{"id":"27a3ac67-fa05-40a8-a7f0-d8ed573a7333","arxiv_id":"2509.00964","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A doubly-dispersive continuous MIMO channel model with matched-filter-like beamforming is derived, but the variational optimality proof is invalid because the unconstrained problem is unbounded.","lead":"The paper derives a doubly-dispersive channel model for continuous-aperture MIMO arrays and proposes calculus-of-variations beamforming designs that mirror matched filtering. A generalist might read it because it targets high-mobility 6G scenarios, but the optimality proof contains a load-bearing gap.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of optimality fails at Lemma 2: unconstrained P3 is unbounded under scaling, so no maximizer exists and the transition (71)→(72) is unjustified.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing concern. I agree that Lemma 2 is false because P3 is unbounded under scaling, and that the transition from the first-order condition (71) to the eigenvalue equation (72) is unjustified. The channel model in Sec. II and the waveform input-output relations in Sec. III appear to be standard extensions of prior work and are not the weak point; the issue is confined to the beamforming optimality proof. The CoV derivation is not a minor gap: Lemma 2's equivalence is false as stated, and Theorems 2 and 4 inherit the flaw. A Lagrange-multiplier derivation might salvage a related result, but that is not what the paper proves. The reader's REJECT verdict is therefore appropriate, though the rejection should be understood as 'optimality not established' rather than 'numerical results are necessarily wrong.' No change to the reader's verdict is needed.","tokens_in":21260,"tokens_out":7310,"duration_ms":95360,"concrete_test":"Analytic check: in the setup of Sec. IV-B take M=1, H(r,s)=1, J_R(r)=1. Then P3 in (61) reduces to max_{J_T} |∫_{S_T} J_T(s) ds|^2. For any nonzero J_T with nonzero integral, scaling J_T → α J_T makes the objective α^2 times as large, so sup P3 = ∞ and no optimizer exists; this directly falsifies Lemma 2. Then, to test whether the proposed solution is even a constrained stationary point, re-derive the first-order condition for P2 with the power constraint via a Lagrange multiplier: (∫_{S_R} H^H(r,s)J_R(r) dr) A_t[J_T] = λ_t J_T(s). Substitute Theorem 2's J_T^* with the Algorithm's choice à = O, λ_t = max SVD(O) for a two-path example; if the identity is not satisfied, the closed-form optimality claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of closed-form optimal beamforming collapses at Lemma 2 (Sec. IV-B, eqs. (61)-(62)). Lemma 2 assumes an optimal solution to the unconstrained functional P3 exists. It does not. The objective is ||A_t[J_T]||_F^2 with A_t[J_T] = ∫∫ J_R^H(r) H(r,s) J_T(s) ds dr. For any nonzero J_T with A_t[J_T] ≠ 0, scaling J_T → α J_T multiplies the objective by α^2, so the supremum over the unconstrained space is +∞. The only finite critical point is J_T = 0, which gives objective 0. Hence no nonzero maximizer exists to be rescaled in (62). The first-order condition (71), obtained by applying Theorem 1 to this nonexistent optimizer, is therefore vacuous. The subsequent statement that the condition 'becomes an eigenvalue problem' in (72) introduces λ_t with no derivation; the correct constrained stationarity condition would require a Lagrange multiplier for the power constraint. Because this false premise supports both Theorem 2 and, symmetrically, Theorem 4, the claimed optimality proof is invalid. The final matched-filter-like formulas may remain useful as heuristics or as solutions to a differently posed eigenproblem, but the paper's central claim of proven closed-form optimality is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a continuous-aperture MIMO (CAPA) channel model for doubly dispersive (DD) channels, derives input-output relations for OFDM, OTFS, and AFDM waveforms, and then formulates transmit/receive beamforming problems that maximize received power. The beamforming section claims to obtain low-complexity, closed-form optimal solutions via calculus of variations, with expressions closely related to classical matched filters (Theorems 2 and 4), and it proposes an iterative algorithm (Algorithm 1) with claimed convergence guarantees. The channel-modeling part (Sections II-III) is a plausible extension of existing DD models to continuous apertures, but the beamforming optimality proofs are the advertised central contribution.","tokens_in":21565,"tokens_out":5887,"duration_ms":70518,"significance":"If the beamforming result were correct, it would be a significant contribution: provably optimal, low-complexity matched-filter-like beamformers for CAPA systems over DD channels would be an important step for holographic MIMO in high-mobility scenarios. The channel-modeling portion (Section II) and the waveform I/O relations (Section III) also have standalone value, as they extend prior discrete-array DD models to continuous apertures under clear far-field and narrowband assumptions. However, the paper's main claim of proven closed-form optimality is not established: the unconstrained problems are unbounded, the eigenvalue step is unjustified, and the resulting expressions are circular fixed-point equations rather than closed forms. The modeling contribution is useful, but the central beamforming claim as stated is unsupported.","major_comments":[{"comment":"Lemma 2 assumes the existence of an optimal solution to the unconstrained functional P3. This assumption is false. The objective is ||A_t[J_T]||_F^2, and for any J_T with A_t[J_T] != 0, scaling J_T -> alpha J_T multiplies the objective by alpha^2. The supremum over the unconstrained space is +infinity, so no nonzero maximizer exists; J_T = 0 is the only finite critical point and gives objective 0. Consequently, the rescaling argument in Eq. (62) is vacuous, and Lemma 2 cannot establish equivalence between P2 and P3. Lemma 4 for the RX side (P5) has exactly the same flaw.","section":"Section IV-B, Lemma 2 (Eqs. (61)-(62))"},{"comment":"The first-order condition (71) is derived by applying Theorem 1 to a nonexistent maximizer and is vacuous. Moreover, the sentence 'The condition then becomes an eigenvalue problem' introduces lambda_t with no derivation. A zero product of the form (integral) * A_t = 0 does not imply an eigenvalue equation (integral) * A_t = lambda_t J_T(s). The correct stationarity condition for the constrained problem P2 would require a Lagrange multiplier for the power constraint. Since Eq. (72) is the basis of Theorem 2, Theorem 2 is unsupported; Theorem 4 relies on the analogous unjustified step (89).","section":"Section IV-B, Eqs. (71)-(72)"},{"comment":"The claimed closed-form solutions are not closed forms. In Eq. (73), the matrix \\tilde{A} is defined (Eq. (72)) as the double integral involving the unknown J_T(s'), so the right-hand side depends on the solution it is supposed to define. Similarly, \\tilde{B} in Eq. (90) depends on the unknown J_R(r'). The quantities \\lambda_t and \\lambda_r are also left unspecified. At best, Eqs. (73) and (90) are fixed-point equations; they cannot be evaluated directly from problem data. This directly contradicts the abstract's claim of 'novel low-complexity, closed-form solutions.'","section":"Section IV-B/C, Theorems 2 and 4 (Eqs. (73) and (90))"},{"comment":"The convergence proof in Eq. (99) states that O(J_T^{(i+1)}, J_R^{(i+1)}) >= O(J_T^{(i)}, J_R^{(i)}) 'follows from the global optimality described in Lemma 2 and Lemma 4.' Since Lemmas 2 and 4 are invalid, no monotonicity is established. Steps 2-3 of Algorithm 1 set \\tilde{A} = \\tilde{B} = the current objective matrix and \\lambda_t = \\lambda_r = max SVD of that matrix; these assignments are not derived from any optimization principle. The numerical convergence in Fig. 6 does not substitute for a proof, especially because the algorithm may converge to a fixed point of an unjustified update.","section":"Section IV-D, Eq. (99) and Algorithm 1"}],"minor_comments":[{"comment":"The y-axis label 'dB7W' appears to be a typo (likely 'dBW' or 'dBm'). Please correct.","section":"Figures 2-6"},{"comment":"The statements that J_T^*(s) and J_R^*(r) are '4-D tensors' and then defining per-element beamformers J_T^*(s_x,s_y), J_R^*(r_x,r_y) are confusing; the beamformers are matrix-valued functions on 2-D surfaces. Also, the coordinate notation (s_x,s_y) should be (s_x,s_z).","section":"Eqs. (96)-(97)"},{"comment":"The displayed equation has a formatting issue with '≜ ˇH_ℓ' inside the sum; please clarify the definition.","section":"Eq. (30)"},{"comment":"The notation max(SVD(O)) is ambiguous. If the largest singular value is intended, please state it explicitly; if the whole matrix is meant, clarify.","section":"Algorithm 1, Step 3"},{"comment":"The proof says the scaled solution achieves a 'higher maximum objective' but should say a higher objective value for the fixed functional; as written the wording is imprecise.","section":"Lemma 1"}],"recommendation":"reject","confidential_remarks":"The channel-modeling and waveform sections may have standalone value, and the references are appropriate. However, the advertised optimal beamforming contribution is central and is not merely under-explained: the optimization problems are unbounded as posed, the eigenvalue step is invalid, and the 'closed-form' results are circular. These issues cannot be repaired by local edits without changing the paper's main claims, so rejection is appropriate. If the authors later resubmit a version that either proves optimality for a correctly constrained formulation or explicitly presents the iterative procedure as a heuristic fixed-point algorithm, the modeling portion could form the basis of a solid paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper has one genuinely useful contribution and one load-bearing proof that doesn't hold. The DDC MIMO channel model in (21), extending discrete doubly-dispersive models to continuous apertures with polarization, is a legitimate and well-derived extension under the standard far-field and narrowband assumptions. The I/O relations for OFDM, OTFS, and AFDM follow cleanly from the prior unified framework in [10], so that section is mostly a translation exercise, but it’s competently done and may be practically useful.\n\nThe problem is Section IV-B. Lemma 2 asserts an optimal solution to the unconstrained functional P3 in (61). As stated, P3 is unbounded: scaling J_T by α scales the objective by α^2, so no nonzero maximizer exists. The only critical point is the zero function, which gives objective zero. Consequently, the necessary condition in (71) is vacuous, and the jump to the eigenvalue equation (72)—introducing λ_t out of nowhere—is unjustified. The same flaw breaks Theorem 4 for the receiver. The final matched-filter-like expressions might be right for a single stream, or as heuristics, but the claim of proven closed-form optimality is not supported.\n\nThere’s a second, lesser issue: the 'closed-form' solutions contain the unknown beamformers through à and B̃, so they are really fixed-point equations, not closed forms. The iterative Algorithm 1 is fine as a heuristic; the convergence proof, however, relies on the false lemmas.\n\nThe simulations are thin on statistical detail and no code is provided. That is a minor point relative to the proof issue.\n\nOverall: the channel model and waveform relations are salvageable, but the central beamforming optimality result is not established. The paper is not a waste of time—the model part could be a building block for future work—but the main claim should be fixed or withdrawn.\n\nI would send it to peer review, because the channel model is new and the proof issue is fixable in principle (e.g., by properly incorporating the power constraint via Lagrange multipliers). A good referee can separate the wheat from the chaff. I wouldn't cite it until the beamforming part is repaired.\n\nBest,\n[You]","headline":"Solid DDC channel model, but the CoV optimality proof collapses: P3 is unbounded and (71)→(72) invents λ_t without justification.","tokens_in":22081,"tokens_out":2937,"would_cite":false,"duration_ms":36108,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives a doubly dispersive continuous MIMO channel model for continuous aperture arrays and claims closed-form transmit and receive beamformers, obtained by calculus of variations, that maximize received power and resemble class","keywords":["continuous aperture arrays","doubly dispersive channels","MIMO","beamforming","calculus of variations","matched filter","OFDM","OTFS"],"falsifier":"Evaluate the paper's own stationarity condition at its claimed optimal transmit beamformer J̄_T(s), using η = J̄_T as the perturbation direction: the directional derivative equals 2‖∫∫ J_R^H H J̄_T dsdr‖²_F, which is positive for any nonzero received-power level. A direct numerical evaluation in the paper's simulation setup would show this derivative is not zero, contradicting Theorem 1's necessary condition and showing the eigenvalue step does not follow from the calculus of variations.","tokens_in":21148,"feed_emoji":"📡","tokens_out":8753,"duration_ms":102818,"temperature":0.7,"pith_summary":"This paper is trying to establish that a MIMO system whose transmit and receive arrays are continuous surfaces—not discrete antenna elements—can be modeled and optimized directly in the continuous domain over doubly dispersive (high-mobility multipath) channels. It derives a continuous channel response from scattering-matrix and dyadic-Green's-function principles, and shows that OFDM, OTFS, and AFDM all share the same effective channel structure once the continuous spatial beamformers are factored out. On that foundation, it claims closed-form transmit and receive beamformers, obtained by calculus of variations, that maximize received power and are structurally the same as classical matched filters. If correct, this would mean optimal CAPA beamforming can be computed with simple quadrature integrals instead of large basis-function expansions or huge discrete antenna arrays, and the same beamformers work across the three waveforms.","feed_headline":"Closed-form beamformers found for continuous-aperture MIMO","feed_subtitle":"The same closed-form transmit/receive design fits OFDM, OTFS, and AFDM.","key_machinery":"The load-bearing machinery is the calculus-of-variations functional derivative on L²(S, C^{3×M}). The paper expands the squared Frobenius norm of the integral operator A[J] = ∫∫ J_R^H H J_T ds dr, sets the first variation to zero, invokes the fundamental lemma of the calculus of variations to obtain an integral stationarity condition, and then converts that condition into an eigenvalue equation because the double integral à is constant in s. The resulting expressions are matched-filter-like integral equations: the optimal TX beamformer is, up to normalization, the channel Hermitian-transposed and integrated against the RX beamformer; the optimal RX beamformer is the same operation with roles","core_discovery":"The paper's central object is a continuous-space channel response between a transmit point s and receive point r, modeled as H(r,s;τ,t) = Σ_{ℓ=1}^L h_ℓ Ξ_ℓ δ(τ−τ_ℓ) e^{−j2πν_ℓ t} e^{j(2π/λ_c)k_{R,ℓ}^T r} e^{j(2π/λ_c)k_{T,ℓ}^T s}. Built from a scattering-matrix response and dyadic Green's functions under far-field and wideband approximations, this response is then used to derive effective channels for OFDM, OTFS, and AFDM, all of the common form H̄ = Σ_ℓ (Ȟ_ℓ ⊗ Ḡ_ℓ), where Ȟ_ℓ is the continuous spatial beamforming matrix and Ḡ_ℓ is the waveform-dependent delay-Doppler matrix. The paper formulates received-power maximization as a functional optimization over beamformer functions J_T(s) and J","pith_inferences":["If the matched-filter form survives a properly constrained derivation, similar closed forms should hold for any waveform whose effective channel factors as Ȟ_ℓ ⊗ Ḡ_ℓ; the proof only uses that factorization, so the design may extend to other DD-robust modulations.","The unboundedness of the unconstrained problem suggests the eigenvalue λ_t is not determined by the stationarity condition alone; a constrained derivation, such as a Rayleigh quotient with the power constraint, would be needed to pin it down, and the paper's closed forms may still emerge as stationary points of that quotient.","A testable consequence of the waveform-independence claim is that in a CAPA link running the same physical beamformers, switching from OFDM to OTFS or AFDM should leave the achieved receive power nearly unchanged, apart from channel-estimation and overhead effects."],"forward_implications":["The beamformer design is independent of the choice among OFDM, OTFS, and AFDM, because the objective depends only on the spatial matrices Ȟ_ℓ; the waveform matrices Ḡ_ℓ factor out.","CAPA systems can approximate very large discrete MIMO arrays with a fraction of the complexity: the paper's simulations show the continuous solution matching a 1089×1089 conventional MIMO array.","Increasing aperture size gives continuous gains in received power, whereas conventional discretized arrays gain only when the aperture grows enough to add another antenna.","The alternating algorithm converges in a few iterations, with per-iteration cost set by the quadrature grid, not the number of antennas."],"supporting_citations":[{"why":"Supplies the scattering-matrix response and dyadic-Green's-function model on which the continuous channel response is built.","marker":"[28]"},{"why":"Supplies the unified delay-Doppler waveform framework for OFDM, OTFS, and AFDM that the paper extends to continuous apertures.","marker":"[10]"},{"why":"Defines OTFS modulation and the inverse discrete Zak transform used for the OTFS input-output relation.","marker":"[11]"},{"why":"Defines AFDM and its chirp parameters, which the paper uses for the AFDM input-output relation.","marker":"[13]"},{"why":"Provides the Fourier-based CAPA beamforming benchmark that the paper compares against and seeks to simplify.","marker":"[24]"},{"why":"Provides the CAPA MIMO beamforming formulation, including the coordinate-system extension the paper invokes.","marker":"[25]"},{"why":"Supplies the calculus of variations results (functional derivative and fundamental lemma) used to derive the matched-filter solutions.","marker":"[44]"}],"fun_headline_variants":["Closed-form beamformers for continuous-aperture MIMO in doubly-dispersive channels","Optimal beamforming for CAPA MIMO: closed-form matches classic matched filter","One closed-form beamformer fits OFDM, OTFS, and AFDM over dispersive channels","Continuous-aperture MIMO: closed-form TX/RX design for DD channels"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The entire optimality argument assumes the unconstrained transmit-beamforming problem has a finite maximizer, but that problem is unbounded: scaling the transmit beamformer multiplies the received-power objective by the square of the scale factor, so the stationarity condition used to derive the matched-filter solution cannot identify a maximizer.","fun_headline_variants_meta":{"raw":{"variants":["Closed-form beamformers for continuous-aperture MIMO in doubly-dispersive channels","Optimal beamforming for CAPA MIMO: closed-form matches classic matched filter","One closed-form beamformer fits OFDM, OTFS, and AFDM over dispersive channels","Continuous-aperture MIMO: closed-form TX/RX design for DD channels"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000238,"raw_usage":{"total_tokens":1378,"prompt_tokens":802,"completion_tokens":576,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":546,"completion_tokens_details":{"reasoning_tokens":484}},"tokens_in":546,"tokens_out":576,"duration_ms":6585,"temperature":1.0,"reasoning_tokens":484,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T13:02:06.607304+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the paper's own stationarity condition at its claimed optimal transmit beamformer J̄_T(s), using η = J̄_T as the perturbation direction: the directional derivative equals 2‖∫∫ J_R^H H J̄_T dsdr‖²_F, which is positive for any nonzero received-power level. A direct numerical evaluation in the paper's simulation setup would show this derivative is not zero, contradicting Theorem 1's necessary condition and showing the eigenvalue step does not follow from the calculus of variations.","supporting_citations":[{"cited_title":"Degrees of freedom in multiple-antenna channels: a signal space approach,","cited_arxiv_id":null,"evidence_quote":"Supplies the scattering-matrix response and dyadic-Green's-function model on which the continuous channel response is built."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the unified delay-Doppler waveform framework for OFDM, OTFS, and AFDM that the paper extends to continuous apertures."},{"cited_title":"Orthogonal Time Frequency Space Modulation,","cited_arxiv_id":null,"evidence_quote":"Defines OTFS modulation and the inverse discrete Zak transform used for the OTFS input-output relation."},{"cited_title":"Affine Frequency Division Multiplexing for Next Generation Wireless Communications,","cited_arxiv_id":null,"evidence_quote":"Defines AFDM and its chirp parameters, which the paper uses for the AFDM input-output relation."},{"cited_title":"Beamforming Optimization for Continuous Aperture Array (CAPA)-Based Communications,","cited_arxiv_id":null,"evidence_quote":"Provides the Fourier-based CAPA beamforming benchmark that the paper compares against and seeks to simplify."},{"cited_title":"Beamforming Design for Continuous Aperture Array (CAPA)-Based MIMO Systems","cited_arxiv_id":"2504.00181","evidence_quote":"Provides the CAPA MIMO beamforming formulation, including the coordinate-system extension the paper invokes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the calculus of variations results (functional derivative and fundamental lemma) used to derive the matched-filter solutions."}],"review_version":1}