{"id":"9455b3e8-f2b8-4d44-bb3a-3d025fd6f581","arxiv_id":"2608.11476","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A reconfigurable surface built from spherical-wave modes of the terminal positions attains the fully connected performance optimum with tunable entries set by geometry, not panel size.","lead":"The paper shows that an extremely large smart surface can match the best possible reflecting performance with only a few hundred adjustable connections instead of hundreds of thousands. The key is to design the surface using spherical waves from the known positions of the base station and users, which provably captures full performance at a cost that does not grow with the surface size.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Exactness at L=2ρ depends on an unstated rank tolerance: with the exact active-subspace rank the claimed 196-entry operating point becomes 484 entries, so the 'provably attains' claim needs an explicit ε caveat.","rationale":"I read Proposition 2 as mathematically sound: the Halmos unitary dilation correctly shows that any unitary M×M surface has the same action on the active subspace as some L×L modal matrix when L≥2ρ and S⊆range(Φ). The load-bearing weakness is not the dilation itself but the gap between the theorem's hypothesis and the construction actually used in simulation. The construction sets ρ to a numerical rank at a relative tolerance, so the condition S⊆range(Φ) is not exactly satisfied; the theorem is applied to an ε-truncated channel instead of the true Green-function channel. This is an internal consistency issue between the exactness claim and the implementation, and it is more direct than the reader's LoS-multipath concern, though both are real. A single SVD spectrum plus two L values settles whether the 196-entry operating point is genuinely exact or merely good at 0 dB; the proposed test is cheap and does not require new theory. Because the paper can repair the claim by using L=2(N+K) or by stating an explicit ε and showing the rate perturbation, the conditional verdict stands, so I do not change the reader's verdict. I mark agreement as partial because the reader identified the LoS model as the weakest assumption, whereas I find the numerical-rank/tolerance issue to be the more immediate threat to the exactness claim even under the paper's own LoS model.","tokens_in":20040,"tokens_out":18251,"duration_ms":175229,"concrete_test":"Reproduce the reference deployment (M=576, N=8, K=3, dBS=0.16dF) and compute the full SVD of A=[G,F*] without thresholding; report the singular-value spectrum and the exact rank. Then run Algorithm 1 at L=2ρ_num=14 and at L=2·rank_exact=22, and compare the achieved sum rates against an independent dense-U(M) unitary optimization (feasible on the 16×16 panel) at SNR=0 dB and 20 dB. If L=14 matches the dense optimum to numerical precision at both SNRs, the threshold is immaterial; if it matches only at 0 dB or only after discarding singular values, the exactness claim should be restated as approximate with an explicit rank tolerance.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2 requires S ⊆ range(Φ) with ρ = dim S, but the basis construction in Sec. IV-B sets ρ to the numerical rank of [G,F*] at a relative tolerance. The stacked atom matrix is an analytic Green-function matrix, and for generic terminal positions its exact rank is min(M, N+K), i.e., 11 in the reference deployment (N=8, K=3), whereas the paper uses rG=4, ρ=7, and L=14. After thresholding, span(Φ_S) is not S, so the Halmos-dilation argument proves equality only for the ε-truncated channel; the discarded singular values are asserted to be negligible rather than shown to be irrelevant at the operating SNR. The headline 'provably attains the fully connected optimum with 196 entries' is therefore a tolerance-dependent numerical statement, not a direct consequence of Prop. 2. Using the exact rank would give L=2(N+K)=22 and 484 entries; the qualitative M-independence survives, but the quantitative claim and the comparison against DFT beamspace change. The paper's footnote 2 transparently notes exactness on the ε-truncated channel, yet the abstract and Prop. 2 do not carry that caveat.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces XL-BD-RIS, a beyond-diagonal RIS architecture for the radiative near field. It models both cascade legs with a free-space Green function, observes that the aperture fields lie in the span of N+K position-dependent spherical-wave atoms, and constructs a compact unitary modal matrix on a subspace of dimension L=2ρ, where ρ is the (numerical) rank of the stacked atom matrix. Proposition 2 claims that this modal representation provably attains the fully-connected unitary-surface optimum with L=2ρ≤2(N+K), independently of the panel size, via a Halmos unitary dilation argument. The paper also develops a WMMSE-Riemannian alternating algorithm with monotone sum-rate convergence and per-iteration cost O(L^3) independent of M, and presents numerical results for a 24×24 panel showing that the fully-connected sum rate is reached with 196 reconfigurable entries, compared with about 12,100 for a DFT beamspace and 331,776 for the fully-connected element-domain surface.","tokens_in":20253,"tokens_out":8072,"duration_ms":82577,"significance":"If the exactness claim is made fully rigorous, the result is significant: it identifies a low-dimensional subspace structure of near-field cascaded channels and converts a prohibitive O(M^2)-entry, O(M^3)-per-iteration problem into an O(L^2)-entry, O(L^3)-per-iteration problem whose dimension depends on the deployment geometry rather than on panel size. The proofs of Propositions 1 and 2 are mathematically sound under the stated exact-rank, LoS, and unitary assumptions, and the WMMSE-Riemannian optimization with its monotonicity result is a solid, standard contribution. The numerical study is extensive and internally consistent, and the comparison against DFT beamspace and group-connected architectures is informative. The main caveats are that the exactness theorem is applied to an ε-truncated channel without an explicit perturbation analysis, and that the claimed equivalence is to the non-reciprocal unitary surface rather than the reciprocal symmetric BD-RIS of the usual literature; these caveats affect the quantitative headline but not the core qualitative idea.","major_comments":[{"comment":"The central exactness claim is not literally what is proved. Proposition 2 assumes S = span([G,F*]) with ρ = dim S and S ⊆ range(Φ), but the basis construction in Eqs. (32)-(33) sets ρ to the numerical rank of Σ at an unspecified relative tolerance. For the reference deployment (M=576, N=8, K=3), the exact rank of the stacked atom matrix is generically min(M,N+K)=11, so the reported ρ=7 corresponds to a truncated subspace, not to the S used in the theorem. Consequently, the Halmos dilation argument proves equality (35) only for the ε-truncated channel, and the abstract's statement that the design 'provably attains the fully connected optimum with about two hundred entries' overstates what Prop. 2 establishes. The discarded singular values are asserted to be negligible rather than shown to be irrelevant at the operating SNR. Please restate Prop. 2 and the abstract with an explicit ε-truncation caveat, and either use the exact rank (which would give L=22 and 484 entries in this deployment) or provide a quantitative bound or numerical certification that the rate gap caused by the discarded singular values is negligible over the SNR range considered.","section":"Sec. IV-B, Prop. 2, footnote 2"},{"comment":"The fully-connected optimum in Proposition 2 is the maximum over the full unitary group U(M), whereas lossless reciprocal BD-RIS, which is the standard physical model cited in the introduction, is constrained to symmetric unitary scattering matrices. The paper explicitly retains the non-reciprocal unitary model and leaves symmetry as an extension, but the abstract and the numerical comparisons still refer to 'fully connected BD-RIS' without this qualification. The equivalence to the fully connected optimum should be scoped to non-reciprocal unitary surfaces, and the paper should discuss how the reciprocal-symmetric constraint changes the result. If the authors intend to claim the reciprocal case, the Halmos dilation construction must be adapted to produce symmetric unitary dilations, which is not demonstrated.","section":"Sec. II-A, Sec. III-C, footnote 1"},{"comment":"The dimensioning rule L=2ρ depends on two unquantified quantities: the relative tolerance used to define rank_epsilon and the geometry constant η_dof in the heuristic formula rG ≈ η_dof D_BS D/(λ d_BS). The paper states that ρ is 'measured rather than assumed,' but the measurement is threshold-dependent, and the threshold is never specified. The claimed M-independence of the reconfigurable budget is therefore only as meaningful as the chosen tolerance. Please specify the tolerance selection rule, report its numerical value in the experiments, and analyze its sensitivity with respect to the attained rates and the exactness guarantee.","section":"Sec. IV-A and IV-B"}],"minor_comments":[{"comment":"The reference deployment reports 'the measured rank of G is rG=4' but does not state the relative tolerance used for the numerical rank. Please provide the tolerance and the singular-value spectrum of [G,F*] so that the rank measurement is reproducible.","section":"Sec. VII-A"},{"comment":"The paraxial estimate rG ≈ D_BS D/(λ d_BS)+1 uses quantities D_BS and the constant η_dof that are not defined in the main text or the figure caption. Please define all symbols and state the parameter values used for the dashed curve.","section":"Sec. VII-D, Fig. 5(a)"},{"comment":"The operating point 'SNR is 0 dB' is only loosely defined. Please specify the reference noise power, the normalization of the cascaded channel, and how the per-user noise variance is set in the simulations.","section":"Sec. VII"},{"comment":"The label 'Modal (oracle)' is potentially confusing; the modal basis is constructed from known terminal positions, not from an oracle channel. Please clarify in the captions what 'oracle' refers to.","section":"Figs. 2 and 3"},{"comment":"There are several typographical artifacts in the text, such as 'coefficients' and 'suffices.' A careful proofreading pass is recommended before a revised submission.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the journal's scope and the core idea is novel and useful. The main obstacle is not the mathematics under the exact-rank assumption but the gap between that assumption and the numerical-rank construction used to produce the headline 196-entry number. This is fixable within the manuscript's scope by an explicit ε-truncation formulation and a perturbation analysis, so I would not reject. I would also ask the authors to present the non-reciprocal unitary limitation more prominently, as the term 'beyond-diagonal RIS' in the current literature usually implies reciprocal symmetric scattering."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know about this paper because it makes a real dent in the scalability problem for fully-connected BD-RIS. The central claim is that in the near field, with both cascade legs modeled as free-space Green functions, the aperture fields live in the span of N+K spherical-wave atoms, and a unitary modal matrix of dimension 2ρ (ρ = rank of the stacked atom matrix) exactly matches the fully-connected optimum. That turns a 24×24 panel's 331,776 reconfigurable entries into 196, with the number set by geometry rather than panel size.\n\nWhat's genuinely new: the active-subspace reduction via Halmos dilation, the geometric basis built from terminal positions alone, and the closed-form explanation of why DFT beamspace fails (spherical wavefronts occupy about (dF/2r)^2 beams). The WMMSE-Riemannian algorithm is standard, but the low-rank line search is a nice touch, and the complexity ladder (diagonal → group → fully-connected → modal) is clean. Proposition 2 is correct under the stated assumptions, and the paper is honest enough to include footnote 2 flagging the ε-truncated channel.\n\nThe main soft spot is exactly that footnote. The abstract and Prop. 2 say \"provably attains\" at L=2ρ, but ρ is the numerical rank at a relative tolerance, not the exact rank. In the reference deployment the exact rank of the stacked atom matrix is 11, while they use ρ=7. So the 196-entry operating point is a tolerance-dependent statement. The equality holds for the truncated channel, and the discarded singular values are asserted—not shown—to be negligible. That's not fatal: using L=2(N+K)=22 gives 484 entries, still M-independent. But the headline number should carry the caveat.\n\nOther soft spots: the fixed mode-forming network that implements Φ is not realized or costed, and the LoS/Green-function model with no random path gains is a strong assumption, though they list it as future work. Numerical results are for a single deployment without error bars.\n\nWho's this for: anyone working on BD-RIS, holographic MIMO, or near-field XL-MIMO. The math is worth a serious referee, and the result, if the hardware gap closes, is significant. I'd send it to review, but insist the rank-tolerance caveat be made prominent.","headline":"Correct core equivalence and a genuinely useful dimensionality reduction, but the headline entry count quietly depends on a numerical rank tolerance that should be stated up front.","tokens_in":20831,"tokens_out":3521,"would_cite":true,"duration_ms":31317,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Near-field geometry makes fully connected beyond-diagonal RIS affordable at extremely large scale by reducing the design to a low-dimensional modal problem.","keywords":["beyond-diagonal RIS","extremely large surfaces","near-field communications","modal decomposition","low-rank optimization","sum rate maximization","spherical-wave atoms","Riemannian optimization"],"falsifier":"Deploy a $24\\times24$ panel at 28 GHz in a scattering-rich indoor environment, or simulate the same geometry with ray-traced multipath, measure the numerical rank $\\rho$ of the stacked spherical-wave atom matrix derived from the actual channel, and compare the sum rate optimized at $L=2\\rho$ with the fully connected optimum: any gap beyond numerical tolerance, or any $\\rho$ significantly larger than $N+K$, would refute the exactness claim.","tokens_in":19817,"feed_emoji":"📶","tokens_out":6405,"duration_ms":57962,"temperature":0.7,"pith_summary":"This paper tries to show that the radiative near field, usually a complication for extremely large surfaces, is exactly what makes fully connected beyond-diagonal reconfigurable intelligent surfaces affordable. It models both legs of the base-station-surface-user cascade with the free-space Green function and proves that all fields on the aperture live in the low-dimensional span of the spherical-wave responses of the terminal positions. An $L\\times L$ unitary modal matrix with $L=2\\rho\\le 2(N+K)$ then provably attains the same maximum sum rate as any $M\\times M$ unitary surface, so the number of tunable entries is set by the geometry, not the panel size. A WMMSE-Riemannian alternating algorithm optimizes beamformers and the modal matrix with monotone convergence and per-iteration cost independent of $M$. This is why a $24\\times24$ panel reaches the fully connected optimum with 196 reconfigurable entries instead of 331,776.","feed_headline":"Near-field geometry makes giant smart surfaces cheap to control","feed_subtitle":"A 24x24 panel reaches the fully connected optimum with 196 tunable entries instead of 331,776.","key_machinery":"The machinery is the active subspace of the aperture, $S=\\operatorname{span}([G,F^*])$ with $\\rho=\\dim S\\le N+K$; the geometric modal basis $\\Phi=[\\Phi_S,\\Phi_\\perp]$ formed by the leading $\\rho$ left singular vectors of the stacked atom matrix plus an arbitrary orthonormal complement; the lossless completion $\\Theta=\\Phi\\Psi\\Phi^H+(I_M-\\Phi\\Phi^H)$, which makes every $\\Psi\\in U(L)$ a valid passive unitary surface; and the Halmos unitary dilation, which supplies the $2\\rho\\times 2\\rho$ block matrix reproducing a given contraction $T=\\Pi_S\\Theta\\Pi_S$ on $S$. Together these turn the $M$-dimensional unitary surface design into an equivalent $L$-dimensional design with $L=2\\rho$, and the paper optimizes the latter by unitary steepest descent on $U(L)$ with a factored low-rank line search.","core_discovery":"The central discovery, stated as Proposition 2, is an exact equivalence: if the active subspace $S=\\operatorname{span}([G,F^*])$ of the sampled aperture channel lies inside the span of the chosen modal basis and $L\\ge 2\\rho$ with $\\rho=\\dim S$, then the maximum sum rate over $L\\times L$ unitary modal matrices equals the maximum over the full $M\\times M$ unitary surface. Because in free-space line of sight the sampled base-station-surface and surface-user matrices are exactly collections of spherical-wave atoms at the $N+K$ terminal positions, $\\rho\\le N+K$, and a basis built from one thin SVD of the stacked atom matrix plus arbitrary orthonormal slack gives $L=2\\rho$. The proof uses the Halmos unitary dilation to realize any contraction on $S$ by a $2\\rho\\times 2\\rho$ unitary, embedded in the surface through a lossless completion that leaves the complementary modes reflected unchanged. Consequently the fully connected optimum is attained with $L^2$ tunable entries independent of $M$; in the reference deployment with $N=8$, $K=3$, $M=576$, the measured active dimension is $\\rho=7$, so $L=14$ and 196 entries reach the anchor, while the mismatched DFT beamspace needs about 12,100 entries.","pith_inferences":["The graceful degradation below $L=2\\rho$ visible in the numerical study suggests an adaptive protocol the paper does not state: start from a geometric estimate of $\\rho$, optimize at $L=2\\rho$, and grow $L$ only when measured rate gains exceed a threshold.","If position uncertainty or diffuse multipath is introduced, replacing the empirical atom stack by an ensemble covariance turns the geometric basis into a Karhunen-Loève basis, and the exactness condition would become an effective-rank condition rather than the hard $N+K$ count used here.","In the far-field limit the active dimension collapses toward $K+1$ and a plane-wave beamspace already spans the subspace, so the practical advantage of the geometric modal design is specific to near-field depth and should vanish as the geometry recedes.","A testable wideband extension is to measure $\\rho$ over the joint space-time aperture; one would expect the required $L$ to grow with bandwidth as the spherical-wave atoms become frequency-dependent."],"forward_implications":["Fully connected BD-RIS optimality becomes reachable at panel-size-independent cost: after a one-time $O(M\\rho^2)$ compression, each optimization iteration costs $O(L^3)$ with $L=2\\rho$.","The reconfigurable entry count tracks the propagation geometry, not the aperture: growing the panel from $16\\times16$ to $24\\times24$ moves the modal operating point only from 144 to 196 entries.","Far-field DFT beamspace is the wrong coordinate system in the near field: a spherical wavefront atom spreads over roughly $(d_F/2r)^2$ beams, producing about a sixty-fold entry-count penalty in the reference deployment.","Element-domain group-connected surfaces cannot match the modal design at the same entry budget: they reach 99% of the fully connected anchor only at 18,432 entries in the reference setting, and their per-iteration cost grows with block size.","The value of beyond-diagonal coupling itself grows with near-field depth: the gap between diagonal phasing and the fully connected optimum widens from 2.5 to 4.4 bit as the base station moves deeper into the near field."],"supporting_citations":[{"why":"Defines the single-, group-, and fully-connected BD-RIS architecture family and its quadratic/cubic cost, the baseline against which the modal design is compared.","marker":"[2]"},{"why":"Supplies the channel-operator diagonalization idea the authors generalize from a single active aperture to the two-leg reflective cascade of a BD-RIS.","marker":"[14]"},{"why":"Supports physical realizability of a general unitary scattering matrix through non-reciprocal elements, justifying the lossless completion model.","marker":"[15]"},{"why":"Provides the Halmos unitary dilation that is the key proof tool of Proposition 2, realizing any contraction on the active subspace by a $2\\rho\\times 2\\rho$ unitary.","marker":"[16]"},{"why":"Supplies the unitary steepest-descent update and geodesic step used to optimize the modal matrix on the unitary group.","marker":"[17]"}],"fun_headline_variants":["Near-field geometry cuts RIS tuning to 196 entries","Giant RIS needs only 196 tunable entries via near-field","Low-rank modal design shrinks RIS footprint to 196 entries","Near-field makes fully connected RIS practical at scale","196 entries replace 331,776 in near-field RIS design"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that both legs of the cascade are deterministic line-of-sight point-source responses with no random path gains, so the aperture fields are exactly spanned by the known terminal-position wave responses—if multipath, scattering, or position uncertainty adds extra aperture degrees of freedom, the active subspace grows beyond $N+K$ and the promised equivalence at twice that dimension no longer follows.","fun_headline_variants_meta":{"raw":{"variants":["Near-field geometry cuts RIS tuning to 196 entries","Giant RIS needs only 196 tunable entries via near-field","Low-rank modal design shrinks RIS footprint to 196 entries","Near-field makes fully connected RIS practical at scale","196 entries replace 331,776 in near-field RIS design"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000743,"raw_usage":{"total_tokens":3389,"prompt_tokens":1093,"completion_tokens":2296,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":709,"completion_tokens_details":{"reasoning_tokens":2214}},"tokens_in":709,"tokens_out":2296,"duration_ms":14516,"temperature":1.0,"reasoning_tokens":2214,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:13:51.840549+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Deploy a $24\\times24$ panel at 28 GHz in a scattering-rich indoor environment, or simulate the same geometry with ray-traced multipath, measure the numerical rank $\\rho$ of the stacked spherical-wave atom matrix derived from the actual channel, and compare the sum rate optimized at $L=2\\rho$ with the fully connected optimum: any gap beyond numerical tolerance, or any $\\rho$ significantly larger than $N+K$, would refute the exactness claim.","supporting_citations":[{"cited_title":"Beyond Diagonal Reconfig- urable Intelligent Surfaces: From Transmitting and Reflecting Modes to Single-, Group-, and Fully-Connected Architectures,","cited_arxiv_id":null,"evidence_quote":"Defines the single-, group-, and fully-connected BD-RIS architecture family and its quadratic/cubic cost, the baseline against which the modal design is compared."},{"cited_title":"Multi-user holo- graphic communications via channel operator diagonalization,","cited_arxiv_id":null,"evidence_quote":"Supplies the channel-operator diagonalization idea the authors generalize from a single active aperture to the two-leg reflective cascade of a BD-RIS."},{"cited_title":"Capacity maximization for MIMO channels assisted by beyond-diagonal RIS,","cited_arxiv_id":null,"evidence_quote":"Supports physical realizability of a general unitary scattering matrix through non-reciprocal elements, justifying the lossless completion model."},{"cited_title":"Normal dilations and extensions of operators,","cited_arxiv_id":null,"evidence_quote":"Provides the Halmos unitary dilation that is the key proof tool of Proposition 2, realizing any contraction on the active subspace by a $2\\rho\\times 2\\rho$ unitary."},{"cited_title":"Steepest descent algorithms for optimization under unitary matrix constraint,","cited_arxiv_id":null,"evidence_quote":"Supplies the unitary steepest-descent update and geodesic step used to optimize the modal matrix on the unitary group."}],"review_version":1}