{"id":"b13d89e1-1f4b-426e-8f3a-52a213fd5e36","arxiv_id":"2501.04924","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A CAPA downlink secrecy-rate optimization is solved by an FP-BCD algorithm with closed-form continuous current patterns, plus a zero-leakage ZF heuristic, though the key equivalence step is flawed.","lead":"This paper designs secure beamforming for continuous aperture array (CAPA) base stations serving multiple users and eavesdroppers, maximizing a weighted secrecy sum-rate. It proposes a fractional-programming block-coordinate-descent method and a zero-forcing heuristic, claiming large gains over discrete MIMO and Fourier-based baselines.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2's equivalence is false: the term log2(1+GΓ) dropped in Eq. (22) is multiplied by b_k in (15a), so it is not independent of the optimization variables; Eq. (22b) is also algebraically wrong. The FP-BCD algorithm therefore does not solve the WSSR problem.","rationale":"I read the manuscript in good faith and focused on the central claim: the FP-BCD method obtains the exact optimal solution to the WSSR problem. The reader's critique targets Lemma 2, and my independent check confirms it is load-bearing. The term log2(1+GΓ) that Lemma 2 discards is in fact multiplied by b_k in the objective, so it is not independent of the optimization variables. This invalidates the equivalence between (15) and (20), and with it the b-update, the β-update, and the claimed optimality of the closed-form current. I additionally noticed a distinct algebraic error from Eq. (22a) to (22b), where 1+GΓ is replaced by 1+GΓ−Γ_k in the numerator; this is not an identity unless Γ_k=0. The two errors compound: even if the intended constant were handled correctly, the reformulated objective would not be equivalent to (15). My scalar counterexample shows the two problems have different maximizers, settling the concern. The ZF method and the complexity comparison appear less affected, so a revision that fixes the equivalence step might produce a valid heuristic algorithm, but the paper as written does not support its headline claim of an exact optimal solution. Hence I agree with the reader's REJECT verdict and recommend no change.","tokens_in":19603,"tokens_out":7617,"duration_ms":64584,"concrete_test":"Evaluate the scalar instance K=Q=1 with H(s)=h, Hbar(s)=g, power P=1, weight α_1=1, and |h|^2=2, |g|^2=1. Let x=|J|^2 ∈ [0,1] and GΓ = P|g|^2 = 1. The true problem (15) has objective F(b,x)=b[log2(1+2x)−log2(1+x)], maximized at x=1, b=1 with value log2(3/2)≈0.585. The reformulated objective (20) in the paper is G(b,x)=b[log2(1+2x)+log2((2−x)/(1+x))], which at x=0.5, b=1 equals 1.0, exceeding the true optimum. Since the two problems have different maximizers, Lemma 2's equivalence is refuted. Alternatively, run Algorithm 1 on this scalar channel and compare its output to the exhaustive optimum of (13); any discrepancy confirms the failure.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that Algorithm 1 exactly solves the WSSR problem (13), with the closed-form current pattern in Theorem 2 being the optimal solution. This claim depends critically on Lemma 2 (Section III-A, Eqs. 20–22), which asserts that problem (15) is equivalent to problem (20). The proof rewrites log2(1/(1+Γ_k)) = log2((1+GΓ)/(1+Γ_k)) − log2(1+GΓ) and then discards the term log2(1+GΓ), claiming it is independent of {J_k(s)} and {b_k}. This is incorrect: the objective (15a) is Σ_k b_k [log2(1+γ_k) + log2(1/(1+Γ_k))], so after the rewrite it becomes Σ_k b_k [log2(1+γ_k) + log2((1+GΓ)/(1+Γ_k))] − log2(1+GΓ)·Σ_k b_k. Since {b_k} are optimization variables with 0 ≤ b_k ≤ α_k, the term −log2(1+GΓ)·Σ_k b_k is not constant and depends on the current iterate. Therefore maximizing (20) is not equivalent to maximizing (15), and the b-update rule (30) does not correctly maximize (20). The subsequent β-update (34) and the current expression (51) inherit this flaw, so the claimed optimality of the BCD algorithm does not follow. Separately, the algebraic equality in (22b), replacing 1+GΓ by 1+GΓ−Γ_k in the numerator of the log, is false unless Γ_k = 0. The ZF-based scheme and the complexity analysis are largely independent of Lemma 2 and may remain valid, but the headline contribution — an exact, approximation-free optimal solution — is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper considers a downlink secure transmission system in which a continuous aperture array (CAPA) serves multiple legitimate users in the presence of multiple eavesdroppers. The authors formulate the weighted secrecy sum-rate (WSSR) maximization as a functional programming problem over the source current density, subject to a transmit power constraint. They propose an FP-based block coordinate descent (BCD) algorithm that they claim obtains the globally optimal current pattern in closed form via a continuous-function inversion theory, and they also propose a zero-forcing (ZF) heuristic with water-filling power allocation. Numerical results compare the proposed schemes with discrete MIMO baselines and a Fourier-based discretization benchmark.","tokens_in":19914,"tokens_out":10964,"duration_ms":93984,"significance":"Should the main optimality claim hold, the paper would establish that a nonconvex functional WSSR problem is exactly solvable with low computational complexity, and it would quantify substantial secrecy gains of CAPA over discrete MIMO. The continuous-function inversion theorem and the observation that the optimal current is a linear combination of channel spatial responses are elegant and potentially useful. However, the central equivalence on which the FP-BCD algorithm rests is flawed, so the main contribution is not established. The ZF heuristic is a reasonable baseline, but its conceptual novelty is limited.","major_comments":[{"comment":"The equivalence claimed in Lemma 2 is false. Starting from Eq. (15a), one has sum_k b_k [log2(1+gamma_k) - log2(1+Gamma_k)] = sum_k b_k [log2(1+gamma_k) + log2((1+G_Gamma)/(1+Gamma_k))] - log2(1+G_Gamma) sum_k b_k. The term -log2(1+G_Gamma) sum_k b_k depends on the optimization variables b_k, so discarding it does not preserve equivalence. In addition, Eq. (22b) is algebraically incorrect: (1+G_Gamma)/(1+Gamma_k) equals 1 + (G_Gamma - Gamma_k)/(1+Gamma_k), not (1+G_Gamma - Gamma_k)/(1+Gamma_k); the two quantities differ unless Gamma_k = 0. Thus problem (20) is not equivalent to problem (15), and the subsequent FP transformations and the optimality claim for Algorithm 1 are unsupported.","section":"Section III-A, Lemma 2 (Eqs. (15a), (19)-(22))"},{"comment":"The b_k update rule in Eq. (30) is the optimal solution for problem (15) for fixed J_k(s), but in the transformed problem (20a) the coefficient of b_k is different, namely log2(1+gamma_k) + log2((1+G_Gamma - Gamma_k)/(1+Gamma_k)) as written, or log2((1+G_Gamma)/(1+Gamma_k)) after the algebraic correction. The threshold condition gamma_k >= Gamma_k therefore does not maximize the transformed objective. Moreover, the convergence proof in Eq. (61) omits the b_k block and states that b_k is a non-negative constant, which contradicts its role as an optimization variable in Algorithm 1. Consequently, the monotonicity chain does not establish convergence of Algorithm 1 to a stationary point of the WSSR problem (13).","section":"Section III-B and III-C, Eq. (30) and Eq. (61)"},{"comment":"The proof of Theorem 1 invokes \"Lemma 3 in [21]\" to conclude that V_k(s) = 0 from the condition that the integral of U_k^*(s) V_k(s) over S_T vanishes for all smooth U_k vanishing on the boundary. Since [21] is an unpublished preprint and the lemma is not stated in the manuscript, the derivation of the optimal current structure in Eq. (41) is not self-contained. The authors should state and prove the needed variational lemma or provide a complete derivation.","section":"Appendix A, proof of Theorem 1"}],"minor_comments":[{"comment":"The logarithm base switches between log2, used in the problem statement, and log, apparently the natural logarithm, in the FP transformations; this should be made consistent.","section":"Lemma 3 and Eqs. (23a)-(26)"},{"comment":"The sentence \"We omit b_k since it is a non-negative constant\" is inaccurate because b_k is an optimization variable updated in Algorithm 1 and must be treated as a block in the BCD analysis.","section":"Section III-C"},{"comment":"Theorem 3 requires the N x N channel correlation matrix H to be invertible, but the paper does not discuss rank deficiency when K + Q exceeds the available spatial degrees of freedom; a regularity condition should be stated.","section":"Theorem 3"},{"comment":"The definition of G_Gamma as P times the sum over q of the integral of |bar_H_q(s)|^2 ds lacks parentheses; the intended grouping should be clarified.","section":"Eq. (21b)"}],"recommendation":"reject","confidential_remarks":"The manuscript relies extensively on preprints from the same research group ([20], [21], [27]) for the channel model, the variational lemma, the baseline, and a closely related single-user secrecy result. The only numerical baseline is [20], and the main proof imports a lemma from [21] without stating it. Even setting aside the technical error in Lemma 2, the novelty and independence of the contributions would need careful vetting. The paper may still be of interest to the CAPA community as a heuristic ZF scheme, but the central claim of an exact optimal solution is not supported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline: the paper's main claim—that the FP-based BCD algorithm exactly solves the multi-user CAPA secure beamforming problem—is unsupported. Lemma 2 is genuinely wrong: the term log2(1+GΓ) that gets discarded in (22) is multiplied by b_k in the objective, so it is not a constant with respect to the optimization variables. The b-update (30) and the claimed optimality rest on this false equivalence. Separately, the algebraic identity in (22b) is also incorrect unless Γ_k=0.\n\nThat said, the paper is not a waste of time. The problem formulation itself—WSSR for multiple LUTs and Eves under a continuous aperture—is new relative to the single-user single-eve [27], and it is a natural extension. The continuous-function inversion theorem (Theorem 2) is a useful tool, and the resulting closed-form current pattern as a linear combination of channel responses is a real simplification. The ZF-based scheme in Section IV, with the water-filling power allocation, appears to stand independent of Lemma 2; the zero-forcing construction (Theorem 3) is simple and correct as far as I can tell. The complexity analysis, comparing against the Fourier discretization, is reasonable.\n\nThe soft spot is the load-bearing one. Because the equivalence in Lemma 2 fails, Algorithm 1 does not solve problem (13). The simulation claims about 'optimal' performance are therefore not grounded, though the qualitative WSSR gains over discrete MIMO might still hold if the algorithm behaves reasonably. The paper gives no code, so I cannot check the numerics independently. Also, the heavy reliance on the same group's prior work ([20], [21], [27]) is not a flaw by itself, but it means the novelty hinges on how much of the derivation is genuinely different; the CoV lemma from [21] is explicitly imported.\n\nThe ZF part and the complexity analysis could survive a careful revision. If the authors can fix Lemma 2—or else reframe the algorithm as a heuristic—the paper would be worth a second look. As is, the central contribution is not established.\n\nRecommendation: I would not accept this in its current form, but I would send it to peer review rather than desk-reject, because the problem is important and a fix is plausible. The ZF section alone might be publishable as a shorter paper.\n\nFor your reading group: maybe. It is a good case study in how a small algebraic slip can undermine a whole optimization claim.","headline":"The FP-BCD optimality claim is undone by a false equivalence in Lemma 2, but the problem formulation and the ZF scheme have real merit.","tokens_in":20545,"tokens_out":2303,"would_cite":false,"duration_ms":21333,"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":"Secure CAPA beamforming collapses to a single matrix inversion, and the paper derives the closed-form current pattern behind it.","keywords":["continuous aperture array","secure beamforming","weighted secrecy sum-rate","fractional programming","block coordinate descent","zero-forcing beamforming","continuous function inversion","physical layer security"],"falsifier":"A direct numerical check settles whether the reformulation is equivalent: take $K=1$, pick any current $J_1(s)$ with $\\gamma_1 < \\Gamma_1$, and choose parameters so that $G_\\Gamma > 2\\Gamma_1$. Problem (15) then has optimum $b_1=0$ because the secrecy rate is negative, while problem (20) is positive at $b_1=\\alpha_1$ since $\\log_2((1+G_\\Gamma-\\Gamma_1)/(1+\\Gamma_1))>0$; the two problems therefore have different optima on this instance.","tokens_in":19321,"feed_emoji":"📡","tokens_out":7612,"duration_ms":70936,"temperature":0.7,"pith_summary":"This paper tries to show that secure downlink beamforming for a continuous aperture array (CAPA) can be solved exactly, rather than approximated by discretizing the radiating surface. The goal is to maximize the weighted secrecy sum-rate (WSSR) for multiple legitimate users when several eavesdroppers are listening. The central claim is that the optimal source current pattern has a closed form: it is a linear combination of the channel spatial responses, so the search over continuous functions reduces to inverting an $N \\times N$ matrix with $N = K + Q$ (users plus eavesdroppers). A complementary zero-forcing construction produces a zero-leakage current pattern with water-filling power allocation that approaches the optimization-based solution at high transmission power. If the argument is right, CAPA secure beamforming avoids the performance loss and high complexity of Fourier-basis discretization, with reported WSSR gains of 77\\% and 117\\% over discrete MIMO at $P = 10^2$ mA$^2$.","feed_headline":"Secure CAPA beamforming reduces to one matrix inversion","feed_subtitle":"A closed-form current pattern replaces Fourier discretization, with reported 77–117% secrecy-rate gains over discrete MIMO.","key_machinery":"The load-bearing object is the continuous inversion identity for kernels of the form $\\Psi(s,s')=\\delta(s-s')+\\sum_{i=1}^{I}\\psi_i^*(s)\\psi_i(s')$. The paper proves that its inverse is $\\delta(s-s')$ minus a finite-rank correction built from the matrix $(I+\\Psi)^{-1}$, where $[\\Psi]_{i,i'}=\\int \\psi_i(s)\\psi_{i'}^*(s)\\,ds$. This identity turns the infinite-dimensional task of inverting an operator into an $I \\times I$ matrix inversion. It is applied to the kernel $W_{\\lambda,k}$ that appears in the first-order optimality condition for the current pattern, with the functions $\\psi_i$ taken as the normalized legitimate and eavesdropping channel responses; combined with fractional-programming auxiliary variables, it yields the closed-form current update in (51).","core_discovery":"On the paper's own terms, the contribution is a continuous-domain counterpart to linear precoding. The transmit current density $J_k(s)$ for each legitimate user is not searched over all functions: the optimal pattern is shown to be a weighted sum of the conjugate channel spatial responses $H_i^*(s)$ of the legitimate users and $\\bar{H}_q^*(s)$ of the eavesdroppers. The weights come from the inverse of a kernel $W_{\\lambda,k}(s,s')$ built from the Dirac delta plus the Gram matrix of those responses. The paper's Theorem 2 supplies a continuous-function inversion identity that converts this operator inversion into an $(K+Q) \\times (K+Q)$ matrix inversion, giving a closed-form current update at every step of a block coordinate descent algorithm based on fractional programming. The same span result drives the zero-forcing construction, whose coefficients are columns of the inverse channel-correlation matrix. The claimed consequence is that optimal secure beamforming for CAPA is achievable without Fourier truncation, with complexity $O(LKN^3)$ after the channel correlation matrix is precomputed.","pith_inferences":["If the span result is correct, a testable extension is that secure CAPA beamforming needs only the Gram matrix of channel responses, not full continuous channel functions, which would make CSI acquisition for CAPA substantially lighter than a literal continuous-channel model suggests.","The same inversion identity would apply to other continuous-aperture functional programming problems, such as joint sensing and communication or energy-splitting designs, whenever the kernel has the structure 'Dirac delta plus finite-rank channel correlations'.","A natural prediction to test experimentally is that the gap between the ZF heuristic and the FP-based optimum shrinks with SNR, so a hybrid scheme could switch from ZF at high SNR to the optimization method only when the number of users or eavesdroppers makes interference management difficult.","The dependence of the ZF method on inverting the full $K+Q$ channel matrix implies each additional eavesdropper consumes one spatial degree of freedom; the optimization method, by contrast, spends power to suppress leakage softly, which explains its reported advantage when many users compete."],"forward_implications":["Because the optimal current pattern lies in the span of the $K+Q$ channel responses, each BCD iteration only needs the precomputed channel correlation matrix $\\mathbf{H}$; no numerical integration over the aperture is required inside the iterations.","The complexity of the optimization-based method is $O(LKN^3)$ after computing $\\mathbf{H}$ with $M$-point quadrature, compared with $O(LKN_F^3)$ for Fourier discretization, where the number of Fourier bases $N_F$ grows rapidly with aperture size and carrier frequency.","The zero-forcing CAPA scheme achieves zero inter-user interference and zero eavesdropper leakage, uses water-filling for power allocation, and approaches the FP-based optimum at high SNR while costing only $O(N^3)$ beyond computing $\\mathbf{H}$.","Reported simulations put CAPA's WSSR gain over discrete MIMO at 77% for the optimization-based method and 117% for the zero-forcing method at transmit power $10^2$ mA$^2$, with the gap widening as the aperture size grows.","The optimization-based method flexibly trades residual leakage against power, whereas the ZF method nulls all leakage; this is why the paper reports that the ZF method degrades as the number of users grows, while the optimization method improves."],"supporting_citations":[{"why":"Supplies the Green's function electromagnetic channel model and the continuous-aperture field representation used in the system model.","marker":"[6]"},{"why":"Provides the Fourier-based discretization benchmark that the proposed methods must outperform and the complexity baseline for the comparison.","marker":"[20]"},{"why":"Introduces the calculus-of-variations approach to functional beamforming for CAPA that the paper extends to secure transmission and whose variation argument is reused in Appendix A.","marker":"[21]"},{"why":"Pioneers secure transmission for CAPA with a single user and a single eavesdropper, the setting the paper generalizes to multiple users and multiple eavesdroppers.","marker":"[27]"},{"why":"Supplies the fractional programming framework used to reformulate the sum-of-functions-of-ratio objective in Lemma 2.","marker":"[28]"},{"why":"Provides the quadratic-transform auxiliary variables $\\eta_k$ used in Lemma 4 to convert the fractional terms into a tractable form.","marker":"[29]"},{"why":"Supplies the water-filling algorithm used for the optimal power allocation in the zero-forcing scheme.","marker":"[31]"}],"fun_headline_variants":["CAPA secure beamforming closed-form via one matrix inversion","Closed-form CAPA beamforming: no Fourier truncation needed","Secure CAPA gains: matrix inversion beats MIMO","CAPA secrecy rate up, complexity down: one inversion","Zero-leakage CAPA beamforming without Fourier series"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a term involving $G_\\Gamma$ can be discarded as a constant during the reformulation, even though it is multiplied by the decision variable $b_k$, so the claimed equivalence of the transformed problems depends on that multiplication being harmless.","fun_headline_variants_meta":{"raw":{"variants":["CAPA secure beamforming closed-form via one matrix inversion","Closed-form CAPA beamforming: no Fourier truncation needed","Secure CAPA gains: matrix inversion beats MIMO","CAPA secrecy rate up, complexity down: one inversion","Zero-leakage CAPA beamforming without Fourier series"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000332,"raw_usage":{"total_tokens":1909,"prompt_tokens":1073,"completion_tokens":836,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":689,"completion_tokens_details":{"reasoning_tokens":769}},"tokens_in":689,"tokens_out":836,"duration_ms":6049,"temperature":1.0,"reasoning_tokens":769,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:24:18.929953+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct numerical check settles whether the reformulation is equivalent: take $K=1$, pick any current $J_1(s)$ with $\\gamma_1 < \\Gamma_1$, and choose parameters so that $G_\\Gamma > 2\\Gamma_1$. Problem (15) then has optimum $b_1=0$ because the secrecy rate is negative, while problem (20) is positive at $b_1=\\alpha_1$ since $\\log_2((1+G_\\Gamma-\\Gamma_1)/(1+\\Gamma_1))>0$; the two problems therefore have different optima on this instance.","supporting_citations":[{"cited_title":"Communicating with large intelligent surfaces: Fundamen- tal limits and models,","cited_arxiv_id":null,"evidence_quote":"Supplies the Green's function electromagnetic channel model and the continuous-aperture field representation used in the system model."},{"cited_title":"Pattern-division multiplexing for multi-user continuous-aperture MIMO,","cited_arxiv_id":null,"evidence_quote":"Provides the Fourier-based discretization benchmark that the proposed methods must outperform and the complexity baseline for the comparison."},{"cited_title":"Beamforming Optimization for Continuous Aperture Array (CAPA)-based Communications","cited_arxiv_id":"2410.13677","evidence_quote":"Introduces the calculus-of-variations approach to functional beamforming for CAPA that the paper extends to secure transmission and whose variation argument is reused in Appendix A."},{"cited_title":"Physical Layer Secu- rity for Continuous-Aperture Array (CAPA) Systems,","cited_arxiv_id":null,"evidence_quote":"Pioneers secure transmission for CAPA with a single user and a single eavesdropper, the setting the paper generalizes to multiple users and multiple eavesdroppers."},{"cited_title":"Fractional Programming for Communication Systems—Part II: Uplink Scheduling via Matching,","cited_arxiv_id":null,"evidence_quote":"Supplies the fractional programming framework used to reformulate the sum-of-functions-of-ratio objective in Lemma 2."},{"cited_title":"Fractional Programming for Communication Systems—Part I: Power Control and Beamforming,","cited_arxiv_id":null,"evidence_quote":"Provides the quadratic-transform auxiliary variables $\\eta_k$ used in Lemma 4 to convert the fractional terms into a tractable form."},{"cited_title":"Practical algorithms for a family of waterfilling solutions,","cited_arxiv_id":null,"evidence_quote":"Supplies the water-filling algorithm used for the optimal power allocation in the zero-forcing scheme."}],"review_version":1}