{"id":"94387834-310e-410a-8a19-63095733fb3a","arxiv_id":"2509.06378","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"The paper formulates a joint optimization of tunable capacitance matrix and subcarrier power allocation for a frequency-dependent BD-IRS in OFDM, claiming rate gains over benchmarks, but the solution contains flawed derivations.","lead":"An OFDM link aided by a beyond-diagonal intelligent reflecting surface is studied, with the surface's frequency-dependent reflection modeled through its tunable capacitor matrix. The paper proposes an algorithm to tune capacitors and subcarrier power to maximize data rate, but key mathematical steps in the algorithm are incorrect as written.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The capacitance recovery formula (25) does not invert the admittance model (2), so the constructed C need not realize the optimized reflection matrices; the claimed feasible solution is unsupported.","rationale":"The reader's REJECT verdict is supported by a concrete, load-bearing flaw: the paper's only mechanism for converting the optimized relaxed solution into a physically realizable BD-IRS configuration is the inversion in Section V-C. If that inversion is algebraically inconsistent with the forward model (2), then the algorithm's output C is not a feasible solution in any meaningful sense, and the numerical comparisons do not establish the claimed performance. The off-diagonal check is enough to show the inversion is not the inverse: substituting the C produced by (25) into (2) yields a different admittance, and the dimensional mismatch (Ω vs S) confirms an error rather than a harmless approximation. The diagonal formula is also structurally wrong, since [A_n]_{m,m} couples all C_{m,k} through the summation. This alone invalidates the central claim, independent of the separate issue in Proposition 2 about the initialization constraint. I therefore agree with the REJECT recommendation, although my primary concern is the recovery formula (25) rather than the specific unproven gap in (P3) that the reader listed as the weakest assumption. A single-subcarrier numerical check would settle the matter conclusively.","tokens_in":10729,"tokens_out":10721,"duration_ms":85954,"concrete_test":"For a single subcarrier n, pick any feasible Φ (e.g., M=2, Φ = diag(e^{jθ}, e^{jφ}) with unit moduli). Compute A_n via (24) with a_0>0. Compute C from (25) entrywise. Substitute this C back into (2) to obtain A'_n, and compute the relative Frobenius error ||A'_n - A_n||_F / ||A_n||_F. If the error is not near zero (or if (2) cannot even be evaluated because units are inconsistent), the inversion in (25) is invalid. This single-subcarrier test isolates the construction from the common-C projection: if (25) fails even for one subcarrier, then the matrices \\tilde C_n fed into (P3) are not the correct per-subcarrier capacitance targets, and the whole feasibility construction collapses.","verdict_should_be":"REJECT","load_bearing_attack":"The central claim rests on the feasibility construction of Section V-C. After solving the relaxed problem (P2), (24) gives the desired per-subcarrier admittance A_n from the optimized Φ_n, and (25) claims to recover the capacitance matrix C that realizes A_n under model (2). This inversion is algebraically wrong. For off-diagonal entries, (2) states [A_n]_{m,k} = -(1/(R+jωL2) + 1/(jωC_{m,k}) + 1/(jωL1)), so the required value of 1/(jωC_{m,k}) is -[A_n]_{m,k} - 1/(R+jωL2) - 1/(jωL1). But (25) sets 1/(jωC_{m,k}) equal to 1/(1/(-[A_n]_{m,k}) - 1/(jωL1) - R - jωL2), which is not the same expression; moreover the terms mix siemens and ohms, as [A_n] has admittance units while R+jωL2 has impedance units. The diagonal formula is even more problematic: [A_n]_{m,m} is a sum over all k of the branch admittances involving C_{m,k}, so C_{m,m} cannot be recovered from [A_n]_{m,m} alone without knowing the off-diagonal C_{m,k}; (25) ignores this. Consequently the C obtained from (P3) does not, in general, reproduce the optimized {Φ_n} on any subcarrier, and the rate reported in the numerical section is not achievable by the modeled BD-IRS circuit. The algorithm may still produce some C, but without a valid inversion step there is no guarantee—and in fact no likelihood—that it realizes the high-quality solution of (P2). The claim of a high-quality feasible solution to (P1) therefore fails at its main constructive step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a broadband OFDM system aided by a beyond-diagonal IRS (BD-IRS) whose reflection matrices are frequency-dependent through a common tunable capacitance matrix C. It formulates a joint optimization problem over C and per-subcarrier power allocations to maximize the achievable rate, relaxes the circuit constraints to convex constraints, solves the relaxed problem via alternating optimization (water-filling and successive convex approximation), and then constructs a feasible C by inverting the optimized per-subcarrier admittance matrices and projecting onto a common nonnegative capacitance matrix. The paper claims that this algorithm finds a high-quality feasible solution and demonstrates its superiority over several benchmarks in numerical results.","tokens_in":11209,"tokens_out":9882,"duration_ms":79151,"significance":"If the algorithm were correct as stated, it would provide a systematic optimization framework for frequency-dependent BD-IRS in wideband OFDM, going beyond the linear-fitting approach of [11] and explicitly handling the coupling across subcarriers through a common C. The system model is clearly stated, and the AO/SCA machinery for the relaxed problem (P2) is standard. However, the central feasibility recovery step contains an algebraic inversion error and a dimensional inconsistency, and the projection step is not accompanied by any performance gap analysis. As printed, the claimed achievable rate is not supported by the derivation, which substantially reduces the paper's significance. The paper does not provide reproducible code or machine-checked proofs.","major_comments":[{"comment":"The inversion of the admittance model (2) is incorrect. For m≠k, (2) gives -[A_n]_{m,k} = 1/(R+j2π f_n L2) + 1/(j2π f_n C_{m,k}) + 1/(j2π f_n L1), so the required capacitance is C_{m,k} = 1/(j2π f_n ( -[A_n]_{m,k} - 1/(R+j2π f_n L2) - 1/(j2π f_n L1))). Equation (25) instead places -R - j2π f_n L2 in the denominator, which is an impedance rather than an admittance, making the expression dimensionally inconsistent. For m=k, [A_n]_{m,m} is the sum over all columns of the branch admittances, so it cannot be inverted to a single C_{m,m} without first subtracting the off-diagonal branch admittances, which are determined by the off-diagonal entries of A_n. As a consequence, the capacitance matrix obtained from (P3) will not, in general, realize the optimized reflection matrices {Φ_n}, and the rate reported from the subsequent power allocation is not an achievable rate of the modeled BD-IRS circuit. This invalidates the central claim of constructing a feasible solution to (P1).","section":"V-C, Eq. (25)"},{"comment":"The claimed equivalence is false. Direct computation shows q_n^H(I_M⊗1_M)q_n = \\sum_{i=1}^M |\\sum_{j=1}^M [Φ_n]_{j,i}|^2, which is a single scalar inequality, whereas Φ_nΦ_n^H ⪯ I_M is a positive-semidefinite matrix constraint. For M=2, Φ_n = diag(√2,0) satisfies q_n^H(I_M⊗1_M)q_n = 2 ≤ M but has largest singular value √2 > 1, violating Φ_nΦ_n^H ⪯ I_M. The proof establishes only the necessity direction; the sufficiency assertion is unproven. Therefore (P4-eqv) is a relaxation, not an equivalent reformulation of (P4), and the SDR-based initialization may produce q_n that do not correspond to physically admissible reflection matrices.","section":"V-D, Proposition 2"},{"comment":"Even if (25) were corrected, the projection step (P3) provides no guarantee about the rate loss incurred by enforcing a common capacitance matrix. The paper does not bound the gap between the relaxed optimum of (P2) and the achievable rate of the constructed C, nor does it analyze the conditioning of the mapping from C to {Φ_n}. The numerical results show only a single set of channel and circuit parameters; the claim of a 'high-quality feasible solution' is therefore an empirical assertion without supporting analysis.","section":"V-C, P3"}],"minor_comments":[{"comment":"In the definition of [\\tilde C_n]_{m,k} for m=k, the parentheses in the denominator are unbalanced; the expression should be checked and written cleanly.","section":"V-C"},{"comment":"The proof of Proposition 2 stops after the necessity direction; even if the equivalence were true, the sufficiency direction should be demonstrated explicitly.","section":"V-D"},{"comment":"The statement that the SCA update 'is guaranteed to reach at least a stationary point of (P2-II)' is not proven; the paper should provide a citation or argument for the adopted SCA variant.","section":"V-B"},{"comment":"It is unclear whether the rate plotted in Fig. 2 is the relaxed objective of (P2) during AO or the achievable rate after the feasibility projection; this should be stated explicitly.","section":"VI"},{"comment":"Benchmark scheme 3 assumes a frequency-independent BD-IRS, but under the circuit model (2)-(3) with a common C the reflection matrices are inevitably frequency-dependent; the implementation of this benchmark should be clarified.","section":"VI"}],"recommendation":"reject","confidential_remarks":"The manuscript contains a fundamental algebraic error in its key feasibility construction, and the missing projection-gap analysis would require new theoretical results. Although the authors might be able to correct the inversion formula, the numerical results would need to be redone and the central claim revalidated. The paper is not suitable for publication in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The problem this paper goes after is real: BD-IRS reflection depends on frequency and is coupled across subcarriers through a common capacitance matrix, and prior work [11] only used a linear fit. Replacing that fit with direct optimization of C is the right ambition. The paper also does something correct early on: Proposition 1, the passive-network constraint Φ_nΦ_n^H ⪯ I, is a valid relaxation, and the AO/SCA loop is described carefully.\n\nBut the constructive step at the end does not hold up. Equation (25) does not invert (2). For off-diagonal entries, (2) says [A_n]_{m,k} = -(1/(R+jωL2) + 1/(jωC_{m,k}) + 1/(jωL1)); solving for C requires subtracting the other two terms from -[A_n]_{m,k}, then inverting. What (25) does is different: it takes 1/(-[A_n]_{m,k}) and subtracts impedance terms. The units are wrong too—[A_n] is admittance, so 1/[A_n] is impedance, and multiplying by 1/(j2πf_n) gives henry, not farad. The diagonal case is worse, because the diagonal admittance is a sum over k of branch admittances, so C_{m,m} cannot be recovered from [A_n]_{m,m} alone. As a result, the common C produced by (P3) is not shown to realize the optimized {Φ_n}, and the claimed 'high-quality feasible solution' has no basis. The numerical section also never says whether the plotted rate is computed with the actual circuit model for the constructed C or with the relaxed Φ_n; if it is the latter, the plots are not achievable rates at all.\n\nThere is also a false equivalence in Proposition 2. The condition q_n^H(I_M⊗1_M)q_n ≤ M bounds the sum of squared column sums of Φ_n, which is much weaker than the spectral norm constraint Φ_nΦ_n^H ⪯ I. This one only affects the initialization, so it is secondary, but it is another sign that the derivations were not checked line by line.\n\nWho wants this paper? Researchers in BD-IRS and wideband optimization will want to know that this formulation exists, and there is a plausible route to a correct algorithm: invert (2) properly, add a consistency constraint for the common C, and re-run the numerics with the actual circuit response. But as printed, the main result is unsupported. I would not desk-reject the topic; a referee should write the report telling the authors exactly where (25) fails and give them a chance to fix it. In current form, though, I would not trust the numbers.","headline":"A well-motivated BD-IRS/OFDM formulation with a sound relaxation, but the capacitance-recovery step (25) and Proposition 2 are not valid, so the paper's central claim of a feasible high-quality solution is unsupported as written.","tokens_in":11639,"tokens_out":8708,"would_cite":false,"duration_ms":71967,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A single capacitance matrix shared across subcarriers is enough to configure a frequency-dependent BD-IRS for OFDM rate maximization.","keywords":["beyond diagonal IRS","BD-IRS","OFDM","frequency-dependent reflection","capacitance matrix optimization","alternating optimization","successive convex approximation","rate maximization"],"falsifier":"Run the proposed algorithm on the paper's setup (for example, M=10, P=30 dBm, 2.4 GHz, 300 MHz) and compare the actual rate obtained from the capacitance matrix recovered by (P3) with the relaxed upper bound of (P2), and with the linear-fitting benchmark. If the gap to the relaxed bound is large, or if a simpler alternative projection matches or beats the proposed rate, the claim of a high-quality feasible solution is weakened.","tokens_in":10548,"feed_emoji":"📡","tokens_out":8302,"duration_ms":60771,"temperature":0.7,"pith_summary":"This paper addresses a practical obstacle to using beyond-diagonal intelligent reflecting surfaces (BD-IRS) in broadband communication: the reflection response of the surface depends on frequency, so the reflection matrices on different OFDM subcarriers are coupled through one shared tunable capacitance matrix. The authors claim that, despite the non-convex coupling, jointly optimizing that capacitance matrix together with per-subcarrier power allocations yields a high-quality feasible configuration. The proposed method relaxes the circuit constraints, alternates water-filling with successive convex approximation, and then recovers a single feasible capacitance matrix by a convex projection. If the claim holds, it provides a systematic circuit-aware design that replaces an earlier linear-fitting approach restricted to a narrow band, and it quantifies the gains of frequency-dependent and non-reciprocal BD-IRS designs.","feed_headline":"One capacitor matrix tunes every OFDM subcarrier of a smart surface","feed_subtitle":"The algorithm sets one circuit capacitance matrix and per-tone powers, beating fitting-based and frequency-flat designs.","key_machinery":"The load-bearing object is the circuit-level map from the tunable capacitance matrix C to the admittance matrix A_n at subcarrier frequency f_n, and from A_n to the reflection matrix Phi_n = (a0 I + A_n)^(-1)(a0 I - A_n). The paper proves the passivity condition Phi_n Phi_n^H <= I, which becomes a convex relaxation of the circuit constraints, then closes the loop with the convex projection (P3) that picks one common C closest in Frobenius norm to the ideal per-tone capacitance matrices derived from (25). Initialization of the alternating loop uses a semi-definite relaxation of a per-subcarrier channel-power maximization.","core_discovery":"The paper's central claim is that the achievable rate of a BD-IRS aided OFDM link can be maximized by selecting one nonnegative capacitance matrix C shared by all subcarriers, even though the ideal capacitance matrix differs from subcarrier to subcarrier. The argument proceeds by proving that every circuit-realizable reflection matrix satisfies Phi_n Phi_n^H <= I, using that passivity condition to form a relaxed problem, solving the relaxed rate-maximization problem by alternating optimization, and then constructing a feasible C by minimizing the total Frobenius distance to the per-subcarrier ideal capacitance matrices. Numerical results in the paper show this design outperforming linear-fitting, frequency-independent, conventional-IRS, and reciprocal-constrained benchmarks.","pith_inferences":["The same alternating-projection pipeline should extend to multi-antenna, multi-user, or OFDMA systems, where coupling through a common C is even more restrictive; that would be a direct stress test of the model.","The Frobenius-norm projection (P3) is a heuristic choice; weighted norms, minimax error, or discrete capacitor-value constraints could beat it on the same rate metric.","Because the frequency dependence is set by the fixed circuit elements R, L1, and L2, jointly optimizing those along with C is a natural extension the paper leaves open.","The reported gains are from one simulated channel model; testing at other carrier frequencies, bandwidths, and measured multipath profiles would show whether the structural advantage persists."],"forward_implications":["A BD-IRS can be configured for wideband OFDM with a single nonnegative capacitance matrix per circuit state, with per-subcarrier powers set by water-filling.","The design works directly from the circuit model, so it is not limited to the narrow band for which the earlier linear-fitting method was calibrated.","Allowing non-symmetric, non-reciprocal capacitance matrices yields a substantial rate gain over the reciprocal constraint in the paper's simulations.","The advantage over the fitting-based benchmark grows with the number of reflecting elements, so the method matters most for moderate-to-large fully connected BD-IRS."],"supporting_citations":[{"why":"supplies the linear-fitting frequency-dependent BD-IRS design that the proposed method is compared against and aims to generalize.","marker":"[11]"},{"why":"introduces the multiport-network BD-IRS architecture whose inter-element connections create the beyond-diagonal reflection structure.","marker":"[1]"},{"why":"gives the admittance-matrix expression (2) that ties each circuit element and capacitance entry to the frequency response.","marker":"[13]"},{"why":"is the network-theory source for the passivity condition A_n + A_n^H >= 0 used to prove Proposition 1.","marker":"[14]"},{"why":"provides the rate-gap parameter Gamma used in the achievable-rate expression for practical modulation and coding.","marker":"[12]"},{"why":"supplies the water-filling solution that solves the power-allocation subproblem (P2-I).","marker":"[16]"},{"why":"provides the successive convex approximation framework that gives the monotonic convergence of the reflection subproblem.","marker":"[17]"},{"why":"gives the semi-definite relaxation technique used to initialize the alternating optimization.","marker":"[20]"}],"fun_headline_variants":["One capacitance matrix for all subcarriers: BD-IRS rate max","Single matrix, every tone: Efficient BD-IRS OFDM design","Frequency-dependent BD-IRS: One capacitance optimizes OFDM","Joint capacitance and power tuning for BD-IRS OFDM"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The algorithm's quality guarantee rests on the unverified assumption that the single capacitance matrix found by the convex projection (P3) achieves an actual rate close to the relaxed optimum of (P2), with no stated bound or condition on that gap.","fun_headline_variants_meta":{"raw":{"variants":["One capacitance matrix for all subcarriers: BD-IRS rate max","Single matrix, every tone: Efficient BD-IRS OFDM design","Frequency-dependent BD-IRS: One capacitance optimizes OFDM","Joint capacitance and power tuning for BD-IRS OFDM"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000868,"raw_usage":{"total_tokens":3732,"prompt_tokens":885,"completion_tokens":2847,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":501,"completion_tokens_details":{"reasoning_tokens":2774}},"tokens_in":501,"tokens_out":2847,"duration_ms":21742,"temperature":1.0,"reasoning_tokens":2774,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:17:12.388176+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the proposed algorithm on the paper's setup (for example, M=10, P=30 dBm, 2.4 GHz, 300 MHz) and compare the actual rate obtained from the capacitance matrix recovered by (P3) with the relaxed upper bound of (P2), and with the linear-fitting benchmark. If the gap to the relaxed bound is large, or if a simpler alternative projection matches or beats the proposed rate, the claim of a high-quality feasible solution is weakened.","supporting_citations":[{"cited_title":"A universal framework for multiport network analysis of reconfigurable intelligent surfaces,","cited_arxiv_id":null,"evidence_quote":"gives the admittance-matrix expression (2) that ties each circuit element and capacitance entry to the frequency response."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"is the network-theory source for the passivity condition A_n + A_n^H >= 0 used to prove Proposition 1."},{"cited_title":"Multi-channel modulation,","cited_arxiv_id":null,"evidence_quote":"provides the rate-gap parameter Gamma used in the achievable-rate expression for practical modulation and coding."},{"cited_title":"A general inner approximation algorithm for nonconvex mathematical programs,","cited_arxiv_id":null,"evidence_quote":"provides the successive convex approximation framework that gives the monotonic convergence of the reflection subproblem."}],"review_version":2}