{"id":"1208310d-b496-4c40-a8e5-75d1ababd72c","arxiv_id":"2505.22138","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The reduced boundary data on a planar string tree are claimed to determine all edge lengths, densities, and angles, but a 90-degree gauge rotation with density swap breaks uniqueness.","lead":"This paper proves an identifiability claim for a tree-shaped network of strings: from boundary measurements at all but one leaf, it says one can recover the varying string densities, all edge lengths, and the angles between edges. The proof combines the boundary control method with a leaf-peeling argument. The claim is too strong: rotating any unmeasured edge by 90 degrees and swapping its two densities leaves all boundary data unchanged.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Gauge symmetry on the unmeasured root edge: swapping its two densities and rotating its local frame by 90° leaves all reduced boundary data invariant, so Theorem 2's uniqueness claim fails.","rationale":"The reader's weakest_assumption correctly locates the gap: Step 4 (Eq. (36)) needs to know which eigenvalue of C_i is the longitudinal one, but the trace/determinant invariants provide only the unordered pair. The present analysis shows this is not a mere technical oversight but an exact gauge symmetry of the model: on any edge whose far boundary is unmeasured, swapping the two densities and rotating the local frame by π/2 gives the same global dynamics and the same reduced response. This directly falsifies the identifiability claim in Theorem 1 (star graph) and therefore the general claim in Theorem 2, whose proof peels the tree down to such star subproblems. The proposed concrete test settles the matter with a simple two-edge example. I agree with the reader's REJECT verdict; no adjustment is needed.","tokens_in":13921,"tokens_out":14523,"duration_ms":154404,"concrete_test":"Explicit counterexample on a two-edge star with e_1 measured and e_2 unmeasured. Set constant densities and lengths: l_1=l_2=1, ρ_1=μ_1=1, ρ_2=2, μ_2=3, α_12=π/6. Compute the reduced TW matrix M_11(λ) (a 2×2 matrix function) at several values λ∈{1,2,4} using the standard impedance/scattering representation for the two-velocity wave equation with Dirichlet conditions at the two leaves. Repeat the computation with parameters ρ_2=3, μ_2=2, α_12=2π/3. If M_11(λ) is identical for both parameter sets at all tested λ and to numerical precision, the non-uniqueness is confirmed and Theorem 1 is false. The same check can be done analytically for constant coefficients, where the vertex scattering matrix makes the invariance explicit.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim fails because of an exact gauge symmetry, already visible in Section 4 Step 4, Eq. (36). The reduced data do not include measurements at the root edge e_n. From Eq. (36) one obtains the symmetric matrix C_i = S_{in} B_n S_{in}^{-1}; its trace and determinant give only the unordered pair {1/√ρ_n(0), 1/√μ_n(0)}. To recover B_n and α_{in} one must decide which eigenvalue belongs to the longitudinal channel. The data provide no such ordering: replacing (ρ_n(x), μ_n(x), α_{in}) by (μ_n(x), ρ_n(x), α_{in}+π/2) on the unmeasured edge leaves the global displacement field, the continuity conditions (3), and the force balance (4) pointwise unchanged, because the rotation S_{π/2} conjugates the diagonal density matrix to the swapped one. Hence every entry M_{ij}(λ) for 1≤i,j≤m−1 is identical for the two systems. If ρ_n and μ_n are distinct, this is a two-fold ambiguity in B_n and a π/2 ambiguity in the angle; if they coincide, B_n is scalar and α_{in} is completely unobservable from C_i. Thus Theorem 1 and the peeling-based Theorem 2 overclaim unique recoverability. A generic ordering assumption (e.g., ρ_i < μ_i on every edge) or full boundary measurements at all leaves is required.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper addresses the inverse problem of recovering a planar tree-like network of elastic strings with two wave speeds from the reduced Titchmarsh-Weyl matrix or the reduced dynamical response operator measured at all but one boundary vertex. The claimed results include unique recovery of the graph (edge lengths, topology, angles between edges) and of the two spatially varying densities on every edge. The proof proceeds by solving a star-graph inverse problem via the boundary control method and then applying a leaf-peeling procedure to arbitrary trees.","tokens_in":14286,"tokens_out":8985,"duration_ms":92595,"significance":"If the central uniqueness theorem were correct, the paper would represent a meaningful step beyond earlier constant-coefficient results and would provide a constructive algorithm with potential numerical implementation. The paper honestly engages with the boundary control method and builds on the authors' prior work. However, the main result as stated is false: the reduced data are invariant under a gauge transformation on the unmeasured root edge that swaps its two densities and rotates its local frame by 90 degrees. This is an exact counterexample, not a mere proof gap. A generic ordering assumption (or an explicit labeling convention) could salvage the result, but the theorems in the current form overclaim.","major_comments":[{"comment":"The recovery of B_n and α_in from C_i = S_{in} B_n S_{in}^{-1} is ambiguous. Equation (36) determines C_i, whose trace and determinant provide only the unordered pair {1/sqrt(ρ_n(0)), 1/sqrt(μ_n(0))}. The reduced data do not identify which eigenvalue belongs to the longitudinal channel. Consequently, replacing (ρ_n, μ_n, α_in) by (μ_n, ρ_n, α_in+π/2) on the unmeasured edge leaves the global PDE system, the continuity conditions (3), and the force balance (4) pointwise invariant, because the rotation S_{π/2} conjugates the diagonal density matrix to the swapped one. Thus every M_{ij}(λ), 1≤i,j≤m−1, is identical for the two systems. If ρ_n(0)=μ_n(0), the angle α_in is completely unobservable. This invalidates Theorem 1 and Theorem 2 as stated. A generic ordering assumption (e.g., ρ_i(x)<μ_i(x) on every edge) or an explicit labeling of the longitudinal versus transverse channel is required.","section":"Section 4, Step 4, Eq. (36)"},{"comment":"The peeling argument propagates the ambiguity described above. When a sheaf is removed, the internal edge e_0 becomes the new boundary of the reduced tree, but its recovered density assignment and angle are known only up to the swap/π/2 transformation. The formulas for the reduced TW matrix fM(λ), in particular equations (44)-(48), use S_{10} and D_0, so the reconstructed data for the smaller tree inherit the same ambiguity. Therefore the iterative procedure cannot break the symmetry, and the uniqueness claim of Theorem 2 fails unless an additional ordering assumption is imposed on every edge.","section":"Section 5, leaf-peeling step"}],"minor_comments":[{"comment":"The name Titchmarsh is misspelled as 'Titchmarch' throughout the manuscript; it should be 'Titchmarsh-Weyl'.","section":"Title and Abstract"},{"comment":"The Fourier transform convention connecting M(k^2) and R(t) is not fully specified; for λ off the real axis the integral should be understood in a distributional sense, and a brief clarification would help the reader.","section":"Section 3, Eq. (16)"},{"comment":"The definitions of pre-sheaf and sheaf are given informally; a precise topological definition (in terms of the subtree consisting of a vertex and its incident edges) would improve readability.","section":"Section 5, Step 1"},{"comment":"The proof of Theorem 3 is only a sketch, relying on controllability results from [29] and on an asserted equivalence between the reduced response and the reduced TW matrix. Since this theorem is a main advertised result, a more complete proof or a precise reference would be expected.","section":"Theorem 3 proof"}],"recommendation":"major_revision","confidential_remarks":"The counterexample presented in the major comments is straightforward and decisive: one can construct two explicit two-velocity star graphs that differ on the unmeasured edge only by swapping densities and rotating the local frame, yet have identical reduced TW matrices. The authors should be asked to add an ordering assumption and to restate Theorems 1-3 accordingly. Without such a fix the paper is not publishable in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper extends the constant-coefficient two-velocity inverse result of Avdonin–Leugering–Mikhaylov to spatially varying densities on trees, using the same boundary-control and leaf-peeling machinery. The extension is natural and the bulk of the argument, especially the local recovery of densities on measured edges, is credible. But the main identifiability theorem is false as stated. There is an exact gauge symmetry on the unmeasured root edge: replace (rho_n(x), mu_n(x)) by (mu_n(x), rho_n(x)) and rotate that edge's local frame by 90 degrees. The PDE on the edge is unchanged in the new variables, and at the vertex the continuity and force-balance terms are unchanged because S_{pi/2} diag(1/rho, 1/mu) S_{pi/2}^{-1} = diag(1/mu, 1/rho). So every entry of the reduced TW matrix and of the reduced response operator is identical. Step 4 (Eq. (36)) recovers only the unordered pair {1/sqrt(rho_n(0)), 1/sqrt(mu_n(0))} from trace and determinant; B_n and alpha_in are ambiguous up to the swap, and when rho_n(0)=mu_n(0) the angle is completely unobservable. Theorems 1–3 therefore overclaim unique recovery. This is a load-bearing flaw, not a cosmetic gap. It can be repaired by adding an ordering or genericity assumption (e.g. rho_i < mu_i on every edge) or by including the root in the measurements. The dynamic-to-spectral reduction in Theorem 3 relies on previously published controllability and spectral-data results; that part is plausible. The citation pattern is fine; the self-citations are standard and not circular. Who should read this: anyone working on inverse problems on quantum graphs or elastic networks. It is a solid technical contribution to a known program, but in the present form the central uniqueness statement needs major revision. I would not desk reject it: a serious referee should be asked to evaluate the corrected statement and the peeling construction. My recommendation: send to peer review, expect rejection or major revision unless the authors add the missing assumption.","headline":"Variable-density extension has a solid core but the main uniqueness theorem fails due to an exact gauge symmetry on the unmeasured root edge.","tokens_in":14702,"tokens_out":8114,"would_cite":false,"duration_ms":78572,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","34B45","35L05","47E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The reduced Titchmarsh-Weyl matrix at all but one boundary vertex of a planar tree of strings determines the tree, its angles, and the two densities on every edge.","keywords":["inverse problem","metric tree","planar network of strings","two-velocity wave equation","Titchmarsh-Weyl matrix","leaf-peeling method","boundary control method"],"falsifier":"Compute the reduced Titchmarsh-Weyl matrix for a three-edge star graph, then repeat the computation after rotating the unobserved edge by 90 degrees in the plane and interchanging its two density functions; exact identity of the two matrices for all $\\lambda$ would show that the angle and the density assignment on that edge are not individually determined by the stated data.","tokens_in":13743,"feed_emoji":"🌳","tokens_out":14908,"duration_ms":143249,"temperature":0.7,"pith_summary":"This paper asks whether boundary measurements at most of the leaves of a planar tree of elastic strings determine the whole system: the shape of the tree, the angles at which branches meet, and the two different densities (longitudinal and transverse) at every point of every edge. The authors' answer is yes for the spectral and dynamical inverse problems alike, provided the data are collected at all but one boundary vertex. The proof shows that the reduced Titchmarsh-Weyl matrix, which records normal derivatives at the observed leaves for the two-component displacement, is enough to peel the tree layer by layer from the leaves inward, recovering each slab's densities before moving to the next vertex. A finite-time version holds for the response operator, with the needed time equal to twice the controllability time of the wave system.","feed_headline":"All but one leaf suffice to recover a whole tree of strings","feed_subtitle":"Boundary spectral data at every leaf except the root determine the angles, lengths, and two densities of all edges.","key_machinery":"The machinery is the reduced Titchmarsh-Weyl matrix function, a matrix of $2\\times 2$ blocks whose entry $M_{ij}(\\lambda)$ maps the boundary value at leaf $j$ of the two-component displacement to its normal derivative at leaf $i$ for solutions of the spectral problem with Dirichlet conditions elsewhere. Around it the proof builds a leaf-peeling reduction: the star-graph step uses the matrix identities $A_k = S_{ki}A_i(S_{ki})^{-1}$ and $C_i = S_{in}B_n(S_{in})^{-1}$, where $S_\\alpha$ is the rotation matrix between edge frames and $B_n$ collects the root-edge impedances $1/\\sqrt{\\varrho_n(0)}$, $1/\\sqrt{\\mu_n(0)}$, to pass from known boundary edges to the unknown interior edge and its incidence angle. The boundary control method supplies the locality needed to recover the density on each boundary edge independently from the diagonal entries $R_{ii}(t)$.","core_discovery":"The central claim, stated as Theorem 2, is that an arbitrary finite tree with a two-velocity wave equation on each edge is determined by the reduced Titchmarsh-Weyl matrix function $M_{ij}(\\lambda)$ for $1\\le i,j\\le m-1$, i.e., by spectral data at every leaf except one chosen as the root. From those data one recovers the combinatorial tree, the edge lengths, the angles between neighbouring edges, and the two positive $C^2$ densities on every edge. The argument treats the root as clamped and works inward: first the boundary-control method extracts each boundary edge's densities from the diagonal of the response function, then the star-graph analysis recovers the transmission coefficients and the impedance of the next inner edge, and a leaf-peeling step eliminates the resolved leaves and produces the reduced data for the remaining tree. Iterating yields the full graph. The same conclusion is transferred to the dynamical setting in Theorem 3, where the reduced response operator on a finite time interval of length twice the controllability time determines the system.","pith_inferences":["The proof's Step 4 determines the excluded edge's incidence angle only up to a 90-degree rotation with a simultaneous swap of the two densities on that edge; unless an ordering or genericity assumption is imposed, the uniqueness statement in Theorem 2 is stronger than the argument supports.","Because Theorem 3 is derived from Theorem 2 through the response-operator/TW correspondence, the same 90-degree-swap ambiguity would transfer to the finite-time dynamical inverse problem.","The ambiguity disappears if the two impedances at the root end of the excluded edge are known to be unequal and their longitudinal/transverse assignment is fixed; alternatively, a single extra measurement at the excluded leaf would break the symmetry.","The layer-by-layer scheme suggests a concrete numerical test: run the star-graph inversion on synthetic data with the excluded edge's endpoint densities nearly equal and watch the recovered angle and density assignment degrade as the two values approach each other."],"forward_implications":["If Theorem 2 is right, an inspection of a tree from all but one boundary point is sufficient: the missing leaf can remain clamped and unobserved.","Planar geometry becomes identifiable: the angles between edges at a vertex, not just the graph's combinatorial type, are encoded in the spectral data.","The dynamical version gives a finite-time procedure: the reduced response operator on $[0,2\\tau]$, where $\\tau$ is the maximal optical distance from the excluded leaf to the others, determines the system.","The recovery is constructive and layer-by-layer; Remark 2 asserts the method leads to a stable, numerically implementable algorithm, with numerical experiments already available for the one-velocity star case."],"supporting_citations":[{"why":"the constant-coefficient planar string-tree inverse problem whose methods and statements this paper extends to spatially varying densities.","marker":"[9]"},{"why":"provides the elastic multi-link model, the well-posedness of the dynamical problem, and the sharp controllability time used in Theorem 3.","marker":"[29]"},{"why":"sets out the boundary control approach connecting controllability to identifiability, used here to recover edge densities from diagonal response data.","marker":"[11]"},{"why":"supplies the correspondence between the response operator and the Titchmarsh-Weyl matrix and between spectral data and the TW function.","marker":"[12]"},{"why":"is the boundary control method survey whose locality principle lets each boundary edge be recovered independently before peeling.","marker":"[14]"},{"why":"gives the one-dimensional variant of the BC method applied on each edge interval to recover the two densities.","marker":"[16]"},{"why":"unifies the inverse-problem data classes and supports the transfer from spectral to dynamical data in the final theorem.","marker":"[17]"},{"why":"reports numerical experiments for a one-velocity star graph that motivate the claimed numerical stability of the leaf-peeling algorithm.","marker":"[2]"}],"fun_headline_variants":["All but one leaf fix a two-velocity tree","One leaf missing, whole two-velocity tree determined","Two-velocity tree fixed by all but one leaf's spectral data","Missing root leaf still yields full two-velocity tree","Inverse two-velocity tree problem solved with one leaf to spare"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes one can tell which of the two impedances at the root end of the excluded edge is the longitudinal one; otherwise a simultaneous 90-degree rotation of that edge and swap of its two densities leaves all the reduced data unchanged.","fun_headline_variants_meta":{"raw":{"variants":["All but one leaf fix a two-velocity tree","One leaf missing, whole two-velocity tree determined","Two-velocity tree fixed by all but one leaf's spectral data","Missing root leaf still yields full two-velocity tree","Inverse two-velocity tree problem solved with one leaf to spare"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001249,"raw_usage":{"total_tokens":5098,"prompt_tokens":901,"completion_tokens":4197,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":517,"completion_tokens_details":{"reasoning_tokens":4115}},"tokens_in":517,"tokens_out":4197,"duration_ms":29764,"temperature":1.0,"reasoning_tokens":4115,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:18:31.627824+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the reduced Titchmarsh-Weyl matrix for a three-edge star graph, then repeat the computation after rotating the unobserved edge by 90 degrees in the plane and interchanging its two density functions; exact identity of the two matrices for all $\\lambda$ would show that the angle and the density assignment on that edge are not individually determined by the stated data.","supporting_citations":[{"cited_title":"Avdonin, G","cited_arxiv_id":null,"evidence_quote":"the constant-coefficient planar string-tree inverse problem whose methods and statements this paper extends to spatially varying densities."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the elastic multi-link model, the well-posedness of the dynamical problem, and the sharp controllability time used in Theorem 3."},{"cited_title":"Avdonin and V","cited_arxiv_id":null,"evidence_quote":"sets out the boundary control approach connecting controllability to identifiability, used here to recover edge densities from diagonal response data."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the correspondence between the response operator and the Titchmarsh-Weyl matrix and between spectral data and the TW function."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"is the boundary control method survey whose locality principle lets each boundary edge be recovered independently before peeling."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the one-dimensional variant of the BC method applied on each edge interval to recover the two densities."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"unifies the inverse-problem data classes and supports the transfer from spectral to dynamical data in the final theorem."},{"cited_title":"Avdonin, B","cited_arxiv_id":null,"evidence_quote":"reports numerical experiments for a one-velocity star graph that motivate the claimed numerical stability of the leaf-peeling algorithm."}],"review_version":1}