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| \fancyhead[L]{\small\sffamily\color{ink}CONTROLLED SPECTRAL RECONSTRUCTION} |
| \fancyhead[R]{\small\sffamily Research extension / v2.0} |
| \fancyfoot[L]{\small\sffamily Exact theorems under explicit data assumptions} |
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| \newcommand{\N}{\mathbb N}\newcommand{\Z}{\mathbb Z}\newcommand{\R}{\mathbb R} |
| \newcommand{\C}{\mathbb C}\newcommand{\Lcal}{\mathcal L} |
| \newcommand{\norm}[1]{\left\|#1\right\|}\newcommand{\dd}{\,d} |
| \newcommand{\rank}{\operatorname{rank}}\newcommand{\spec}{\sigma} |
| \newcommand{\chapterpage}[1]{\clearpage\section{#1}} |
| \begin{document} |
| \begin{titlepage} |
| {\sffamily\color{accent}\large EVE / MATHEMATICAL DEVELOPMENT AND ADVERSARIAL AUDIT\par} |
| \vspace{16mm} |
| {\Huge\bfseries\color{ink}Controlled Spectral\par Reconstruction\par} |
| \vspace{7mm} |
| {\Large Four complete extensions beyond\par decoupled phase-profile inversion\par} |
| \vspace{10mm} |
| {\large A standalone research note for Maciej Nowicki\par} |
| Eve / AI-assisted mathematical development\par |
| 5 October 2026 \quad Version 2.0\par |
| \vspace{9mm} |
| \textbf{Strongest quantitative result.} For integer length-tail counts satisfying |
| $H(u)\le Bu^{-D}$, $0<D<1$, the optimal worst-case sorted-length error after $Q$ adaptive threshold queries is |
| \[ |
| \boxed{R_Q^{\rm ad}\asymp_D B^{1/D}Q^{-1/D},\qquad |
| R_Q^{\rm non}\asymp_D B^{1/D}Q^{-1}.} |
| \] |
| Matching algorithms and lower bounds establish the exact power-law advantage of adaptation. |
|
|
| \textbf{Three further completed statements.} The magnetic ground-band Hessian determines a metric graph within the simple 3-vertex-connected class; $r+1$ nonzero coupling spectra remove an unknown rank-at-most-$r$ additive interaction; and two clean energy scans identify an unknown analytic soft branch and an infinite length sequence up to one scalar calibration. |
|
|
| \vfill |
| \begin{tabular}{@{}p{.60\textwidth}p{.33\textwidth}@{}}\toprule |
| Verification item & Status\\\midrule |
| Displayed mathematical derivations & Complete under stated assumptions\\ |
| Research and adversarial review routes & Six agents, with cross-checks\\ |
| Exact and numerical certificates & Reproducible scripts supplied\\ |
| Formal proof-assistant verification & Not performed\\ |
| External human peer review & Not performed\\ |
| Priority and major-breakthrough rating & Unverified\\\bottomrule |
| \end{tabular} |
| \end{titlepage} |
| \tableofcontents |
|
|
| \chapterpage{The actual advance and the data it requires} |
| The starting mechanism turns a clean low-energy phase count into the tail count of a summable multiset of lengths. Exact reencoding alone is a limited inverse result: it assumes a known component operator, separated components, continuous phase access, and perfect data. |
|
|
| This extension addresses four harder questions. Can a connected object be reconstructed from much less than its full spectrum? Can unknown interactions be removed without eigenvectors? Can an unknown component response be calibrated from spectral data? How many finite measurements are necessary, and what does adaptation buy? |
|
|
| The answer to each is affirmative in a specified model. The models and conclusions are different; they are not interchangeable physical realizations of a single universal theorem. |
|
|
| \begin{longtable}{@{}>{\raggedright\arraybackslash}p{.19\textwidth}>{\raggedright\arraybackslash}p{.35\textwidth}>{\raggedright\arraybackslash}p{.39\textwidth}@{}}\toprule |
| Research area & Exact input & Completed conclusion\\\midrule |
| \endhead |
| Metric quantum graphs & Multivariable Hessian of the lowest magnetic eigenvalue at zero, with the integer flux lattice & Topology and all lengths within the simple 3-vertex-connected class, including nonplanar graphs\\[4pt] |
| Inverse operator spectra & $r+1$ complete unordered spectra of $T-u_iC$, known distinct nonzero $u_i$, $\rank C\le r$ & Complete spectrum of the unknown $T$, without eigenvectors or commutativity\\[4pt] |
| Blind soft-branch calibration & Two complete clean cutoff scans, an infinite string, a common real-analytic strictly increasing branch & Branch and length sequence up to scalar gauge; one normalization removes it\\[4pt] |
| Information-based complexity & At most $Q$ exact integer tail-count queries; known envelope $Bu^{-D}$ & Sharp adaptive rate $Q^{-1/D}$ and sharp nonadaptive rate $Q^{-1}$\\\bottomrule |
| \end{longtable} |
|
|
| \textbf{Status of the requested 10/10.} This note establishes the displayed model results with analytic proofs and independently checked examples. It does not establish a verified 10/10 rating for novelty, significance, generality, or practical robustness. Those are separate claims. Closely related magnetic Hessian, Torelli, two-spectrum, Fredholm-product, and adaptive approximation ideas already exist. The exact combinations and specialized minimax rate require a deeper priority review. No result here solves the original coarse one-channel Lapidus--Maier inverse problem or the Riemann hypothesis. |
|
|
| The strongest candidate for a focused new paper is the sharp query theorem. Its hypothesis, algorithm, matching obstruction, and comparison with nonadaptive measurements are all explicit. The graph theorem supplies a strong exact-data compression statement; its proof deliberately credits the established metric Torelli theorem. |
|
|
| \chapterpage{The phase profile and its exact loss identity} |
| Let $\Lcal=(\ell_j)$ be a finite or countably infinite positive multiset with $\sum_j\ell_j<\infty$. Sort lengths nonincreasingly and pad finite sequences by zeros for comparison. Write |
| \[ |
| H_{\Lcal}(u)=\#\{j:\ell_j\ge u\},\qquad u>0. |
| \] |
| For the normalized twisted circle Dirac operator |
| \[ |
| D_{a,j}=-\frac{i}{2\pi}\frac{\dd}{\dd s},\qquad |
| u(\ell_j)=e^{2\pi ia}u(0), |
| \] |
| the eigenvalues are $(n+a)/\ell_j$, $n\in\Z$. If $x_0\ell_1<1/2$, its direct-sum absolute-frequency count at energy $x_0$, for $0<a<1/2$, is |
| \[ |
| C_{\Lcal}(a)=H_{\Lcal}(a/x_0). |
| \] |
| Indeed $n=0$ contributes exactly when $a\le x_0\ell_j$, whereas every other mode has absolute frequency greater than $1/(2\ell_1)>x_0$. |
|
|
| Every length is a crossing location divided by $x_0$. This reconstruction determines the multiset, including multiplicities; it does not determine spatial placement in an ambient interval. A query at a positive phase is exactly a query at a positive length threshold. |
|
|
| \begin{lemma}[Count/sequence isometry]\label{lem:isometry} |
| For two summable sorted length sequences $\Lcal=(\ell_j)$ and $\mathcal M=(m_j)$, |
| \[ |
| d_1(\Lcal,\mathcal M):=\sum_j|\ell_j-m_j| |
| =\int_0^\infty |H_{\Lcal}(u)-H_{\mathcal M}(u)|\dd u. |
| \] |
| \end{lemma} |
| \begin{proof} |
| At a fixed threshold $u$, the indicators $\mathbf1_{u\le\ell_j}$ form an initial block in the sorted index $j$, as do the indicators for $m_j$. Consequently |
| \[ |
| \sum_j|\mathbf1_{u\le\ell_j}-\mathbf1_{u\le m_j}| |
| =|H_{\Lcal}(u)-H_{\mathcal M}(u)|. |
| \] |
| Tonelli's theorem integrates this identity, and each indicator difference integrates to $|\ell_j-m_j|$. |
| \end{proof} |
|
|
| This elementary integer-atomic identity is the basis of the sharp finite-query result. Related count/quantile and transportation identities are classical; the proof here fixes the exact loss and the unequal-cardinality convention needed below. |
|
|
| \chapterpage{A connected graph from one ground band's quadratic jet} |
| Let $\Gamma$ be a finite connected compact metric graph with positive edge lengths $\ell_e$, ordinary continuity and Kirchhoff vertex conditions, and no electric potential. Its combinatorial graph is promised to be simple and 3-vertex-connected. The latter means that it has at least four vertices and remains connected after removal of any one or two vertices. |
|
|
| Orient the edges and let $B_\Gamma$ be the vertex-edge incidence matrix. Choose an actual integral basis of |
| \[ |
| H_1(\Gamma,\Z)=\ker B_\Gamma\cap\Z^E, |
| \] |
| and place its row vectors in a $g\times E$ matrix $C$, where $g=E-V+1$. Define |
| \[ |
| L=\sum_e\ell_e,\qquad Q=C\operatorname{diag}(\ell_e)C^T. |
| \] |
| The magnetic flux coordinates are radian periods |
| $\theta\in\R^g/(2\pi\Z)^g$. Their full integral period lattice is part of the data; its marking may be changed by an integral unimodular basis transformation. |
|
|
| Let $\lambda_0(\theta)$ be the lowest eigenvalue of the magnetic Kirchhoff Laplacian $-(\dd/\dd x-iA_e)^2$ with these periods. This Laplacian has the ordinary physical frequency normalization, unlike the $1/(2\pi)$ Dirac normalization above. |
|
|
| \begin{theorem}[Graph reconstruction]\label{thm:graph} |
| Within the promised simple 3-vertex-connected class, the Hessian |
| \[ |
| \mathsf H=\nabla^2\lambda_0(0) |
| \] |
| and the integer flux-period lattice uniquely determine the metric graph, up to metric-graph isomorphism. In the chosen cycle coordinates, |
| \[ |
| \boxed{\mathsf H=\frac2LQ^{-1},\qquad 2\mathsf H^{-1}=LQ.} |
| \] |
| There is no planarity or rational-independence assumption on the lengths. |
| \end{theorem} |
|
|
| \subsection{Direct calculation of the Hessian} |
| Use the harmonic constant-per-edge gauge |
| \[ |
| A(\theta)=C^TQ^{-1}\theta. |
| \] |
| Then $B_\Gamma A=0$ and $C\operatorname{diag}(\ell_e)A=\theta$. At a vertex, the magnetic correction to the Kirchhoff derivative sum is its common function value times the oriented divergence of $A$, which is zero. Thus the operator domain is independent of $\theta$ in this gauge. |
|
|
| The zero-flux ground eigenvalue is simple, and its normalized eigenfunction is $\psi_0=L^{-1/2}$. The operator expansion on this fixed domain is |
| \[ |
| H_A=H_0+2iA\frac{\dd}{\dd x}+A^2. |
| \] |
| The first-order term annihilates $\psi_0$. Analytic perturbation at a simple isolated eigenvalue therefore has no first-order mixing contribution to its quadratic coefficient: |
| \[ |
| \lambda_0(\theta) |
| =\frac1L\sum_e\ell_eA_e(\theta)^2+O(|\theta|^3) |
| =\frac1L\theta^TQ^{-1}\theta+O(|\theta|^3). |
| \] |
| Complex conjugation identifies the spectra at $\theta$ and $-\theta$. The analytic ground branch is even, so all odd total-degree terms vanish and the remainder improves to $O(|\theta|^4)$. Differentiation gives the displayed Hessian. |
|
|
| For a circle of circumference $\ell$, the near-zero ground eigenvalue is $\theta^2/\ell^2$; its Hessian is $2/\ell^2$, which checks the factor. If normalized phase coordinates $a=\theta/(2\pi)$ are used, the correct reconstruction Gram is $8\pi^2(\nabla_a^2\lambda_0(0))^{-1}$. |
|
|
| \subsection{The integral lattice and the established Torelli step} |
| The cycle quadratic form is $q_\ell(c)=\sum_e\ell_ec_e^2$. Its Gram in the chosen integral basis is $Q$. Set |
| \[ |
| s_e=L\ell_e. |
| \] |
| The form on the same cycle lattice for these scaled lengths has Gram $Q_s=LQ=2\mathsf H^{-1}$. The known flux lattice supplies its dual integral cohomology lattice, and hence the integral homology lattice by the evaluation pairing. |
|
|
| The metric Torelli theorem of Caporaso--Viviani \cite{CV}, Theorems 4.1.9--4.1.10, determines the 3-edge-connected cyclic-equivalence class from this polarized integral cycle lattice. In the simple 3-vertex-connected class, there are no bridges or separating edge pairs, and Whitney rigidity removes the cycle-matroid ambiguity. It therefore determines the graph with all lengths $s_e$. This is an application of an established theorem, not a new proof of tropical Torelli. |
|
|
| All edges survive in this class. Hence |
| \[ |
| \sum_es_e=L^2, |
| \qquad |
| \boxed{L=\sqrt{\sum_es_e},\qquad |
| \ell_e=\frac{s_e}{\sqrt{\sum_fs_f}}.} |
| \] |
| This removes the unknown total-length factor and completes the proof of Theorem~\ref{thm:graph}. |
|
|
| The data comprise $g(g+1)/2$ real Hessian entries, rather than infinitely many eigenvalues. They can be specified by the directional limits |
| \[ |
| q(v)=\lim_{t\to0}\lambda_0(tv)/t^2 |
| \] |
| for $v=e_i$ and $v=e_i+e_j$, $i<j$, followed by polarization. Each limit still requires ideal asymptotic data. The theorem does not replace those limits by finitely many noisy eigenvalue samples. |
|
|
| \subsection{Why the comparison class is essential} |
| Start with a bridgeless cyclic graph, for example a graph in the promised class, with total length $L$ and cycle Gram $Q$. Scale all its edges by $0<r<1$ and attach a tree of total length $L(1/r-r)$. The new graph has |
| \[ |
| L'=L/r,\qquad Q'=rQ, |
| \] |
| and therefore $L'Q'=LQ$: its entire ground-band Hessian is identical. The Hessian cannot itself certify the absence of attached trees. With independently known $L$, different attached trees of the same total length remain invisible at quadratic order. |
|
|
| Degree-two subdivisions, grouped lengths in separating edge pairs, and Whitney twists also limit unrestricted reconstruction. For arbitrary connected graphs, the Hessian determines the 3-edge-connected cyclic class with scaled lengths $L\ell_e$; it does not determine every bridge or every attachment. |
|
|
| \chapterpage{Removing an unknown finite-rank interaction from spectra} |
| Let $T=T^*\ge bI>0$ on a Hilbert space and assume $T^{-1}$ is trace class. Let $C=C^*\ge0$ be a bounded unknown operator of rank at most $r$. Choose distinct known nonzero real settings $u_1,\ldots,u_{r+1}$ for which |
| \[ |
| E_i=T-u_iC>0. |
| \] |
| Here and below strict positivity means a positive lower spectral bound. Complete spectra include multiplicities, but no common eigenbasis or branch labeling is supplied. |
|
|
| \begin{theorem}[Spectral-only rank-bounded correction]\label{thm:rank} |
| The $r+1$ complete spectra of $E_i$ determine $\spec(T)$, with multiplicities. Define from each measured spectrum |
| \[ |
| D_i(z)=\det(I+zE_i^{-1})=\prod_j(1+z/e_{i,j}), |
| \] |
| and define |
| \[ |
| \alpha_i=\lim_{s\to+\infty}\frac{D_1(s)}{D_i(s)},\qquad |
| w_i=\prod_{k\ne i}\frac{-u_k}{u_i-u_k}. |
| \] |
| All limits are finite and positive, and |
| \[ |
| \boxed{D_T(z)=\det(I+zT^{-1}) |
| =\frac{\sum_iw_i\alpha_iD_i(z)}{\sum_iw_i\alpha_i}.} |
| \] |
| The denominator is positive. The eigenvalues of $T$ are the negatives of the zeros of $D_T$, with their zero multiplicities. |
| \end{theorem} |
|
|
| \subsection{Domains, products, and their normalization} |
| Bounded perturbation keeps the self-adjoint domain of $T$. The identity |
| \[ |
| E_i^{-1}-T^{-1}=u_iE_i^{-1}CT^{-1} |
| \] |
| shows that $E_i^{-1}$ is trace class. Its spectral product converges absolutely and locally uniformly, and defines an entire function without an unknown exponential factor. |
|
|
| For an allowed coupling $u$, write |
| \[ |
| \kappa(u)=\det(I-uCT^{-1}) |
| =\det(I-uT^{-1/2}CT^{-1/2})>0. |
| \] |
| Positivity follows from $T-uC>0$ and the finite-dimensional nontrivial part of the last identity-minus-finite-rank operator. All determinants used here concern bounded identity-plus-trace-class operators. |
|
|
| The bounded operator identity |
| \[ |
| (I+zE_u^{-1})(I-uCT^{-1})=I+zT^{-1}-uCT^{-1} |
| \] |
| and determinant multiplicativity imply |
| \[ |
| \kappa(u)D_u(z)=\mathcal F(u,z),\qquad |
| \mathcal F(u,z)=\det(I+zT^{-1}-uCT^{-1}). |
| \] |
| Outside the negative spectrum of $T$, factorization gives |
| \[ |
| \mathcal F(u,z)=D_T(z)\det(I-uC(T+z)^{-1}). |
| \] |
| The second determinant is a polynomial in $u$ of degree at most $r$, by restriction to the finite-dimensional interaction range. Entire continuation in $z$ extends this polynomial property through all apparent poles. Thus $\mathcal F(\cdot,z)$ has degree at most $r$ for every $z\in\C$. |
|
|
| \subsection{Recovering the missing normalization from spectra} |
| For $s\ge0$, |
| \[ |
| \norm{C(T+s)^{-1}}_1\le\frac{\norm C_1}{b+s}\longrightarrow0. |
| \] |
| Continuity of the Fredholm determinant in trace norm gives |
| \[ |
| \frac{D_1(s)}{D_i(s)} |
| =\frac{\kappa(u_i)}{\kappa(u_1)} |
| \frac{\det(I-u_1C(T+s)^{-1})} |
| {\det(I-u_iC(T+s)^{-1})} |
| \longrightarrow\frac{\kappa(u_i)}{\kappa(u_1)}. |
| \] |
| Consequently $\alpha_i=\kappa(u_i)/\kappa(u_1)>0$ is determined by complete spectra alone. |
|
|
| Lagrange interpolation at $u=0$ now gives |
| \[ |
| D_T(z)=\mathcal F(0,z)=\sum_iw_i\kappa(u_i)D_i(z). |
| \] |
| At $z=0$ all spectral products equal one. Hence $\sum_iw_i\alpha_i=1/\kappa(u_1)>0$, and normalization yields the formula in Theorem~\ref{thm:rank}. Its zeros give the claimed spectrum. |
|
|
| \subsection{Minimality proved for ranks one and two} |
| One nonzero spectrum cannot determine an unknown rank-one correction: $E$ can arise from many different $T=E+uC$. For rank two, compare |
| \[ |
| T_A=\operatorname{diag}(11,12),\quad C_A=\operatorname{diag}(2,2), |
| \] |
| \[ |
| T_B=\operatorname{diag}(10,13),\quad C_B=\operatorname{diag}(1,3). |
| \] |
| At $u=1$ both observed spectra are $\{9,10\}$; at $u=2$ both are $\{7,8\}$. The unperturbed spectra differ. Thus two settings are insufficient for rank two, while three suffice. The bound $r+1$ is proved minimal here only for $r=1,2$. |
|
|
| The result does not reconstruct the interaction or the eigenvectors of $T$. Complete spectral products and their large-$s$ ratios are ideal infinite data. Finite-band spectral measurements require a separate quantitative truncation theory. Two-spectrum inverse problems and spectral-product normalizations have extensive prior art \cite{DH,SW}; the present proof is a general rank-bounded cancellation statement. |
|
|
| \chapterpage{An exact fixed-cutoff realization of the interaction theorem} |
| On a soft space $\mathcal K$ and a hard space $\mathcal Q$, consider |
| \[ |
| H_{a,u}^{\rm phys}= |
| \begin{pmatrix}aT&\sqrt{au}\,W\\\sqrt{au}\,W^*&F\end{pmatrix}, |
| \qquad a>0,\quad u>0, |
| \] |
| where $F=F^*\ge fI>0$ is independent of $a$ and $W$ is bounded. Coupling at scale $\sqrt a$ survives division by the soft scale $a$. |
|
|
| \begin{proposition}[Exact Schur counting identity]\label{prop:schur} |
| Fix $0<x_0<f$ and let |
| \[ |
| C_{x_0}=W(F-x_0)^{-1}W^*,\qquad E_u^{(x_0)}=T-uC_{x_0}. |
| \] |
| If $T$ has compact resolvent, the count strictly below $x_0$ satisfies |
| \[ |
| \boxed{M_{a,u}(x_0)=N_{E_u^{(x_0)}}(<x_0/a).} |
| \] |
| \end{proposition} |
| \begin{proof} |
| The hard block of $H_{a,u}^{\rm phys}-x_0$ is strictly positive. Completing the square in its quadratic form, or using a bounded invertible triangular congruence, identifies its negative subspace dimension with that of |
| \[ |
| aT-x_0-auW(F-x_0)^{-1}W^* |
| =a(E_u^{(x_0)}-x_0/a). |
| \] |
| The latter count is finite because it is a bounded perturbation of a compact-resolvent operator bounded below. Equality of negative dimensions proves the identity, including the strict endpoint convention. |
| \end{proof} |
|
|
| \begin{corollary}[Exact critically coupled tomography] |
| Suppose $T^{-1}$ is trace class, $\rank W\le r$, and for known distinct positive $u_i$ one has $E_{u_i}^{(x_0)}>x_0I$. Then the $r+1$ full scans $a\mapsto M_{a,u_i}(x_0)$ over $0<a<1$ determine $\spec(T)$. |
| \end{corollary} |
| \begin{proof} |
| Every effective eigenvalue $e_{i,j}>x_0$ gives a crossing $a=x_0/e_{i,j}\in(0,1)$ with its multiplicity. The scans supply complete spectra of $E_{u_i}^{(x_0)}$. Apply Theorem~\ref{thm:rank}, since $C_{x_0}\ge0$ has rank at most $r$. |
| \end{proof} |
|
|
| For geometric eigenvalues $t_j=1/\ell_j$, reciprocal inversion supplies the length multiset. If the effective gap condition is omitted, eigenvalues below $x_0$ are only counted in aggregate and the scan does not supply the complete products. If the hard block depends on $a$, the self-energy generally depends on $a$ too; the exact identity no longer gives a fixed additive pencil. The natural hard sector of a shifted Dirac ladder has this dependence. A canonical graph or Dirac realization of this constant-hard-block model is not asserted. |
|
|
| For completeness, the rescaled resolvent has effective soft limit $T-uWF^{-1}W^*$. Its compressed error at a fixed positive resolvent shift is $O(a)$, and its full embedded error is $O(\sqrt a)$, by direct block inversion. This is a generalized resolvent limit; the hard space disappears from the limiting inverse. The exact count identity above is stronger for this specified model and does not depend on taking that limit. |
|
|
| \chapterpage{Blind analytic calibration from two clean energies} |
| Here the unknown is a common isolated soft-branch response $\eta(a)$ as well as the lengths. Assume $\eta$ is real analytic on a neighborhood of $[0,a_*]$, $\eta(0)=0$, and $\eta'(a)>0$ there. The observed clean counts are |
| \[ |
| C_{x_i}(a)=\#\{j:\eta(a)\le x_i\ell_j\}, |
| \qquad 0<a<a_*,\quad i=1,2, |
| \] |
| for distinct known positive $x_1,x_2$. All crossings are covered: $x_i\ell_1<\eta(a_*)$. Other component branches are assumed to be absent below these cutoffs. |
|
|
| \begin{theorem}[Two-cutoff blind identifiability]\label{thm:blind} |
| Let $\Lcal$ and $\widetilde{\Lcal}$ be infinite positive summable strings, and let $\eta,\widetilde\eta$ satisfy the stated analytic and coverage conditions. If both pairs of exact count curves agree, then for one $c>0$, |
| \[ |
| \widetilde\eta(a)=c\eta(a),\qquad |
| \widetilde\ell_j=c\ell_j\quad\text{for every sorted index }j. |
| \] |
| The derivative calibration $\eta'(0)=\widetilde\eta'(0)$ forces $c=1$. A known common positive total length or longest length also removes the scalar gauge. |
| \end{theorem} |
| \begin{proof} |
| Strict monotonicity and complete coverage identify the sorted crossing locations, including multiplicities, separately at each energy. They imply |
| \[ |
| \eta^{-1}(x_i\ell_j)=\widetilde\eta^{-1}(x_i\widetilde\ell_j). |
| \] |
| The function $h=\widetilde\eta\circ\eta^{-1}$ is analytic on a connected interval containing zero and all observed arguments, since $\eta'$ is strictly positive. Therefore |
| \[ |
| \frac{h(x_1\ell_j)}{x_1} |
| =\widetilde\ell_j |
| =\frac{h(x_2\ell_j)}{x_2}. |
| \] |
| An infinite summable positive string has infinitely many distinct lengths tending to zero. The analytic function |
| \[ |
| f(t)=h(x_1t)/x_1-h(x_2t)/x_2 |
| \] |
| has infinitely many zeros accumulating at the interior point zero, so it vanishes identically near zero. Write $h(z)=\sum_{n\ge1}c_nz^n$. Coefficient comparison gives |
| \[ |
| c_n(x_1^{n-1}-x_2^{n-1})=0. |
| \] |
| Since $x_1\ne x_2$, all coefficients except $c_1$ vanish. Thus $h(z)=cz$ locally, with $c=h'(0)>0$. Analytic continuation on its connected interval gives this equality globally, yielding both stated relations. Finally $c=\widetilde\eta'(0)/\eta'(0)$, and any of the asserted calibrations forces $c=1$. |
| \end{proof} |
|
|
| This is exact identifiability, not a stable analytic continuation theorem. It uses an infinite string, two energies, a common analytic branch, and complete crossing coverage. Finite strings supply only finitely many analytic constraints and do not support this identity-theorem argument. |
|
|
| \subsection{One clean energy still fails after natural calibrations} |
| Take an infinite reference string with branch $\eta(a)=a$ and write $a_j=x_0\ell_j$, $a_1=\max_j a_j$. It has lengths strictly smaller than the longest. Define |
| \[ |
| c_*= |
| \frac{\sum_j a_j^3(a_1-a_j)}{\sum_j a_j^2(a_1-a_j)},\qquad |
| p(a)=a^2(a-a_1)(a-c_*). |
| \] |
| Both sums converge and the denominator is positive. For sufficiently small nonzero $\varepsilon$, put |
| \[ |
| \widetilde\eta(a)=a+\varepsilon p(a),\qquad |
| \widetilde\ell_j=\widetilde\eta(a_j)/x_0. |
| \] |
| This branch is strictly increasing and positive on the fixed phase interval, with $\widetilde\eta'(0)=1$. Also $p(a_1)=0$ and |
| \[ |
| \sum_jp(a_j)=0. |
| \] |
| Thus the longest length and total length are unchanged. The crossings at the one cutoff remain exactly $a_j$, but the other lengths change. For an infinite string, the polynomial cannot vanish at every distinct $a_j$. |
|
|
| Replacing $a^2$ by $a^{k+1}$ and choosing the corresponding moment ratio also preserves every calibrated Taylor coefficient through order $k$. One cutoff remains insufficient under any fixed finite Taylor-jet calibration. Two distinct cutoffs supply the missing analytic compatibility constraint. |
|
|
| \subsection{Real analyticity is essential} |
| There is a smooth two-cutoff counterexample even when every derivative at zero, the total length, and the longest length are known. Take $x_1=1$, $x_2=2$, $\ell_j=10^{-j}$, and $\eta(a)=a$ on $[0,1]$. Choose four disjoint smooth bumps supported close to $0.01,0.02,0.001,0.002$, respectively, avoiding every other point of $\{\ell_j,2\ell_j\}$. Give them heights $1,2,-1,-2$ and call their sum $b(a)$. For sufficiently small nonzero $\varepsilon$, |
| \[ |
| \widetilde\eta(a)=a+\varepsilon b(a),\qquad |
| \widetilde\ell_2=0.01+\varepsilon,\quad |
| \widetilde\ell_3=0.001-\varepsilon, |
| \] |
| with all other lengths unchanged. The new branch is strictly increasing, agrees with $a$ near zero, and satisfies |
| \[ |
| \widetilde\eta(x_i\ell_j)=x_i\widetilde\ell_j |
| \quad(i=1,2;\ j\ge1). |
| \] |
| Both complete count scans are therefore identical. Total length, longest length, and all Taylor coefficients at zero agree, while two geometric lengths differ. This shows why the analytic hypothesis in Theorem~\ref{thm:blind} cannot be replaced by smoothness. |
|
|
| \chapterpage{Sharp finite-query reconstruction: statement and model} |
| Fix $0<D<1$ and $B>0$. Let $\mathcal F_{B,D}$ be the class of finite or infinite positive multisets obeying |
| \[ |
| H_{\Lcal}(u)\le Bu^{-D}\quad\text{for every }u>0. |
| \] |
| Include the empty multiset for convenience. Put $U=B^{1/D}$. Every admissible length is at most $U$, and after sorting, |
| \[ |
| \ell_j\le(B/j)^{1/D},\qquad |
| \sum_j\ell_j\le\int_0^U Bu^{-D}\dd u=\frac U{1-D}. |
| \] |
| The oracle returns the exact integer $H(u)$ at a chosen positive real threshold. The loss is the $d_1$ of Lemma~\ref{lem:isometry}. An adaptive procedure may choose a threshold from its past observations. A nonadaptive procedure fixes every threshold before seeing them. The budget counts calls, with arbitrarily precise real threshold access. |
|
|
| Let $R_Q^{\rm ad}$ and $R_Q^{\rm non}$ denote the deterministic minimax losses with at most $Q$ queries. Estimators may output a summable multiset; the upper constructions below actually remain in the envelope class. |
|
|
| \begin{theorem}[Sharp adaptivity separation]\label{thm:queries} |
| There are positive constants depending only on $D$ such that, for all $Q\ge1$, |
| \[ |
| \boxed{R_Q^{\rm ad}\asymp_D U(Q+1)^{-1/D},\qquad |
| R_Q^{\rm non}\asymp_D U(Q+1)^{-1}.} |
| \] |
| The same orders hold for randomized algorithms evaluated in worst-case expected loss. Arbitrary count noise bounded in magnitude by $\eta<1/2$ leaves these rates unchanged after nearest-integer rounding. |
| \end{theorem} |
|
|
| This first minimax statement does not assume a fixed common total length; its lower examples may have different totals. A separate corollary below proves the same rates with the prescribed total $L=U/2$. Stochastic count noise without the stated hard magnitude bound, threshold-location errors, and a minimum accessible threshold are different models and are not covered by the rounding assertion. |
|
|
| For physical phase profiles, substitute $u=a/x_0$. Choose the low cutoff small enough that the entire admissible class lies in the isolated soft window and that the required thresholds lie in the allowed phase range. The known zero count at $2U$ in the algorithm below is supplied by the envelope itself; it need not be measured. |
|
|
| \chapterpage{Adaptive algorithm and a proof of its optimal upper rate} |
| Choose a tolerance $0<\tau\le U$. Work on the dyadic root interval $[0,2U)$, whose upper count is known to be zero. The endpoint $0$ is never queried. |
|
|
| \begin{enumerate} |
| \item Bisect the interval adjoining zero until its endpoint $a$ satisfies |
| \[ |
| T(a):=\frac B{1-D}a^{1-D}\le\tau. |
| \] |
| Each split queries its positive midpoint and creates a positive sibling with known multiplicity. |
| \item A positive interval $[a_i,b_i)$ has known count $c_i=H(a_i)-H(b_i)$. Bisect it whenever $c_i(b_i-a_i)>\tau$; one midpoint query determines both child counts. |
| \item At every final positive leaf, output $c_i$ copies of its lower endpoint $a_i$. Discard all lengths below the final origin endpoint. |
| \end{enumerate} |
|
|
| The lower-endpoint estimator is coordinatewise no greater than the sorted true sequence, and therefore belongs to $\mathcal F_{B,D}$. The certificate |
| \[ |
| d_1(\Lcal,\widehat\Lcal) |
| \le\sum_{\rm leaves}c_i(b_i-a_i)+T(a) |
| \le(q+1)\tau |
| \] |
| is observable after $q$ queries. The origin-tail bound follows by integrating the count envelope; it is conservative because it also includes part of the contribution from longer lengths. |
|
|
| \subsection{How many cells can be refined at a given width?} |
| At dyadic width $\delta$, every positive cell has left endpoint at least $\delta$. Let $N_\delta$ be the number of positive cells of that width that are refined. |
|
|
| If $\delta\ge\tau$, every refined cell contains at least one atom. Split the axis at |
| \[ |
| r=(B\delta)^{1/(D+1)}. |
| \] |
| For $\delta\le U$, one has $r\ge\delta$. At most $2r/\delta$ cells lie below or intersect $r$; at most $H(r)\le Br^{-D}$ occupied cells lie entirely above it. Hence |
| \[ |
| N_\delta\le3B^{1/(D+1)}\delta^{-D/(D+1)}. |
| \] |
| Widths greater than $U$ contain no occupied positive cells. |
|
|
| If $\delta<\tau$, refinement requires multiplicity $c>\tau/\delta$. Split instead at |
| \[ |
| r=(B\delta^2/\tau)^{1/(D+1)}. |
| \] |
| When $r\ge\delta$, the same space/count argument gives |
| \[ |
| N_\delta\le\frac{2r}{\delta}+\frac{Br^{-D}}{\tau/\delta} |
| \le3B^{1/(D+1)}\tau^{-1/(D+1)}\delta^{(1-D)/(D+1)}. |
| \] |
| When $r<\delta$, all positive cells lie above $\delta$ and their entire available count is bounded by $B\delta^{-D}<\tau/\delta$. No such cell can be refined. Thus the last bound has no additive term that could accumulate over infinitely many scales. |
|
|
| \subsection{Summing the tree levels and inverting the budget} |
| The coarse-width bounds grow geometrically toward $\delta\approx\tau$, while the fine-width bounds decrease geometrically afterward. Both level sums are bounded by |
| \[ |
| K_D B^{1/(D+1)}\tau^{-D/(D+1)} |
| =K_D(U/\tau)^{D/(D+1)} |
| \] |
| for a finite constant $K_D$ depending only on $D$. The origin chain has only order $1+\log(U/\tau)$ steps, which is absorbed into the same power by enlarging $K_D$. Therefore |
| \[ |
| \boxed{q\le K_D(U/\tau)^{D/(D+1)}.} |
| \] |
| For large enough $Q$, choose |
| \[ |
| \tau=U(K_D/Q)^{(D+1)/D}. |
| \] |
| Then $q\le Q$ and the certificate gives |
| \[ |
| d_1\le(Q+1)\tau\le C_DUQ^{-1/D}. |
| \] |
| For the finitely many smaller budgets, the empty-string estimate and the total-length bound suffice after increasing $C_D$. This proves the adaptive upper rate in Theorem~\ref{thm:queries}. |
|
|
| The improvement uses unit integer multiplicities. A positive cell consumes at least one whole atom, and a refined fine cell consumes more than $\tau/\delta$ atoms. Arbitrary continuously weighted atoms would not supply this packing constraint. The algorithm is an envelope-aware specialization of established adaptive monotone approximation, rather than a new interval-refinement principle \cite{GGM}. |
|
|
| \chapterpage{Matching lower bounds and optimal nonadaptive grids} |
| \subsection{Adaptive lower bound} |
| Let $N=2Q+2$ and $w=\tfrac12(B/N)^{1/D}$. Inside $[w,2w]$, place $N$ disjoint intervals of width $h=w/(2N)$, separated by equal gaps, and put one unknown length in each. Every resulting string is admissible: below $2w$ the count is at most $N=B(2w)^{-D}$, and above $2w$ it is zero. |
|
|
| Execute a deterministic adaptive procedure on the midpoint configuration. Count a threshold on a closed interval's endpoint as touching that interval; a threshold can touch at most one. At least $N-Q\ge N/2$ intervals are never touched. Move their atoms arbitrarily close to the left endpoints, strictly inside the intervals, in one alternative and close to the right endpoints in another, leaving touched atoms unchanged. All observations, and hence the procedure's next queries and final output, remain identical. The alternatives have distance arbitrarily close to |
| \[ |
| (N-Q)h\ge w/4. |
| \] |
| The triangle inequality forces error at least $w/8$ on one of them. This is a constant times $U(Q+1)^{-1/D}$. |
|
|
| For randomized adaptive procedures, choose independent equally likely left/right positions in each interval. Conditional on any transcript, untouched bits remain independent and unbiased; at most $Q$ intervals were touched. On each untouched interval, the true tail count takes two equiprobable values differing by one. Any estimated count function has expected integrated error at least $h/2$ there. Summing and averaging gives the same lower order for worst-case expected loss. |
|
|
| The lower proof also works for infinite true strings. Use half the envelope for the uncertain finite cluster and append a common infinite tail bounded by the other half, with all its lengths smaller than $w/2$. Common tail counts cancel in the loss. A sufficiently truncated sequence of constant multiples of $j^{-1/D}$ can give geometric exponent exactly $D$. |
|
|
| \subsection{A graded nonadaptive grid reaches order \texorpdfstring{$Q^{-1}$}{1/Q}} |
| Put $M=Q+1$, $p=2/(1-D)$, and choose the $Q$ thresholds |
| \[ |
| u_k=U(k/M)^p,\qquad k=1,\ldots,M-1. |
| \] |
| Discard the origin bin. Put each counted length at its bin's lower endpoint. The final bin is $[u_{M-1},U]$ and contains exactly $H(u_{M-1})$ atoms; $H(U)$ need not vanish. |
|
|
| Let $\alpha=(D+1)/2>D$. For a length $u$ in the $k$th positive bin, the mean value theorem gives |
| \[ |
| u_{k+1}-u_k\le\frac{p2^{p-1}}M U^{1-\alpha}u^\alpha. |
| \] |
| The envelope supplies |
| \[ |
| \sum_j\ell_j^\alpha |
| =\alpha\int_0^Uu^{\alpha-1}H(u)\dd u |
| \le\frac\alpha{\alpha-D}U^\alpha. |
| \] |
| Thus all positive-bin errors sum to at most $C_DU/M$. The discarded tail costs at most |
| \[ |
| \frac B{1-D}u_1^{1-D}=\frac U{1-D}M^{-2}. |
| \] |
| This proves the nonadaptive upper rate. |
|
|
| Any fixed $Q$ thresholds cut $[0,U]$ into at most $Q+1$ gaps. A gap has width at least $U/(Q+1)$. A single length near either end gives identical observations, so worst-case error is at least $U/[2(Q+1)]$. For randomized nonadaptive procedures, a uniform single length on $[0,U]$ gives expected error |
| \[ |
| \frac1{4U}\sum_i(\Delta u_i)^2\ge\frac U{4(Q+1)} |
| \] |
| for every fixed threshold realization. Averaging yields the same lower bound. An envelope-slack common tail again extends it to infinite true strings. This completes all claims in Theorem~\ref{thm:queries}. |
|
|
| \subsection{The rates remain sharp with a known fixed total length} |
| \begin{corollary}[Fixed-mass minimax separation]\label{cor:mass} |
| Restrict the true strings to $\mathcal F_{B,D}$ with the known total length $L=U/2$. Both rates in Theorem~\ref{thm:queries} remain sharp, for deterministic and randomized expected loss. Estimates can also be required to have that exact total and satisfy the envelope. |
| \end{corollary} |
| \begin{proof} |
| For the adaptive lower bound, put $N=2Q+2$ and |
| \[ |
| w=\frac U{16}(2N+1)^{-1/D},\qquad h=\frac w{4N}. |
| \] |
| Choose $N$ lower centers $c_i=w+(i-1/2)w/(2N)$ and their upper partners $3w-c_i$. Each pair chooses one of the two configurations |
| \[ |
| (c_i-h/2,\,3w-c_i+h/2),\qquad |
| (c_i+h/2,\,3w-c_i-h/2). |
| \] |
| The $2N$ uncertainty intervals are disjoint inside $[w,2w]$. Each pair has constant sum $3w$. Add the common background length $L-3Nw$; all candidates have total $L$. Since $3Nw<3U/32$, the background is greater than $13U/32$ and is the largest length. Every small atom is at most $2w\le U(2N+1)^{-1/D}$, so every sorted coordinate satisfies $\ell_j\le Uj^{-1/D}$ and the envelope holds. |
|
|
| One threshold can touch at most one pair bit. At least $N-Q$ bits remain unresolved. Flipping each of these changes two lengths by $h$, giving distance $2h(N-Q)\ge w/4$ and deterministic minimax error at least $w/8$. Independent unbiased bits give expected loss at least $h(N-Q)\ge w/8$ for the same reason as before. This yields the adaptive lower order with constants depending only on $D$. |
|
|
| For the nonadaptive lower bound, take two lengths $\{t,L-t\}$ with |
| \[ |
| t\in[v,2v],\qquad v=\frac U8\,2^{-1/D}. |
| \] |
| The smaller length is at most $U2^{-1/D}$ and the larger at most $U$, so every candidate is admissible. A threshold introduces at most two breakpoints in $t$; $Q$ queries therefore give at most $2Q+1$ cells. The largest cell has width at least $v/(2Q+1)$, and two candidates in it have $d_1$ arbitrarily close to twice that width. This gives a deterministic lower bound $v/(2Q+1)$. Uniform $t$ gives a randomized expected-loss lower bound $v/[2(2Q+1)]$ by the sum-of-squared-cell-width argument. Thus the nonadaptive lower order remains $U/Q$. |
|
|
| For the upper bounds, the earlier lower-endpoint estimates $\widehat\ell_j$ are sorted, lie below the true sorted sequence, and have mass $\widehat L\le L$. Let $z_j=Uj^{-1/D}$, $Z=\sum_jz_j>L$, and set |
| \[ |
| \beta=\frac{L-\widehat L}{Z-\widehat L},\qquad |
| \ell_j^{\rm corrected}=(1-\beta)\widehat\ell_j+\beta z_j. |
| \] |
| The corrected sequence is sorted, obeys the envelope, and has total exactly $L$. Its distance from the original estimate is $L-\widehat L\le d_1(\Lcal,\widehat\Lcal)$. The triangle inequality increases either upper bound by at most a factor of two. This proves the corollary. |
| \end{proof} |
|
|
| The corollary concerns the particular known mass $U/2$; it is not a universal assertion for every possible total. At an extremal total, a constrained class could have much less freedom. It establishes that the sharp adaptive gain is not merely an artifact of uncertainty about total length. |
|
|
| \subsection{Measurement precision is a separate resource} |
| The displayed adaptive method accesses origin thresholds of order |
| \[ |
| u_{\min}\asymp_D UQ^{-(D+1)/(D(1-D))}. |
| \] |
| Its call complexity assumes these thresholds can be set exactly. Count magnitudes may grow polynomially in $Q$. Noise of magnitude less than $1/2$ is removed because counts are integers, but phase calibration errors move the thresholds and are not removed by rounding. A fixed positive lower phase limit leaves an unresolved tail floor. |
|
|
| \begin{proposition}[Sharp count-amplitude noise transition] |
| For adversarial count errors of magnitude at least $1/2$, the envelope-only model has a minimax loss floor of at least $U/6$, even with unlimited adaptive queries. |
| \end{proposition} |
| \begin{proof} |
| The single-length strings $\{U/3\}$ and $\{2U/3\}$ both satisfy the envelope, have $d_1=U/3$, and their counts differ by at most one at every threshold. The common response $(H_1(u)+H_2(u))/2$ lies within $1/2$ of both exact counts for every $u$. Every adaptive query therefore has identical allowable observations in the two cases. Any common output has error at least half their distance for one case. Together with the rounding theorem for errors below $1/2$, this proves the claimed transition. It is a worst-case real-valued response-error model, without a prescribed common total length. |
| \end{proof} |
| For the fixed-mass class of Corollary~\ref{cor:mass}, the same transition holds with a smaller positive floor: the two-atom candidates at $t=v$ and $t=2v$ have distance $2v$ and counts differing by at most one everywhere, so the midpoint response forces minimax error at least $v$ even with unlimited queries. |
|
|
| \chapterpage{A verified obstruction to robust bounded-band harmonic inversion} |
| One useful oriented phase harmonic from the original model is |
| \[ |
| K_{\Lcal}(x)=\sum_j(1-\cos(2\pi x\ell_j)). |
| \] |
| Exact analytic data can determine lengths. This does not imply a well-conditioned inverse when only $0\le x\le X$ is available with noise. |
|
|
| \begin{theorem}[No uniform H\"older inverse on a bounded band]\label{thm:noholder} |
| Fix $X>0$. Even among strings of common total length $1$ with $H(u)\le u^{-1/2}$, there is no estimate |
| \[ |
| d_1(\Lcal,\mathcal M)\le C\norm{K_{\Lcal}-K_{\mathcal M}}_{L^\infty(0,X)}^\gamma |
| \] |
| valid for all such finite strings, for any fixed $C<\infty$ and $\gamma>0$. The failure also holds for infinite strings with a fixed common tail, common total length, and a fixed envelope constant enlarged to $2$. |
| \end{theorem} |
| \begin{proof} |
| Fix an integer $m\ge1$, and put $r=2m+1$, $N=2^{r-1}$, $J=2^r$. Partition the integers $1,\ldots,J$ into $A$ and $B$ by the parity of the binary digit sum of $k-1$. The generating identity |
| \[ |
| \sum_{k\in A}e^{tk}-\sum_{k\in B}e^{tk} |
| =e^t\prod_{j=0}^{r-1}(1-e^{2^jt})=O(t^r) |
| \] |
| shows that their power sums agree through degree $r-1=2m$, including cardinality and first moment. Let $S$ be their common first-moment sum. |
|
|
| Choose $\delta=1/[J(N+1)^2]$ and $0<\rho\le1$. Use the small lengths $\rho\delta k$, $k\in A$ or $k\in B$, and in both strings add the same length $1-\rho\delta S$. The total length is exactly one. The common length is at least $1/2$, and every small length is at most $(N+1)^{-2}$. Hence the sorted sequence satisfies $\ell_j\le j^{-2}$, which is equivalent to the envelope $H(u)\le u^{-1/2}$. |
|
|
| The two small multisets are different. Their sorted distance is $c_m\rho$ for a fixed $c_m>0$. The common large component cancels exactly from the difference of $K$ curves. Taylor expansion through degree $2m$ cancels all polynomial terms, and the global cosine remainder gives |
| \[ |
| \norm{K_A-K_B}_{L^\infty(0,X)} |
| \le\frac{(2\pi X\rho\delta)^{2m+2}}{(2m+2)!} |
| \left(\sum_{k\in A}k^{2m+2}+\sum_{k\in B}k^{2m+2}\right) |
| =C_{m,X}\rho^{2m+2}. |
| \] |
| Given $\gamma>0$, choose $m$ so that $\gamma(2m+2)>1$. Then $d_1/\norm{K_A-K_B}_\infty^\gamma$ diverges as $\rho\downarrow0$. This rules out every asserted H\"older estimate. |
|
|
| For infinite strings append a fixed common summable $j^{-2}$ tail with total less than $1/4$ and count envelope at most $u^{-1/2}$, and subtract its total from the common large component. Reducing the large component preserves the finite part's envelope. The combined envelope is at most $2u^{-1/2}$, while common tail and background terms cancel from both differences. The same proof applies. |
| \end{proof} |
|
|
| This proof uses classical Prouhet cancellation. It is an information limit, not an assertion that exact analytic inversion fails. Fourier continuation has extensive stability and instability literature \cite{IN}; the specific construction respects unit multiplicities and fixed total length. |
|
|
| The accompanying exact-arithmetic certificates use rational Taylor bounds, rather than subtraction of nearly equal floating-point cosines. Separately optimized examples at $X=1$, total length $1$, and envelope $u^{-1/2}$ give: |
| \[ |
| \begin{array}{c@{\qquad}c@{\qquad}c} |
| m & d_1 & \sup_{0\le x\le1}|K_A(x)-K_B(x)|\text{ upper bound}\\\hline |
| 2 & 1/108 & 4.26\times10^{-11}\\ |
| 3 & 1/1331 & 1.39\times10^{-25}\\ |
| 6 & 2.01\times10^{-7}\text{ approximately} & 6.31\times10^{-101} |
| \end{array} |
| \] |
| These upper bounds are rounded upward for display. A tiny certified discrepancy does not make the two strings identical; it makes them indistinguishable under a corresponding bounded noise radius. |
|
|
| \section{Audit record, prior art, and a defensible completion claim} |
| \subsection{What has actually been checked} |
| The graph Hessian was derived independently by three routes and matched against the precise lattice convention in metric Torelli. The rank correction was checked independently for determinant normalization, positivity, polynomial degree, trace classes, and the Schur endpoint convention. The query theorem received independent adversarial reviews of both scale sums, budget inversion, endpoint counts, and randomized lower bounds. Blind calibration was checked using sorted crossing correspondence and the analytic identity theorem. |
|
|
| The release includes exact rational calculations for cycle forms, edge-length recovery within supplied topologies, noncommuting finite-rank determinant identities, and moment-cancellation counterexamples. Independent magnetic eigenvalue calculations test the quadratic normalization numerically. These finite checks support the written proofs; they do not formally verify infinite-dimensional arguments, the metric Torelli theorem, or every graph topology. |
|
|
| \begin{center} |
| \begin{tabular}{@{}lrr@{}}\toprule |
| Verification suite & Exact checks & Numerical checks\\\midrule |
| Adaptive queries and integer rounding & 10,622 & 10\\ |
| Graph forms, interacting determinants, magnetic FEM & 891 & 126\\ |
| Moment cancellation and rational noise certificates & 592 & 0\\\midrule |
| New checks in this release & 12,105 & 136\\\bottomrule |
| \end{tabular} |
| \end{center} |
| The graph suite covers 60 rational metrics on five supplied graph topologies, including $K_{3,3}$. Its independent magnetic finite-element calculation has worst observed relative quadratic-coefficient error below $4.0\times10^{-5}$. Exact interaction normalization succeeds in 45 noncommuting finite examples; unnormalized interpolation fails in the negative controls. The 10 high-precision query-suite calculations concern an illustrative infinite string and are numerical, not interval-certified. |
|
|
| No external human referee and no formal proof assistant has certified the note. Test counts are reproducibility details, not a significance score. The scripts and their results record exact checks separately from numerical approximations. |
|
|
| \subsection{Closest established ideas} |
| \begin{enumerate} |
| \item \textbf{Graph reconstruction.} Caporaso--Viviani \cite{CV} supply the metric Torelli theorem. Rueckriemen \cite{Rueckriemen} derives graph information from full Bloch spectra, including complete reconstruction for planar 3-connected graphs. Theorem~\ref{thm:graph} uses the quadratic jet and covers the promised nonplanar class through metric Torelli. Colbois--Provenzano--Savo \cite{CPS} already supplies a magnetic-ground-state-Hessian plus Torelli strategy on surfaces. Older twisted transition-operator work relates band Hessians to harmonic graph geometry \cite{KSS}. The graph adaptation needs priority review; the Hessian/Torelli mechanism is established. |
| \item \textbf{Interaction removal.} Dobosevych--Hryniv \cite{DH} studies rank-one perturbation spectra with characteristic products and normalization at infinity. Silva--Weder \cite{SW} reconstructs structured Jacobi operators from two boundary spectra. The finite-rank determinant polynomial and Lagrange interpolation are classical. The present exact rank-bounded correction and fixed-cutoff composition are the candidate extensions. |
| \item \textbf{Finite queries.} Gaillard--Gerchinovitz--de Montbrun \cite{GGM} studies adaptive monotone approximation and GreedyBox, tracing interval refinement to Novak. Their setting is bounded real-valued monotone functions. The unbounded integer staircase with a prescribed power envelope and the sharp exponent $1/D$ are the specific candidate contribution here. |
| \item \textbf{Bounded-band limits.} Prouhet cancellation, analytic continuation, and Fourier inverse stability have long histories. The no-H\"older construction is a calibrated obstruction for this geometry class, not a new general instability principle. |
| \end{enumerate} |
|
|
| Targeted primary-source searches did not locate every displayed combination in exactly these hypotheses. That is not proof of priority. Information-based complexity, weak-$\ell^p$ sequence recovery, adaptive quantization, graph Albanese theory, and finite-rank inverse pencils remain relevant citation networks to inspect before submitting a novelty claim. |
|
|
| \subsection{The concrete path beyond this note} |
| For the query theorem, the next meaningful advance is a precise threshold-error model with a matching minimax law and a realizable phase-measurement protocol. For graph reconstruction, it is stable recovery of the integral graph cell from noisy Hessian entries, with explicit separation assumptions and certified topology selection. For interacting spectra, it is a canonical quantum-graph or Dirac implementation of the fixed pencil, plus controlled finite-spectrum truncation. For blind calibration, it is a quantitative theorem with a specified analytic radius and coefficient bounds. |
|
|
| These questions are now concrete and testable. They are not additional completed theorems in this release. A verified major breakthrough would require a substantial novel conclusion, a defensible primary-literature comparison, and independent external assessment. The honest completed claim is four rigorous model extensions, one sharp minimax separation, and explicit limits preventing overstatement. |
|
|
| \clearpage |
| \begin{thebibliography}{99} |
| \bibitem{CV} Lucia Caporaso and Filippo Viviani, \emph{Torelli theorem for graphs and tropical curves}, Duke Math. J. 153 (2010), 129--171. \url{https://arxiv.org/abs/0901.1389}. Theorems 4.1.9--4.1.10. |
| \bibitem{Rueckriemen} Ralf Rueckriemen, \emph{Recovering quantum graphs from their Bloch spectrum}, Ann. Inst. Fourier 63 (2013), 1149--1176. \url{https://arxiv.org/abs/1101.6002}. |
| \bibitem{CPS} Bruno Colbois, Luigi Provenzano and Alessandro Savo, \emph{Magnetic ground states and the conformal class of a surface}, preprint (2025). \url{https://arxiv.org/abs/2503.16940}. Theorems 3--6. |
| \bibitem{DH} Oles Dobosevych and Rostyslav Hryniv, \emph{Direct and inverse spectral problems for rank-one perturbations of self-adjoint operators}, Integral Equations Operator Theory 93 (2021), 18. \url{https://doi.org/10.1007/s00020-021-02630-y}. |
| \bibitem{SW} Luis O. Silva and Ricardo Weder, \emph{On the two spectra inverse problem for semi-infinite Jacobi matrices}, Math. Phys. Anal. Geom. 9 (2006), 263--290. \url{https://arxiv.org/abs/math-ph/0511078}. |
| \bibitem{KSS} Motoko Kotani, Tomoyuki Shirai and Toshikazu Sunada, \emph{Asymptotic behavior of the transition probability of a random walk on an infinite graph}, J. Funct. Anal. 159 (1998), 664--689. \url{https://doi.org/10.1006/jfan.1998.3322}. Theorem 2. |
| \bibitem{GGM} Pierre Gaillard, S\'ebastien Gerchinovitz and \'Etienne de Montbrun, \emph{Adaptive approximation of monotone functions}, preprint (2023). \url{https://arxiv.org/abs/2309.07530}. |
| \bibitem{IN} Mikhail Isaev and Roman G. Novikov, \emph{H\"older-logarithmic stability in Fourier synthesis}, Inverse Problems 36 (2020), 125003. \url{https://arxiv.org/abs/2005.01414}. |
| \end{thebibliography} |
| \end{document} |
|
|
