{"id":"28969ea6-a59d-483b-9745-f9bec95a4b9e","arxiv_id":"2505.05161","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A self-review of the authors' Boundary Control method for discrete inverse problems, tying together Jacobi matrices, moment problems, Toda lattices, de Branges spaces, and Krein strings.","lead":"This paper reviews the authors' program of applying the Boundary Control method to inverse problems for discrete dynamical systems, covering Jacobi matrices, moment problems, Toda lattices, de Branges spaces, and Krein strings. It is a map of a specialized mathematical toolkit, but it contains no new proofs and relies almost entirely on the authors' own prior papers.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sign error in the printed b_k recovery formula (2.20) and missing minus sign in (2.19) mean the reconstruction algorithm as written yields the negative of the Jacobi off-diagonal coefficients; the central claim is not correctly demonstrated in the text.","rationale":"The paper's central claim is that the Boundary Control method extends to discrete dynamical systems and that the response operator determines the Jacobi coefficients. The explicit determinant formulas (2.16) and (2.20) are the concrete mechanism by which this recovery is supposed to work, and they constitute the only self-contained algorithmic content in the review. My check shows that (2.19) is missing a minus sign that follows directly from the preceding identities, and consequently (2.20) as printed has the wrong sign for b_k. This is a load-bearing defect because a reader following the text would recover the negative of the off-diagonal coefficients in simple cases. The defect is local and clearly repairable: the correct sign pattern can be obtained from the derivation itself, and the cited papers (e.g., [42]) presumably contain the correct formulas. This is why the appropriate disposition remains conditional acceptance with mandatory correction, rather than rejection or an upgraded verdict. The reader's weakest assumption concerned the unproved characterization in Theorem 3 and the typo in (2.17); my analysis agrees partially, but identifies a more specific and more damaging error in the same section. I do not see a problem with the broader mathematical strategy of the review, and I credit the paper for being a useful consolidation of the authors' prior work, even though it contains no new theorems and relies heavily on self-citations. The proposed concrete test is minimal: it reproduces the sign error explicitly and independently of any spectral theory, so it settles the concern directly.","tokens_in":39639,"tokens_out":19943,"duration_ms":162714,"concrete_test":"Recompute the b1 recovery for the explicit 2x2 case a0=a1=1, b1=1, b2=0. Directly solve the recurrence (1.2) with the delta control to obtain the response vector (r0,r1,r2)=(1,1,1); construct C2=[[2,1],[1,1]] via (2.6). Then evaluate (2.20) for k=1 and compare with the true b1=1. Separately, re-derive (2.19) from (2.13) and (2.14) using the inductively verified identity w_{n,n}=(Π_{j=0}^{n-1}a_j) Σ_{k=1}^{n} b_k; the missing minus sign will appear. Correct the signs in (2.19) and (2.20) and re-check that the corrected formulas agree with the direct computation. This single test isolates whether the printed reconstruction algorithm, as opposed to the cited underlying papers, is internally correct.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 2.4, the factorization method is the operative algorithm by which the response operator determines the Jacobi coefficients. Two printed formulas are inconsistent with the derivation that precedes them. From (2.13)-(2.14), the Goursat boundary condition gives w_{n,n} = (Π_{j=0}^{n-1} a_j) Σ_{k=1}^{n} b_k by induction. Substituting this into (2.14) for k=T-1 yields q_{T-1,T} = -(Π_{j=0}^{T-1} a_j)^{-1} Σ_{k=1}^{T-1} b_k; the minus sign is missing in (2.19). Consequently, equating with (2.18) gives Σ_{k=1}^{T-1} b_k = det C_{T,T-1}/det C_{T-1}, so b_k should equal det C_{k+1,k}/det C_k - det C_{k,k-1}/det C_{k-1}, but (2.20) prints the opposite signs. A concrete check: take a0=a1=1, b1=1, b2=0. Directly solving (1.2) gives response vector (r0,r1,r2)=(1,1,1), so C2=[[2,1],[1,1]], det C1=1, det C_{2,1}=1. The correct b1 is 1, while (2.20) as printed returns -1. The typo in (2.17) noted by the reader belongs to the same block: the displayed matrix is (T-1)xT but is multiplied by a (T-1)-vector, and the right-hand side should be -q_{T,T} times the last column of C_T, not -q_{T,T} times the unknown vector. These are repairable sign and dimension errors, but as printed the text does not correctly implement the central reconstruction claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript is a review of the authors' program extending the Boundary Control method to inverse problems for discrete dynamical systems generated by Jacobi matrices. After introducing the discrete wave equation (1.2) and the response operator, the paper presents the connecting operator C_T, Krein-type equations, and a factorization method for recovering the Jacobi coefficients from response data. It then applies the discrete BC machinery to classical (Hamburger, Stieltjes, Hausdorff) moment problems, to finite and semi-infinite Toda lattices, to the construction of de Branges spaces, to Weyl functions, and to continuous-time systems including Krein-Stieltjes strings. The final section discusses numerical simulations. The paper is largely a survey of the authors' previous works, with many theorems quoted without proofs.","tokens_in":40085,"tokens_out":9045,"duration_ms":79159,"significance":"If the formulas are corrected, the paper would provide a useful unified survey of a coherent research program: the explicit determinantal reconstruction formulas and the characterization theorem establish a dynamic-to-spectral dictionary that connects several inverse problems in one framework. The positive features include explicit algebraic formulas that can be checked by hand, a clear hierarchy of finite-dimensional truncations, and concrete approximation statements in Propositions 10-12. However, the central factorization algorithm in Section 2.4 contains sign errors and a dimension mismatch that affect later sections, so the paper in its present form does not correctly demonstrate the central reconstruction claim.","major_comments":[{"comment":"","section":"Section 2.4, Eqs. (2.19)-(2.20)"},{"comment":"","section":"Section 2.4, Eq. (2.17)"},{"comment":"","section":"Sections 3 and 4"}],"minor_comments":[{"comment":"","section":"Section 2.1-2.2"},{"comment":"","section":"Section 7.2, Eq. (7.8)"},{"comment":"","section":"Section 2.5, Theorem 3"},{"comment":"","section":"Section 2.4, after Eq. (2.12)"},{"comment":"","section":"Section 3.1, Eq. (3.1)"},{"comment":"","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is essentially a survey of the authors' own previously published results, with a very high proportion of self-citations. This is transparent and is not by itself a reason for rejection, but the editors may wish to consider whether the journal's policies on self-citation in review articles are satisfied. The technical errors in Section 2.4 are local and repairable; the mathematical program appears coherent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know before you open this. It is a review paper — the abstract says so — consolidating about a decade of the Mikhaylovs' work on the Boundary Control method for discrete inverse problems, and it contains no new theorems; all substantive results are quoted from their own prior papers ([39]–[61]) with proofs omitted. And the factorization formula for the Jacobi off-diagonal coefficients, (2.20), is printed with the wrong sign.\n\nThe paper has real value as a map. It connects the discrete BC method to Jacobi matrices, classical moment problems, Toda lattices, de Branges spaces, Weyl functions, the Krein-Stieltjes string, and numerical approximation, and the structure is clear. Some collected results are genuinely attractive, such as the Weyl m-function as the generating function of the response vector (Theorem 14) and the Section 7.3 observation that the naive point-mass response converges to δ(t) while the corrected discrete response converges to −δ'(t). The underlying program looks solid, and several cited papers are in well-regarded journals.\n\nThe soft spots, in proportion. The reader's report caught a typo in (2.17); the stress-test check holds up and goes further. Section 2.4's factorization method miscarries on signs. From (2.13)–(2.14), the Goursat relation gives q_{T−1,T} = −(Π a_j)^{−1} Σ_{k=1}^{T−1} b_k, but (2.19) drops the minus sign. With the sign restored, equating to (2.18) gives Σ b_k = det C_{T,T−1}/det C_{T−1}, hence b_k = det C_{k+1,k}/det C_k − det C_{k,k−1}/det C_{k−1}, the opposite of (2.20). Concrete check: a0=a1=1, b1=1, b2=0 gives response vector (1,1,1), so det C_{2,1}=det C_1=1 and (2.20) as printed yields b1=−1, where the true value is 1. The (2.17) RHS should be the last column of C_T, not the unknown vector. These are repairable sign and dimension slips rather than a collapse of the framework, but they sit in the operative algorithm of the review, and Sections 3.3 and 4 reuse (2.20), so the error propagates. A referee should require the corrected formulas and a worked example.\n\nThe other structural note is the verification burden. With all theorems attributed to self-citations and no proofs in the text, the review can only be checked at the formula level; the sign error shows that level of checking is worthwhile. The self-citation density is transparent given the review framing, so I would not call it a flaw, but it means there is no independent machinery in the text to catch slips.\n\nWho benefits: readers wanting an entry point to the BC method in discrete inverse problems, or a bird's-eye view of this research program. It deserves a serious referee, but I would not accept it until Section 2.4 is corrected. I would cite the original papers rather than this review.","headline":"Useful and honest review of the BC method for discrete inverse problems, but the printed b_k factorization formula (2.20) has a sign error, so Section 2.4 needs correction before publication.","tokens_in":40581,"tokens_out":12379,"would_cite":false,"duration_ms":103235,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","39A12","47B36","34A55","47A57"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Boundary Control method extends to discrete systems: the response operator of a discrete wave equation is complete inverse data, giving Jacobi matrices, moments, Toda lattices, de Branges spaces, Weyl functions, Krein strings.","keywords":["Boundary Control method","discrete dynamical systems","Jacobi matrices","moment problems","Toda lattices","de Branges spaces","Weyl functions","Krein-Stieltjes strings"],"falsifier":"Take a vector $(r_0,\\dots,r_{2T-2})$ whose $C_T$ is positive definite, run the factorization formulas (2.16) and (2.20) to obtain coefficients, simulate the forward system (1.2), and compare the computed response vector to the input; any mismatch would disprove the characterization. In particular, one can search for a positive-definite $C_T$ for which the recovered coefficients are not real or do not reproduce the given response.","tokens_in":39411,"feed_emoji":"🧮","tokens_out":7082,"duration_ms":62600,"temperature":0.7,"pith_summary":"This review argues that the Boundary Control method, developed for continuous wave equations, carries over to discrete dynamical systems. The central object is a Jacobi matrix paired with a discrete wave equation driven by a boundary control; the measured response operator is shown to contain all information needed to reconstruct the matrix. From that reconstruction the authors obtain solutions to classical moment problems, Toda lattices, Weyl functions, de Branges spaces, and Krein-Stieltjes strings, as well as numerical schemes. If the approach is correct, it turns a broad family of inverse problems into explicit algebraic procedures based on the connecting operator.","feed_headline":"Boundary control method now solves discrete inverse problems","feed_subtitle":"The response operator of a discrete wave equation carries full information to recover Jacobi matrices, moment-problem solutions, and…","key_machinery":"The load-bearing object is the connecting operator $C_T = (W^T)^* W^T$, where $W^T$ maps boundary controls to states of the discrete system at time $T$. For the discrete wave equation (1.2), its matrix entries are determined by the response vector through (2.6), so the operator is known from inverse data alone. Krein-type equations (2.11) and factorization of $C_T$ yield the Jacobi coefficients via the determinant formulas (2.16) and (2.20). The same machinery gives spectral representations in Chebyshev polynomials, the moment-to-response change of basis (3.20), and a generating-function formula for Weyl functions.","core_discovery":"The paper's central claim is that the dynamic inverse problem for a discrete hyperbolic system is well posed: from the response operator $R_{2n}$ of system (1.2), one can recover the Jacobi coefficients $\\{a_0,\\dots,a_{n-1}\\}$ and $\\{b_1,\\dots,b_{n-1}\\}$. The mechanism is the connecting operator $C_T$, built from the response vector by (2.6); Theorem 3 characterizes genuine response vectors by positivity of $C_T$, and the factorization formulas (2.16) and (2.20) give the coefficients explicitly. The same operator, via spectral representations (3.17)-(3.19) and the moment-response relation (3.20), connects dynamics to spectral data, yielding existence and uniqueness criteria for Hamburger, Stieltjes, and Hausdorff moment problems, evolution of moments in Toda lattices, Weyl-function expansions, de Branges spaces, and Krein-Stieltjes strings.","pith_inferences":["A natural test not pursued in the review is whether the determinant reconstruction still succeeds when $C_T$ is positive semidefinite but not definite; the theory only covers the definite case.","The response-vector representation of the Weyl function suggests numerical algorithms that estimate moments from boundary measurements without first reconstructing the matrix.","The corrected-response convergence result for point-mass strings indicates that choosing the discrete observable that matches the continuous one is essential; a similar correction may be needed in other discretizations.","The complex-Jacobi result that only squares $a_k^2$ are recoverable hints at gauge freedom in non-self-adjoint discrete inverse problems beyond the usual sign choice."],"forward_implications":["Recovering a Jacobi matrix from its response operator is reduced to linear algebra: build $C_T$ from the response vector, check positivity, and read off the coefficients from determinants.","Existence and uniqueness for Hamburger, Stieltjes, and Hausdorff moment problems are characterized by positivity of Hankel matrices $S_N^0$ and $S_N^1$, and indeterminacy by finiteness of certain limits involving $C_N$.","The Toda lattice can be solved for unbounded initial data by evolving moments of the spectral measure through formulas (4.4)-(4.5) and then reconstructing the Jacobi matrix at each time.","The Weyl function of a Jacobi operator is the generating function of its response vector; the same holds for finite blocks.","The dynamic inverse problem for finite Jacobi matrices with continuous time and for Krein-Stieltjes strings is solvable from the response function, with an explicit characterization of inverse data."],"supporting_citations":[{"why":"Supplies the core dynamic inverse problem for Jacobi matrices, including Theorem 3 (positivity characterization) and the determinant formulas (2.16), (2.20).","marker":"[42]"},{"why":"Supplies the moment-problem results: the moment-response relation (3.20) and the existence and indeterminacy criteria of Theorems 7-9.","marker":"[44]"},{"why":"Supplies Theorem 14, the representation of Weyl functions as generating functions of response vectors.","marker":"[45]"},{"why":"Supplies the continuous-time finite Jacobi inverse problem: Krein equations (7.7), representation of $C_T$ by (7.5), and characterization Theorem 17.","marker":"[51]"},{"why":"Supplies the Krein-Stieltjes string system and the point-mass approximation results, Propositions 10-12.","marker":"[48]"},{"why":"Supplies the infinite-dimensional de Branges construction and the closability and eigenvalue theorems, Theorems 12-13.","marker":"[58]"},{"why":"Supplies the discrete Schrödinger special case and the determinant-one characterization in Theorem 4.","marker":"[39]"},{"why":"Supplies the discrete parabolic inverse problem and the Hankel representation of the connecting operator, Theorems 18-19.","marker":"[61]"}],"fun_headline_variants":["Discrete inverse problems yield to boundary control","Boundary control goes discrete, solving inverse problems","From response to Jacobi: discrete inverse solved","Moment problems cracked by discrete boundary control","Discrete wave equations: inverse problems now solvable"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The reconstruction algorithms rest on the criterion that a vector is a genuine response vector exactly when the connecting matrix $C_T$ built from it is positive definite, together with the determinant formulas that recover the coefficients; the text cites this criterion to another paper rather than proving it here.","fun_headline_variants_meta":{"raw":{"variants":["Discrete inverse problems yield to boundary control","Boundary control goes discrete, solving inverse problems","From response to Jacobi: discrete inverse solved","Moment problems cracked by discrete boundary control","Discrete wave equations: inverse problems now solvable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000356,"raw_usage":{"total_tokens":1854,"prompt_tokens":788,"completion_tokens":1066,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":404,"completion_tokens_details":{"reasoning_tokens":996}},"tokens_in":404,"tokens_out":1066,"duration_ms":9813,"temperature":1.0,"reasoning_tokens":996,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:10:42.062451+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a vector $(r_0,\\dots,r_{2T-2})$ whose $C_T$ is positive definite, run the factorization formulas (2.16) and (2.20) to obtain coefficients, simulate the forward system (1.2), and compare the computed response vector to the input; any mismatch would disprove the characterization. In particular, one can search for a positive-definite $C_T$ for which the recovered coefficients are not real or do not reproduce the given response.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the core dynamic inverse problem for Jacobi matrices, including Theorem 3 (positivity characterization) and the determinant formulas (2.16), (2.20)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the moment-problem results: the moment-response relation (3.20) and the existence and indeterminacy criteria of Theorems 7-9."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 14, the representation of Weyl functions as generating functions of response vectors."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the continuous-time finite Jacobi inverse problem: Krein equations (7.7), representation of $C_T$ by (7.5), and characterization Theorem 17."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Krein-Stieltjes string system and the point-mass approximation results, Propositions 10-12."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the infinite-dimensional de Branges construction and the closability and eigenvalue theorems, Theorems 12-13."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the discrete Schrödinger special case and the determinant-one characterization in Theorem 4."},{"cited_title":"Mikhaylov, V.S","cited_arxiv_id":null,"evidence_quote":"Supplies the discrete parabolic inverse problem and the Hankel representation of the connecting operator, Theorems 18-19."}],"review_version":1}