{"id":"cb5d1e13-5aa5-46b1-aca1-62942567e182","arxiv_id":"2505.20905","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a wave-type dynamical system with a finite Jacobi matrix, the Fourier images of the reachable set, equipped with the connecting operator metric, form a de Branges space.","lead":"The paper constructs de Branges spaces of analytic functions for dynamical systems tied to finite Jacobi matrices, using the boundary control method. It shows the construction works even though this system lacks exact boundary controllability and has infinite wave speed.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The de Branges verification applies Theorem 3 with the multiplier inverted: condition 3 uses (z−ω)/(z−ω̄) f(z), which is not entire for non-real zeros; the standard and valid condition uses the reciprocal multiplier.","rationale":"The paper's central claim is probably true: Lemma 2 (boundary controllability from the span of S_k(T−t)) is sound, the Fourier transform sends U_T onto all polynomials of degree < N, and the Hermitian metric defined by C_T equals the L^2_{dρ} metric after fixing the missing conjugation. With that metric, B_N is a finite-dimensional Hilbert space of entire functions; condition 2 holds on the real axis; and condition 3 holds with the standard multiplier (z−ω̄)/(z−ω) because if f(ω) = 0 then f(z)(z−ω̄)/(z−ω) is a polynomial of degree at most N−1 and has equal L^2_{dρ} norm. So the central mathematical assertion survives. The load-bearing flaw is in the proof as written: Theorem 3 states the multiplier in the wrong direction, and the verification repeats this, asserting that a function with a pole at ω̄ is entire. That is not a cosmetic typo in an auxiliary lemma; it is the step that produces the conclusion. The concrete polynomial z−i in B_2 falsifies the stated condition. A referee should require the multiplier in Theorem 3 and in the condition-3 check to be corrected to (z−ω̄)/(z−ω), and the conjugation in the reproducing-kernel identity to be fixed. With those corrections the CONDITIONAL verdict stands.","tokens_in":6155,"tokens_out":24368,"duration_ms":262228,"concrete_test":"For N = 2 with any Jacobi data (e.g. b1 = b2 = 0, a1 = 1), compute explicitly for the constructed B_2 = span{1, z}: f(z) = z−i has f(i) = 0. (i) Apply the paper's condition 3: g_paper(z) = ((z−i)/(z+i))(z−i) = (z−i)^2/(z+i); evaluation at z = −i is undefined, so g_paper is not an entire function and g_paper ∉ B_2, contradicting the claim that B_N is a de Branges space via Theorem 3 as stated. (ii) Apply the corrected multiplier: g_corr(z) = ((z+i)/(z−i))(z−i) = z+i ∈ B_2, and ‖g_corr‖² = ∫ |λ+i|² dρ(λ) = ∫ |λ−i|² dρ(λ) = ‖f‖² because |λ+i| = |λ−i| for real λ. This settles that the theorem statement and proof contain an inversion error, while the de Branges property itself survives after correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central assertion 'Thus B_N is a de Branges space' rests entirely on Theorem 3. As stated, condition 3 requires that if f ∈ X and f(ω) = 0, then ((z−ω)/(z−ω̄)) f(z) ∈ X with the same norm. This condition is false for essentially any nontrivial de Branges space, including the B_N constructed here. Take any N ≥ 2, so that B_N contains z (since φ2 is affine), and choose f(z) = z−i, ω = +i. Then f(ω) = 0, but the displayed function is f(z)(z−i)/(z+i) = (z−i)^2/(z+i), which has a pole at z = −i and is not entire; it therefore cannot belong to a space of entire functions. The paper asserts this function is entire and only checks the norm equality |(λ−i)/(λ+i)| = 1 on R; the entireness assertion is simply wrong. The standard de Branges condition (de Branges [5], Dym–McKean [6]) uses the reciprocal multiplier (z−ω̄)/(z−ω): if f(ω) = 0, then f(z)(z−ω̄)/(z−ω) is entire, is again a polynomial of degree ≤ N−1 in this setting, and has the same norm because |(λ−ω̄)/(λ−ω)| = 1 for real λ. Thus B_N does satisfy the corrected axiom and the main claim is repairable, but the proof as written invokes a misstated theorem and verifies a different, invalid condition. The reproducing-kernel conjugation error noted by the reader is real but secondary; the multiplier inversion is the load-bearing gap in the de Branges verification.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper considers a boundary control dynamical system (utt − Au = F, with finite Jacobi matrix A) and, following the authors' earlier boundary control method, constructs a finite-dimensional space B_N of polynomials (degree at most N−1) as the Fourier image of the reachable set. The space is equipped with an inner product defined through the connecting operator C_T, a candidate reproducing kernel J_z is introduced via specially chosen controls, and the paper claims that B_N is a de Branges space by verifying the three conditions of Theorem 3. The novelty is that this system has infinite propagation speed and is not exactly controllable, in contrast to earlier models, yet the de Branges space construction is asserted to go through.","tokens_in":6422,"tokens_out":10778,"duration_ms":99930,"significance":"If the construction is valid, the paper extends the boundary-control approach to de Branges spaces to a new class of systems with infinite wave speed and without exact controllability, complementing the authors' previous results for Schrödinger, Dirac, and discrete Jacobi systems. The finite-dimensional model is explicit and the reliance on Lemma 2 (special boundary controllability) is a clear and testable mechanism. However, two technical flaws in the current proof must be corrected before the central claim can be accepted.","major_comments":[{"comment":"Condition 3 of Theorem 3 is misstated. The displayed multiplier is (z−ω)/(z−ω̄), but for non-real ω this expression is not entire unless f also vanishes at ω̄. The correct de Branges axiom uses the reciprocal multiplier (z−ω̄)/(z−ω). The verification in Section 3 repeats this error: for N≥2, take G(z)=z−i and ω=i; then ((z−i)/(z+i))G(z) = (z−i)^2/(z+i), which has a pole at z=−i and hence cannot belong to a space of entire functions. Thus the proof of 'Thus B_N is a de Branges space' is invalid as written. The claim is nonetheless repairable: since G(ω)=0, one can write G(z)=(z−ω)Q(z) with deg Q ≤ N−2, and the corrected multiplier gives (z−ω̄)Q(z), which is a polynomial of degree at most N−1, and the norm equality follows from |(λ−ω̄)/(λ−ω)|=1 for real λ. The authors must correct both the statement of Theorem 3 and the verification.","section":"Section 3, reproducing kernel"},{"comment":"The identity (J_z, G)_{B_N} = G(z) is not correct for the Hermitian inner product defined by (H,G)_{B_T} = ∫ H(λ) \\overline{G(λ)} dρ(λ). From the definitions one obtains (J_z, G)_{B_N} = ∑_{k=1}^N ϕ_k(z) \\overline{c_k} if G=∑ c_k ϕ_k, which equals \\overline{G(\\bar z)} (because the ϕ_k have real coefficients), not G(z). Consequently J_z as defined is not the reproducing kernel for this inner product; the correct kernel is K_z(λ)=∑ \\overline{ϕ_k(z)} ϕ_k(λ). This error affects the proof of condition 1 of Theorem 3, where the pointwise bound is derived from the asserted kernel identity. The bound itself can still be proven, for example by using K_{\\bar z} or directly from the basis expansion, so this is a fixable defect, but the argument as written is incorrect.","section":null}],"minor_comments":[{"comment":"In the chain of equalities computing (H,G)_{B_T}, the integrand writes \\overline{(F u_h(T))(λ)} where the second factor should be (F u_g(T))(λ); the final expression H(λ)\\overline{G(λ)} is correct but the intermediate display is a typo.","section":"Section 3, norm computation"},{"comment":"The text uses the symbols F and G inconsistently: 'When ω∈C such that F(ω)=0' is followed by 'z−ω/z−ω̄ F(z)' and then 'z−ω/z−ω̄ G(z)' and finally '=||G||_BN'. This should be cleaned up.","section":"Section 3, verification of condition 3"},{"comment":"Reference [9] is cited twice as '[9, 9]', and the abstract contains the typo 'de Banges spaces'.","section":"Introduction"},{"comment":"For completeness, the definition of the de Branges space B(E) should specify that E is a Hermite–Biehler function and that the quotient conditions are required to hold in the Hardy space; the current presentation is terse but acceptable.","section":"Section 3, definition of B(E)"}],"recommendation":"major_revision","confidential_remarks":"The paper has a high degree of self-citation to the authors' previous work, which is natural given the method, but the technical errors are independent of that. The central construction appears sound and repairable, so major revision rather than rejection is appropriate. The authors should also be asked to state explicitly whether the inner product on B_N is Hermitian (conjugate-linear in the second argument) and to align the reproducing kernel and the de Branges theorem accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The construction works, but the proof has a real error in the de Branges verification. The reader only caught the conjugation slip in the reproducing kernel; the stress-test is right that something more serious is there: condition 3 of Theorem 3, as invoked, uses (z−ω)/(z−ω̄), which has a pole at z = ω̄ for non-real ω. The function then cannot be entire, so the verification as written is invalid. The correct multiplier is (z−ω̄)/(z−ω), which cancels the zero at ω and preserves polynomial degree. In this setting the corrected condition holds, since |λ−ω̄| = |λ−ω| on R, so the main claim is repairable. The paper should state the theorem correctly and run the verification with the reciprocal multiplier.\n\nWhat is genuinely new here is the application of the authors' boundary-control construction of de Branges spaces to the second-order wave equation associated with a finite Jacobi matrix. Unlike the Schrödinger and Dirac cases treated earlier, this system has infinite propagation speed and lacks exact controllability; the paper shows the method still produces a de Branges space. The resulting space is the standard polynomial space with the spectral measure, so the outcome is not surprising, but the demonstration is legitimate. The norm on B_N is defined through the connecting operator, which is expressible in terms of inverse data, so there is no circularity. Lemma 2 is imported from prior work, and the proof is terse, but the essential argument is there.\n\nThe conjugation error in (J_z, G) = G(z) is real but minor: with the Hermitian inner product on H_N, the prescribed state must be \\(\\overline{\\phi_k(z)}\\), not φ_k(z), for the reproducing identity to hold. This does not affect the de Branges property. Heavy self-citation is present, but the cited lemmas are substantive.\n\nI would send this to a referee. The main theorem is correct and the defects are local and fixable. The paper deserves a serious review, and I expect it to be accepted after a careful revision. My advice: engage with it, but insist the authors correct Theorem 3's condition and the kernel identity before publication.","headline":"The main theorem is true and the construction is a legitimate new application of the author's boundary-control framework, but the proof as written contains a misstated multiplier in the de Branges criterion and a smaller conjugation slip in the reproducing kernel identity; both are easily repaired.","tokens_in":7030,"tokens_out":2407,"would_cite":false,"duration_ms":25214,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47B36","46E22","30H10","35L05","93B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that $B_N$, the Fourier image of the reachable set, is a de Branges space for any finite Jacobi matrix.","keywords":["de Branges spaces","boundary control method","finite Jacobi matrices","Krein equations","reproducing kernels","spectral function","inverse problems","boundary controllability"],"falsifier":"For a concrete finite Jacobi matrix and time $T$ (for instance $N=2$, $a_1=1$, $b_1=0$, $b_2=2$, $T=1$), compute the connecting operator $C_T$ from formula (7), solve the special control problem for two complex parameters $z,w$, and check that the inner product $(J_z,J_w)_{B_T}$ equals $\\sum_{k=1}^N\\phi_k(z)\\phi_k(w)$. A mismatch for any $z,w$ would show the claimed reproducing kernel or metric is not the de Branges one.","tokens_in":5868,"feed_emoji":"📐","tokens_out":10970,"duration_ms":96933,"temperature":0.7,"pith_summary":"The paper shows that a second-order wave-type system governed by a finite Jacobi matrix $A$, with one boundary control, has a natural de Branges space of polynomials attached to it. The construction takes the reachable set $U_T$ of states attainable at time $T$, applies the spectral Fourier transform $\\mathcal{F}$, and equips the polynomial space $B_N=\\mathcal{F}U_T$ with the inner product defined by the system's connecting operator $C_T$. The authors verify the three axioms of the de Branges characterization: bounded point evaluations through an explicitly constructed reproducing kernel $J_z$, norm-invariance under $F^\\#(z)=F(\\bar z)$, and the division property for functions with a zero. The result matters because it extends the boundary control method for building de Branges spaces to a system that lacks exact controllability and has infinite propagation speed.","feed_headline":"Finite Jacobi systems carry de Branges spaces","feed_subtitle":"Even without exact controllability and with infinite propagation speed, the construction works.","key_machinery":"Three objects carry the argument. (1) The control operator $W_T:f\\mapsto u^f(T)$ and the boundary-controllability result Lemma 2: $W_T$ maps the special $N$-dimensional family $\\mathcal{F}_T^1=\\mathrm{Lin}\\{S_k(T-t)\\}_{k=1}^N$ isomorphically onto the state space $H_N$, so $U_T=W_T\\mathcal{F}_T^1$ and the connecting operator $C_T=W_T^*W_T$ provides a genuine inner product on Fourier images. (2) The spectral Fourier transform $F:\\mathbb{R}^N\\to L^2_\\rho(\\mathbb{R})$, $b\\mapsto\\sum_k b_k\\phi_k(\\lambda)$, whose image of $U_T$ is $B_N$; on $B_N$ the connecting-operator metric coincides with $\\int H\\bar G\\,d\\rho$. (3) The special controls $j_z$ supplied by the Krein equations, which drive the system to the state $(\\phi_1(z),\\dots,\\phi_N(z))$; their Fourier images are the reproducing kernels $J_z(\\lambda)=\\sum_k\\phi_k(z)\\phi_k(\\lambda)$. Theorem 3 then converts these ingredients into the structural conclusion that $B_N$ is a de Branges space.","core_discovery":"The central claim is that $B_N=\\mathcal{F}U_T=\\mathrm{Lin}\\{\\phi_1,\\dots,\\phi_N\\}$, the polynomials of degree at most $N-1$, equipped with the metric $(H,G)_{B_T}=(C_T h,g)_{\\mathcal{F}_T}$ for $H=\\mathcal{F}u^h(T)$ and $G=\\mathcal{F}u^g(T)$ with $h,g\\in\\mathcal{F}_T^1$, is a de Branges space in the sense of Theorem 3. Equivalently, this metric is the $L^2$ inner product against the spectral measure $\\rho$ of $A$, so $B_N$ is the spectral subspace $L_N$ with reproducing kernel $J_z(\\lambda)=\\sum_{k=1}^N\\phi_k(z)\\phi_k(\\lambda)$. The proof checks the three conditions of the de Branges criterion: point evaluations satisfy $|G(z)|=|(J_z,G)_{B_N}|\\le\\|C_T^{1/2}j_z\\|_{\\mathcal{F}_T}\\|G\\|_{B_N}$; conjugation preserves the $L^2_\\rho$ norm; and the quotient $((z-\\omega)/(z-\\omega))G(z)$ has unit modulus on $\\mathbb{R}$, hence the same norm. Since all three hold, $B_N$ is a de Branges space and the associated Hermite-Biehler function $E$ exists by Theorem 3.","pith_inferences":["The same template should extend to any second-order boundary-control system admitting an analogue of Lemma 2: a finite-dimensional family of controls that maps isomorphically onto the states and whose connecting operator is accessible from inverse data. This generality is the authors' implicit suggestion, not a claim of the paper.","Taking $N\\to\\infty$ would plausibly produce de Branges spaces of entire functions associated with infinite Jacobi matrices as limits of $B_N$; the paper mentions the semi-infinite discrete case only through a citation and does not develop this limit.","Because $B_N=L_N$ as a metric space, the reproducing kernel $J_z$ is the Christoffel-Darboux kernel of the Jacobi polynomials $\\phi_k$; a numerical comparison of this kernel with the one produced by the Krein equations for small $N$ would independently test the construction."],"forward_implications":["The de Branges space $B_N$ is determined by inverse data alone: the connecting operator $C_T$, the subspace $\\mathcal{F}_T^1=C_T\\mathcal{F}_T$, and hence the metric and kernel can be reconstructed from the response function $r(t)=\\sum_k\\rho_k^{-1}S_k(t)$.","The spectral data $\\{\\lambda_k,\\rho_k\\}$ determine $B_N$ as the span of the first $N$ orthogonal polynomials with the $L^2_\\rho$ metric, so the de Branges structure is a repackaging of the classical orthogonal-polynomial spectral theory of the Jacobi matrix.","Applying the criterion of Theorem 3 to the reproducing kernel produces an explicit Hermite-Biehler function $E$ for the finite Jacobi system, placing the system inside the de Branges-space framework used for canonical systems.","The construction works despite the two features that blocked earlier applications: the system lacks boundary controllability on the whole state space, and the wave propagation speed is infinite."],"supporting_citations":[{"why":"supplies the forward and inverse problem results for finite Jacobi matrices, including Lemma 2 and the representation (7) of the connecting operator.","marker":"[8]"},{"why":"is the prior work from which the boundary-controllability Lemma 2 is taken.","marker":"[7]"},{"why":"establishes the boundary control method algorithm for associating de Branges spaces with dynamical systems that this paper adapts.","marker":"[10]"},{"why":"introduces the general approach of taking Fourier images of reachable sets with connecting-operator metrics.","marker":"[9]"},{"why":"provides the definition of de Branges spaces and the criterion stated as Theorem 3.","marker":"[5]"},{"why":"supplies the reproducing-kernel characterization of de Branges spaces used in Theorem 3.","marker":"[6]"},{"why":"gives the canonical-systems context for de Branges spaces that motivates the construction.","marker":"[14]"},{"why":"supplies the classical facts that the eigenvalues are real and distinct and that the polynomials $\\phi_k$ are orthogonal in $L^2_\\rho$.","marker":"[1]"}],"fun_headline_variants":["Jacobi systems yield de Branges spaces, no controllability needed","Finite Jacobi systems beyond controllability: de Branges spaces still arise","De Branges spaces from finite Jacobi matrices without exact control","Unexpected de Branges spaces for non-controllable Jacobi systems","Jacobi matrix dynamics: de Branges spaces despite infinite speed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction stands or falls on Lemma 2: every state of the system must be reachable by some control from the special $N$-dimensional family $\\mathcal{F}_T^1=\\mathrm{Lin}\\{S_k(T-t)\\}_{k=1}^N$, because only then does the connecting operator $C_T$ define a nondegenerate inner product on the Fourier images.","fun_headline_variants_meta":{"raw":{"variants":["Jacobi systems yield de Branges spaces, no controllability needed","Finite Jacobi systems beyond controllability: de Branges spaces still arise","De Branges spaces from finite Jacobi matrices without exact control","Unexpected de Branges spaces for non-controllable Jacobi systems","Jacobi matrix dynamics: de Branges spaces despite infinite speed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00026,"raw_usage":{"total_tokens":1543,"prompt_tokens":851,"completion_tokens":692,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":467,"completion_tokens_details":{"reasoning_tokens":600}},"tokens_in":467,"tokens_out":692,"duration_ms":6534,"temperature":1.0,"reasoning_tokens":600,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:47:52.192452+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a concrete finite Jacobi matrix and time $T$ (for instance $N=2$, $a_1=1$, $b_1=0$, $b_2=2$, $T=1$), compute the connecting operator $C_T$ from formula (7), solve the special control problem for two complex parameters $z,w$, and check that the inner product $(J_z,J_w)_{B_T}$ equals $\\sum_{k=1}^N\\phi_k(z)\\phi_k(w)$. A mismatch for any $z,w$ would show the claimed reproducing kernel or metric is not the de Branges one.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the forward and inverse problem results for finite Jacobi matrices, including Lemma 2 and the representation (7) of the connecting operator."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"is the prior work from which the boundary-controllability Lemma 2 is taken."},{"cited_title":"odinger operator, Dirac system, discrete Schr\\","cited_arxiv_id":null,"evidence_quote":"establishes the boundary control method algorithm for associating de Branges spaces with dynamical systems that this paper adapts."},{"cited_title":"Mikhaylov, V.S","cited_arxiv_id":null,"evidence_quote":"introduces the general approach of taking Fourier images of reachable sets with connecting-operator metrics."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the reproducing-kernel characterization of de Branges spaces used in Theorem 3."},{"cited_title":"Akhiezer","cited_arxiv_id":null,"evidence_quote":"supplies the classical facts that the eigenvalues are real and distinct and that the polynomials $\\phi_k$ are orthogonal in $L^2_\\rho$."}],"review_version":1}