{"id":"3b0a85f7-99a0-4a80-8809-130ff740e62f","arxiv_id":"2607.29060","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For network-level ISAC, sensing resources should be concentrated on transceivers with favorable angular geometry (geometric water-filling), and a low-complexity variance-guided algorithm realizes this under communication-rate constraints.","lead":"This paper derives how a cellular network should spread OFDM subcarriers and symbols across distributed phones so that cooperative sensing of a target is as accurate as possible, based on each phone's angle and distance to the target. It proposes a 'geometric water-filling' principle and a low-complexity algorithm, validated in simulations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Prop. 3's optimality proof is a non sequitur: a fixed gradient direction does not imply proportional allocation under a budget constraint, and Eq. (26) approximates the wrong quantity.","rationale":"The paper's two-UE analysis (Prop. 1, Appendix A) is clean and the Fig. 4 sweep is consistent with the closed form. The multi-UE claim is the load-bearing part of the abstract and conclusion, and it is where the argument breaks. The reader identified the asymptotic i.i.d. assumption as the weak point; I agree this is fragile, but a more fundamental break occurs even inside that model: the derivation computes an approximate fixed gradient direction and then asserts optimality of proportional allocation without an optimality proof. That assertion is not a consequence of convexity or of the gradient direction; it is a category error between a descent direction and a constrained minimizer. The numerical validation of VGPA against baselines (Figs. 6-9) shows the heuristic is plausible and often better than uniform/random, but it does not independently verify optimality, since no exhaustive search or true minimizer is included and no code/data are provided. I therefore concur with the reader's REJECT: the central theorem as stated should not be accepted; a revision that reframes the multi-UE allocation as a heuristic, or supplies a genuine KKT/optimality proof, could make the paper viable.","tokens_in":20025,"tokens_out":11342,"duration_ms":116263,"concrete_test":"For a fixed K=3 or K=4 system with angles/distances drawn from a specified p(θ,d) satisfying the paper's i.i.d. assumption, compute the true expected CRLB over a fine grid on the feasible simplex ∑ N_k σ²_k = E_total, including boundary planes where one or more σ²_k → 0. Find the numerical minimizer and compare with (i) the Prop. 3 proportional-to-gradient allocation obtained from Eq. (38), and (ii) the KKT allocation where marginal CRLB reductions per resource are equal. Also compare the exact E[∂CRLB/∂σ²_k] (or full CRLB) with the approximation in Eq. (26)/(37) to test whether the 'fixed' direction survives. If the numerical minimizer differs from the proportional allocation—or if Eq. (37) fails near σ²_k→0—Prop. 3 is refuted.","verdict_should_be":"REJECT","load_bearing_attack":"The central multi-transmitter result (Prop. 3, §III-C.2) is not established. Even granting the i.i.d./LLN asymptotic model and the 'fixed' gradient direction in (38), the inference from 'the gradient direction is constant' to 'allocate σ² proportionally to that direction' is invalid for a differentiable objective under a linear budget constraint. A fixed gradient direction makes E[CRLB] depend on σ² only through one linear combination; the constrained minimizer is then found at a boundary point or at a KKT point where marginal reductions per unit resource are equalized (∇E[CRLB] ∝ (N1,...,NK)), not at the proportional interior allocation. The two-UE Proposition 1 itself satisfies the KKT condition, not proportionality to the gradient direction. In addition, Eq. (26) replaces E[A/B] by E[A]/E[B], and the resulting componentwise expression (37) is not the derivative of the true CRLB: it remains finite as σ²_k→0 whereas the actual CRLB diverges, so the claimed invariant direction is partly an artifact of the approximation. VGPA may be a reasonable heuristic, but the paper does not prove—and the statement can fail in simple instances—that proportional-to-gradient allocation is the constrained minimizer.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a two-stage framework for allocating time-frequency resources among distributed OFDMA transceivers in a network-level ISAC system, with the goal of minimizing the CRLB of target position estimation. For the two-transmitter case, Proposition 1 gives a closed-form optimal allocation of frequency apertures (variance of subcarrier indices) under a total variance budget; this part is a standard convex optimization result. For the multi-transmitter case, the paper uses an asymptotic law-of-large-numbers argument to claim that the gradient of the expected CRLB has a fixed direction, and then states Proposition 3, asserting that allocating aperture proportions along this gradient minimizes the expected CRLB. The authors call this 'geometric water-filling.' They then formulate the discrete subcarrier assignment as an NP-hard integer partitioning problem and propose the Variance-Guided Partitioning Algorithm (VGPA), a two-stage heuristic combining variance-guided initialization with Metropolis-based local search. Numerical simulations compare VGPA against random and uniform allocation in two-UE, multi-UE, Gaussian-mixture, and street-canyon scenarios.","tokens_in":20311,"tokens_out":8554,"duration_ms":89781,"significance":"If the multi-transmitter optimality claim were correct, the geometric water-filling principle would be a genuinely useful design insight for network-level ISAC, and the paper would make a strong theoretical contribution. The two-UE closed-form result is clean and appears correct given the stated CRLB model; it is a worthwhile contribution. The proposed VGPA is low-complexity and the numerical comparisons suggest that it can outperform geometry-agnostic baselines. However, the central multi-transmitter theorem is not established: the asymptotic derivation uses mathematically invalid expectation approximations, and the inference from a fixed gradient direction to proportional allocation is a non sequitur. As a result, the paper's headline claim of optimal resource allocation is not supported. The numerical evaluation of VGPA is also circular in that the algorithm is constructed to enforce the very ratios that Proposition 3 claims to be optimal. The practical heuristic may still be useful, but the paper as submitted does not provide a sound theoretical foundation for it.","major_comments":[{"comment":"The inference from a fixed gradient direction to proportional allocation is invalid. If ∇E[CRLB] = c·h with h constant, then the objective is, to first order, linear in the aperture vector. Under a linear budget constraint Σ N_k σ²_k ≤ E_total, the constrained minimizer of a linear function is an extreme point (put the entire budget into the most sensitive coordinate, subject to bounds), not the proportional interior point σ² ∝ h. The two-UE solution in Appendix A, Eq. (46), satisfies σ²_1/σ²_2 = sqrt(Γ₁N₂/(Γ₂N₁)), which is not proportional to the gradient of CRLB. Thus Proposition 3 is not merely unproved; it is false as stated. No proof or appendix is provided for it.","section":"§III-C.2, Eq. (38), Proposition 3"},{"comment":"Equation (26) replaces E[(A−B)/(ρ C)] with (E[A]−E[B])/(ρ E[C]), where A = ∂g₁/∂v_k · g₂, B = g₁·∂g₂/∂v_k, and C = g₂². This is mathematically invalid unless strong higher-order moment and correlation conditions are established, which the paper does not do. Chebyshev's LLN does not justify replacing the expectation of a ratio by the ratio of expectations. Every subsequent expression, including the non-positive derivative in Proposition 2 and the fixed gradient in Eq. (38), inherits this error. Eq. (37) is therefore not the derivative of the true expected CRLB.","section":"§III-C.1, Eq. (26)"},{"comment":"The factorization leading to Eqs. (27) and (28) is not justified. For example, E[f(k,k′)(f(k,k′)−f(k′,k)) v_k v_k′] is rewritten as E[f(k,k′)(f(k,k′)−f(k′,k))] E[v_k v_k′], which requires independence between the angular terms and the distance-dependent terms v_k, v_k′; the paper only assumes a joint PDF p(θ,d), not independence. Moreover, θ_{k1} is sometimes treated as a deterministic index (e.g., Eqs. (29)-(33)) and sometimes as random (as in the stated expectation over geometry). The quantity E[∂CRLB/∂σ²_{k1}] is not well-defined under the current notation.","section":"§III-C.1, Eqs. (27)-(31)"},{"comment":"The numerical evaluation does not test the optimality claim. VGPA is explicitly constructed to match the GWF ratios: Algorithm 1 uses the variance ratios α from the geometric water-filling strategy and Eq. (40) projects the partition onto those ratios. Comparing VGPA against random and uniform allocation therefore only shows that this particular heuristic beats those baselines; it does not validate Proposition 3. The abstract's claim that the scheme 'approaches the theoretical performance lower bound' is supported only in the two-UE simulations (Fig. 5); no comparable multi-UE lower-bound comparison is presented.","section":"§V, Figs. 6-9"}],"minor_comments":[{"comment":"The definition of E_total is badly typeset: the expression 'NX n=1 ... NX m=1' is garbled, making the summation range and the variance definition unclear. Please correct the notation.","section":"Eq. (12)"},{"comment":"The text says 'we first optimize the objective with respect to σ²_1 and σ²_1'; the second variable should be σ²_2.","section":"§III-B, after Eq. (13)"},{"comment":"Typo: 'the minimized communication rate of each UE shuld be satisfied' should be 'should'.","section":"§IV-A"},{"comment":"The text refers to 'Fig. 6(a)' and 'Fig. 6(b)' when discussing the results shown in Fig. 7. Please renumber the cross-references.","section":"§V, paragraph before Fig. 7"},{"comment":"The figure annotation appears to give α* = 0.632 while the text reports α = 0.607. Please reconcile the values.","section":"Fig. 4"}],"recommendation":"reject","confidential_remarks":"The two-UE result is solid, but the multi-transmitter optimality theorem is the main advertised contribution, and it is unsound. The flaws are not local presentation issues: Proposition 3 is false as stated, and the asymptotic derivation is mathematically invalid. A revised paper could potentially be reframed around the two-UE result and the VGPA as a heuristic, but that would be a substantially different contribution. I therefore recommend rejection in the current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this paper deserves a serious referee, but not for the theorem it headlines. What is actually new is the two-UE aperture allocation and the geometry-aware framing over prior work [21]. The two-UE derivation is clean Lagrange multiplier work, and the simulation confirms the closed form. The time-frequency isomorphism is a useful organizing idea, and VGPA is a reasonable low-complexity heuristic that handles communication-rate constraints in the experiments.\n\nThe soft spot is exactly where the reader put it. Proposition 3 is not established. The asymptotic passage replaces E[A/B] by E[A]/E[B] in (26), then treats the gradient direction as fixed. Even granting the i.i.d. model, a fixed gradient direction does not imply proportional allocation. The paper's own two-UE case demonstrates this: the optimum satisfies the KKT condition Γ_k/σ_k^4 ∝ N_k, giving σ_1^2/σ_2^2 = sqrt(Γ1 N2/(Γ2 N1)), not σ^2 proportional to ∇CRLB. A genuinely fixed linear gradient on a simplex would push to a corner, not to an interior proportional point. So the 'geometric water-filling principle' is currently a heuristic, not a theorem. The K>2 simulations compare against the paper's own 'optimal' allocation rather than an independent lower bound or exhaustive search, so they do not close that gap.\n\nWhat I would not do is discard the paper. The two-UE result is correct and useful; the algorithm design is practical; the street-canyon and Gaussian-mixture experiments are the right kind of robustness checks. The multi-UE section can be reframed as an asymptotic heuristic, with Proposition 3 replaced by a claim like 'under the LLN approximation, a reasonable strategy is to assign apertures according to sensitivity, and VGPA achieves that target.' That version is publishable after revision. The paper should also disclose the expectation-ratio approximation and stop calling the heuristic a proof. Missing code and data are minor here; this is an algorithm paper for TWC, not a benchmark submission.\n\nBottom line: accept for review. A good referee should push for the multi-UE rewrite, but a desk reject would discard a sound two-UE contribution and a practical heuristic that the simulations show works.","headline":"The two-UE closed form is solid and VGPA is a sensible heuristic, but the multi-UE 'proportional-to-gradient' optimality claim is not proven and as stated contradicts the paper's own KKT solution.","tokens_in":692,"tokens_out":719,"would_cite":false,"duration_ms":75607,"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":"For cooperative OFDMA sensing, the optimal time-frequency resource allocation distributes each transceiver's aperture along the gradient direction of the Cramér–Rao lower bound.","keywords":["network-level ISAC","cooperative sensing","resource allocation","Cramér–Rao lower bound","OFDMA","geometric water-filling","frequency aperture","localization accuracy"],"falsifier":"For a fixed small network (e.g., K=4 transceivers with a chosen, non-uniform coupled angle–distance distribution), numerically grid-search the continuous aperture split under a total-aperture constraint to locate the exact CRLB minimizer, and compare it with the allocation proportional to the gradient in eq. (38) computed from the assumed PDF. If the exact minimizer does not lie on that gradient direction, or if the expected-CRLB landscape has a minimum inconsistent with that direction, Proposition 3 fails. A lighter check: evaluate ∂CRLB/∂σ_k^2 at several feasible aperture points and test whe","tokens_in":19892,"feed_emoji":"📡","tokens_out":7876,"duration_ms":73208,"temperature":0.7,"pith_summary":"This paper asks how a multi-antenna base station should divide time-frequency resources — which subcarriers and symbol slots — among many user devices that jointly illuminate a target for sensing. The central claim is that the optimal division is not fixed by channel quality alone: it is controlled by each transceiver's aperture, defined as the variance of its allocated subcarrier or symbol indices, and the best allocation follows the gradient of the sensing Cramér–Rao lower bound with respect to those apertures. For two transmitters the paper gives a closed-form solution; for many transmitters it proves, under a statistical model treating UE geometry as i.i.d. draws from a fixed joint distribution, that distributing apertures proportionally to this gradient minimizes the expected CRLB, a rule it names geometric water-filling. Because delay and Doppler estimation are mathematically isomorphic, the same rule covers both ranging and velocity sensing. The paper then converts the continuous optimal apertures into discrete OFDMA subcarrier sets with a linear-complexity Variance-Guided Partitioning Algorithm, and numerical results show near-lower-bound sensing performance while honoring each UE's communication-rate requirement.","feed_headline":"Pour sensing resources into well-placed transceivers","feed_subtitle":"New proof shows subcarrier and symbol spread should follow the sensing-error gradient, not just channel quality.","key_machinery":"The central object is the aperture, the variance of the subcarrier indices (frequency aperture) or symbol indices (time aperture) assigned to a UE; it enters the Fisher information matrix and therefore sets the CRLB. The proof rests on three pieces: the two-transmitter closed form, which exposes the geometric water-filling intuition that resources are cross-leveraged between partners; the asymptotic multi-transmitter gradient, computed by modeling UE angles and distances as i.i.d. draws from a fixed joint PDF and applying the law of large numbers, which yields a gradient direction independent of the operating point (eq. (38)); and the variance-stride relationship (Prop. 4), which lets an alg","core_discovery":"In cooperative OFDMA sensing, the frequency-domain aperture of each transmitter — the variance of its assigned subcarrier indices — is the resource that controls localization accuracy, and the whole allocation problem reduces to distributing this variance under a total-aperture budget. For K=2 the optimal apertures have a closed form that reveals cross-linked geometric gains: each node's allocation is scaled by the partner's angular quality. For K>2, using a law-of-large-numbers approximation with i.i.d. UE angles and distances, the paper derives the expected gradient of the CRLB with respect to the apertures, proves the CRLB decreases with every aperture unless the UE sits at angle pi (coll","pith_inferences":["A natural adaptive extension would estimate the joint angle–distance distribution online from tracking data and recompute the aperture gradient, turning water-filling into a control policy for mobile targets or users.","The same variance–geometry coupling should generalize beyond apertures to other sensing resource dimensions such as power or antenna selection, since the FIM depends only on weighted sums of steering-vector outer products.","For finite small networks, an iterative scheme that re-evaluates the exact CRLB gradient at the current allocation point would likely beat the fixed asymptotic direction; comparing the two would measure how much the law-of-large-numbers assumption actually costs.","When the i.i.d. assumption fails — as in clustered or street-canyon deployments — the fixed-gradient theorem is not strictly applicable, and the paper's own street-canyon results suggest that a geometry-only (angle-based) allocation already captures most of the gain, pointing to a robustness hierarchy worth formalizing."],"forward_implications":["In OFDMA network-level ISAC, sensing resources should be allocated to transceivers in proportion to their geometric sensitivity, not merely their channel SNR.","Transceivers whose angle to the target is collinear with the target-receiver axis should receive minimal aperture, since their contribution to the CRLB vanishes.","The same gradient-based allocation rule applies to velocity estimation as to ranging, because delay and Doppler estimation are mathematically isomorphic.","The Variance-Guided Partitioning Algorithm offers a scalable, roughly linear-complexity way to turn continuous optimal apertures into discrete subcarrier sets while satisfying per-UE communication-rate constraints.","According to the paper's numerical results, uniform aperture allocation can underperform even random allocation when transceiver geometry is asymmetric, because it wastes budget on geometrically blind nodes."],"fun_headline_variants":["Proof: sensing error falls with every aperture except collinear","Resource allocation: follow the CRLB gradient, not channel quality","Sensing apertures: the geometric key to accuracy","Cooperative OFDMA: allocate variance, not power"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof for multi-transmitter networks assumes UE angles and distances are independent draws from a fixed probability distribution and that the network is large enough for the law of large numbers to replace random sums by expectations; if that does not hold at the actual network size, allocating apertures proportionally to the fixed gradient is not proven to minimize the CRLB.","fun_headline_variants_meta":{"raw":{"variants":["Proof: sensing error falls with every aperture except collinear","Resource allocation: follow the CRLB gradient, not channel quality","Sensing apertures: the geometric key to accuracy","Cooperative OFDMA: allocate variance, not power"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00055,"raw_usage":{"total_tokens":2488,"prompt_tokens":800,"completion_tokens":1688,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":544,"completion_tokens_details":{"reasoning_tokens":1621}},"tokens_in":544,"tokens_out":1688,"duration_ms":12173,"temperature":1.0,"reasoning_tokens":1621,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T14:24:00.417504+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed small network (e.g., K=4 transceivers with a chosen, non-uniform coupled angle–distance distribution), numerically grid-search the continuous aperture split under a total-aperture constraint to locate the exact CRLB minimizer, and compare it with the allocation proportional to the gradient in eq. (38) computed from the assumed PDF. If the exact minimizer does not lie on that gradient direction, or if the expected-CRLB landscape has a minimum inconsistent with that direction, Proposition 3 fails. A lighter check: evaluate ∂CRLB/∂σ_k^2 at several feasible aperture points and test whe","supporting_citations":[],"review_version":1}