+\\infty$, then every perturbed residualized datum $Ry+e$ with $\\|e\\|_2<\\Gamma/2$\nselects $q_*$ uniquely by nearest-ray distance.","y-\\widehat a(q)Rq\\|_2^2.\n\\]\nThe true mask has $\\widehat a(q_*)=a_*$ and $J(q_*)=0$.\nIt is the unique zero-residual positive-amplitude mask exactly when no other\nresidualized mask lies on its positive ray. Define the closed rays\n$C_q=\\{aRq:a\\ge0\\}$. If\n\\[\n \\Gamma=\\inf_{q\\ne q_*}\\operatorname{dist}(Ry,C_q)>0,\n\\]\nwith the convention $\\inf\\varnothing=+\\infty$, then every perturbed residualized datum $Ry+e$ with $\\|e\\|_2<\\Gamma/2$\nselects $q_*$ uniquely by nearest-ray distance.","(q)Rq\\|_2^2.\n\\]\nThe true mask has $\\widehat a(q_*)=a_*$ and $J(q_*)=0$.\nIt is the unique zero-residual positive-amplitude mask exactly when no other\nresidualized mask lies on its positive ray. Define the closed rays\n$C_q=\\{aRq:a\\ge0\\}$. If\n\\[\n \\Gamma=\\inf_{q\\ne q_*}\\operatorname{dist}(Ry,C_q)>0,\n\\]\nwith the convention $\\inf\\varnothing=+\\infty$, then every perturbed residualized datum $Ry+e$ with $\\|e\\|_2<\\Gamma/2$\nselects $q_*$ uniquely by nearest-ray distance.","sk has $\\widehat a(q_*)=a_*$ and $J(q_*)=0$.\nIt is the unique zero-residual positive-amplitude mask exactly when no other\nresidualized mask lies on its positive ray. Define the closed rays\n$C_q=\\{aRq:a\\ge0\\}$. If\n\\[\n \\Gamma=\\inf_{q\\ne q_*}\\operatorname{dist}(Ry,C_q)>0,\n\\]\nwith the convention $\\inf\\varnothing=+\\infty$, then every perturbed residualized datum $Ry+e$ with $\\|e\\|_2<\\Gamma/2$\nselects $q_*$ uniquely by nearest-ray distance.","Differentiate the quadratic function $\\|Ry-aRq\\|_2^2$ with respect to $a$.\nIts unique stationary point is the profile formula and is the global\nminimum. At the true mask $Ry=a_*Rq_*$, giving the stated values.\nA wrong mask gives an admissible zero residual precisely when\n$Ry=aRq$ for some $a>0$. This is the positive-ray condition.\nFor the noise statement, the distance to any nonempty closed set is\n1-Lipschitz by the triangle inequality. The true ray has perturbed distance\nat most $\\|e\\|_2$, and every wrong ray has distance at least\n$\\Gamma-\\|e\\|_2>\\|e\\|_2$. Thus the true ray wins strictly.\nA squared objective gap must be square-rooted before it is used in this\nnorm-distance condition. For positive constraints the optimum on a closed\nray uses","Differentiate the quadratic function $\\|Ry-aRq\\|_2^2$ with respect to $a$.\nIts unique stationary point is the profile formula and is the global\nminimum. At the true mask $Ry=a_*Rq_*$, giving the stated values.\nA wrong mask gives an admissible zero residual precisely when\n$Ry=aRq$ for some $a>0$. This is the positive-ray condition.\nFor the noise statement, the distance to any nonempty closed set is\n1-Lipschitz by the triangle inequality. The true ray has perturbed distance\nat most $\\|e\\|_2$, and every wrong ray has distance at least\n$\\Gamma-\\|e\\|_2>\\|e\\|_2$. Thus the true ray wins strictly.\nA squared objective gap must be square-rooted before it is used in this\nnorm-distance condition. For positive constraints the optimum on a closed\nray uses $\\max(0,\\widehat a(q","Differentiate the quadratic function $\\|Ry-aRq\\|_2^2$ with respect to $a$.\nIts unique stationary point is the profile formula and is the global\nminimum. At the true mask $Ry=a_*Rq_*$, giving the stated values.\nA wrong mask gives an admissible zero residual precisely when\n$Ry=aRq$ for some $a>0$. This is the positive-ray condition.\nFor the noise statement, the distance to any nonempty closed set is\n1-Lipschitz by the triangle inequality. The true ray has perturbed distance\nat most $\\|e\\|_2$, and every wrong ray has distance at least\n$\\Gamma-\\|e\\|_2>\\|e\\|_2$. Thus the true ray wins strictly.\nA squared objective gap must be square-rooted before it is used in this\nnorm-distance condition. For positive constraints the optimum on a closed\nray uses $\\max(0,\\widehat a(q))$, rather than an unconditionally signed\namplitude.","stationary point is the profile formula and is the global\nminimum. At the true mask $Ry=a_*Rq_*$, giving the stated values.\nA wrong mask gives an admissible zero residual precisely when\n$Ry=aRq$ for some $a>0$. This is the positive-ray condition.\nFor the noise statement, the distance to any nonempty closed set is\n1-Lipschitz by the triangle inequality. The true ray has perturbed distance\nat most $\\|e\\|_2$, and every wrong ray has distance at least\n$\\Gamma-\\|e\\|_2>\\|e\\|_2$. Thus the true ray wins strictly.\nA squared objective gap must be square-rooted before it is used in this\nnorm-distance condition. For positive constraints the optimum on a closed\nray uses $\\max(0,\\widehat a(q))$, rather than an unconditionally signed\namplitude.","obal\nminimum. At the true mask $Ry=a_*Rq_*$, giving the stated values.\nA wrong mask gives an admissible zero residual precisely when\n$Ry=aRq$ for some $a>0$. This is the positive-ray condition.\nFor the noise statement, the distance to any nonempty closed set is\n1-Lipschitz by the triangle inequality. The true ray has perturbed distance\nat most $\\|e\\|_2$, and every wrong ray has distance at least\n$\\Gamma-\\|e\\|_2>\\|e\\|_2$. Thus the true ray wins strictly.\nA squared objective gap must be square-rooted before it is used in this\nnorm-distance condition. For positive constraints the optimum on a closed\nray uses $\\max(0,\\widehat a(q))$, rather than an unconditionally signed\namplitude.","ent, the distance to any nonempty closed set is\n1-Lipschitz by the triangle inequality. The true ray has perturbed distance\nat most $\\|e\\|_2$, and every wrong ray has distance at least\n$\\Gamma-\\|e\\|_2>\\|e\\|_2$. Thus the true ray wins strictly.\nA squared objective gap must be square-rooted before it is used in this\nnorm-distance condition. For positive constraints the optimum on a closed\nray uses $\\max(0,\\widehat a(q))$, rather than an unconditionally signed\namplitude.","\\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\n\\begin{theorem}[SG-17\\quad Mask error is a support geometry calculation]\\label{SG-17}","\\begin{align*}\n \\|aq-\\widetilde a\\widetilde q\\|_2^2\n &=|a-\\widetilde a|^2|\\operatorname{supp}q\\cap\\operatorname{supp}\\widetilde q|\\\\\n &\\quad+|a|^2|\\operatorname{supp}q\\setminus\\operatorname{supp}\\widetilde q|\n +|\\widetilde a|^2|\\operatorname{supp}\\widetilde q\\setminus\\operatorname{supp}q|.\n\\end{align*}\nFor $a=\\widetilde a=1$ the squared norm is the cardinality of the symmetric\nsupport difference.","&\\quad+|a|^2|\\operatorname{supp}q\\setminus\\operatorname{supp}\\widetilde q|\n +|\\widetilde a|^2|\\operatorname{supp}\\widetilde q\\setminus\\operatorname{supp}q|.\n\\end{align*}\nFor $a=\\widetilde a=1$ the squared norm is the cardinality of the symmetric\nsupport difference.","Partition the coordinates into common support, first-only support,\nsecond-only support, and neither. The difference equals respectively\n$a-\\widetilde a$, $a$, $-\\widetilde a$, and zero. Sum the squared values.\nThe same pointwise partition gives a continuous-measure formula when support\ncardinalities are replaced by support measures.","Let $\\mathcal U_{K,L}$ consist of real periodic signals $f=s+j$, where\n$s\\in\\mathcal S_K$ and $j$ is right-continuous, piecewise constant, mean zero,\nand has at most $L$ distinct active jumps. An active jump has nonzero amplitude.\nFor an actual jump count $r$, write\n\\[\n Dj=\\sum_{\\ell=1}^r a_\\ell\\delta_{\\tau_\\ell},\\quad\n z_\\ell=e^{-i\\tau_\\ell},\\quad\\sum_\\ell a_\\ell=0.\n\\]\nThe sum-zero condition is required by periodicity. A single nonzero active\njump cannot occur, so the nonconstant real class starts at $r=2$. The case\n$r=0$ is allowed and will be handled explicitly. The smooth-edge\nrepresentation is unique with this normalization: a piecewise-constant\ntrigonometric polynomial is constant, and mean zero fixes that","Let $\\mathcal U_{K,L}$ consist of real periodic signals $f=s+j$, where\n$s\\in\\mathcal S_K$ and $j$ is right-continuous, piecewise constant, mean zero,\nand has at most $L$ distinct active jumps. An active jump has nonzero amplitude.\nFor an actual jump count $r$, write\n\\[\n Dj=\\sum_{\\ell=1}^r a_\\ell\\delta_{\\tau_\\ell},\\quad\n z_\\ell=e^{-i\\tau_\\ell},\\quad\\sum_\\ell a_\\ell=0.\n\\]\nThe sum-zero condition is required by periodicity. A single nonzero active\njump cannot occur, so the nonconstant real class starts at $r=2$. The case\n$r=0$ is allowed and will be handled explicitly. The smooth-edge\nrepresentation is unique with this normalization: a piecewise-constant\ntrigonometric polynomial is constant, and mean zero fixes that constant.","ontinuous, piecewise constant, mean zero,\nand has at most $L$ distinct active jumps. An active jump has nonzero amplitude.\nFor an actual jump count $r$, write\n\\[\n Dj=\\sum_{\\ell=1}^r a_\\ell\\delta_{\\tau_\\ell},\\quad\n z_\\ell=e^{-i\\tau_\\ell},\\quad\\sum_\\ell a_\\ell=0.\n\\]\nThe sum-zero condition is required by periodicity. A single nonzero active\njump cannot occur, so the nonconstant real class starts at $r=2$. The case\n$r=0$ is allowed and will be handled explicitly. The smooth-edge\nrepresentation is unique with this normalization: a piecewise-constant\ntrigonometric polynomial is constant, and mean zero fixes that constant.","wise constant, mean zero,\nand has at most $L$ distinct active jumps. An active jump has nonzero amplitude.\nFor an actual jump count $r$, write\n\\[\n Dj=\\sum_{\\ell=1}^r a_\\ell\\delta_{\\tau_\\ell},\\quad\n z_\\ell=e^{-i\\tau_\\ell},\\quad\\sum_\\ell a_\\ell=0.\n\\]\nThe sum-zero condition is required by periodicity. A single nonzero active\njump cannot occur, so the nonconstant real class starts at $r=2$. The case\n$r=0$ is allowed and will be handled explicitly. The smooth-edge\nrepresentation is unique with this normalization: a piecewise-constant\ntrigonometric polynomial is constant, and mean zero fixes that constant.","\\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\n\\begin{theorem}[SG-18\\quad Differentiation isolates an exponential moment packet]\\label{SG-18}","\\[\n Dj=\\sum_\\ell a_\\ell\\delta_{\\tau_\\ell},\\qquad\n m_n:=2\\pi in\\widehat f(n)=\\sum_{\\ell=1}^ra_\\ell z_\\ell^n.\n\\]\nFor a non-bandlimited integrable background the same formula holds with\nan additive defect $e_n=2\\pi in\\widehat s(n)$.","For a smooth periodic test function $\\varphi$, integrate by parts over each\nconstant segment of $j$. Boundary terms telescope at each jump to\n$a_\\ell\\varphi(\\tau_\\ell)$, which is exactly the displayed distribution.\nEvaluate the distribution on $e^{-inx}$:\n\\[\n \\langle Dj,e^{-inx}\\rangle\n =-\\int j(x)(-in)e^{-inx}\\,dx\n =2\\pi in\\widehat j(n).\n\\]\nThe Dirac sum gives $\\sum_\\ell a_\\ell e^{-in\\tau_\\ell}$.\nFor $n>K$, the smooth polynomial contributes zero. Without bandlimiting it\ncontributes the stated defect. At $n=0$ the derivative moment is zero, in\nagreement with $\\sum a_\\ell=0$; it cannot determine the signal mean.","smooth periodic test function $\\varphi$, integrate by parts over each\nconstant segment of $j$. Boundary terms telescope at each jump to\n$a_\\ell\\varphi(\\tau_\\ell)$, which is exactly the displayed distribution.\nEvaluate the distribution on $e^{-inx}$:\n\\[\n \\langle Dj,e^{-inx}\\rangle\n =-\\int j(x)(-in)e^{-inx}\\,dx\n =2\\pi in\\widehat j(n).\n\\]\nThe Dirac sum gives $\\sum_\\ell a_\\ell e^{-in\\tau_\\ell}$.\nFor $n>K$, the smooth polynomial contributes zero. Without bandlimiting it\ncontributes the stated defect. At $n=0$ the derivative moment is zero, in\nagreement with $\\sum a_\\ell=0$; it cannot determine the signal mean.","erms telescope at each jump to\n$a_\\ell\\varphi(\\tau_\\ell)$, which is exactly the displayed distribution.\nEvaluate the distribution on $e^{-inx}$:\n\\[\n \\langle Dj,e^{-inx}\\rangle\n =-\\int j(x)(-in)e^{-inx}\\,dx\n =2\\pi in\\widehat j(n).\n\\]\nThe Dirac sum gives $\\sum_\\ell a_\\ell e^{-in\\tau_\\ell}$.\nFor $n>K$, the smooth polynomial contributes zero. Without bandlimiting it\ncontributes the stated defect. At $n=0$ the derivative moment is zero, in\nagreement with $\\sum a_\\ell=0$; it cannot determine the signal mean.",")$, which is exactly the displayed distribution.\nEvaluate the distribution on $e^{-inx}$:\n\\[\n \\langle Dj,e^{-inx}\\rangle\n =-\\int j(x)(-in)e^{-inx}\\,dx\n =2\\pi in\\widehat j(n).\n\\]\nThe Dirac sum gives $\\sum_\\ell a_\\ell e^{-in\\tau_\\ell}$.\nFor $n>K$, the smooth polynomial contributes zero. Without bandlimiting it\ncontributes the stated defect. At $n=0$ the derivative moment is zero, in\nagreement with $\\sum a_\\ell=0$; it cannot determine the signal mean.","\\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\n\\begin{theorem}[SG-19\\quad Hankel rank detects the actual order]\\label{SG-19}","Writing $V_h=(z_\\ell^i)_{i,\\ell}$ and $V_w=(z_\\ell^j)_{j,\\ell}$,\n\\[\n H_{h,w}=V_h\\operatorname{diag}(c_\\ell)V_w^{\\mathsf T},\n \\qquad\\operatorname{rank}H_{h,w}=r.\n\\]\nFor $r=0$ the Hankel matrix is zero. The transpose is algebraic and is not\na conjugate transpose.","Expand a matrix entry to obtain the factorization. The leading $r\\times r$\nVandermonde submatrix has determinant\n$\\prod_{\\ell<k}(z_k-z_\\ell)$, which is nonzero. Thus both rectangular\nVandermonde factors have full column rank, and their transposes have full\nrow rank. The diagonal factor is invertible. Its product with $V_w^T$ is\nsurjective onto $\\mathbb C^r$, and subsequent multiplication by the injective\n$V_h$ has image dimension $r$. For an empty exponential sum every entry is\nzero and the rank is zero.","matrix entry to obtain the factorization. The leading $r\\times r$\nVandermonde submatrix has determinant\n$\\prod_{\\ell<k}(z_k-z_\\ell)$, which is nonzero. Thus both rectangular\nVandermonde factors have full column rank, and their transposes have full\nrow rank. The diagonal factor is invertible. Its product with $V_w^T$ is\nsurjective onto $\\mathbb C^r$, and subsequent multiplication by the injective\n$V_h$ has image dimension $r$. For an empty exponential sum every entry is\nzero and the rank is zero.","factorization. The leading $r\\times r$\nVandermonde submatrix has determinant\n$\\prod_{\\ell<k}(z_k-z_\\ell)$, which is nonzero. Thus both rectangular\nVandermonde factors have full column rank, and their transposes have full\nrow rank. The diagonal factor is invertible. Its product with $V_w^T$ is\nsurjective onto $\\mathbb C^r$, and subsequent multiplication by the injective\n$V_h$ has image dimension $r$. For an empty exponential sum every entry is\nzero and the rank is zero.","\\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\n\\begin{theorem}[SG-20\\quad Exact Prony annihilation and location recovery]\\label{SG-20}","The unique monic annihilator\n\\[\n Q(z)=\\prod_{\\ell=1}^r(z-z_\\ell)=\\sum_{k=0}^{r-1}q_kz^k+z^r\n\\]\nis recovered from\n\\[\n H_{r,r}q=-v,\\qquad v=(m_{n_0+r+i})_{i=0}^{r-1}.\n\\]\nIts roots are exactly the nodes. Once the nodes are known, $r$ consecutive\nmoments determine the coefficients uniquely by a Vandermonde solve.\nFor unit-modulus nodes the corresponding locations are\n$\\tau_\\ell=-\\arg z_\\ell\\pmod{2\\pi}$.","For every available recurrence index,\n\\[\n \\sum_{k=0}^rq_km_{n+k}\n =\\sum_\\ell c_\\ell z_\\ell^{n-n_0}Q(z_\\ell)=0,\n \\qquad q_r=1.\n\\]\nSG-19 makes $H_{r,r}$ invertible. Thus the system has one solution and the\ntrue polynomial supplies it. Factoring the resulting polynomial recovers the\ndistinct nodes up to a permutation. The amplitude matrix for consecutive\nmoments is a square Vandermonde multiplied by an invertible diagonal node\npower matrix, so it too is invertible. The argument convention identifies\nlocations modulo a period, as required. Polynomial root extraction is a\nfinite algebraic step; the theorem does not claim that arbitrary real\nlocations have rational closed forms.","\\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\n\\begin{theorem}[SG-21\\quad Canonical periodic integration of jump data]\\label{SG-21}","\\[\n j(x)=\\sum_\\ell a_\\ell G_T(x-\\tau_\\ell)\n\\]\nis the unique mean-zero right-continuous piecewise-constant packet with those\njumps. Its distributional derivative is $\\sum_\\ell a_\\ell\\delta_{\\tau_\\ell}$.","The sawtooth derivative is $DG_T=\\delta_0-T^{-1}dx$: its slope is $-1/T$\naway from the endpoint and its endpoint jump is one. Summing translated\ncopies gives the prescribed Dirac masses and a constant-density term\n$-T^{-1}\\sum a_\\ell$, which vanishes. Consequently $j$ is constant between\nthe jump points. Each translated sawtooth has mean zero, so $j$ does too.\nAny other packet with the same jumps has distributional derivative difference\nzero. Since both are piecewise constant, their difference is the same\nconstant on every segment and across every jump. The mean-zero condition\nsets it to zero almost everywhere, and right continuity fixes the endpoint\nvalues. This explicit formula also supplies the perturbation estimate later.","\\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\n\\begin{theorem}[SG-22\\quad Finite exact unknown-order reconstruction]\\label{SG-22}","Those data uniquely determine the actual active order $r$, every jump\nlocation and amplitude up to permutation, the canonical edge packet $j$,\nthe smooth coefficients of $s$, and the complete signal $f$ as an $L^1$ class\nand as its selected right-continuous representative. The high-frequency\nbudget is $2L$ complex coefficients. The low budget is $2K+1$ coefficients,\nwith conjugate symmetry redundancy for a real signal.","Use SG-18 to form $2L$ exact high derivative moments. With $n_0=K+1$,\nform $H_{L,L}$ using the first $2L-1$ of them. SG-19 says its rank is the\nactual $r$. If $r=0$, $j=0$ and the low coefficients directly give $s$.\nOtherwise the first $2r$ moments are available. SG-20 recovers the nodes and\namplitudes. SG-21 constructs the unique normalized edge.\nFor each $|n|\\le K$, subtract its coefficient from $\\widehat f(n)$ to obtain\n$\\widehat s(n)$; all other smooth coefficients are zero by the model.\nTheir finite trigonometric sum gives $s$. Every step is uniquely determined,\nso another signal in the class with those data must have the same edge and\nbackground. Finally add $s+j$. Values at jump points are selected by the","Use SG-18 to form $2L$ exact high derivative moments. With $n_0=K+1$,\nform $H_{L,L}$ using the first $2L-1$ of them. SG-19 says its rank is the\nactual $r$. If $r=0$, $j=0$ and the low coefficients directly give $s$.\nOtherwise the first $2r$ moments are available. SG-20 recovers the nodes and\namplitudes. SG-21 constructs the unique normalized edge.\nFor each $|n|\\le K$, subtract its coefficient from $\\widehat f(n)$ to obtain\n$\\widehat s(n)$; all other smooth coefficients are zero by the model.\nTheir finite trigonometric sum gives $s$. Every step is uniquely determined,\nso another signal in the class with those data must have the same edge and\nbackground. Finally add $s+j$. Values at jump points are selected by the\nstated convention, rather than inferred from","Use SG-18 to form $2L$ exact high derivative moments. With $n_0=K+1$,\nform $H_{L,L}$ using the first $2L-1$ of them. SG-19 says its rank is the\nactual $r$. If $r=0$, $j=0$ and the low coefficients directly give $s$.\nOtherwise the first $2r$ moments are available. SG-20 recovers the nodes and\namplitudes. SG-21 constructs the unique normalized edge.\nFor each $|n|\\le K$, subtract its coefficient from $\\widehat f(n)$ to obtain\n$\\widehat s(n)$; all other smooth coefficients are zero by the model.\nTheir finite trigonometric sum gives $s$. Every step is uniquely determined,\nso another signal in the class with those data must have the same edge and\nbackground. Finally add $s+j$. Values at jump points are selected by the\nstated convention, rather than inferred from measure-zero Fou","Use SG-18 to form $2L$ exact high derivative moments. With $n_0=K+1$,\nform $H_{L,L}$ using the first $2L-1$ of them. SG-19 says its rank is the\nactual $r$. If $r=0$, $j=0$ and the low coefficients directly give $s$.\nOtherwise the first $2r$ moments are available. SG-20 recovers the nodes and\namplitudes. SG-21 constructs the unique normalized edge.\nFor each $|n|\\le K$, subtract its coefficient from $\\widehat f(n)$ to obtain\n$\\widehat s(n)$; all other smooth coefficients are zero by the model.\nTheir finite trigonometric sum gives $s$. Every step is uniquely determined,\nso another signal in the class with those data must have the same edge and\nbackground. Finally add $s+j$. Values at jump points are selected by the\nstated convention, rather than inferred from measure-zero Fourier data.","e SG-18 to form $2L$ exact high derivative moments. With $n_0=K+1$,\nform $H_{L,L}$ using the first $2L-1$ of them. SG-19 says its rank is the\nactual $r$. If $r=0$, $j=0$ and the low coefficients directly give $s$.\nOtherwise the first $2r$ moments are available. SG-20 recovers the nodes and\namplitudes. SG-21 constructs the unique normalized edge.\nFor each $|n|\\le K$, subtract its coefficient from $\\widehat f(n)$ to obtain\n$\\widehat s(n)$; all other smooth coefficients are zero by the model.\nTheir finite trigonometric sum gives $s$. Every step is uniquely determined,\nso another signal in the class with those data must have the same edge and\nbackground. Finally add $s+j$. Values at jump points are selected by the\nstated convention, rather than inferred from measure-zero Fourier data.","high derivative moments. With $n_0=K+1$,\nform $H_{L,L}$ using the first $2L-1$ of them. SG-19 says its rank is the\nactual $r$. If $r=0$, $j=0$ and the low coefficients directly give $s$.\nOtherwise the first $2r$ moments are available. SG-20 recovers the nodes and\namplitudes. SG-21 constructs the unique normalized edge.\nFor each $|n|\\le K$, subtract its coefficient from $\\widehat f(n)$ to obtain\n$\\widehat s(n)$; all other smooth coefficients are zero by the model.\nTheir finite trigonometric sum gives $s$. Every step is uniquely determined,\nso another signal in the class with those data must have the same edge and\nbackground. Finally add $s+j$. Values at jump points are selected by the\nstated convention, rather than inferred from measure-zero Fourier data.","k is the\nactual $r$. If $r=0$, $j=0$ and the low coefficients directly give $s$.\nOtherwise the first $2r$ moments are available. SG-20 recovers the nodes and\namplitudes. SG-21 constructs the unique normalized edge.\nFor each $|n|\\le K$, subtract its coefficient from $\\widehat f(n)$ to obtain\n$\\widehat s(n)$; all other smooth coefficients are zero by the model.\nTheir finite trigonometric sum gives $s$. Every step is uniquely determined,\nso another signal in the class with those data must have the same edge and\nbackground. Finally add $s+j$. Values at jump points are selected by the\nstated convention, rather than inferred from measure-zero Fourier data.","vers the nodes and\namplitudes. SG-21 constructs the unique normalized edge.\nFor each $|n|\\le K$, subtract its coefficient from $\\widehat f(n)$ to obtain\n$\\widehat s(n)$; all other smooth coefficients are zero by the model.\nTheir finite trigonometric sum gives $s$. Every step is uniquely determined,\nso another signal in the class with those data must have the same edge and\nbackground. Finally add $s+j$. Values at jump points are selected by the\nstated convention, rather than inferred from measure-zero Fourier data.","\\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\n\\begin{theorem}[SG-23\\quad Hankel perturbation from Fourier noise]\\label{SG-23}","\\[\n \\|E\\|_2\\le\\|E\\|_F\n \\le\\sqrt{hw}\\max_{0\\le i<h,\\ 0\\le j<w}|\\epsilon_{n_0+i+j}|.\n\\]\nIn particular, an error bound $|\\eta_n|\\le\\eta$ implies\n\\[\n \\|E\\|_2\\le2\\pi\\sqrt{hw}\\eta\\max_{i,j}|n_0+i+j|.\n\\]","Differentiate the noisy Fourier coefficients to obtain the exact moment\nerror identity. The spectral norm is at most the Frobenius norm. Its square\nis the sum of $hw$ squared entry magnitudes, each bounded by the squared\nmaximum. Substitute $|\\epsilon_n|=2\\pi|n||\\eta_n|$.\nRepeated moments occur at multiple matrix entries; the Frobenius bound\ncorrectly includes those repetitions. Smooth background tails can be included\nin $\\epsilon_n$ by the same calculation.","urier coefficients to obtain the exact moment\nerror identity. The spectral norm is at most the Frobenius norm. Its square\nis the sum of $hw$ squared entry magnitudes, each bounded by the squared\nmaximum. Substitute $|\\epsilon_n|=2\\pi|n||\\eta_n|$.\nRepeated moments occur at multiple matrix entries; the Frobenius bound\ncorrectly includes those repetitions. Smooth background tails can be included\nin $\\epsilon_n$ by the same calculation.","\\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\n\\begin{theorem}[SG-24\\quad Annihilator coefficient perturbation]\\label{SG-24}","The perturbed system is invertible, and its solution satisfies\n\\[\n \\|\\widetilde q-q\\|_2\n \\le\\frac{\\|H^{-1}\\|_2}{1-\\|H^{-1}\\|_2\\|E\\|_2}\n       (\\|e\\|_2+\\|E\\|_2\\|q\\|_2).\n\\]","Factor $\\widetilde H=H(I+H^{-1}E)$. The Neumann series converges in operator\nnorm under the supplied inequality, so\n$\\|\\widetilde H^{-1}\\|_2\\le\n\\|H^{-1}\\|_2/(1-\\|H^{-1}\\|_2\\|E\\|_2)$.\nSubtract the equations to obtain\n$\\widetilde H(\\widetilde q-q)=-e-Eq$. Apply the inverse bound and the\ntriangle inequality. The estimate assumes that the same true order is used;\nit does not itself choose the order from noisy data.","\\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\n\\begin{theorem}[SG-25\\quad Conditional simple-root disk localization]\\label{SG-25}","If for every $\\ell$\n\\[\n \\sup_{|z-z_\\ell|=\\rho}|E_Q(z)|\n <\\min_{|z-z_\\ell|=\\rho}|Q(z)|,\n\\]\nthen $\\widetilde Q$ has exactly one root, counted with multiplicity, in each\ndisk $|z-z_\\ell|<\\rho$. All its roots are thereby matched with error less\nthan $\\rho$. A sufficient explicit bound on each circle is\n\\[\n \\sum_{k=0}^{r-1}|\\widetilde q_k-q_k|(|z_\\ell|+\\rho)^k\n <\\rho\\prod_{j\\ne\\ell}(|z_\\ell-z_j|-\\rho).\n\\]","Each disk contains exactly one root of $Q$. The strict boundary inequality\nis the hypothesis of Rouche's theorem, which preserves the number of zeros\ninside. There are $r$ disjoint disks, each containing one zero of the degree\n$r$ perturbed polynomial, so they account for every zero.\nFor the explicit version, apply the triangle inequality to the coefficient\nerror polynomial and use $|z|\\le|z_\\ell|+\\rho$. On the circle,\n$|z-z_\\ell|=\\rho$ and\n$|z-z_j|\\ge|z_\\ell-z_j|-\\rho$, giving the lower product bound.\nFor unit true nodes and a recovered root inside a disk with $\\rho<1$, radial\nnormalization changes its chord error to at most $2\\rho$ by the triangle\ninequality. Its angular location error is then at most $2\\arcsin\\rho$.\nThe disk rad","Each disk contains exactly one root of $Q$. The strict boundary inequality\nis the hypothesis of Rouche's theorem, which preserves the number of zeros\ninside. There are $r$ disjoint disks, each containing one zero of the degree\n$r$ perturbed polynomial, so they account for every zero.\nFor the explicit version, apply the triangle inequality to the coefficient\nerror polynomial and use $|z|\\le|z_\\ell|+\\rho$. On the circle,\n$|z-z_\\ell|=\\rho$ and\n$|z-z_j|\\ge|z_\\ell-z_j|-\\rho$, giving the lower product bound.\nFor unit true nodes and a recovered root inside a disk with $\\rho<1$, radial\nnormalization changes its chord error to at most $2\\rho$ by the triangle\ninequality. Its angular location error is then at most $2\\arcsin\\rho$.\nThe disk radius is a complex-root error, not directly an angular error.","ly one root of $Q$. The strict boundary inequality\nis the hypothesis of Rouche's theorem, which preserves the number of zeros\ninside. There are $r$ disjoint disks, each containing one zero of the degree\n$r$ perturbed polynomial, so they account for every zero.\nFor the explicit version, apply the triangle inequality to the coefficient\nerror polynomial and use $|z|\\le|z_\\ell|+\\rho$. On the circle,\n$|z-z_\\ell|=\\rho$ and\n$|z-z_j|\\ge|z_\\ell-z_j|-\\rho$, giving the lower product bound.\nFor unit true nodes and a recovered root inside a disk with $\\rho<1$, radial\nnormalization changes its chord error to at most $2\\rho$ by the triangle\ninequality. Its angular location error is then at most $2\\arcsin\\rho$.\nThe disk radius is a complex-root error, not directly an angular error.","e strict boundary inequality\nis the hypothesis of Rouche's theorem, which preserves the number of zeros\ninside. There are $r$ disjoint disks, each containing one zero of the degree\n$r$ perturbed polynomial, so they account for every zero.\nFor the explicit version, apply the triangle inequality to the coefficient\nerror polynomial and use $|z|\\le|z_\\ell|+\\rho$. On the circle,\n$|z-z_\\ell|=\\rho$ and\n$|z-z_j|\\ge|z_\\ell-z_j|-\\rho$, giving the lower product bound.\nFor unit true nodes and a recovered root inside a disk with $\\rho<1$, radial\nnormalization changes its chord error to at most $2\\rho$ by the triangle\ninequality. Its angular location error is then at most $2\\arcsin\\rho$.\nThe disk radius is a complex-root error, not directly an angular error.","r$ disjoint disks, each containing one zero of the degree\n$r$ perturbed polynomial, so they account for every zero.\nFor the explicit version, apply the triangle inequality to the coefficient\nerror polynomial and use $|z|\\le|z_\\ell|+\\rho$. On the circle,\n$|z-z_\\ell|=\\rho$ and\n$|z-z_j|\\ge|z_\\ell-z_j|-\\rho$, giving the lower product bound.\nFor unit true nodes and a recovered root inside a disk with $\\rho<1$, radial\nnormalization changes its chord error to at most $2\\rho$ by the triangle\ninequality. Its angular location error is then at most $2\\arcsin\\rho$.\nThe disk radius is a complex-root error, not directly an angular error.","urbed polynomial, so they account for every zero.\nFor the explicit version, apply the triangle inequality to the coefficient\nerror polynomial and use $|z|\\le|z_\\ell|+\\rho$. On the circle,\n$|z-z_\\ell|=\\rho$ and\n$|z-z_j|\\ge|z_\\ell-z_j|-\\rho$, giving the lower product bound.\nFor unit true nodes and a recovered root inside a disk with $\\rho<1$, radial\nnormalization changes its chord error to at most $2\\rho$ by the triangle\ninequality. Its angular location error is then at most $2\\arcsin\\rho$.\nThe disk radius is a complex-root error, not directly an angular error.","ly the triangle inequality to the coefficient\nerror polynomial and use $|z|\\le|z_\\ell|+\\rho$. On the circle,\n$|z-z_\\ell|=\\rho$ and\n$|z-z_j|\\ge|z_\\ell-z_j|-\\rho$, giving the lower product bound.\nFor unit true nodes and a recovered root inside a disk with $\\rho<1$, radial\nnormalization changes its chord error to at most $2\\rho$ by the triangle\ninequality. Its angular location error is then at most $2\\arcsin\\rho$.\nThe disk radius is a complex-root error, not directly an angular error.","\\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\n\\begin{theorem}[SG-26\\quad Amplitude stability with perturbed nodes]\\label{SG-26}","\\[\n \\|\\widehat a-a\\|_2\n \\le\\frac{\\|e\\|_2+\\|\\Delta V\\|_2\\|a\\|_2}\n {\\sigma_{\\min}(V)-\\|\\Delta V\\|_2}.\n\\]\nFor unit nodes in rows with nonnegative powers $n$, matched chord error\n$|\\widehat z_\\ell-z_\\ell|\\le\\delta$ gives the entry bound\n$|\\widehat z_\\ell^n-z_\\ell^n|\\le n\\delta$.","The minimum singular value of $\\widetilde V$ is at least\n$\\sigma_{\\min}(V)-\\|\\Delta V\\|_2>0$. Thus its least-squares inverse exists.\nSince $\\widetilde V a=m+\\Delta V a$,\n\\[\n \\widehat a-a=\\widetilde V^\\dagger(e-\\Delta V a).\n\\]\nApply $\\|\\widetilde V^\\dagger\\|_2=1/\\sigma_{\\min}(\\widetilde V)$.\nThe power bound follows by factoring\n$\\widehat z^n-z^n=(\\widehat z-z)\\sum_{k=0}^{n-1}\\widehat z^{n-1-k}z^k$;\neach summand has modulus one. The statement assumes aligned nodes and unit\nnormalization. Arbitrary unnormalized roots require extra modulus factors.","Decimation reorganizes the differentiated moments, and it also maps each\nphysical node $z$ to $z^N$. This can improve a selected geometry or create a\ncollision. The physical phase is retained in the offset coefficients. The\ncentral recovery step uses those coefficients rather than choosing a principal\nroot by convention. The shifted-exponential ingredient is classical\n\\cite{vexpa}; the contracts below specify what is sufficient for its use in\nthe complete unknown-order pipeline.","\\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\n\\begin{theorem}[SG-27\\quad Shared-node offset channels]\\label{SG-27}","\\[\n m_{(q_0+q)N+r}=\\sum_\\ell b_{\\ell,r}u_\\ell^q+e_{(q_0+q)N+r},\n \\quad u_\\ell=z_\\ell^N,\\quad b_{\\ell,r}=a_\\ell u_\\ell^{q_0}z_\\ell^r.\n\\]\nFor each matched edge $b_{\\ell,r+1}=z_\\ell b_{\\ell,r}$.\nIf $|\\widehat s(n)|\\le C_s(1+|n|)^{-p-2}$ with $p\\ge0$, then for\nnonnegative $q,r$\n\\[\n |e_{(q_0+q)N+r}|\\le2\\pi C_s(q_0N)^{-p-1}.\n\\]","Split the exponent as $(q_0+q)N+r=q_0N+r+qN$ to obtain the channel\nidentity. Incrementing the offset multiplies only $z_\\ell^r$ by $z_\\ell$.\nFor the tail bound, use\n$|e_n|=2\\pi|n||\\widehat s(n)|\\le2\\pi C_s(1+|n|)^{-p-1}$ and\n$n\\ge q_0N>0$. The channel identity is valid even when aliased nodes collide,\nbut a Prony solve on that channel then observes combined coefficients and\ncannot separate those nodes using the simple distinct-node theorem.","\\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\n\\begin{theorem}[SG-28\\quad Order recovery with a supplied singular threshold]\\label{SG-28}","Exactly $r$ singular values of $H$ exceed $t$. If $r=0$, a supplied\nthreshold $t>\\|E\\|_2$ selects zero. If $r=k$, all $k$ singular values exceed\nthe threshold and no nonexistent $\\sigma_{k+1}$ is used.","The singular-value perturbation inequality \\cite{stewart} gives\n$|\\sigma_j(H)-\\sigma_j(H_0)|\\le\\|E\\|_2$ for $1\\le j\\le k$.\nIt holds equally over the complex field: in the min--max characterization\n\\[\n \\sigma_j(A)=\\min_{\\dim V=w-j+1}\\max_{x\\in V,\\,\\|x\\|=1}\\|Ax\\|.\n\\]\nthe triangle inequality bounds each restricted maximum for $A+E$ by that\nfor $A$ plus $\\|E\\|_2$. Minimize and then interchange the two matrices.\nFor $j\\le r$ this yields $\\sigma_j(H)\\ge\\sigma_r(H_0)-\\epsilon>t$.\nFor $j>r$, when such indices exist, it gives $\\sigma_j(H)\\le\\epsilon<t$.\nFor rank zero the same upper bound applies to every singular value. For full\nrectangular rank the lower bound already accounts for every singular value.\nThe theorem assumes a threshold satisfying the inequalities; it does not\nprove that every heuristi","The singular-value perturbation inequality \\cite{stewart} gives\n$|\\sigma_j(H)-\\sigma_j(H_0)|\\le\\|E\\|_2$ for $1\\le j\\le k$.\nIt holds equally over the complex field: in the min--max characterization\n\\[\n \\sigma_j(A)=\\min_{\\dim V=w-j+1}\\max_{x\\in V,\\,\\|x\\|=1}\\|Ax\\|.\n\\]\nthe triangle inequality bounds each restricted maximum for $A+E$ by that\nfor $A$ plus $\\|E\\|_2$. Minimize and then interchange the two matrices.\nFor $j\\le r$ this yields $\\sigma_j(H)\\ge\\sigma_r(H_0)-\\epsilon>t$.\nFor $j>r$, when such indices exist, it gives $\\sigma_j(H)\\le\\epsilon<t$.\nFor rank zero the same upper bound applies to every singular value. For full\nrectangular rank the lower bound already accounts for every singular value.\nThe theorem assumes a threshold satisfying the inequalities; it does not\nprove that every heuristic gap selector fin","The singular-value perturbation inequality \\cite{stewart} gives\n$|\\sigma_j(H)-\\sigma_j(H_0)|\\le\\|E\\|_2$ for $1\\le j\\le k$.\nIt holds equally over the complex field: in the min--max characterization\n\\[\n \\sigma_j(A)=\\min_{\\dim V=w-j+1}\\max_{x\\in V,\\,\\|x\\|=1}\\|Ax\\|.\n\\]\nthe triangle inequality bounds each restricted maximum for $A+E$ by that\nfor $A$ plus $\\|E\\|_2$. Minimize and then interchange the two matrices.\nFor $j\\le r$ this yields $\\sigma_j(H)\\ge\\sigma_r(H_0)-\\epsilon>t$.\nFor $j>r$, when such indices exist, it gives $\\sigma_j(H)\\le\\epsilon<t$.\nFor rank zero the same upper bound applies to every singular value. For full\nrectangular rank the lower bound already accounts for every singular value.\nThe theorem assumes a threshold satisfying the inequalities; it does not\nprove that every heuristic gap selector finds it.","ue perturbation inequality \\cite{stewart} gives\n$|\\sigma_j(H)-\\sigma_j(H_0)|\\le\\|E\\|_2$ for $1\\le j\\le k$.\nIt holds equally over the complex field: in the min--max characterization\n\\[\n \\sigma_j(A)=\\min_{\\dim V=w-j+1}\\max_{x\\in V,\\,\\|x\\|=1}\\|Ax\\|.\n\\]\nthe triangle inequality bounds each restricted maximum for $A+E$ by that\nfor $A$ plus $\\|E\\|_2$. Minimize and then interchange the two matrices.\nFor $j\\le r$ this yields $\\sigma_j(H)\\ge\\sigma_r(H_0)-\\epsilon>t$.\nFor $j>r$, when such indices exist, it gives $\\sigma_j(H)\\le\\epsilon<t$.\nFor rank zero the same upper bound applies to every singular value. For full\nrectangular rank the lower bound already accounts for every singular value.\nThe theorem assumes a threshold satisfying the inequalities; it does not\nprove that every heuristic gap selector finds it.","ality \\cite{stewart} gives\n$|\\sigma_j(H)-\\sigma_j(H_0)|\\le\\|E\\|_2$ for $1\\le j\\le k$.\nIt holds equally over the complex field: in the min--max characterization\n\\[\n \\sigma_j(A)=\\min_{\\dim V=w-j+1}\\max_{x\\in V,\\,\\|x\\|=1}\\|Ax\\|.\n\\]\nthe triangle inequality bounds each restricted maximum for $A+E$ by that\nfor $A$ plus $\\|E\\|_2$. Minimize and then interchange the two matrices.\nFor $j\\le r$ this yields $\\sigma_j(H)\\ge\\sigma_r(H_0)-\\epsilon>t$.\nFor $j>r$, when such indices exist, it gives $\\sigma_j(H)\\le\\epsilon<t$.\nFor rank zero the same upper bound applies to every singular value. For full\nrectangular rank the lower bound already accounts for every singular value.\nThe theorem assumes a threshold satisfying the inequalities; it does not\nprove that every heuristic gap selector finds it.","a_j(H_0)|\\le\\|E\\|_2$ for $1\\le j\\le k$.\nIt holds equally over the complex field: in the min--max characterization\n\\[\n \\sigma_j(A)=\\min_{\\dim V=w-j+1}\\max_{x\\in V,\\,\\|x\\|=1}\\|Ax\\|.\n\\]\nthe triangle inequality bounds each restricted maximum for $A+E$ by that\nfor $A$ plus $\\|E\\|_2$. Minimize and then interchange the two matrices.\nFor $j\\le r$ this yields $\\sigma_j(H)\\ge\\sigma_r(H_0)-\\epsilon>t$.\nFor $j>r$, when such indices exist, it gives $\\sigma_j(H)\\le\\epsilon<t$.\nFor rank zero the same upper bound applies to every singular value. For full\nrectangular rank the lower bound already accounts for every singular value.\nThe theorem assumes a threshold satisfying the inequalities; it does not\nprove that every heuristic gap selector finds it.","t holds equally over the complex field: in the min--max characterization\n\\[\n \\sigma_j(A)=\\min_{\\dim V=w-j+1}\\max_{x\\in V,\\,\\|x\\|=1}\\|Ax\\|.\n\\]\nthe triangle inequality bounds each restricted maximum for $A+E$ by that\nfor $A$ plus $\\|E\\|_2$. Minimize and then interchange the two matrices.\nFor $j\\le r$ this yields $\\sigma_j(H)\\ge\\sigma_r(H_0)-\\epsilon>t$.\nFor $j>r$, when such indices exist, it gives $\\sigma_j(H)\\le\\epsilon<t$.\nFor rank zero the same upper bound applies to every singular value. For full\nrectangular rank the lower bound already accounts for every singular value.\nThe theorem assumes a threshold satisfying the inequalities; it does not\nprove that every heuristic gap selector finds it.","The native implementation estimates order from singular-value ratios and\nvalidation heuristics. A largest adjacent ratio is not implied to select the\ntrue order by SG-28. An exact witness makes the distinction concrete. Take\nfour-by-four Hankel matrices of moments beginning at $n_0=4$ from the nodes\n$(1,i,-1)$ and real amplitudes $(100,-99,-1)$. The four-point Fourier columns\nare orthogonal, so the true singular values are $(400,396,4,0)$. Add the\nmoment perturbation $0.1(-i)^n$. It contributes a fourth orthogonal column and\nproduces singular values $(400,396,4,0.4)$ with perturbation norm $0.4$.\nThe hypothesis $0.4<4/2$ holds. A threshold in $(0.4,3.6)$ selects the true\norder three, while the largest adjacent ratio $396/4=99$ selects order two.\nThus a valid quantitative threshold theorem and a successful practical\nheuristic remain different claims.","timates order from singular-value ratios and\nvalidation heuristics. A largest adjacent ratio is not implied to select the\ntrue order by SG-28. An exact witness makes the distinction concrete. Take\nfour-by-four Hankel matrices of moments beginning at $n_0=4$ from the nodes\n$(1,i,-1)$ and real amplitudes $(100,-99,-1)$. The four-point Fourier columns\nare orthogonal, so the true singular values are $(400,396,4,0)$. Add the\nmoment perturbation $0.1(-i)^n$. It contributes a fourth orthogonal column and\nproduces singular values $(400,396,4,0.4)$ with perturbation norm $0.4$.\nThe hypothesis $0.4<4/2$ holds. A threshold in $(0.4,3.6)$ selects the true\norder three, while the largest adjacent ratio $396/4=99$ selects order two.\nThus a valid quantitative threshold theorem and a successful practical\nheuristic remain different claims.","validation heuristics. A largest adjacent ratio is not implied to select the\ntrue order by SG-28. An exact witness makes the distinction concrete. Take\nfour-by-four Hankel matrices of moments beginning at $n_0=4$ from the nodes\n$(1,i,-1)$ and real amplitudes $(100,-99,-1)$. The four-point Fourier columns\nare orthogonal, so the true singular values are $(400,396,4,0)$. Add the\nmoment perturbation $0.1(-i)^n$. It contributes a fourth orthogonal column and\nproduces singular values $(400,396,4,0.4)$ with perturbation norm $0.4$.\nThe hypothesis $0.4<4/2$ holds. A threshold in $(0.4,3.6)$ selects the true\norder three, while the largest adjacent ratio $396/4=99$ selects order two.\nThus a valid quantitative threshold theorem and a successful practical\nheuristic remain different claims.","der by SG-28. An exact witness makes the distinction concrete. Take\nfour-by-four Hankel matrices of moments beginning at $n_0=4$ from the nodes\n$(1,i,-1)$ and real amplitudes $(100,-99,-1)$. The four-point Fourier columns\nare orthogonal, so the true singular values are $(400,396,4,0)$. Add the\nmoment perturbation $0.1(-i)^n$. It contributes a fourth orthogonal column and\nproduces singular values $(400,396,4,0.4)$ with perturbation norm $0.4$.\nThe hypothesis $0.4<4/2$ holds. A threshold in $(0.4,3.6)$ selects the true\norder three, while the largest adjacent ratio $396/4=99$ selects order two.\nThus a valid quantitative threshold theorem and a successful practical\nheuristic remain different claims.","distinction concrete. Take\nfour-by-four Hankel matrices of moments beginning at $n_0=4$ from the nodes\n$(1,i,-1)$ and real amplitudes $(100,-99,-1)$. The four-point Fourier columns\nare orthogonal, so the true singular values are $(400,396,4,0)$. Add the\nmoment perturbation $0.1(-i)^n$. It contributes a fourth orthogonal column and\nproduces singular values $(400,396,4,0.4)$ with perturbation norm $0.4$.\nThe hypothesis $0.4<4/2$ holds. A threshold in $(0.4,3.6)$ selects the true\norder three, while the largest adjacent ratio $396/4=99$ selects order two.\nThus a valid quantitative threshold theorem and a successful practical\nheuristic remain different claims.","Take\nfour-by-four Hankel matrices of moments beginning at $n_0=4$ from the nodes\n$(1,i,-1)$ and real amplitudes $(100,-99,-1)$. The four-point Fourier columns\nare orthogonal, so the true singular values are $(400,396,4,0)$. Add the\nmoment perturbation $0.1(-i)^n$. It contributes a fourth orthogonal column and\nproduces singular values $(400,396,4,0.4)$ with perturbation norm $0.4$.\nThe hypothesis $0.4<4/2$ holds. A threshold in $(0.4,3.6)$ selects the true\norder three, while the largest adjacent ratio $396/4=99$ selects order two.\nThus a valid quantitative threshold theorem and a successful practical\nheuristic remain different claims.","s of moments beginning at $n_0=4$ from the nodes\n$(1,i,-1)$ and real amplitudes $(100,-99,-1)$. The four-point Fourier columns\nare orthogonal, so the true singular values are $(400,396,4,0)$. Add the\nmoment perturbation $0.1(-i)^n$. It contributes a fourth orthogonal column and\nproduces singular values $(400,396,4,0.4)$ with perturbation norm $0.4$.\nThe hypothesis $0.4<4/2$ holds. A threshold in $(0.4,3.6)$ selects the true\norder three, while the largest adjacent ratio $396/4=99$ selects order two.\nThus a valid quantitative threshold theorem and a successful practical\nheuristic remain different claims.","udes $(100,-99,-1)$. The four-point Fourier columns\nare orthogonal, so the true singular values are $(400,396,4,0)$. Add the\nmoment perturbation $0.1(-i)^n$. It contributes a fourth orthogonal column and\nproduces singular values $(400,396,4,0.4)$ with perturbation norm $0.4$.\nThe hypothesis $0.4<4/2$ holds. A threshold in $(0.4,3.6)$ selects the true\norder three, while the largest adjacent ratio $396/4=99$ selects order two.\nThus a valid quantitative threshold theorem and a successful practical\nheuristic remain different claims.","\\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\n\\begin{theorem}[SG-29\\quad Exact phase from adjacent offset products]\\label{SG-29}","\\[\n S(c)=R|b|^2z\\ne0,\\qquad\\Phi(c):=S(c)/|S(c)|=z.\n\\]\nTwo channels already give $z=c_1/c_0$ exactly. More channels provide repeated\nadjacent relations, without an unconditional deterministic averaging gain.","Each summand is\n$\\overline{bz^r}\\,bz^{r+1}=|b|^2z$ because $|z|=1$.\nThere are $R$ summands and $R\\ge1$, so normalization returns $z$.\nFor two channels direct division cancels $b$. No statistical independence is\nassumed, so one cannot improve a worst-case noise order merely by increasing\n$R$; the next bound shows the cancellation of the number of products.","\\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\n\\begin{theorem}[SG-30\\quad Deterministic offset-phase perturbation]\\label{SG-30}","The phase sum is nonzero and\n\\[\n |\\Phi(c)-z|\\le\\frac{2\\Delta}{A}\n =\\frac{4\\epsilon}{|b|}+2\\left(\\frac\\epsilon{|b|}\\right)^2.\n\\]","Expand each product:\n\\[\n \\overline{c_r}c_{r+1}=|b|^2z\n +\\overline{bz^r}e_{r+1}+\\overline{e_r}bz^{r+1}\n +\\overline{e_r}e_{r+1}.\n\\]\nThe three error terms have total magnitude at most\n$2|b|\\epsilon+\\epsilon^2$. Summing gives $S(c)=Az+E$ with $|E|\\le\\Delta$.\nTherefore $|S(c)|\\ge A-\\Delta>0$.\nFor any nonzero $v$ and unit $z$,\n\\[\n \\left|\\frac v{|v|}-z\\right|\n \\le\\left|\\frac v{|v|}-\\frac vA\\right|+\\left|\\frac vA-z\\right|\n =\\frac{\\bigl|A-|v|\\bigr|}{A}+\\frac{|v-Az|}{A}\n \\le\\frac{2|v-Az|}{A}.\n\\]\nApply this inequality to $v=S(c)$ and substitute the definitions.\nThe number of products cancels, as appropriate for coherent adversarial errors.","}e_{r+1}.\n\\]\nThe three error terms have total magnitude at most\n$2|b|\\epsilon+\\epsilon^2$. Summing gives $S(c)=Az+E$ with $|E|\\le\\Delta$.\nTherefore $|S(c)|\\ge A-\\Delta>0$.\nFor any nonzero $v$ and unit $z$,\n\\[\n \\left|\\frac v{|v|}-z\\right|\n \\le\\left|\\frac v{|v|}-\\frac vA\\right|+\\left|\\frac vA-z\\right|\n =\\frac{\\bigl|A-|v|\\bigr|}{A}+\\frac{|v-Az|}{A}\n \\le\\frac{2|v-Az|}{A}.\n\\]\nApply this inequality to $v=S(c)$ and substitute the definitions.\nThe number of products cancels, as appropriate for coherent adversarial errors.","\\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\n\\begin{theorem}[SG-31\\quad A zero offset residual tests a progression]\\label{SG-31}","Then $\\nu(c,\\zeta)=0$ if and only if $c_r=c_0\\zeta^r$ for\n$0\\le r\\le R$.","The denominator is positive because it includes $|c_0|^2$.\nA sum of nonnegative squared magnitudes is zero precisely when each recurrence\n$c_{r+1}=\\zeta c_r$ holds. Induction gives the progression. The reverse\nimplication follows by substitution. This is a consistency check for the\nfitted offsets. Another physical model or structured background could produce\nthe same progression; zero residual alone does not certify the true edge\nmodel or a true-location error bound. Unused offset channels provide a\nstronger practical check because they are not part of the fit, but no\nuniversal probability guarantee is asserted here.","\\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\n\\begin{theorem}[SG-32\\quad Unique branch selection under a strict margin]\\label{SG-32}","If\n\\[\n |\\widetilde z-z|+|\\rho-z|<\\sin(\\pi/N),\n\\]\nthen $\\rho$ is the unique root of $\\overline u$ nearest to $\\widetilde z$.\nIt equals the branch an oracle chooses by the true-node neighborhood.","The $N$ candidates lie on the unit circle at angles spaced by $2\\pi/N$.\nThe smallest distance between distinct candidates is $2\\sin(\\pi/N)$.\nThe triangle inequality gives\n$|\\widetilde z-\\rho|\\le|\\widetilde z-z|+|z-\\rho|<\\sin(\\pi/N)$.\nFor any other candidate $\\rho'$,\n$|\\widetilde z-\\rho'|\\ge|\\rho'-\\rho|-|\\rho-\\widetilde z|\n>\\sin(\\pi/N)$. Thus the nearest candidate is unique. The same spacing\nargument places every wrong root farther from the true node than the\nassociated root under the given margin. The oracle and data-based choices\ntherefore coincide. This is conditional on the margin; the true-node terms\nare not available directly to the algorithm.","\\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\n\\begin{theorem}[SG-33\\quad Associated roots divide the angular error]\\label{SG-33}","\\[\n |\\rho-z|\\le2\\sin\\left(\\frac1N\\arcsin\\frac\\eta2\\right)\n \\le\\frac2N\\arcsin\\frac\\eta2.\n\\]\nFor a nonzero raw estimate, $|\\widehat u-u|\\le\\epsilon$ implies\n$|\\widehat u/|\\widehat u|-u|\\le2\\epsilon$. Thus when the raw error tends to\nzero, correct lifting gains a factor $1/N$ in chord and angular accuracy.","If $\\theta\\in(-\\pi,\\pi)$ is the shortest angular difference, the unit-circle\nchord identity gives $|\\overline u-u|=2\\sin(|\\theta|/2)$.\nHence $|\\theta|\\le2\\arcsin(\\eta/2)$. On the associated branch the root\nangle changes by $\\theta/N$. Convert to chord distance and use $\\sin t\\le t$.\nFor normalization, add and subtract $\\widehat u$:\n\\[\n |\\widehat u/|\\widehat u|-u|\n \\le|1-|\\widehat u||+|\\widehat u-u|\n \\le2|\\widehat u-u|.\n\\]\nThe shortest-angle ambiguity at an antipodal estimate is excluded by\n$\\eta<2$. The gain applies to an associated branch, so a branch-selection\nargument such as SG-32 is needed to transfer it to an estimator.","s $|\\overline u-u|=2\\sin(|\\theta|/2)$.\nHence $|\\theta|\\le2\\arcsin(\\eta/2)$. On the associated branch the root\nangle changes by $\\theta/N$. Convert to chord distance and use $\\sin t\\le t$.\nFor normalization, add and subtract $\\widehat u$:\n\\[\n |\\widehat u/|\\widehat u|-u|\n \\le|1-|\\widehat u||+|\\widehat u-u|\n \\le2|\\widehat u-u|.\n\\]\nThe shortest-angle ambiguity at an antipodal estimate is excluded by\n$\\eta<2$. The gain applies to an associated branch, so a branch-selection\nargument such as SG-32 is needed to transfer it to an estimator.","\\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\n\\begin{theorem}[SG-34\\quad An explicit exact self-calibration budget]\\label{SG-34}","The Fourier coefficients at the high indices\n\\[\n \\mathcal I_0=\\{(q_0+q)N:0\\le q\\le2L_{\\max}\\},\\qquad\n \\mathcal I_1=\\{(q_0+q)N+1:0\\le q<L_{\\max}\\}\n\\]\ndetermine the unknown edge order, every physical node, and every jump amplitude\nexactly. The high-frequency budget is $3L_{\\max}+1$. Low modes\n$-K\\le n\\le K$ then give the complete signal. No supplied jump neighborhood\nis required.","All high indices exceed $K$, so the derivative moments have no smooth defect.\nThe base channel is an exponential sum in the distinct $u_\\ell$ with\ncoefficients $b_{\\ell,0}=a_\\ell u_\\ell^{q_0}\\ne0$.\nIts $2L_{\\max}+1$ moments form an $(L_{\\max}+1)$ square Hankel matrix;\nSG-19 gives its rank $r$. If $r=0$, the edge is zero and the low modes finish\nthe recovery. Otherwise SG-20 recovers the $u_\\ell$ and then the base\ncoefficients. The adjacent channel supplies at least $r$ moments of\n$\\sum_\\ell b_{\\ell,0}z_\\ell u_\\ell^q$. Using the same recovered nodes, its\nVandermonde system recovers $b_{\\ell,1}=b_{\\ell,0}z_\\ell$.\nDivision gives $z_\\ell=b_{\\ell,1}/b_{\\ell,0}$, and\n$a_\\ell=b_{\\ell,0}u_\\ell^{-q_0}$.\nSG-21 constructs $j$,","All high indices exceed $K$, so the derivative moments have no smooth defect.\nThe base channel is an exponential sum in the distinct $u_\\ell$ with\ncoefficients $b_{\\ell,0}=a_\\ell u_\\ell^{q_0}\\ne0$.\nIts $2L_{\\max}+1$ moments form an $(L_{\\max}+1)$ square Hankel matrix;\nSG-19 gives its rank $r$. If $r=0$, the edge is zero and the low modes finish\nthe recovery. Otherwise SG-20 recovers the $u_\\ell$ and then the base\ncoefficients. The adjacent channel supplies at least $r$ moments of\n$\\sum_\\ell b_{\\ell,0}z_\\ell u_\\ell^q$. Using the same recovered nodes, its\nVandermonde system recovers $b_{\\ell,1}=b_{\\ell,0}z_\\ell$.\nDivision gives $z_\\ell=b_{\\ell,1}/b_{\\ell,0}$, and\n$a_\\ell=b_{\\ell,0}u_\\ell^{-q_0}$.\nSG-21 constructs $j$, and subtraction from the low modes constructs $s$.\nThe two index sets are disjoint because $N\\ge2$ and their resid","All high indices exceed $K$, so the derivative moments have no smooth defect.\nThe base channel is an exponential sum in the distinct $u_\\ell$ with\ncoefficients $b_{\\ell,0}=a_\\ell u_\\ell^{q_0}\\ne0$.\nIts $2L_{\\max}+1$ moments form an $(L_{\\max}+1)$ square Hankel matrix;\nSG-19 gives its rank $r$. If $r=0$, the edge is zero and the low modes finish\nthe recovery. Otherwise SG-20 recovers the $u_\\ell$ and then the base\ncoefficients. The adjacent channel supplies at least $r$ moments of\n$\\sum_\\ell b_{\\ell,0}z_\\ell u_\\ell^q$. Using the same recovered nodes, its\nVandermonde system recovers $b_{\\ell,1}=b_{\\ell,0}z_\\ell$.\nDivision gives $z_\\ell=b_{\\ell,1}/b_{\\ell,0}$, and\n$a_\\ell=b_{\\ell,0}u_\\ell^{-q_0}$.\nSG-21 constructs $j$, and subtraction from the low modes constructs $s$.\nThe two index sets are disjoint because $N\\ge2$ and their residues are zero\nand one. Their sizes add to $3L_{\\max}+1$.","All high indices exceed $K$, so the derivative moments have no smooth defect.\nThe base channel is an exponential sum in the distinct $u_\\ell$ with\ncoefficients $b_{\\ell,0}=a_\\ell u_\\ell^{q_0}\\ne0$.\nIts $2L_{\\max}+1$ moments form an $(L_{\\max}+1)$ square Hankel matrix;\nSG-19 gives its rank $r$. If $r=0$, the edge is zero and the low modes finish\nthe recovery. Otherwise SG-20 recovers the $u_\\ell$ and then the base\ncoefficients. The adjacent channel supplies at least $r$ moments of\n$\\sum_\\ell b_{\\ell,0}z_\\ell u_\\ell^q$. Using the same recovered nodes, its\nVandermonde system recovers $b_{\\ell,1}=b_{\\ell,0}z_\\ell$.\nDivision gives $z_\\ell=b_{\\ell,1}/b_{\\ell,0}$, and\n$a_\\ell=b_{\\ell,0}u_\\ell^{-q_0}$.\nSG-21 constructs $j$, and subtraction from the low modes constructs $s$.\nThe two index sets are disjoint because $N\\ge2$ and their residues are zero\nand one. Their sizes add to $3L_{\\max}+1$. This is an upper b","All high indices exceed $K$, so the derivative moments have no smooth defect.\nThe base channel is an exponential sum in the distinct $u_\\ell$ with\ncoefficients $b_{\\ell,0}=a_\\ell u_\\ell^{q_0}\\ne0$.\nIts $2L_{\\max}+1$ moments form an $(L_{\\max}+1)$ square Hankel matrix;\nSG-19 gives its rank $r$. If $r=0$, the edge is zero and the low modes finish\nthe recovery. Otherwise SG-20 recovers the $u_\\ell$ and then the base\ncoefficients. The adjacent channel supplies at least $r$ moments of\n$\\sum_\\ell b_{\\ell,0}z_\\ell u_\\ell^q$. Using the same recovered nodes, its\nVandermonde system recovers $b_{\\ell,1}=b_{\\ell,0}z_\\ell$.\nDivision gives $z_\\ell=b_{\\ell,1}/b_{\\ell,0}$, and\n$a_\\ell=b_{\\ell,0}u_\\ell^{-q_0}$.\nSG-21 constructs $j$, and subtraction from the low modes constructs $s$.\nThe two index sets are disjoint because $N\\ge2$ and their residues are zero\nand one. Their sizes add to $3L_{\\max}+1$. This is an upper bound for the\nconstruction and e","All high indices exceed $K$, so the derivative moments have no smooth defect.\nThe base channel is an exponential sum in the distinct $u_\\ell$ with\ncoefficients $b_{\\ell,0}=a_\\ell u_\\ell^{q_0}\\ne0$.\nIts $2L_{\\max}+1$ moments form an $(L_{\\max}+1)$ square Hankel matrix;\nSG-19 gives its rank $r$. If $r=0$, the edge is zero and the low modes finish\nthe recovery. Otherwise SG-20 recovers the $u_\\ell$ and then the base\ncoefficients. The adjacent channel supplies at least $r$ moments of\n$\\sum_\\ell b_{\\ell,0}z_\\ell u_\\ell^q$. Using the same recovered nodes, its\nVandermonde system recovers $b_{\\ell,1}=b_{\\ell,0}z_\\ell$.\nDivision gives $z_\\ell=b_{\\ell,1}/b_{\\ell,0}$, and\n$a_\\ell=b_{\\ell,0}u_\\ell^{-q_0}$.\nSG-21 constructs $j$, and subtraction from the low modes constructs $s$.\nThe two index sets are disjoint because $N\\ge2$ and their residues are zero\nand one. Their sizes add to $3L_{\\max}+1$. This is an upper bound for the\nconstruction and excludes the low reconstruction coefficients.","efect.\nThe base channel is an exponential sum in the distinct $u_\\ell$ with\ncoefficients $b_{\\ell,0}=a_\\ell u_\\ell^{q_0}\\ne0$.\nIts $2L_{\\max}+1$ moments form an $(L_{\\max}+1)$ square Hankel matrix;\nSG-19 gives its rank $r$. If $r=0$, the edge is zero and the low modes finish\nthe recovery. Otherwise SG-20 recovers the $u_\\ell$ and then the base\ncoefficients. The adjacent channel supplies at least $r$ moments of\n$\\sum_\\ell b_{\\ell,0}z_\\ell u_\\ell^q$. Using the same recovered nodes, its\nVandermonde system recovers $b_{\\ell,1}=b_{\\ell,0}z_\\ell$.\nDivision gives $z_\\ell=b_{\\ell,1}/b_{\\ell,0}$, and\n$a_\\ell=b_{\\ell,0}u_\\ell^{-q_0}$.\nSG-21 constructs $j$, and subtraction from the low modes constructs $s$.\nThe two index sets are disjoint because $N\\ge2$ and their residues are zero\nand one. Their sizes add to $3L_{\\max}+1$. This is an upper bound for the\nconstruction and excludes the low reconstruction coefficients.","channel is an exponential sum in the distinct $u_\\ell$ with\ncoefficients $b_{\\ell,0}=a_\\ell u_\\ell^{q_0}\\ne0$.\nIts $2L_{\\max}+1$ moments form an $(L_{\\max}+1)$ square Hankel matrix;\nSG-19 gives its rank $r$. If $r=0$, the edge is zero and the low modes finish\nthe recovery. Otherwise SG-20 recovers the $u_\\ell$ and then the base\ncoefficients. The adjacent channel supplies at least $r$ moments of\n$\\sum_\\ell b_{\\ell,0}z_\\ell u_\\ell^q$. Using the same recovered nodes, its\nVandermonde system recovers $b_{\\ell,1}=b_{\\ell,0}z_\\ell$.\nDivision gives $z_\\ell=b_{\\ell,1}/b_{\\ell,0}$, and\n$a_\\ell=b_{\\ell,0}u_\\ell^{-q_0}$.\nSG-21 constructs $j$, and subtraction from the low modes constructs $s$.\nThe two index sets are disjoint because $N\\ge2$ and their residues are zero\nand one. Their sizes add to $3L_{\\max}+1$. This is an upper bound for the\nconstruction and excludes the low reconstruction coefficients.","0}\\ne0$.\nIts $2L_{\\max}+1$ moments form an $(L_{\\max}+1)$ square Hankel matrix;\nSG-19 gives its rank $r$. If $r=0$, the edge is zero and the low modes finish\nthe recovery. Otherwise SG-20 recovers the $u_\\ell$ and then the base\ncoefficients. The adjacent channel supplies at least $r$ moments of\n$\\sum_\\ell b_{\\ell,0}z_\\ell u_\\ell^q$. Using the same recovered nodes, its\nVandermonde system recovers $b_{\\ell,1}=b_{\\ell,0}z_\\ell$.\nDivision gives $z_\\ell=b_{\\ell,1}/b_{\\ell,0}$, and\n$a_\\ell=b_{\\ell,0}u_\\ell^{-q_0}$.\nSG-21 constructs $j$, and subtraction from the low modes constructs $s$.\nThe two index sets are disjoint because $N\\ge2$ and their residues are zero\nand one. Their sizes add to $3L_{\\max}+1$. This is an upper bound for the\nconstruction and excludes the low reconstruction coefficients.","_{\\max}+1)$ square Hankel matrix;\nSG-19 gives its rank $r$. If $r=0$, the edge is zero and the low modes finish\nthe recovery. Otherwise SG-20 recovers the $u_\\ell$ and then the base\ncoefficients. The adjacent channel supplies at least $r$ moments of\n$\\sum_\\ell b_{\\ell,0}z_\\ell u_\\ell^q$. Using the same recovered nodes, its\nVandermonde system recovers $b_{\\ell,1}=b_{\\ell,0}z_\\ell$.\nDivision gives $z_\\ell=b_{\\ell,1}/b_{\\ell,0}$, and\n$a_\\ell=b_{\\ell,0}u_\\ell^{-q_0}$.\nSG-21 constructs $j$, and subtraction from the low modes constructs $s$.\nThe two index sets are disjoint because $N\\ge2$ and their residues are zero\nand one. Their sizes add to $3L_{\\max}+1$. This is an upper bound for the\nconstruction and excludes the low reconstruction coefficients.","-19 gives its rank $r$. If $r=0$, the edge is zero and the low modes finish\nthe recovery. Otherwise SG-20 recovers the $u_\\ell$ and then the base\ncoefficients. The adjacent channel supplies at least $r$ moments of\n$\\sum_\\ell b_{\\ell,0}z_\\ell u_\\ell^q$. Using the same recovered nodes, its\nVandermonde system recovers $b_{\\ell,1}=b_{\\ell,0}z_\\ell$.\nDivision gives $z_\\ell=b_{\\ell,1}/b_{\\ell,0}$, and\n$a_\\ell=b_{\\ell,0}u_\\ell^{-q_0}$.\nSG-21 constructs $j$, and subtraction from the low modes constructs $s$.\nThe two index sets are disjoint because $N\\ge2$ and their residues are zero\nand one. Their sizes add to $3L_{\\max}+1$. This is an upper bound for the\nconstruction and excludes the low reconstruction coefficients.","and the low modes finish\nthe recovery. Otherwise SG-20 recovers the $u_\\ell$ and then the base\ncoefficients. The adjacent channel supplies at least $r$ moments of\n$\\sum_\\ell b_{\\ell,0}z_\\ell u_\\ell^q$. Using the same recovered nodes, its\nVandermonde system recovers $b_{\\ell,1}=b_{\\ell,0}z_\\ell$.\nDivision gives $z_\\ell=b_{\\ell,1}/b_{\\ell,0}$, and\n$a_\\ell=b_{\\ell,0}u_\\ell^{-q_0}$.\nSG-21 constructs $j$, and subtraction from the low modes constructs $s$.\nThe two index sets are disjoint because $N\\ge2$ and their residues are zero\nand one. Their sizes add to $3L_{\\max}+1$. This is an upper bound for the\nconstruction and excludes the low reconstruction coefficients.","-20 recovers the $u_\\ell$ and then the base\ncoefficients. The adjacent channel supplies at least $r$ moments of\n$\\sum_\\ell b_{\\ell,0}z_\\ell u_\\ell^q$. Using the same recovered nodes, its\nVandermonde system recovers $b_{\\ell,1}=b_{\\ell,0}z_\\ell$.\nDivision gives $z_\\ell=b_{\\ell,1}/b_{\\ell,0}$, and\n$a_\\ell=b_{\\ell,0}u_\\ell^{-q_0}$.\nSG-21 constructs $j$, and subtraction from the low modes constructs $s$.\nThe two index sets are disjoint because $N\\ge2$ and their residues are zero\nand one. Their sizes add to $3L_{\\max}+1$. This is an upper bound for the\nconstruction and excludes the low reconstruction coefficients.","oefficients. The adjacent channel supplies at least $r$ moments of\n$\\sum_\\ell b_{\\ell,0}z_\\ell u_\\ell^q$. Using the same recovered nodes, its\nVandermonde system recovers $b_{\\ell,1}=b_{\\ell,0}z_\\ell$.\nDivision gives $z_\\ell=b_{\\ell,1}/b_{\\ell,0}$, and\n$a_\\ell=b_{\\ell,0}u_\\ell^{-q_0}$.\nSG-21 constructs $j$, and subtraction from the low modes constructs $s$.\nThe two index sets are disjoint because $N\\ge2$ and their residues are zero\nand one. Their sizes add to $3L_{\\max}+1$. This is an upper bound for the\nconstruction and excludes the low reconstruction coefficients.","of\n$\\sum_\\ell b_{\\ell,0}z_\\ell u_\\ell^q$. Using the same recovered nodes, its\nVandermonde system recovers $b_{\\ell,1}=b_{\\ell,0}z_\\ell$.\nDivision gives $z_\\ell=b_{\\ell,1}/b_{\\ell,0}$, and\n$a_\\ell=b_{\\ell,0}u_\\ell^{-q_0}$.\nSG-21 constructs $j$, and subtraction from the low modes constructs $s$.\nThe two index sets are disjoint because $N\\ge2$ and their residues are zero\nand one. Their sizes add to $3L_{\\max}+1$. This is an upper bound for the\nconstruction and excludes the low reconstruction coefficients.","\\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\n\\begin{theorem}[SG-35\\quad Conditional asymptotic rates after calibrated lifting]\\label{SG-35}","cted 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\n\\begin{theorem}[SG-35\\quad Conditional asymptotic rates after calibrated lifting]\\label{SG-35}","The threshold-selected order is eventually correct. The aligned lifted nodes\nand locations satisfy\n\\[\n |\\widehat z_\\ell-z_\\ell|=O(N^{-p-2}),\\qquad\n d_{\\mathbb T}(\\widehat\\tau_\\ell,\\tau_\\ell)=O(N^{-p-2}).\n\\]\nThe recovered amplitudes satisfy\n$|\\widehat a_\\ell-a_\\ell|=O(N^{-p-1})$.","Uniform local inverse bounds transfer the moment error to aliased node and\nmatched coefficient errors of order $N^{-p-1}$. The supplied rank gap and\nthreshold select the correct order by SG-28. Nonvanishing base coefficients\nallow SG-30 to give the same order for the coarse phase error.\nThe strict branch margin selects the associated root by SG-32. Normalize the\naliased estimate and use SG-33: its root error is $N^{-1}$ times an error of\norder $N^{-p-1}$, giving $N^{-p-2}$. On the unit circle,\n$d_{\\mathbb T}=2\\arcsin(|\\widehat z-z|/2)$, which has the same order for\nsmall errors. For the amplitude use\n$\\widehat a=\\widehat b_0\\,\\overline u^{-q_0}$ and $a=b_0u^{-q_0}$.\nSince $q_0$ is fixed and both powers have unit modulus, SG-26's telescoping\npower estimate bounds the power error by $q_0|\\overline u-u|$.\nTogeth","verse bounds transfer the moment error to aliased node and\nmatched coefficient errors of order $N^{-p-1}$. The supplied rank gap and\nthreshold select the correct order by SG-28. Nonvanishing base coefficients\nallow SG-30 to give the same order for the coarse phase error.\nThe strict branch margin selects the associated root by SG-32. Normalize the\naliased estimate and use SG-33: its root error is $N^{-1}$ times an error of\norder $N^{-p-1}$, giving $N^{-p-2}$. On the unit circle,\n$d_{\\mathbb T}=2\\arcsin(|\\widehat z-z|/2)$, which has the same order for\nsmall errors. For the amplitude use\n$\\widehat a=\\widehat b_0\\,\\overline u^{-q_0}$ and $a=b_0u^{-q_0}$.\nSince $q_0$ is fixed and both powers have unit modulus, SG-26's telescoping\npower estimate bounds the power error by $q_0|\\overline u-u|$.\nTogether with the coefficient error this gives the stated amplitude order.\nThe smooth-tail assumption in SG-27 supplies the moment defect.\nThe conclusion is uniform only on the selected scales and parameter families\nfor which the inverse, rank, and branch bounds are uniform.","error to aliased node and\nmatched coefficient errors of order $N^{-p-1}$. The supplied rank gap and\nthreshold select the correct order by SG-28. Nonvanishing base coefficients\nallow SG-30 to give the same order for the coarse phase error.\nThe strict branch margin selects the associated root by SG-32. Normalize the\naliased estimate and use SG-33: its root error is $N^{-1}$ times an error of\norder $N^{-p-1}$, giving $N^{-p-2}$. On the unit circle,\n$d_{\\mathbb T}=2\\arcsin(|\\widehat z-z|/2)$, which has the same order for\nsmall errors. For the amplitude use\n$\\widehat a=\\widehat b_0\\,\\overline u^{-q_0}$ and $a=b_0u^{-q_0}$.\nSince $q_0$ is fixed and both powers have unit modulus, SG-26's telescoping\npower estimate bounds the power error by $q_0|\\overline u-u|$.\nTogether with the coefficient error this gives the stated amplitude order.\nThe smooth-tail assumption in SG-27 supplies the moment defect.\nThe conclusion is uniform only on the selected scales and parameter families\nfor which the inverse, rank, and branch bounds are uniform.","de and\nmatched coefficient errors of order $N^{-p-1}$. The supplied rank gap and\nthreshold select the correct order by SG-28. Nonvanishing base coefficients\nallow SG-30 to give the same order for the coarse phase error.\nThe strict branch margin selects the associated root by SG-32. Normalize the\naliased estimate and use SG-33: its root error is $N^{-1}$ times an error of\norder $N^{-p-1}$, giving $N^{-p-2}$. On the unit circle,\n$d_{\\mathbb T}=2\\arcsin(|\\widehat z-z|/2)$, which has the same order for\nsmall errors. For the amplitude use\n$\\widehat a=\\widehat b_0\\,\\overline u^{-q_0}$ and $a=b_0u^{-q_0}$.\nSince $q_0$ is fixed and both powers have unit modulus, SG-26's telescoping\npower estimate bounds the power error by $q_0|\\overline u-u|$.\nTogether with the coefficient error this gives the stated amplitude order.\nThe smooth-tail assumption in SG-27 supplies the moment defect.\nThe conclusion is uniform only on the selected scales and parameter families\nfor which the inverse, rank, and branch bounds are uniform.","s of order $N^{-p-1}$. The supplied rank gap and\nthreshold select the correct order by SG-28. Nonvanishing base coefficients\nallow SG-30 to give the same order for the coarse phase error.\nThe strict branch margin selects the associated root by SG-32. Normalize the\naliased estimate and use SG-33: its root error is $N^{-1}$ times an error of\norder $N^{-p-1}$, giving $N^{-p-2}$. On the unit circle,\n$d_{\\mathbb T}=2\\arcsin(|\\widehat z-z|/2)$, which has the same order for\nsmall errors. For the amplitude use\n$\\widehat a=\\widehat b_0\\,\\overline u^{-q_0}$ and $a=b_0u^{-q_0}$.\nSince $q_0$ is fixed and both powers have unit modulus, SG-26's telescoping\npower estimate bounds the power error by $q_0|\\overline u-u|$.\nTogether with the coefficient error this gives the stated amplitude order.\nThe smooth-tail assumption in SG-27 supplies the moment defect.\nThe conclusion is uniform only on the selected scales and parameter families\nfor which the inverse, rank, and branch bounds are uniform.","se coefficients\nallow SG-30 to give the same order for the coarse phase error.\nThe strict branch margin selects the associated root by SG-32. Normalize the\naliased estimate and use SG-33: its root error is $N^{-1}$ times an error of\norder $N^{-p-1}$, giving $N^{-p-2}$. On the unit circle,\n$d_{\\mathbb T}=2\\arcsin(|\\widehat z-z|/2)$, which has the same order for\nsmall errors. For the amplitude use\n$\\widehat a=\\widehat b_0\\,\\overline u^{-q_0}$ and $a=b_0u^{-q_0}$.\nSince $q_0$ is fixed and both powers have unit modulus, SG-26's telescoping\npower estimate bounds the power error by $q_0|\\overline u-u|$.\nTogether with the coefficient error this gives the stated amplitude order.\nThe smooth-tail assumption in SG-27 supplies the moment defect.\nThe conclusion is uniform only on the selected scales and parameter families\nfor which the inverse, rank, and branch bounds are uniform.","for the coarse phase error.\nThe strict branch margin selects the associated root by SG-32. Normalize the\naliased estimate and use SG-33: its root error is $N^{-1}$ times an error of\norder $N^{-p-1}$, giving $N^{-p-2}$. On the unit circle,\n$d_{\\mathbb T}=2\\arcsin(|\\widehat z-z|/2)$, which has the same order for\nsmall errors. For the amplitude use\n$\\widehat a=\\widehat b_0\\,\\overline u^{-q_0}$ and $a=b_0u^{-q_0}$.\nSince $q_0$ is fixed and both powers have unit modulus, SG-26's telescoping\npower estimate bounds the power error by $q_0|\\overline u-u|$.\nTogether with the coefficient error this gives the stated amplitude order.\nThe smooth-tail assumption in SG-27 supplies the moment defect.\nThe conclusion is uniform only on the selected scales and parameter families\nfor which the inverse, rank, and branch bounds are uniform.","rror.\nThe strict branch margin selects the associated root by SG-32. Normalize the\naliased estimate and use SG-33: its root error is $N^{-1}$ times an error of\norder $N^{-p-1}$, giving $N^{-p-2}$. On the unit circle,\n$d_{\\mathbb T}=2\\arcsin(|\\widehat z-z|/2)$, which has the same order for\nsmall errors. For the amplitude use\n$\\widehat a=\\widehat b_0\\,\\overline u^{-q_0}$ and $a=b_0u^{-q_0}$.\nSince $q_0$ is fixed and both powers have unit modulus, SG-26's telescoping\npower estimate bounds the power error by $q_0|\\overline u-u|$.\nTogether with the coefficient error this gives the stated amplitude order.\nThe smooth-tail assumption in SG-27 supplies the moment defect.\nThe conclusion is uniform only on the selected scales and parameter families\nfor which the inverse, rank, and branch bounds are uniform.","33: its root error is $N^{-1}$ times an error of\norder $N^{-p-1}$, giving $N^{-p-2}$. On the unit circle,\n$d_{\\mathbb T}=2\\arcsin(|\\widehat z-z|/2)$, which has the same order for\nsmall errors. For the amplitude use\n$\\widehat a=\\widehat b_0\\,\\overline u^{-q_0}$ and $a=b_0u^{-q_0}$.\nSince $q_0$ is fixed and both powers have unit modulus, SG-26's telescoping\npower estimate bounds the power error by $q_0|\\overline u-u|$.\nTogether with the coefficient error this gives the stated amplitude order.\nThe smooth-tail assumption in SG-27 supplies the moment defect.\nThe conclusion is uniform only on the selected scales and parameter families\nfor which the inverse, rank, and branch bounds are uniform.","\\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\n\\begin{theorem}[SG-36\\quad Resolved synthesis splits the two error sources]\\label{SG-36}","\\[\n f-\\widehat f_M=(s-A_Ms)+(I-A_M)(j-\\widehat j),\n\\]\nand therefore\n\\[\n \\|f-\\widehat f_M\\|\\le\\|s-A_Ms\\|+(1+C_M)\\|j-\\widehat j\\|.\n\\]\nIf $\\widehat j=j$, the complete error equals the smooth approximation error.","Substitute $f=s+j$ into the definition and collect the two terms. Linearity\nof $A_M$ is used in that collection. The triangle inequality and\n$\\|I-A_M\\|\\le1+\\|A_M\\|$ give the estimate. With the exact edge, its error\nterm vanishes as an object identity. The theorem holds for the declared norm;\na misplaced discontinuity need not be small in supremum norm even when its\nlocation error is small. In $L^1$, the next theorem quantifies that error.\nA nonlinear clipping operator is not an instance of this linear identity.","\\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\n\\begin{theorem}[SG-37\\quad Sharp additive L1 control for normalized jump packets]\\label{SG-37}","\\[\n \\|j-\\widehat j\\|_1\n \\le\\frac T4\\sum_\\ell|a_\\ell-\\widehat a_\\ell|\n +2\\sum_\\ell|\\widehat a_\\ell|d_\\ell(1-d_\\ell/T).\n\\]\nIn particular the simpler location term $2\\sum_\\ell|\\widehat a_\\ell|d_\\ell$\nis valid. The printed location coefficient one in the inherited\nself-calibration manuscript is not valid for mean-zero normalization.","SG-21 writes both packets as translated sawtooth sums. Subtract in the form\n\\[\n j-\\widehat j=\n \\sum_\\ell(a_\\ell-\\widehat a_\\ell)G_T(\\cdot-\\tau_\\ell)\n +\\sum_\\ell\\widehat a_\\ell\n [G_T(\\cdot-\\tau_\\ell)-G_T(\\cdot-\\widehat\\tau_\\ell)].\n\\]\nDirect integration gives $\\|G_T\\|_1=T/4$.\nFor a shift by $d\\in[0,T/2]$, on an interval of length $T-d$ the difference\nhas magnitude $d/T$, and on an interval of length $d$ it has magnitude\n$1-d/T$. Hence its $L^1$ norm is\n$(T-d)d/T+d(1-d/T)=2d(1-d/T)$.\nApply the triangle inequality term by term. Periodic translation invariance\nmakes the calculation depend only on circular shift distance.\nTo refute coefficient one, take\n\\[\n j=\\mathbf1_{[0,T/2)}-\\tfrac12,\\qquad\n \\widehat j=\\mathbf1_{[T/4,T/2)}-\\tfrac14.\n\\]\nBoth have jumps $(1,-1)$, with one jump shifted by $T/4$. Their norm\ndifference is $3T/8$, while the inherited bound gives $T/4$.\nThe sharper location term is attained in this example. Th","SG-21 writes both packets as translated sawtooth sums. Subtract in the form\n\\[\n j-\\widehat j=\n \\sum_\\ell(a_\\ell-\\widehat a_\\ell)G_T(\\cdot-\\tau_\\ell)\n +\\sum_\\ell\\widehat a_\\ell\n [G_T(\\cdot-\\tau_\\ell)-G_T(\\cdot-\\widehat\\tau_\\ell)].\n\\]\nDirect integration gives $\\|G_T\\|_1=T/4$.\nFor a shift by $d\\in[0,T/2]$, on an interval of length $T-d$ the difference\nhas magnitude $d/T$, and on an interval of length $d$ it has magnitude\n$1-d/T$. Hence its $L^1$ norm is\n$(T-d)d/T+d(1-d/T)=2d(1-d/T)$.\nApply the triangle inequality term by term. Periodic translation invariance\nmakes the calculation depend only on circular shift distance.\nTo refute coefficient one, take\n\\[\n j=\\mathbf1_{[0,T/2)}-\\tfrac12,\\qquad\n \\widehat j=\\mathbf1_{[T/4,T/2)}-\\tfrac14.\n\\]\nBoth have jumps $(1,-1)$, with one jump shifted by $T/4$. Their norm\ndifference is $3T/8$, while the inherited bound gives $T/4$.\nThe sharper location term is attained in this example. The amplitude\ncoefficient $T/4$ is also atta","SG-21 writes both packets as translated sawtooth sums. Subtract in the form\n\\[\n j-\\widehat j=\n \\sum_\\ell(a_\\ell-\\widehat a_\\ell)G_T(\\cdot-\\tau_\\ell)\n +\\sum_\\ell\\widehat a_\\ell\n [G_T(\\cdot-\\tau_\\ell)-G_T(\\cdot-\\widehat\\tau_\\ell)].\n\\]\nDirect integration gives $\\|G_T\\|_1=T/4$.\nFor a shift by $d\\in[0,T/2]$, on an interval of length $T-d$ the difference\nhas magnitude $d/T$, and on an interval of length $d$ it has magnitude\n$1-d/T$. Hence its $L^1$ norm is\n$(T-d)d/T+d(1-d/T)=2d(1-d/T)$.\nApply the triangle inequality term by term. Periodic translation invariance\nmakes the calculation depend only on circular shift distance.\nTo refute coefficient one, take\n\\[\n j=\\mathbf1_{[0,T/2)}-\\tfrac12,\\qquad\n \\widehat j=\\mathbf1_{[T/4,T/2)}-\\tfrac14.\n\\]\nBoth have jumps $(1,-1)$, with one jump shifted by $T/4$. Their norm\ndifference is $3T/8$, while the inherited bound gives $T/4$.\nThe sharper location term is attained in this example. The amplitude\ncoefficient $T/4$ is also attained by changing opposite anti","SG-21 writes both packets as translated sawtooth sums. Subtract in the form\n\\[\n j-\\widehat j=\n \\sum_\\ell(a_\\ell-\\widehat a_\\ell)G_T(\\cdot-\\tau_\\ell)\n +\\sum_\\ell\\widehat a_\\ell\n [G_T(\\cdot-\\tau_\\ell)-G_T(\\cdot-\\widehat\\tau_\\ell)].\n\\]\nDirect integration gives $\\|G_T\\|_1=T/4$.\nFor a shift by $d\\in[0,T/2]$, on an interval of length $T-d$ the difference\nhas magnitude $d/T$, and on an interval of length $d$ it has magnitude\n$1-d/T$. Hence its $L^1$ norm is\n$(T-d)d/T+d(1-d/T)=2d(1-d/T)$.\nApply the triangle inequality term by term. Periodic translation invariance\nmakes the calculation depend only on circular shift distance.\nTo refute coefficient one, take\n\\[\n j=\\mathbf1_{[0,T/2)}-\\tfrac12,\\qquad\n \\widehat j=\\mathbf1_{[T/4,T/2)}-\\tfrac14.\n\\]\nBoth have jumps $(1,-1)$, with one jump shifted by $T/4$. Their norm\ndifference is $3T/8$, while the inherited bound gives $T/4$.\nThe sharper location term is attained in this example. The amplitude\ncoefficient $T/4$ is also attained by changing opposite antipodal jump\namplitudes: the diffe","SG-21 writes both packets as translated sawtooth sums. Subtract in the form\n\\[\n j-\\widehat j=\n \\sum_\\ell(a_\\ell-\\widehat a_\\ell)G_T(\\cdot-\\tau_\\ell)\n +\\sum_\\ell\\widehat a_\\ell\n [G_T(\\cdot-\\tau_\\ell)-G_T(\\cdot-\\widehat\\tau_\\ell)].\n\\]\nDirect integration gives $\\|G_T\\|_1=T/4$.\nFor a shift by $d\\in[0,T/2]$, on an interval of length $T-d$ the difference\nhas magnitude $d/T$, and on an interval of length $d$ it has magnitude\n$1-d/T$. Hence its $L^1$ norm is\n$(T-d)d/T+d(1-d/T)=2d(1-d/T)$.\nApply the triangle inequality term by term. Periodic translation invariance\nmakes the calculation depend only on circular shift distance.\nTo refute coefficient one, take\n\\[\n j=\\mathbf1_{[0,T/2)}-\\tfrac12,\\qquad\n \\widehat j=\\mathbf1_{[T/4,T/2)}-\\tfrac14.\n\\]\nBoth have jumps $(1,-1)$, with one jump shifted by $T/4$. Their norm\ndifference is $3T/8$, while the inherited bound gives $T/4$.\nThe sharper location term is attained in this example. The amplitude\ncoefficient $T/4$ is also attained by changing opposite antipodal jump\namplitudes: the difference packet has constant magnitude","-21 writes both packets as translated sawtooth sums. Subtract in the form\n\\[\n j-\\widehat j=\n \\sum_\\ell(a_\\ell-\\widehat a_\\ell)G_T(\\cdot-\\tau_\\ell)\n +\\sum_\\ell\\widehat a_\\ell\n [G_T(\\cdot-\\tau_\\ell)-G_T(\\cdot-\\widehat\\tau_\\ell)].\n\\]\nDirect integration gives $\\|G_T\\|_1=T/4$.\nFor a shift by $d\\in[0,T/2]$, on an interval of length $T-d$ the difference\nhas magnitude $d/T$, and on an interval of length $d$ it has magnitude\n$1-d/T$. Hence its $L^1$ norm is\n$(T-d)d/T+d(1-d/T)=2d(1-d/T)$.\nApply the triangle inequality term by term. Periodic translation invariance\nmakes the calculation depend only on circular shift distance.\nTo refute coefficient one, take\n\\[\n j=\\mathbf1_{[0,T/2)}-\\tfrac12,\\qquad\n \\widehat j=\\mathbf1_{[T/4,T/2)}-\\tfrac14.\n\\]\nBoth have jumps $(1,-1)$, with one jump shifted by $T/4$. Their norm\ndifference is $3T/8$, while the inherited bound gives $T/4$.\nThe sharper location term is attained in this example. The amplitude\ncoefficient $T/4$ is also attained by changing opposite antipodal jump\namplitudes: the difference packet has constant magnitude half the amplitude\nchange on the","ts as translated sawtooth sums. Subtract in the form\n\\[\n j-\\widehat j=\n \\sum_\\ell(a_\\ell-\\widehat a_\\ell)G_T(\\cdot-\\tau_\\ell)\n +\\sum_\\ell\\widehat a_\\ell\n [G_T(\\cdot-\\tau_\\ell)-G_T(\\cdot-\\widehat\\tau_\\ell)].\n\\]\nDirect integration gives $\\|G_T\\|_1=T/4$.\nFor a shift by $d\\in[0,T/2]$, on an interval of length $T-d$ the difference\nhas magnitude $d/T$, and on an interval of length $d$ it has magnitude\n$1-d/T$. Hence its $L^1$ norm is\n$(T-d)d/T+d(1-d/T)=2d(1-d/T)$.\nApply the triangle inequality term by term. Periodic translation invariance\nmakes the calculation depend only on circular shift distance.\nTo refute coefficient one, take\n\\[\n j=\\mathbf1_{[0,T/2)}-\\tfrac12,\\qquad\n \\widehat j=\\mathbf1_{[T/4,T/2)}-\\tfrac14.\n\\]\nBoth have jumps $(1,-1)$, with one jump shifted by $T/4$. Their norm\ndifference is $3T/8$, while the inherited bound gives $T/4$.\nThe sharper location term is attained in this example. The amplitude\ncoefficient $T/4$ is also attained by changing opposite antipodal jump\namplitudes: the difference packet has constant magnitude half the amplitude\nchange on the two half-periods.","wtooth sums. Subtract in the form\n\\[\n j-\\widehat j=\n \\sum_\\ell(a_\\ell-\\widehat a_\\ell)G_T(\\cdot-\\tau_\\ell)\n +\\sum_\\ell\\widehat a_\\ell\n [G_T(\\cdot-\\tau_\\ell)-G_T(\\cdot-\\widehat\\tau_\\ell)].\n\\]\nDirect integration gives $\\|G_T\\|_1=T/4$.\nFor a shift by $d\\in[0,T/2]$, on an interval of length $T-d$ the difference\nhas magnitude $d/T$, and on an interval of length $d$ it has magnitude\n$1-d/T$. Hence its $L^1$ norm is\n$(T-d)d/T+d(1-d/T)=2d(1-d/T)$.\nApply the triangle inequality term by term. Periodic translation invariance\nmakes the calculation depend only on circular shift distance.\nTo refute coefficient one, take\n\\[\n j=\\mathbf1_{[0,T/2)}-\\tfrac12,\\qquad\n \\widehat j=\\mathbf1_{[T/4,T/2)}-\\tfrac14.\n\\]\nBoth have jumps $(1,-1)$, with one jump shifted by $T/4$. Their norm\ndifference is $3T/8$, while the inherited bound gives $T/4$.\nThe sharper location term is attained in this example. The amplitude\ncoefficient $T/4$ is also attained by changing opposite antipodal jump\namplitudes: the difference packet has constant magnitude half the amplitude\nchange on the two half-periods.","ubtract in the form\n\\[\n j-\\widehat j=\n \\sum_\\ell(a_\\ell-\\widehat a_\\ell)G_T(\\cdot-\\tau_\\ell)\n +\\sum_\\ell\\widehat a_\\ell\n [G_T(\\cdot-\\tau_\\ell)-G_T(\\cdot-\\widehat\\tau_\\ell)].\n\\]\nDirect integration gives $\\|G_T\\|_1=T/4$.\nFor a shift by $d\\in[0,T/2]$, on an interval of length $T-d$ the difference\nhas magnitude $d/T$, and on an interval of length $d$ it has magnitude\n$1-d/T$. Hence its $L^1$ norm is\n$(T-d)d/T+d(1-d/T)=2d(1-d/T)$.\nApply the triangle inequality term by term. Periodic translation invariance\nmakes the calculation depend only on circular shift distance.\nTo refute coefficient one, take\n\\[\n j=\\mathbf1_{[0,T/2)}-\\tfrac12,\\qquad\n \\widehat j=\\mathbf1_{[T/4,T/2)}-\\tfrac14.\n\\]\nBoth have jumps $(1,-1)$, with one jump shifted by $T/4$. Their norm\ndifference is $3T/8$, while the inherited bound gives $T/4$.\nThe sharper location term is attained in this example. The amplitude\ncoefficient $T/4$ is also attained by changing opposite antipodal jump\namplitudes: the difference packet has constant magnitude half the amplitude\nchange on the two half-periods.","$\\|G_T\\|_1=T/4$.\nFor a shift by $d\\in[0,T/2]$, on an interval of length $T-d$ the difference\nhas magnitude $d/T$, and on an interval of length $d$ it has magnitude\n$1-d/T$. Hence its $L^1$ norm is\n$(T-d)d/T+d(1-d/T)=2d(1-d/T)$.\nApply the triangle inequality term by term. Periodic translation invariance\nmakes the calculation depend only on circular shift distance.\nTo refute coefficient one, take\n\\[\n j=\\mathbf1_{[0,T/2)}-\\tfrac12,\\qquad\n \\widehat j=\\mathbf1_{[T/4,T/2)}-\\tfrac14.\n\\]\nBoth have jumps $(1,-1)$, with one jump shifted by $T/4$. Their norm\ndifference is $3T/8$, while the inherited bound gives $T/4$.\nThe sharper location term is attained in this example. The amplitude\ncoefficient $T/4$ is also attained by changing opposite antipodal jump\namplitudes: the difference packet has constant magnitude half the amplitude\nchange on the two half-periods.","tude $d/T$, and on an interval of length $d$ it has magnitude\n$1-d/T$. Hence its $L^1$ norm is\n$(T-d)d/T+d(1-d/T)=2d(1-d/T)$.\nApply the triangle inequality term by term. Periodic translation invariance\nmakes the calculation depend only on circular shift distance.\nTo refute coefficient one, take\n\\[\n j=\\mathbf1_{[0,T/2)}-\\tfrac12,\\qquad\n \\widehat j=\\mathbf1_{[T/4,T/2)}-\\tfrac14.\n\\]\nBoth have jumps $(1,-1)$, with one jump shifted by $T/4$. Their norm\ndifference is $3T/8$, while the inherited bound gives $T/4$.\nThe sharper location term is attained in this example. The amplitude\ncoefficient $T/4$ is also attained by changing opposite antipodal jump\namplitudes: the difference packet has constant magnitude half the amplitude\nchange on the two half-periods.","ength $d$ it has magnitude\n$1-d/T$. Hence its $L^1$ norm is\n$(T-d)d/T+d(1-d/T)=2d(1-d/T)$.\nApply the triangle inequality term by term. Periodic translation invariance\nmakes the calculation depend only on circular shift distance.\nTo refute coefficient one, take\n\\[\n j=\\mathbf1_{[0,T/2)}-\\tfrac12,\\qquad\n \\widehat j=\\mathbf1_{[T/4,T/2)}-\\tfrac14.\n\\]\nBoth have jumps $(1,-1)$, with one jump shifted by $T/4$. Their norm\ndifference is $3T/8$, while the inherited bound gives $T/4$.\nThe sharper location term is attained in this example. The amplitude\ncoefficient $T/4$ is also attained by changing opposite antipodal jump\namplitudes: the difference packet has constant magnitude half the amplitude\nchange on the two half-periods.","T$. Hence its $L^1$ norm is\n$(T-d)d/T+d(1-d/T)=2d(1-d/T)$.\nApply the triangle inequality term by term. Periodic translation invariance\nmakes the calculation depend only on circular shift distance.\nTo refute coefficient one, take\n\\[\n j=\\mathbf1_{[0,T/2)}-\\tfrac12,\\qquad\n \\widehat j=\\mathbf1_{[T/4,T/2)}-\\tfrac14.\n\\]\nBoth have jumps $(1,-1)$, with one jump shifted by $T/4$. Their norm\ndifference is $3T/8$, while the inherited bound gives $T/4$.\nThe sharper location term is attained in this example. The amplitude\ncoefficient $T/4$ is also attained by changing opposite antipodal jump\namplitudes: the difference packet has constant magnitude half the amplitude\nchange on the two half-periods.","1-d/T)=2d(1-d/T)$.\nApply the triangle inequality term by term. Periodic translation invariance\nmakes the calculation depend only on circular shift distance.\nTo refute coefficient one, take\n\\[\n j=\\mathbf1_{[0,T/2)}-\\tfrac12,\\qquad\n \\widehat j=\\mathbf1_{[T/4,T/2)}-\\tfrac14.\n\\]\nBoth have jumps $(1,-1)$, with one jump shifted by $T/4$. Their norm\ndifference is $3T/8$, while the inherited bound gives $T/4$.\nThe sharper location term is attained in this example. The amplitude\ncoefficient $T/4$ is also attained by changing opposite antipodal jump\namplitudes: the difference packet has constant magnitude half the amplitude\nchange on the two half-periods.","riance\nmakes the calculation depend only on circular shift distance.\nTo refute coefficient one, take\n\\[\n j=\\mathbf1_{[0,T/2)}-\\tfrac12,\\qquad\n \\widehat j=\\mathbf1_{[T/4,T/2)}-\\tfrac14.\n\\]\nBoth have jumps $(1,-1)$, with one jump shifted by $T/4$. Their norm\ndifference is $3T/8$, while the inherited bound gives $T/4$.\nThe sharper location term is attained in this example. The amplitude\ncoefficient $T/4$ is also attained by changing opposite antipodal jump\namplitudes: the difference packet has constant magnitude half the amplitude\nchange on the two half-periods.","\\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\n\\begin{theorem}[SG-38\\quad A conditional global L1 reconstruction rate]\\label{SG-38}","\\[\n \\|j-\\widehat j\\|_1=O(N^{-p-1}),\\qquad\n \\|f-\\widehat f_M\\|_1\\le b_M+O((1+C_M)N^{-p-1}).\n\\]\nFor Fourier truncation, $C_M=O(1+\\log(M+1))$. If also\n$|\\widehat s(n)|\\le C_s(1+|n|)^{-p-2}$, its smooth remainder satisfies\n$b_M=O(M^{-p-1})$. Taking $M$ comparable to $N$ yields\n$O((1+\\log N)N^{-p-1})$.","SG-35 gives an amplitude error of order $N^{-p-1}$ and a location error of\norder $N^{-p-2}$. If the fitted amplitudes are complex or do not sum to zero,\ntake their real parts and subtract their common arithmetic mean. Since the\ntrue real amplitudes sum to zero, this changes each amplitude error by at\nmost another maximum amplitude error, preserving its order. Apply SG-21 to\nthe resulting real zero-sum data to define $\\widehat j$. Insert these errors into SG-37, with finitely many bounded\namplitudes, to obtain the edge norm bound. SG-36 then gives the general\nsynthesis estimate.\nFor Fourier truncation, its convolution norm is bounded by the normalized\n$L^1$ norm of $D_M$. On $[-\\pi,\\pi]$ use\n$|D_M(t)|\\le\\min(2M+1,\\pi/|t|)$, with the removab","SG-35 gives an amplitude error of order $N^{-p-1}$ and a location error of\norder $N^{-p-2}$. If the fitted amplitudes are complex or do not sum to zero,\ntake their real parts and subtract their common arithmetic mean. Since the\ntrue real amplitudes sum to zero, this changes each amplitude error by at\nmost another maximum amplitude error, preserving its order. Apply SG-21 to\nthe resulting real zero-sum data to define $\\widehat j$. Insert these errors into SG-37, with finitely many bounded\namplitudes, to obtain the edge norm bound. SG-36 then gives the general\nsynthesis estimate.\nFor Fourier truncation, its convolution norm is bounded by the normalized\n$L^1$ norm of $D_M$. On $[-\\pi,\\pi]$ use\n$|D_M(t)|\\le\\min(2M+1,\\pi/|t|)$, with the removable value at zero.\nSplit the integral at $","ude error of order $N^{-p-1}$ and a location error of\norder $N^{-p-2}$. If the fitted amplitudes are complex or do not sum to zero,\ntake their real parts and subtract their common arithmetic mean. Since the\ntrue real amplitudes sum to zero, this changes each amplitude error by at\nmost another maximum amplitude error, preserving its order. Apply SG-21 to\nthe resulting real zero-sum data to define $\\widehat j$. Insert these errors into SG-37, with finitely many bounded\namplitudes, to obtain the edge norm bound. SG-36 then gives the general\nsynthesis estimate.\nFor Fourier truncation, its convolution norm is bounded by the normalized\n$L^1$ norm of $D_M$. On $[-\\pi,\\pi]$ use\n$|D_M(t)|\\le\\min(2M+1,\\pi/|t|)$, with the removable value at zero.\nSplit the integral at $1/(M+1)$ to obtain the logarithmic bound.\nThe smooth Fourier tail has supremum bound\n$2C_s\\sum_{n>M}(1+n)^{-p-2}=O(M^{-p-1})$ by an integral comparison;\nmultiplication by $2\\pi$ gives the $L^1$ estimate.\nThis conclusion accounts for the norm growth of partial sums instead of\nassuming a uniformly bounded Fourier truncation operato…",", this changes each amplitude error by at\nmost another maximum amplitude error, preserving its order. Apply SG-21 to\nthe resulting real zero-sum data to define $\\widehat j$. Insert these errors into SG-37, with finitely many bounded\namplitudes, to obtain the edge norm bound. SG-36 then gives the general\nsynthesis estimate.\nFor Fourier truncation, its convolution norm is bounded by the normalized\n$L^1$ norm of $D_M$. On $[-\\pi,\\pi]$ use\n$|D_M(t)|\\le\\min(2M+1,\\pi/|t|)$, with the removable value at zero.\nSplit the integral at $1/(M+1)$ to obtain the logarithmic bound.\nThe smooth Fourier tail has supremum bound\n$2C_s\\sum_{n>M}(1+n)^{-p-2}=O(M^{-p-1})$ by an integral comparison;\nmultiplication by $2\\pi$ gives the $L^1$ estimate.\nThis conclusion accounts for the norm growth of partial sums instead of\nassuming a uniformly bounded Fourier truncation operator on $L^1$.","each amplitude error by at\nmost another maximum amplitude error, preserving its order. Apply SG-21 to\nthe resulting real zero-sum data to define $\\widehat j$. Insert these errors into SG-37, with finitely many bounded\namplitudes, to obtain the edge norm bound. SG-36 then gives the general\nsynthesis estimate.\nFor Fourier truncation, its convolution norm is bounded by the normalized\n$L^1$ norm of $D_M$. On $[-\\pi,\\pi]$ use\n$|D_M(t)|\\le\\min(2M+1,\\pi/|t|)$, with the removable value at zero.\nSplit the integral at $1/(M+1)$ to obtain the logarithmic bound.\nThe smooth Fourier tail has supremum bound\n$2C_s\\sum_{n>M}(1+n)^{-p-2}=O(M^{-p-1})$ by an integral comparison;\nmultiplication by $2\\pi$ gives the $L^1$ estimate.\nThis conclusion accounts for the norm growth of partial sums instead of\nassuming a uniformly bounded Fourier truncation operator on $L^1$.","itude error by at\nmost another maximum amplitude error, preserving its order. Apply SG-21 to\nthe resulting real zero-sum data to define $\\widehat j$. Insert these errors into SG-37, with finitely many bounded\namplitudes, to obtain the edge norm bound. SG-36 then gives the general\nsynthesis estimate.\nFor Fourier truncation, its convolution norm is bounded by the normalized\n$L^1$ norm of $D_M$. On $[-\\pi,\\pi]$ use\n$|D_M(t)|\\le\\min(2M+1,\\pi/|t|)$, with the removable value at zero.\nSplit the integral at $1/(M+1)$ to obtain the logarithmic bound.\nThe smooth Fourier tail has supremum bound\n$2C_s\\sum_{n>M}(1+n)^{-p-2}=O(M^{-p-1})$ by an integral comparison;\nmultiplication by $2\\pi$ gives the $L^1$ estimate.\nThis conclusion accounts for the norm growth of partial sums instead of\nassuming a uniformly bounded Fourier truncation operator on $L^1$.","most another maximum amplitude error, preserving its order. Apply SG-21 to\nthe resulting real zero-sum data to define $\\widehat j$. Insert these errors into SG-37, with finitely many bounded\namplitudes, to obtain the edge norm bound. SG-36 then gives the general\nsynthesis estimate.\nFor Fourier truncation, its convolution norm is bounded by the normalized\n$L^1$ norm of $D_M$. On $[-\\pi,\\pi]$ use\n$|D_M(t)|\\le\\min(2M+1,\\pi/|t|)$, with the removable value at zero.\nSplit the integral at $1/(M+1)$ to obtain the logarithmic bound.\nThe smooth Fourier tail has supremum bound\n$2C_s\\sum_{n>M}(1+n)^{-p-2}=O(M^{-p-1})$ by an integral comparison;\nmultiplication by $2\\pi$ gives the $L^1$ estimate.\nThis conclusion accounts for the norm growth of partial sums instead of\nassuming a uniformly bounded Fourier truncation operator on $L^1$.","real zero-sum data to define $\\widehat j$. Insert these errors into SG-37, with finitely many bounded\namplitudes, to obtain the edge norm bound. SG-36 then gives the general\nsynthesis estimate.\nFor Fourier truncation, its convolution norm is bounded by the normalized\n$L^1$ norm of $D_M$. On $[-\\pi,\\pi]$ use\n$|D_M(t)|\\le\\min(2M+1,\\pi/|t|)$, with the removable value at zero.\nSplit the integral at $1/(M+1)$ to obtain the logarithmic bound.\nThe smooth Fourier tail has supremum bound\n$2C_s\\sum_{n>M}(1+n)^{-p-2}=O(M^{-p-1})$ by an integral comparison;\nmultiplication by $2\\pi$ gives the $L^1$ estimate.\nThis conclusion accounts for the norm growth of partial sums instead of\nassuming a uniformly bounded Fourier truncation operator on $L^1$.","ely many bounded\namplitudes, to obtain the edge norm bound. SG-36 then gives the general\nsynthesis estimate.\nFor Fourier truncation, its convolution norm is bounded by the normalized\n$L^1$ norm of $D_M$. On $[-\\pi,\\pi]$ use\n$|D_M(t)|\\le\\min(2M+1,\\pi/|t|)$, with the removable value at zero.\nSplit the integral at $1/(M+1)$ to obtain the logarithmic bound.\nThe smooth Fourier tail has supremum bound\n$2C_s\\sum_{n>M}(1+n)^{-p-2}=O(M^{-p-1})$ by an integral comparison;\nmultiplication by $2\\pi$ gives the $L^1$ estimate.\nThis conclusion accounts for the norm growth of partial sums instead of\nassuming a uniformly bounded Fourier truncation operator on $L^1$.","al\nsynthesis estimate.\nFor Fourier truncation, its convolution norm is bounded by the normalized\n$L^1$ norm of $D_M$. On $[-\\pi,\\pi]$ use\n$|D_M(t)|\\le\\min(2M+1,\\pi/|t|)$, with the removable value at zero.\nSplit the integral at $1/(M+1)$ to obtain the logarithmic bound.\nThe smooth Fourier tail has supremum bound\n$2C_s\\sum_{n>M}(1+n)^{-p-2}=O(M^{-p-1})$ by an integral comparison;\nmultiplication by $2\\pi$ gives the $L^1$ estimate.\nThis conclusion accounts for the norm growth of partial sums instead of\nassuming a uniformly bounded Fourier truncation operator on $L^1$.","mate.\nFor Fourier truncation, its convolution norm is bounded by the normalized\n$L^1$ norm of $D_M$. On $[-\\pi,\\pi]$ use\n$|D_M(t)|\\le\\min(2M+1,\\pi/|t|)$, with the removable value at zero.\nSplit the integral at $1/(M+1)$ to obtain the logarithmic bound.\nThe smooth Fourier tail has supremum bound\n$2C_s\\sum_{n>M}(1+n)^{-p-2}=O(M^{-p-1})$ by an integral comparison;\nmultiplication by $2\\pi$ gives the $L^1$ estimate.\nThis conclusion accounts for the norm growth of partial sums instead of\nassuming a uniformly bounded Fourier truncation operator on $L^1$.","t|)$, with the removable value at zero.\nSplit the integral at $1/(M+1)$ to obtain the logarithmic bound.\nThe smooth Fourier tail has supremum bound\n$2C_s\\sum_{n>M}(1+n)^{-p-2}=O(M^{-p-1})$ by an integral comparison;\nmultiplication by $2\\pi$ gives the $L^1$ estimate.\nThis conclusion accounts for the norm growth of partial sums instead of\nassuming a uniformly bounded Fourier truncation operator on $L^1$.","\\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\n\\begin{theorem}[SG-39\\quad Shared-location multichannel recovery]\\label{SG-39}","The stacked system uniquely recovers the common annihilator\n$Q(z)=\\prod_\\ell(z-z_\\ell)$. Its roots recover the common locations.\nWith $r$ consecutive moments per channel, a channelwise Vandermonde solve\nrecovers every amplitude, including amplitudes that vanish in that channel.","The polynomial $Q$ annihilates every exponential term and therefore every\nchannel, as in SG-20. Full column rank of the stacked recurrence matrix means\nthat a consistent coefficient vector is unique; the true $Q$ provides one.\nIts distinct roots give the nodes. Once known, the same nonsingular\nVandermonde system solves each channel independently. No individual channel\nhas to see all nodes, but the stated stacked rank is essential. Merely\ncollecting channels does not prove that rank condition automatically.","\\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\n\\begin{theorem}[SG-40\\quad Confluent annihilation and the leading offset phase]\\label{SG-40}","\\[\n \\prod_{\\ell=1}^r(E-z_\\ell)^{d+1}m=0.\n\\]\nFor offset $v\\ge0$, writing a decimated channel as\n\\[\n m_{(q_0+q)N+v}=\\sum_\\ell u_\\ell^q\n                   \\sum_{h=0}^d\\gamma_{\\ell,v,h}q^h,\n \\qquad u_\\ell=z_\\ell^N,\n\\]\nthe highest-degree coefficients satisfy\n$\\gamma_{\\ell,1,d}/\\gamma_{\\ell,0,d}=z_\\ell$.\nThis ratio applies to separated, matched confluent terms with nonzero leading\ncoefficient. The displayed annihilator is not asserted to have minimal order\nwhen leading coefficients vanish.","For one term, $(E-z)(z^nP(n))=z^{n+1}(P(n+1)-P(n))$.\nA finite difference lowers a nonconstant polynomial's degree by one, and\nannihilates a constant. After $d+1$ differences the term vanishes. The shift\nfactors commute, so the product annihilates every term in the sum.\nIf $P_\\ell(n)$ has leading coefficient $c_\\ell\\ne0$ at degree $d$, substitution\nof $(q_0+q)N+v$ has leading $q$ coefficient $c_\\ell N^d$.\nThe remaining exponential prefactor is $z_\\ell^{q_0N+v}$; hence\n$\\gamma_{\\ell,v,d}=c_\\ell N^dz_\\ell^{q_0N+v}$ and the ratio is $z_\\ell$.\nRepeated roots require confluent recovery and stability arguments; the\nsimple-root disk theorem SG-25 cannot be used to claim one perturbed root\nper physical node or a Lipschitz individual-root bound at a","For one term, $(E-z)(z^nP(n))=z^{n+1}(P(n+1)-P(n))$.\nA finite difference lowers a nonconstant polynomial's degree by one, and\nannihilates a constant. After $d+1$ differences the term vanishes. The shift\nfactors commute, so the product annihilates every term in the sum.\nIf $P_\\ell(n)$ has leading coefficient $c_\\ell\\ne0$ at degree $d$, substitution\nof $(q_0+q)N+v$ has leading $q$ coefficient $c_\\ell N^d$.\nThe remaining exponential prefactor is $z_\\ell^{q_0N+v}$; hence\n$\\gamma_{\\ell,v,d}=c_\\ell N^dz_\\ell^{q_0N+v}$ and the ratio is $z_\\ell$.\nRepeated roots require confluent recovery and stability arguments; the\nsimple-root disk theorem SG-25 cannot be used to claim one perturbed root\nper physical node or a Lipschitz individual-root bound at a multiple root.","))$.\nA finite difference lowers a nonconstant polynomial's degree by one, and\nannihilates a constant. After $d+1$ differences the term vanishes. The shift\nfactors commute, so the product annihilates every term in the sum.\nIf $P_\\ell(n)$ has leading coefficient $c_\\ell\\ne0$ at degree $d$, substitution\nof $(q_0+q)N+v$ has leading $q$ coefficient $c_\\ell N^d$.\nThe remaining exponential prefactor is $z_\\ell^{q_0N+v}$; hence\n$\\gamma_{\\ell,v,d}=c_\\ell N^dz_\\ell^{q_0N+v}$ and the ratio is $z_\\ell$.\nRepeated roots require confluent recovery and stability arguments; the\nsimple-root disk theorem SG-25 cannot be used to claim one perturbed root\nper physical node or a Lipschitz individual-root bound at a multiple root.","lowers a nonconstant polynomial's degree by one, and\nannihilates a constant. After $d+1$ differences the term vanishes. The shift\nfactors commute, so the product annihilates every term in the sum.\nIf $P_\\ell(n)$ has leading coefficient $c_\\ell\\ne0$ at degree $d$, substitution\nof $(q_0+q)N+v$ has leading $q$ coefficient $c_\\ell N^d$.\nThe remaining exponential prefactor is $z_\\ell^{q_0N+v}$; hence\n$\\gamma_{\\ell,v,d}=c_\\ell N^dz_\\ell^{q_0N+v}$ and the ratio is $z_\\ell$.\nRepeated roots require confluent recovery and stability arguments; the\nsimple-root disk theorem SG-25 cannot be used to claim one perturbed root\nper physical node or a Lipschitz individual-root bound at a multiple root.","s a constant. After $d+1$ differences the term vanishes. The shift\nfactors commute, so the product annihilates every term in the sum.\nIf $P_\\ell(n)$ has leading coefficient $c_\\ell\\ne0$ at degree $d$, substitution\nof $(q_0+q)N+v$ has leading $q$ coefficient $c_\\ell N^d$.\nThe remaining exponential prefactor is $z_\\ell^{q_0N+v}$; hence\n$\\gamma_{\\ell,v,d}=c_\\ell N^dz_\\ell^{q_0N+v}$ and the ratio is $z_\\ell$.\nRepeated roots require confluent recovery and stability arguments; the\nsimple-root disk theorem SG-25 cannot be used to claim one perturbed root\nper physical node or a Lipschitz individual-root bound at a multiple root.","\\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\n\\begin{theorem}[SG-41\\quad Model mismatch enters the same sparse stability bound]\\label{SG-41}","\\[\n \\|\\widehat\\alpha-\\alpha_0\\|_2\\le2\\epsilon/\\gamma.\n\\]\nIf an actual edge $j$ obeys $\\|j-D\\alpha_0\\|_2\\le\\delta$, then\n\\[\n \\|j-D\\widehat\\alpha\\|_2\\le\\delta+2\\|D\\|_2\\epsilon/\\gamma.\n\\]\nThe theorem applies to finite image dictionaries under these printed\nsparsity, residual, and restricted-injectivity hypotheses.","The nuisance $e+\\eta$ has exactly the role of the residualized noise in\nSG-15, so its proof gives the coefficient bound. Add and subtract\n$D\\alpha_0$ in the edge difference and apply the operator norm bound to the\ncoefficient error. A best approximation can be substituted when it exists;\notherwise any admissible approximate minimizer gives the same statement.\nThe theorem does not show that a greedy finite dictionary selector outputs\nan $s$-sparse vector with the specified tolerance. Nor does it establish\nrestricted injectivity for every geometric dictionary; both remain\nhypotheses to verify for a concrete model.","e of the residualized noise in\nSG-15, so its proof gives the coefficient bound. Add and subtract\n$D\\alpha_0$ in the edge difference and apply the operator norm bound to the\ncoefficient error. A best approximation can be substituted when it exists;\notherwise any admissible approximate minimizer gives the same statement.\nThe theorem does not show that a greedy finite dictionary selector outputs\nan $s$-sparse vector with the specified tolerance. Nor does it establish\nrestricted injectivity for every geometric dictionary; both remain\nhypotheses to verify for a concrete model.","\\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\n\\begin{theorem}[SG-42\\quad Ideal nested-span refitting decreases residual]\\label{SG-42}","\\[\n \\|f-P_{k+1}f\\|\\le\\|f-P_kf\\|,\n \\qquad f-P_kf\\perp V_k.\n\\]\nIf $\\bigcup_kV_k$ is dense, then $P_kf\\to f$ in norm.","Orthogonal projection minimizes distance to its subspace. Since $P_kf$\nbelongs to the larger subspace $V_{k+1}$, its residual is a feasible upper\nbound on the next minimum. Orthogonality follows from the defining projection\nidentity, or by varying the quadratic norm along every vector in $V_k$.\nFor convergence, fix $\\epsilon>0$ and choose $v\\in V_K$ within $\\epsilon$\nof $f$ by density. For all $k\\ge K$, minimum distance is at most\n$\\|f-v\\|<\\epsilon$. Thus the residual converges to zero.\nThis theorem assumes dense actual selected spans. A countably dense candidate\npool alone does not show that a particular greedy selection sequence becomes\ndense.","ance to its subspace. Since $P_kf$\nbelongs to the larger subspace $V_{k+1}$, its residual is a feasible upper\nbound on the next minimum. Orthogonality follows from the defining projection\nidentity, or by varying the quadratic norm along every vector in $V_k$.\nFor convergence, fix $\\epsilon>0$ and choose $v\\in V_K$ within $\\epsilon$\nof $f$ by density. For all $k\\ge K$, minimum distance is at most\n$\\|f-v\\|<\\epsilon$. Thus the residual converges to zero.\nThis theorem assumes dense actual selected spans. A countably dense candidate\npool alone does not show that a particular greedy selection sequence becomes\ndense.","\\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\n\\begin{theorem}[SG-43\\quad Ridge refitting has a different stationarity law]\\label{SG-43}","The minimizer is unique and\n\\[\n (M^{\\mathsf T}M+\\lambda I)c=M^{\\mathsf T}y,\n \\qquad M^{\\mathsf T}r=\\lambda c.\n\\]\nAdding candidate columns cannot increase the minimized ridge objective,\nbut exact orthogonality $M^{\\mathsf T}r=0$ is not the ridge stationarity law.\nEven a complete pixel basis need not give zero residual when $\\lambda>0$.","The Hessian is twice $M^TM+\\lambda I$, which is positive definite because\n$v^T(M^TM+\\lambda I)v=\\|Mv\\|^2+\\lambda\\|v\\|^2>0$ for nonzero $v$.\nSetting the gradient to zero gives the unique minimizer and rearrangement gives\n$M^Tr=\\lambda c$. When columns are added, extending the previous coefficient\nvector by zero is a feasible point with exactly the previous objective, so the\nnew optimum is no larger. This concerns the ridge objective, not an automatic\nmonotonicity theorem for the unpenalized residual alone.\nFor the full pixel basis $M=I$, $c=y/(1+\\lambda)$ and\n$r=\\lambda y/(1+\\lambda)$, which is nonzero for $y\\ne0$.\nThe native default $\\lambda=10^{-9}$ makes this residual small, not\nmathematically zero. Numerical tolerances must","The Hessian is twice $M^TM+\\lambda I$, which is positive definite because\n$v^T(M^TM+\\lambda I)v=\\|Mv\\|^2+\\lambda\\|v\\|^2>0$ for nonzero $v$.\nSetting the gradient to zero gives the unique minimizer and rearrangement gives\n$M^Tr=\\lambda c$. When columns are added, extending the previous coefficient\nvector by zero is a feasible point with exactly the previous objective, so the\nnew optimum is no larger. This concerns the ridge objective, not an automatic\nmonotonicity theorem for the unpenalized residual alone.\nFor the full pixel basis $M=I$, $c=y/(1+\\lambda)$ and\n$r=\\lambda y/(1+\\lambda)$, which is nonzero for $y\\ne0$.\nThe native default $\\lambda=10^{-9}$ makes this residual small, not\nmathematically zero. Numerical tolerances must remain numerical tolerances.","$.\nSetting the gradient to zero gives the unique minimizer and rearrangement gives\n$M^Tr=\\lambda c$. When columns are added, extending the previous coefficient\nvector by zero is a feasible point with exactly the previous objective, so the\nnew optimum is no larger. This concerns the ridge objective, not an automatic\nmonotonicity theorem for the unpenalized residual alone.\nFor the full pixel basis $M=I$, $c=y/(1+\\lambda)$ and\n$r=\\lambda y/(1+\\lambda)$, which is nonzero for $y\\ne0$.\nThe native default $\\lambda=10^{-9}$ makes this residual small, not\nmathematically zero. Numerical tolerances must remain numerical tolerances.","ing the gradient to zero gives the unique minimizer and rearrangement gives\n$M^Tr=\\lambda c$. When columns are added, extending the previous coefficient\nvector by zero is a feasible point with exactly the previous objective, so the\nnew optimum is no larger. This concerns the ridge objective, not an automatic\nmonotonicity theorem for the unpenalized residual alone.\nFor the full pixel basis $M=I$, $c=y/(1+\\lambda)$ and\n$r=\\lambda y/(1+\\lambda)$, which is nonzero for $y\\ne0$.\nThe native default $\\lambda=10^{-9}$ makes this residual small, not\nmathematically zero. Numerical tolerances must remain numerical tolerances.","ro gives the unique minimizer and rearrangement gives\n$M^Tr=\\lambda c$. When columns are added, extending the previous coefficient\nvector by zero is a feasible point with exactly the previous objective, so the\nnew optimum is no larger. This concerns the ridge objective, not an automatic\nmonotonicity theorem for the unpenalized residual alone.\nFor the full pixel basis $M=I$, $c=y/(1+\\lambda)$ and\n$r=\\lambda y/(1+\\lambda)$, which is nonzero for $y\\ne0$.\nThe native default $\\lambda=10^{-9}$ makes this residual small, not\nmathematically zero. Numerical tolerances must remain numerical tolerances.","gives\n$M^Tr=\\lambda c$. When columns are added, extending the previous coefficient\nvector by zero is a feasible point with exactly the previous objective, so the\nnew optimum is no larger. This concerns the ridge objective, not an automatic\nmonotonicity theorem for the unpenalized residual alone.\nFor the full pixel basis $M=I$, $c=y/(1+\\lambda)$ and\n$r=\\lambda y/(1+\\lambda)$, which is nonzero for $y\\ne0$.\nThe native default $\\lambda=10^{-9}$ makes this residual small, not\nmathematically zero. Numerical tolerances must remain numerical tolerances.","columns are added, extending the previous coefficient\nvector by zero is a feasible point with exactly the previous objective, so the\nnew optimum is no larger. This concerns the ridge objective, not an automatic\nmonotonicity theorem for the unpenalized residual alone.\nFor the full pixel basis $M=I$, $c=y/(1+\\lambda)$ and\n$r=\\lambda y/(1+\\lambda)$, which is nonzero for $y\\ne0$.\nThe native default $\\lambda=10^{-9}$ makes this residual small, not\nmathematically zero. Numerical tolerances must remain numerical tolerances.","\\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\n\\begin{theorem}[SG-44\\quad Clipping is nonexpansive toward a bounded truth]\\label{SG-44}","\\[\n \\|C(x)-C(y)\\|_2\\le\\|x-y\\|_2,\\qquad\n \\|C(x)-f\\|_2\\le\\|x-f\\|_2.\n\\]\nClipping is nonlinear; it is not the bounded linear synthesis operator in\nSG-36. Its truth-error guarantee depends on the stated box constraint.","For real scalars the clipping function is nondecreasing with slope between\nzero and one on each of its three affine pieces. A direct case split at zero\nand one gives $|C(x)-C(y)|\\le|x-y|$ for every pair.\nSquare and sum the scalar inequalities. Since $C(f)=f$, substitute $y=f$ to\nobtain the second estimate. Nonlinearity is explicit: $C(1+1)=1$ whereas\n$C(1)+C(1)=2$. If the truth lies outside the box, clipping can increase its\nerror; for instance $x=f=2$ becomes $C(x)=1$.","asing with slope between\nzero and one on each of its three affine pieces. A direct case split at zero\nand one gives $|C(x)-C(y)|\\le|x-y|$ for every pair.\nSquare and sum the scalar inequalities. Since $C(f)=f$, substitute $y=f$ to\nobtain the second estimate. Nonlinearity is explicit: $C(1+1)=1$ whereas\n$C(1)+C(1)=2$. If the truth lies outside the box, clipping can increase its\nerror; for instance $x=f=2$ becomes $C(x)=1$.","between\nzero and one on each of its three affine pieces. A direct case split at zero\nand one gives $|C(x)-C(y)|\\le|x-y|$ for every pair.\nSquare and sum the scalar inequalities. Since $C(f)=f$, substitute $y=f$ to\nobtain the second estimate. Nonlinearity is explicit: $C(1+1)=1$ whereas\n$C(1)+C(1)=2$. If the truth lies outside the box, clipping can increase its\nerror; for instance $x=f=2$ becomes $C(x)=1$.","\\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\n\\begin{theorem}[SG-45\\quad An exact residual archive preserves the object]\\label{SG-45}","The synthesis $q+r=f$ is exact. This identity supplies no sparsity, compression,\nor recovery-from-fewer-measurements conclusion. In finite pixel space the\nfull pixel basis likewise spans every image in the ideal unregularized model.","The first statement is the defining subtraction identity. Retaining $r$\ncan carry every component omitted from $q$, so its data cost is not bounded by\na small number of packet coefficients. For an image $f\\in\\mathbb R^m$, the\nstandard pixel vectors satisfy $f=\\sum_{j=1}^mf_je_j$ and form a complete\nfinite basis. Those coefficients are all the pixels themselves.\nA program using the truth residual thus certifies faithful storage of its\navailable input, rather than uniqueness under a smaller measurement map.","\\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\n\\begin{theorem}[SG-46\\quad Bounded identities extend from a dense core]\\label{SG-46}","Then $A=B$ on $X$. Consequently a finite packet identity established on a\ndense core extends to a completion when both represented operators have\nbounded extensions. An infinite projector sum additionally requires a\nspecified convergent synthesis; bounded finite identities alone do not\nsupply that convergence.","For $x\\in X$ choose $x_n\\in D$ with $x_n\\to x$.\nContinuity gives $Ax=\\lim_nAx_n=\\lim_nBx_n=Bx$.\nThis proves uniqueness of bounded extensions. For an infinite packet sum one\nwould need partial sums to converge to the declared operator, in a stated\noperator or strong topology. Neither density nor the finite identity by\nitself proves such a limit. In SG-42 density of the actual selected spans\nprovides a specific strong convergence theorem; a capped dictionary store or\nan arbitrary greedy prefix does not inherit it automatically.","Take a bandlimited background\n\\[\n s(x)=\\frac14+\\frac25\\cos(2x)-\\frac13\\sin(3x)+\\frac15\\cos(5x)\n\\]\nand a mean-zero edge with locations and jumps\n\\[\n (\\tau_\\ell)=(0,\\pi/2,\\pi,3\\pi/2),\\qquad\n (a_\\ell)=(1,-2,-1,2).\n\\]\nThe physical nodes are $(1,-i,-1,i)$. Their derivative moments are\n\\[\n m_n=1-2(-i)^n-(-1)^n+2i^n,\n\\]\nso in residues modulo four they are\n\\[\n m_{4q}=0,\\quad m_{4q+1}=2+4i,\\quad\n m_{4q+2}=0,\\quad m_{4q+3}=2-4i.\n\\]\nSince the smooth degree is five, the first clean consecutive moment is\n$n_0=6$. With an order bound four, coefficients at $n=6,\\ldots,13$ determine\nthe edge. The Hankel factorization has four distinct nodes and four nonzero\namplitudes, hence rank four. Its recurrence is\n$m_{n+4}-m_n=0$, with monic polynomial $Q(z)=z^4-1$.\nRecovering those roots and solving the Vander","Take a bandlimited background\n\\[\n s(x)=\\frac14+\\frac25\\cos(2x)-\\frac13\\sin(3x)+\\frac15\\cos(5x)\n\\]\nand a mean-zero edge with locations and jumps\n\\[\n (\\tau_\\ell)=(0,\\pi/2,\\pi,3\\pi/2),\\qquad\n (a_\\ell)=(1,-2,-1,2).\n\\]\nThe physical nodes are $(1,-i,-1,i)$. Their derivative moments are\n\\[\n m_n=1-2(-i)^n-(-1)^n+2i^n,\n\\]\nso in residues modulo four they are\n\\[\n m_{4q}=0,\\quad m_{4q+1}=2+4i,\\quad\n m_{4q+2}=0,\\quad m_{4q+3}=2-4i.\n\\]\nSince the smooth degree is five, the first clean consecutive moment is\n$n_0=6$. With an order bound four, coefficients at $n=6,\\ldots,13$ determine\nthe edge. The Hankel factorization has four distinct nodes and four nonzero\namplitudes, hence rank four. Its recurrence is\n$m_{n+4}-m_n=0$, with monic polynomial $Q(z)=z^4-1$.\nRecovering those roots and solving the Vandermonde system returns the displayed\njumps. SG-21 then constructs the edge, and coefficients at $-5,\\ldots,5$\nreturn t","Take a bandlimited background\n\\[\n s(x)=\\frac14+\\frac25\\cos(2x)-\\frac13\\sin(3x)+\\frac15\\cos(5x)\n\\]\nand a mean-zero edge with locations and jumps\n\\[\n (\\tau_\\ell)=(0,\\pi/2,\\pi,3\\pi/2),\\qquad\n (a_\\ell)=(1,-2,-1,2).\n\\]\nThe physical nodes are $(1,-i,-1,i)$. Their derivative moments are\n\\[\n m_n=1-2(-i)^n-(-1)^n+2i^n,\n\\]\nso in residues modulo four they are\n\\[\n m_{4q}=0,\\quad m_{4q+1}=2+4i,\\quad\n m_{4q+2}=0,\\quad m_{4q+3}=2-4i.\n\\]\nSince the smooth degree is five, the first clean consecutive moment is\n$n_0=6$. With an order bound four, coefficients at $n=6,\\ldots,13$ determine\nthe edge. The Hankel factorization has four distinct nodes and four nonzero\namplitudes, hence rank four. Its recurrence is\n$m_{n+4}-m_n=0$, with monic polynomial $Q(z)=z^4-1$.\nRecovering those roots and solving the Vandermonde system returns the displayed\njumps. SG-21 then constructs the edge, and coefficients at $-5,\\ldots,5$\nreturn the smooth background.","and a mean-zero edge with locations and jumps\n\\[\n (\\tau_\\ell)=(0,\\pi/2,\\pi,3\\pi/2),\\qquad\n (a_\\ell)=(1,-2,-1,2).\n\\]\nThe physical nodes are $(1,-i,-1,i)$. Their derivative moments are\n\\[\n m_n=1-2(-i)^n-(-1)^n+2i^n,\n\\]\nso in residues modulo four they are\n\\[\n m_{4q}=0,\\quad m_{4q+1}=2+4i,\\quad\n m_{4q+2}=0,\\quad m_{4q+3}=2-4i.\n\\]\nSince the smooth degree is five, the first clean consecutive moment is\n$n_0=6$. With an order bound four, coefficients at $n=6,\\ldots,13$ determine\nthe edge. The Hankel factorization has four distinct nodes and four nonzero\namplitudes, hence rank four. Its recurrence is\n$m_{n+4}-m_n=0$, with monic polynomial $Q(z)=z^4-1$.\nRecovering those roots and solving the Vandermonde system returns the displayed\njumps. SG-21 then constructs the edge, and coefficients at $-5,\\ldots,5$\nreturn the smooth background.","\\tau_\\ell)=(0,\\pi/2,\\pi,3\\pi/2),\\qquad\n (a_\\ell)=(1,-2,-1,2).\n\\]\nThe physical nodes are $(1,-i,-1,i)$. Their derivative moments are\n\\[\n m_n=1-2(-i)^n-(-1)^n+2i^n,\n\\]\nso in residues modulo four they are\n\\[\n m_{4q}=0,\\quad m_{4q+1}=2+4i,\\quad\n m_{4q+2}=0,\\quad m_{4q+3}=2-4i.\n\\]\nSince the smooth degree is five, the first clean consecutive moment is\n$n_0=6$. With an order bound four, coefficients at $n=6,\\ldots,13$ determine\nthe edge. The Hankel factorization has four distinct nodes and four nonzero\namplitudes, hence rank four. Its recurrence is\n$m_{n+4}-m_n=0$, with monic polynomial $Q(z)=z^4-1$.\nRecovering those roots and solving the Vandermonde system returns the displayed\njumps. SG-21 then constructs the edge, and coefficients at $-5,\\ldots,5$\nreturn the smooth background.","-(-1)^n+2i^n,\n\\]\nso in residues modulo four they are\n\\[\n m_{4q}=0,\\quad m_{4q+1}=2+4i,\\quad\n m_{4q+2}=0,\\quad m_{4q+3}=2-4i.\n\\]\nSince the smooth degree is five, the first clean consecutive moment is\n$n_0=6$. With an order bound four, coefficients at $n=6,\\ldots,13$ determine\nthe edge. The Hankel factorization has four distinct nodes and four nonzero\namplitudes, hence rank four. Its recurrence is\n$m_{n+4}-m_n=0$, with monic polynomial $Q(z)=z^4-1$.\nRecovering those roots and solving the Vandermonde system returns the displayed\njumps. SG-21 then constructs the edge, and coefficients at $-5,\\ldots,5$\nreturn the smooth background.","four they are\n\\[\n m_{4q}=0,\\quad m_{4q+1}=2+4i,\\quad\n m_{4q+2}=0,\\quad m_{4q+3}=2-4i.\n\\]\nSince the smooth degree is five, the first clean consecutive moment is\n$n_0=6$. With an order bound four, coefficients at $n=6,\\ldots,13$ determine\nthe edge. The Hankel factorization has four distinct nodes and four nonzero\namplitudes, hence rank four. Its recurrence is\n$m_{n+4}-m_n=0$, with monic polynomial $Q(z)=z^4-1$.\nRecovering those roots and solving the Vandermonde system returns the displayed\njumps. SG-21 then constructs the edge, and coefficients at $-5,\\ldots,5$\nreturn the smooth background.","ent is\n$n_0=6$. With an order bound four, coefficients at $n=6,\\ldots,13$ determine\nthe edge. The Hankel factorization has four distinct nodes and four nonzero\namplitudes, hence rank four. Its recurrence is\n$m_{n+4}-m_n=0$, with monic polynomial $Q(z)=z^4-1$.\nRecovering those roots and solving the Vandermonde system returns the displayed\njumps. SG-21 then constructs the edge, and coefficients at $-5,\\ldots,5$\nreturn the smooth background.","The same object demonstrates internal phase recovery. Choose $N=3$ and\n$q_0=2$. The nodes are permuted by cubing, so the aliased nodes remain distinct.\nThe sufficient self-calibration sets for $L_{\\max}=4$ are\n\\[\n \\mathcal I_0=\\{6,9,12,15,18,21,24,27,30\\},\\qquad\n \\mathcal I_1=\\{7,10,13,16\\}.\n\\]\nFor each matched term the base coefficient is\n$b_{\\ell,0}=a_\\ell z_\\ell^6$, and the adjacent coefficient is\n$b_{\\ell,1}=b_{\\ell,0}z_\\ell$. Their ratio is the original physical node,\nnot the principal cube root. The total high budget is thirteen, agreeing with\n$3L_{\\max}+1$. The un-decimated construction uses eight high coefficients;\nthe polyphase construction trades additional channel information for its\nchosen decimation geometry and branch recovery. Neither budge","The same object demonstrates internal phase recovery. Choose $N=3$ and\n$q_0=2$. The nodes are permuted by cubing, so the aliased nodes remain distinct.\nThe sufficient self-calibration sets for $L_{\\max}=4$ are\n\\[\n \\mathcal I_0=\\{6,9,12,15,18,21,24,27,30\\},\\qquad\n \\mathcal I_1=\\{7,10,13,16\\}.\n\\]\nFor each matched term the base coefficient is\n$b_{\\ell,0}=a_\\ell z_\\ell^6$, and the adjacent coefficient is\n$b_{\\ell,1}=b_{\\ell,0}z_\\ell$. Their ratio is the original physical node,\nnot the principal cube root. The total high budget is thirteen, agreeing with\n$3L_{\\max}+1$. The un-decimated construction uses eight high coefficients;\nthe polyphase construction trades additional channel information for its\nchosen decimation geometry and branch recovery. Neither budget is claimed","The same object demonstrates internal phase recovery. Choose $N=3$ and\n$q_0=2$. The nodes are permuted by cubing, so the aliased nodes remain distinct.\nThe sufficient self-calibration sets for $L_{\\max}=4$ are\n\\[\n \\mathcal I_0=\\{6,9,12,15,18,21,24,27,30\\},\\qquad\n \\mathcal I_1=\\{7,10,13,16\\}.\n\\]\nFor each matched term the base coefficient is\n$b_{\\ell,0}=a_\\ell z_\\ell^6$, and the adjacent coefficient is\n$b_{\\ell,1}=b_{\\ell,0}z_\\ell$. Their ratio is the original physical node,\nnot the principal cube root. The total high budget is thirteen, agreeing with\n$3L_{\\max}+1$. The un-decimated construction uses eight high coefficients;\nthe polyphase construction trades additional channel information for its\nchosen decimation geometry and branch recovery. Neither budget is claimed\nminimal. Both reconstruct the same canonical object exactly.","ame object demonstrates internal phase recovery. Choose $N=3$ and\n$q_0=2$. The nodes are permuted by cubing, so the aliased nodes remain distinct.\nThe sufficient self-calibration sets for $L_{\\max}=4$ are\n\\[\n \\mathcal I_0=\\{6,9,12,15,18,21,24,27,30\\},\\qquad\n \\mathcal I_1=\\{7,10,13,16\\}.\n\\]\nFor each matched term the base coefficient is\n$b_{\\ell,0}=a_\\ell z_\\ell^6$, and the adjacent coefficient is\n$b_{\\ell,1}=b_{\\ell,0}z_\\ell$. Their ratio is the original physical node,\nnot the principal cube root. The total high budget is thirteen, agreeing with\n$3L_{\\max}+1$. The un-decimated construction uses eight high coefficients;\nthe polyphase construction trades additional channel information for its\nchosen decimation geometry and branch recovery. Neither budget is claimed\nminimal. Both reconstruct the same canonical object exactly.","ufficient self-calibration sets for $L_{\\max}=4$ are\n\\[\n \\mathcal I_0=\\{6,9,12,15,18,21,24,27,30\\},\\qquad\n \\mathcal I_1=\\{7,10,13,16\\}.\n\\]\nFor each matched term the base coefficient is\n$b_{\\ell,0}=a_\\ell z_\\ell^6$, and the adjacent coefficient is\n$b_{\\ell,1}=b_{\\ell,0}z_\\ell$. Their ratio is the original physical node,\nnot the principal cube root. The total high budget is thirteen, agreeing with\n$3L_{\\max}+1$. The un-decimated construction uses eight high coefficients;\nthe polyphase construction trades additional channel information for its\nchosen decimation geometry and branch recovery. Neither budget is claimed\nminimal. Both reconstruct the same canonical object exactly.","lete canonical\nsignal. Under a supplied noncolliding decimation, adjacent offsets internally\nrecover physical phase with a stated finite budget. Under uniform local\nconditioning and a strict branch margin, smooth Fourier tails yield the\nprinted conditional location and amplitude rates. Native edge synthesis\nremoves the edge's global-truncation error, and SG-37 supplies a valid\nmean-zero periodic $L^1$ perturbation estimate.","\\url{https://doi.org/10.1016/j.sigpro.2020.107722}.\n\\bibitem{batenkov} D. Batenkov.\nComplete algebraic reconstruction of piecewise-smooth functions from Fourier data.\n\\emph{Mathematics of Computation} 84 (2015), 2329--2350.\n\\url{https://doi.org/10.1090/S0025-5718-2015-02948-2}.\n\\bibitem{sparse} D. L. Donoho and M. Elad.\nOptimally sparse representation in general (nonorthogonal) dictionaries\nvia $\\ell^1$ minimization. \\emph{PNAS} 100 (2003), 2197--2202.\n\\url{https://doi.org/10.1073/pnas.0437847100}.\n\\bibitem{rouche} J. Lebl. \\emph{Guide to Cultivating Complex Analysis:\nWorking the Complex Field}, version 1.9, July 11, 2026,\nSection 5.4.2. \\url{https://www.jirka.org/ca/ca.pdf}.\n\\bibitem{fri} M. Vetterli, P. Marziliano, and T. Blu.\nSampling Signals with Finite Rate of Innovation.\n\\emph{IEEE Transactions on Signal Processing} 50 (2002), 1417--1428.\n\\url{https://doi.org/10.1109/TSP.2002.1003065}.\n\\end{thebibliography}\nThe finite-rate paper is cited for historical attribution. Exact sampling\nhypotheses from it are not imported: the consecutive-moment theorem is\nproved directly in this manu…"]}