{"id":"12172415-c456-4be5-be85-60c1a8044d0e","arxiv_id":"2412.16096","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Friedrichs extension of a lower semibounded minimal linear relation from a discrete symplectic system is exactly the maximal relation whose elements vanish at zero and are limit-orthogonal to the columns of a recessive solution.","lead":"The paper proves an explicit description of the Friedrichs extension for a class of linear relations generated by discrete symplectic systems with weights depending linearly on the spectral parameter. The characterization uses recessive solutions and generalizes earlier results for banded matrices and Hamiltonian systems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main theorem is plausible, but the proof of Lemma 3.2 defines Λ_k as a sum from j=0 involving ~X_j^{-1}, while ~X_j is only guaranteed invertible eventually; this makes the dominant solution undefined on part of the half-line and leaves the central argument incomplete.","rationale":"The paper proves a conditional characterization: under Hypothesis 2.6, disconjugacy on [0,∞), and eventual controllability, the Friedrichs extension is described by recessive-solution boundary conditions. The reader identified eventual controllability as the weakest assumption, which is a fair scope limitation, but it is explicitly assumed and not an internal error. My stress-test found a more concrete proof gap: Lemma 3.2 defines the dominant solution using ~X_j^{-1} for all j starting at 0, while the stated assumptions only guarantee invertibility eventually. This makes the construction undefined on an initial segment unless extra normalization or a shifted starting point is used. The gap is likely repairable, but it is load-bearing because Lemma 3.2 is the step that places the recessive-solution columns in the Friedrichs domain, and Theorem 3.4 depends on it. The reader's concern about Theorem 2.5 being unproved is also valid and contributes to the conditional verdict. I therefore keep the reader's CONDITIONAL verdict unchanged, with partial agreement on the specific weakest point.","tokens_in":19373,"tokens_out":29216,"duration_ms":236587,"concrete_test":"Construct a 2×2 (n=1) example with B_k singular at k=0 but satisfying Hypothesis 2.6 and disconjugacy, e.g., A_k ≡ 1, C_k ≡ 0, D_k ≡ 1, B_0=0 and B_k=1 for k≥1, with W_k=1, and check whether the recessive solution satisfies X_0=0; if so, the sum from j=0 in Λ_k is undefined, confirming the gap, and then verify whether redefining Λ_k with the sum started at k_0=1 restores the Cauchy argument in Lemma 3.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3.2 (Step 1) defines Λ_k = Σ_{j=0}^{k-1} −~X_j^{-1}(λ) B_j ~X_{j+1}^{*-1}(λ) and the dominant solution X^hat_k = ~X_k Λ_k, U^hat_k = ~U_k Λ_k + ~X_k^{*-1}. These formulas require ~X_j(λ) to be invertible for every j in the range. However, the paper's own discussion (Section 2, before Theorem 2.5) states that nonoscillatory + eventual controllability only implies invertibility of X_k for all sufficiently large k; disconjugacy on [0,∞) does not force X_0 to be nonsingular. Thus the sum from j=0 is not well-defined in general, and the dominant solution may not exist near 0. The rest of Lemma 3.2 builds truncations and a Cauchy sequence from this dominant solution, so this gap directly affects the conclusion that columns of the recessive solution lie in dom TF. The problem is likely repairable by starting the sum at a large k0 and working only on the tail, but as written the proof is incomplete. This is compounded by the fact that Theorem 2.5 (the existence and eigenvalue characterization of the recessive solution) is stated without proof and Corollary 2.8 (extension to λ<ν) is dispatched by reference.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the Friedrichs extension of the minimal linear relation associated with a discrete symplectic system with a special linear dependence on the spectral parameter, under the strong Atkinson condition and a disconjugacy/eventual-controllability hypothesis. The main result (Theorem 3.4) characterizes the domain of the Friedrichs extension T_F as the set of pairs in T_max satisfying a Dirichlet condition at 0 and a finite number (d-n) of limit conditions at infinity determined by a recessive solution of the system. The proof is organized into three lemmas: Lemma 3.1 shows x_0=0 for all elements of T_F, Lemma 3.2 establishes square summability and membership in dom T_F for columns of the recessive solution, and Lemma 3.3 shows that the boundary-condition relation U is a self-adjoint extension; Theorem 2.10 is then used to identify U with T_F. The paper generalizes prior results for Jacobi operators, even-order Sturm-Liouville difference equations, and linear Hamiltonian difference systems to a broader discrete symplectic framework.","tokens_in":19611,"tokens_out":16182,"duration_ms":123820,"significance":"If the proof gap discussed below is repaired, the paper is a valuable contribution: it extends the known characterizations of Friedrichs extensions to a general class of discrete symplectic systems with linear spectral parameter dependence, covering limit point and limit circle cases uniformly. The main theorem gives explicit, checkable boundary conditions and the proof is clearly structured with detailed computations. The paper also acknowledges prior work, and the central result is a natural generalization rather than a dramatic departure. The principal weakness is an incomplete construction in Lemma 3.2 that appears repairable; once fixed, the results should be of interest to researchers in spectral theory and difference equations.","major_comments":[{"comment":"The dominant solution is defined via the matrix sequence Λ_k = Σ_{j=0}^{k-1} -~X_j^{-1}(λ) B_j ~X_{j+1}^{*-1}(λ), and the related expressions for X̂_k and Û_k require ~X_j(λ) to be invertible for every j in the range of the sum. However, the assumptions of the lemma (disconjugacy on [0,∞)_Z and eventual controllability) only guarantee invertibility of ~X_j(λ) for all sufficiently large j, as stated in Section 2 before Theorem 2.5; they do not preclude ~X_0(λ) being singular. Consequently, Λ_k and the associated dominant solution are not well-defined on the initial segment of the half-line, and the approximating sequence Z^{[m]} used in Step 1 is not defined for all k. This issue is load-bearing because the subsequent application of the Patching lemma and the convergence argument in Step 3 establish the square summability of the recessive solution, which is essential for Lemma 3.3 and Theorem 3.4. The gap appears repairable by starting the sum at a sufficiently large index k_0, defining the dominant solution on [k_0,∞), and extending it to [0,k_0) via the backward recursion (possible since S_k(λ) is invertible), but as written the proof is incomplete.","section":"§3, Lemma 3.2, Step 1"}],"minor_comments":[{"comment":"Theorem 2.5 is stated without proof and is used as a black box in Corollary 2.8 and Lemma 3.2. Since it is central to the eigenvalue characterization of the recessive solution, please include a proof or give a precise theorem number and a summary of the argument from the cited sources.","section":"§2, Theorem 2.5"},{"comment":"Lemma 3.2 is stated for all λ ≤ ν, but the proof of the Cauchy property in Step 3 relies on the inequality ⟨f,z⟩ ≥ (c−λ)⟨z,z⟩ with c−λ > 0, which the proof of Theorem 2.9 only guarantees for λ < ν. If λ = ν is to be included, a separate argument is needed; the main theorem only uses λ < ν, so this is not a fatal issue.","section":"§3, Lemma 3.2"},{"comment":"The 'Patching lemma' from [48, Lemma 3.1] is invoked without a statement; a brief statement of the lemma or a description of how it applies to the truncated sequences would improve readability.","section":"§3, Lemma 3.2, Step 2"},{"comment":"Equation (1.3) contains a minor typographical issue: 'z0 = 0 = lim ... for all rws P dom Tmax' should probably read 'z0 = 0 and lim ... = 0 for all rws P dom Tmax' to avoid ambiguity.","section":"§1, equation (1.3)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript header indicates that it has already been accepted at Journal of Spectral Theory. If that is the case, I suggest the editors ensure that the gap in Lemma 3.2 is addressed in the final version, since the current proof is incomplete as written. The paper leans heavily on prior results by the same author and collaborators, especially [48] and [10], which carry much of the technical load; this is a self-containedness concern rather than a correctness issue. Including a proof of Theorem 2.5 or a detailed pointer would strengthen the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem is the first characterization of the Friedrichs extension for minimal linear relations generated by discrete symplectic systems with linear spectral dependence and block-diagonal weights. That genuinely goes beyond the banded-matrix result of Došlý–Hasil and the Hamiltonian-system results of Ren–Xu, Zhang–Sun–Yang, and the continuous-system work of Marletta–Zettl and Šimon Hilscher–Zemánek. The boundary conditions are stated cleanly for all deficiency indices d, and the limit-point and limit-circle cases come out as natural corollaries. No hidden fitting, no circular argument: the proof really does derive the extension from recessive solutions.\n\nThe proof has real merits. Lemma 3.2 is a substantial computation, and Lemma 3.3's construction of the matrix Θ satisfying the full-rank condition is neat. The final inclusion argument using self-adjointness of both TF and U is economical and correct. I also credit the author for being upfront that the result relies on his earlier work with Clark; that reliance is substantial, but it is real prior work, not self-citation inflation.\n\nSoft spots, in order of importance. First, the stress-test concern about Lemma 3.2 is legitimate but minor. The sum defining Λ_k starts at j=0, while ~X_j is only guaranteed invertible eventually. The fix is easy: start the sum at a large k0 and extend the dominant solution backward using invertibility of S_k, or normalize the recessive solution so that X_0 is nonsingular. As written, the proof is incomplete at that line, but not badly. Second, Theorem 2.5 is stated without proof and Corollary 2.8 is dispatched by reference. That is acceptable if the citations are precise, but in a journal review I would want the author to either include a proof of Theorem 2.5 or give the exact statement with page numbers. Third, eventual controllability is doing real work: it guarantees invertibility of X_k and the existence of the recessive solution, and it can fail when B_k is singular. The paper does not discuss what happens then. That limits scope, but it does not undermine the theorem under the stated assumptions.\n\nWho gets value from this? Specialists in discrete symplectic systems and spectral theory of linear relations. It is not a paper for a general audience. If it crossed my desk, I would send it to a serious referee and ask for the Lemma 3.2 indexing fix and a small note on controllability. That is a conditional accept, not a rejection.","headline":"A genuinely new extension of the Friedrichs-extension characterization to discrete symplectic systems, with a solid but slightly under-explained proof; worth a careful referee, not a desk reject.","tokens_in":20175,"tokens_out":5705,"would_cite":true,"duration_ms":56428,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47A06","47A20","47B39","39A06","39A12"],"pacs":[],"model":"deepseek-v4-flash","headline":"For discrete symplectic systems, the Friedrichs extension is fixed by a zero initial block and recessive-solution limits at infinity.","keywords":["discrete symplectic system","Friedrichs extension","minimal linear relation","recessive solution","spectral parameter","self-adjoint extension","limit point and limit circle case"],"falsifier":"Compute the relation $U$ of (3.2) for a concrete system satisfying Hypothesis 2.6 with $(S_\\nu)$ disconjugate on $[0,\\infty)_\\mathbb{Z}$ but not eventually controllable—for instance, one with $B_k=0$ on a finite interval so that the $x$-part can vanish there—and compare it with the Friedrichs extension obtained through the abstract limit characterization (2.2). If the two sets differ, or if $U$ is not well defined because the recessive solution's $X$-block is not eventually invertible, the theorem's eventual-controllability hypothesis is essential.","tokens_in":19100,"feed_emoji":"📐","tokens_out":16538,"duration_ms":131057,"temperature":0.7,"pith_summary":"The paper aims to pin down the Friedrichs extension—the distinguished self-adjoint extension of a semibounded minimal linear relation that has the same lower bound—for a broad class of discrete symplectic systems with linear dependence on the spectral parameter. It proves that the extension consists exactly of those maximal-relation pairs whose first $n$ components vanish at the initial point and for which $d-n$ limits at infinity, formed against the columns of a recessive solution, vanish, where $d$ is the number of linearly independent square-summable solutions. This mirrors, in the discrete symplectic setting, the classical role of principal and recessive solutions in singular differential-operator theory, and it unifies earlier treatments of banded difference matrices and Hamiltonian difference systems. The payoff is explicit boundary conditions for the Friedrichs extension without assuming a particular limit-point or limit-circle case.","feed_headline":"Recessive solutions fix the Friedrichs extension's domain","feed_subtitle":"For discrete symplectic systems, one zero at the origin plus limits at infinity picks out the distinguished extension.","key_machinery":"The machinery revolves around the recessive solution of the homogeneous system $(S_\\lambda)$, the discrete counterpart of a principal solution at a singular endpoint. It is a conjoined basis $\\tilde Z(\\lambda) = (\\tilde X(\\lambda); \\tilde U(\\lambda))$ whose $n\\times n$ block $\\tilde X_k(\\lambda)$ is eventually nonsingular and which is dominated by every other normalized conjoined basis: $X_k^{-1}(\\lambda)\\tilde X_k(\\lambda)\\to 0$ as $k\\to\\infty$, equivalently the accumulated sum $\\sum_j (-\\tilde X_j^{-1}(\\lambda)B_j\\tilde X_{j+1}^{*-1}(\\lambda))$ diverges in the sense that its smallest eigenvalue tends to infinity. This object carries the argument because its columns are shown to be square-summable and to belong to the domain of the Friedrichs extension, and because the limits $z_k^* J \\tilde z_k^{[i_j]}(\\lambda)$ are exactly the boundary functionals that the self-adjoint-extension theorem needs. The remaining ingredients—disconjugacy, eventual controllability, and the definiteness condition—guarantee that such a recessive solution exists and that the minimal relation is bounded below.","core_discovery":"On the paper's own terms, the central discovery is Theorem 3.4: under the definiteness hypothesis (Hypothesis 2.6), if $\\nu \\in \\mathbb{R}$ is such that system $(S_\\nu)$ is disconjugate on $[0,\\infty)_\\mathbb{Z}$ and eventually controllable, then for any $\\lambda < \\nu$ the Friedrichs extension $T_F$ of the minimal linear relation $T_{\\min}$ is $$T_F = \\{ \\{z,f\\} \\in T_{\\max} \\mid x_0 = 0 \\text{ and } \\lim_{k\\to\\infty} z_k^* J \\tilde $z_k^{{[i_j]}}$(\\$\\lambda$) = 0 \\text{ for } j=1,\\ldots,d-n \\},$$ where $x_0$ is the first $n$-vector block of $z$ at $k=0$, $d$ is the number of linearly independent square-summable solutions, and $\\tilde z^{[i_j]}(\\lambda)$ are $d-n$ selected columns of a recessive solution of $(S_\\lambda)$. The proof shows these columns are square summable and lie in the domain of $T_F$, then uses the general self-adjoint-extension description to show that the relation $U$ defined by these conditions is self-adjoint and contains $T_F$. In the limit point case $d=n$ this reduces to $T_F = \\{ \\{z,f\\} \\in T_{\\max} \\mid x_0 = 0 \\}$, and in the limit circle case $d=2n$ exactly $n$ conditions at infinity are needed.","pith_inferences":["A testable weakening is to replace eventual controllability by the explicit requirement that $X_k(\\lambda)$ is eventually invertible for the specific recessive solution; if that alone suffices, the controllability hypothesis is stronger than needed.","The selection of the indices $i_1,\\ldots,i_{d-n}$ in (3.3) is noncanonical; a stable numerical recipe for choosing them, and a check that the resulting limits do not depend on that choice, would make the characterization directly computable.","Remark 3.5(ii) flags the square summability of recessive columns as a topic for further study; if it holds for all nonoscillatory eventually controllable systems, the same boundary-formula method could extend beyond the block-weight setting treated here.","Finite-truncation experiments with Dirichlet conditions should approximate the same recessive-solution limits as the interval grows, offering a concrete numerical check of the identity $T_F=U$."],"forward_implications":["Explicit boundary conditions: once a recessive solution of $(S_\\lambda)$ is known for any $\\lambda<\\nu$, the Friedrichs extension is determined by $x_0=0$ and $d-n$ limits at infinity.","In the limit point case $d=n$, the condition at infinity disappears and $T_F$ is simply $\\{\\{z,f\\}\\in T_{\\max}: x_0=0\\}$.","In the limit circle case $d=2n$, exactly $n$ conditions at infinity involving all columns of the recessive solution are needed, matching the $n$ initial conditions.","The same boundary description holds for every $\\lambda<\\nu$, so the recessive solutions at different spectral parameters define the same extension.","Because $T_{\\min}$ is bounded below whenever $(S_\\nu)$ is disconjugate and eventually controllable, the theorem also gives a uniform semiboundedness and self-adjoint-extension picture for a continuum of spectral parameters."],"supporting_citations":[{"why":"Provides the abstract characterization of the Friedrichs extension of a semibounded linear relation and the identity relating $T_F$ to $T_{\\min}-\\lambda I$.","marker":"[6]"},{"why":"Establishes that $T_{\\min}^*=T_{\\max}$ and links the strong Atkinson condition to uniqueness of representatives.","marker":"[10]"},{"why":"Origin of the principal and recessive solution concept whose time-reversed analogue gives the recessive solution theorem.","marker":"[12]"},{"why":"Supplies the framework of discrete symplectic systems, disconjugacy, and the recessive-solution existence result used as Theorem 2.5.","marker":"[13]"},{"why":"Gives the banded-matrix difference-operator result that this paper generalizes.","marker":"[14]"},{"why":"Connects disconjugacy to positivity of the associated quadratic functional, used in Corollary 2.8 and Theorem 2.9.","marker":"[34]"},{"why":"Provides the characterization of all self-adjoint extensions by boundary conditions used to construct $U$ in Lemma 3.3.","marker":"[48]"},{"why":"Supplies the Wronskian-type identity used to prove that the boundary matrix $\\Upsilon$ has full rank.","marker":"[49]"}],"fun_headline_variants":["Recessive solutions determine Friedrichs extension domain","Boundary conditions at 0 and infinity pick Friedrichs extension","Discrete symplectic: recessive solutions define Friedrichs extension","One zero at zero, limits at infinity: the Friedrichs extension"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that, far enough along the half-line, the system is eventually controllable: no nontrivial solution can have its first $n$ components vanish over an entire finite interval, which is what makes the leading $n\\times n$ block of every conjoined basis eventually invertible and secures a recessive solution; disconjugacy alone does not imply this, and it can fail when the matrices $B_k$ are singular.","fun_headline_variants_meta":{"raw":{"variants":["Recessive solutions determine Friedrichs extension domain","Boundary conditions at 0 and infinity pick Friedrichs extension","Discrete symplectic: recessive solutions define Friedrichs extension","One zero at zero, limits at infinity: the Friedrichs extension"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000258,"raw_usage":{"total_tokens":1589,"prompt_tokens":962,"completion_tokens":627,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":578,"completion_tokens_details":{"reasoning_tokens":559}},"tokens_in":578,"tokens_out":627,"duration_ms":7065,"temperature":1.0,"reasoning_tokens":559,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:47:55.254474+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the relation $U$ of (3.2) for a concrete system satisfying Hypothesis 2.6 with $(S_\\nu)$ disconjugate on $[0,\\infty)_\\mathbb{Z}$ but not eventually controllable—for instance, one with $B_k=0$ on a finite interval so that the $x$-part can vanish there—and compare it with the Friedrichs extension obtained through the abstract limit characterization (2.2). If the two sets differ, or if $U$ is not well defined because the recessive solution's $X$-block is not eventually invertible, the theorem's eventual-controllability hypothesis is essential.","supporting_citations":[{"cited_title":"Proceedings of the Sixth Colloquium on the Qualitative T heory of Differential Equations (Szeged, Hungary, 1999)","cited_arxiv_id":null,"evidence_quote":"Origin of the principal and recessive solution concept whose time-reversed analogue gives the recessive solution theorem."},{"cited_title":"Došlý and P","cited_arxiv_id":null,"evidence_quote":"Gives the banded-matrix difference-operator result that this paper generalizes."},{"cited_title":"Proceedings of The 9’th Colloquium on the Qualitative T heory of Differential Equations","cited_arxiv_id":null,"evidence_quote":"Connects disconjugacy to positivity of the associated quadratic functional, used in Corollary 2.8 and Theorem 2.9."},{"cited_title":"Zemánek and S","cited_arxiv_id":null,"evidence_quote":"Supplies the Wronskian-type identity used to prove that the boundary matrix $\\Upsilon$ has full rank."}],"review_version":1}