{"id":"dee86b75-80c8-4f75-86d3-f17fcfead4cf","arxiv_id":"2507.03449","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Movable antenna positions and beamforming are jointly optimized to expand the achievable secrecy rate region of a two-user multicast-plus-confidential system beyond fixed-antenna baselines.","lead":"This paper shows that equipping a base station with movable antennas can improve the tradeoff between a secure message rate and a public multicast rate in a two-user wireless system, compared with fixed-position antennas. A smart generalist should read it as a concrete demonstration that moving antennas is a new design knob for sending public and secret data at the same time.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 1's null-steering construction requires even N; Theorem 1 is unproven for odd N.","rationale":"The reader's weakest_assumption already targets Lemma 1's Diophantine construction; my analysis sharpens this to a concrete parity obstruction. The proof of Theorem 1 is algebraically sound conditional on Lemma 1, but Lemma 1's construction of the null at user 2 requires r_{c,2} d ≈ 1/2 and simultaneously N r_{c,2} d ≈ 0 mod 1, which forces N even. Since the theorem is stated for all N and the simulations include odd N (Fig. 3c), the analytical support is incomplete. This is a genuine proof gap, not a mere lack of elegance: the printed construction fails for odd N, and no alternative construction is supplied. The other concerns (finite-region limitation, typo in r_{i,k}, SDR rank-count precision) are secondary: the finite-region limitation is explicitly acknowledged in footnote 1, the typo is easily corrected, and the rank-one conclusion still follows from a correct constraint count. Thus the most load-bearing issue is the parity-condition gap in Lemma 1.","tokens_in":10276,"tokens_out":29639,"duration_ms":309602,"concrete_test":"For N=3 with randomly chosen distinct user and steering angles, attempt to find an integer d satisfying conditions (14) as printed. Compute g_{c,2}(T) from (13) for the best such d; if the minimum of |sin(Nπ r_{c,2} d)| does not approach 0 as the approximation tolerance shrinks, the null cannot be achieved with this construction. Then test whether replacing the target r_{c,2} d = 1/2 with r_{c,2} d = 1/N restores the null and preserves the other gains; if it does, the proof gap is patchable, but the printed Lemma 1 remains incomplete for odd N.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central analytical claim (Theorem 1) rests on Lemma 1, which asserts the existence of a ULA spacing d such that the confidential beam gain is N at user 1 and 0 at user 2 while the multicast beam gain is N at both users. The proof selects d so that conditions (14) hold, in particular 2π r_{c,2} d → π (mod 2π) and 2π N r_{c,2} d → 0 (mod 2π). These two limits are consistent only when N is even: r_{c,2} d → 1/2 implies N r_{c,2} d → N/2, which is an integer only for even N. For odd N, the gain at user 2 tends to 1/N rather than 0, so the null is not obtained. Lemma 1 and Theorem 1 are stated for arbitrary N, and the numerical section includes odd N (N=3,5), so the proof as written does not cover odd N. Additionally, the argument requires the r_{i,k} values to be rationally independent, which holds only generically, while the theorem is stated without this qualification. Consequently, the claimed dominance of T* over all other APVs is not established for odd N or for degenerate LoS geometries.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies a downlink MISO physical-layer service integration (PHY-SI) system in which a base station equipped with N movable antennas transmits a multicast message to two users and a confidential message to user 1, with user 2 acting as a potential eavesdropper. The aim is to maximize the secrecy rate subject to a multicast rate constraint, a power constraint, and antenna-position constraints. In the single-MA case, the authors derive a closed-form optimal power allocation and compare the resulting rate region with time-sharing, concluding that PHY-SI may lose to time-sharing with one MA. In the LoS case, Lemma 1 constructs an APV and array-response beamformers for an arbitrarily large movement region, and Theorem 1 claims that this APV dominates every other APV in both multicast and secrecy rates. The general problem is then addressed by a two-layer algorithm: an SDR-based inner beamforming solver and a discrete-sampling outer APV search. Numerical results compare the proposed scheme with FPA and PSO benchmarks and study the effects of sampling resolution, region size, and number of antennas.","tokens_in":10476,"tokens_out":12881,"duration_ms":162809,"significance":"If the analytical claims are fully established, the paper makes a useful conceptual contribution: movable antennas can simultaneously configure a multicast beam toward both users and a null toward the eavesdropper, which is a genuinely different capability from single-service MA designs. The LoS dominance argument is parameter-free and does not rely on fitted data, and the single-MA power-allocation derivation in Eqs. (5)-(6) is algebraically sound. The numerical study is informative and includes a PSO benchmark, which strengthens the empirical comparison. However, the proof of Lemma 1 contains a coordinate inconsistency and an unproven parity condition, and the SDR rank-one argument in Section IV-A rests on a miscounted constraint number. These gaps affect the two main theoretical claims, so the paper needs substantial revision before the results can be accepted.","major_comments":[{"comment":"The definition of r_{i,k} in Eq. (13) is inconsistent with the array response in Eq. (8) when the antennas are placed at t_n = [(n-1)d, 0]^T. Substituting this position into Eq. (8) gives a phase difference proportional to (sin θ_k cos φ_k - sin θ_i cos φ_i), with no term involving (cos θ_k - cos θ_i). Since the Diophantine conditions in Eq. (14) are imposed on the printed r_{i,k}, the proof of Eq. (12) does not currently follow from the actual beam gains. The authors should correct the definition of r_{i,k} and re-derive the limiting gains in Eq. (13).","section":"Section III-B, Eq. (13)"},{"comment":"The null-steering construction in Lemma 1 requires 2π r_{c,2} d → π (mod 2π) and 2π N r_{c,2} d → 0 (mod 2π). These two congruences are compatible only when N is even: if r_{c,2} d ≡ 1/2 mod 1, then N r_{c,2} d ≡ N/2 mod 1, which is 0 only for even N. For odd N the limiting confidential beam gain at user 2 is 1/N rather than 0, so Eq. (12) is not achieved by the stated construction. Theorem 1 is therefore unproven for odd N, even though the numerical section includes N=3 and N=5. The theorem should be restricted to even N, or a different construction valid for all N must be supplied.","section":"Lemma 1, Eq. (14), and Theorem 1"},{"comment":"The rank bound is stated as rank^2(Z*) + rank^2(Γ*) ≤ 3, where '3' is claimed to be the number of linear equalities and inequalities in problem (22). Problem (22) actually contains four affine constraints: the Charnes-Cooper equality, the two multicast-rate inequalities, and the trace constraint. The standard rank result would therefore give rank^2(Z*) + rank^2(Γ*) ≤ 4, which permits, for example, rank(Z*) = 2 and rank(Γ*) = 0 and does not force both matrices to be rank-one. Consequently, the claimed global optimality of the SDR-based inner solution is not established by the cited argument; either a corrected proof of rank-one recovery or a reformulation of the inner solver as a heuristic is needed.","section":"Section IV-A, after Eq. (22)"}],"minor_comments":[{"comment":"The proof calls d a positive integer, but d also acts as a physical antenna spacing in the phase term 2π r_{i,k} d; the authors should state that d is an integer multiple of the wavelength, or otherwise normalize the spacing variable.","section":"Lemma 1 proof"},{"comment":"The proof uses the assumption that all r_{i,k} are irrational and rationally independent, but Lemma 1 and Theorem 1 are stated without this qualification. The statements should either include the generic-geometry assumption or prove the result for all angle tuples.","section":"Lemma 1 and Theorem 1"},{"comment":"The optimal power allocation in Eq. (5) gives P_c^* = (P|h_2(t)|^2 - τ_{ms} σ^2) / ((τ_{ms}+1)|h_2(t)|^2), which is nonnegative only when P|h_2(t)|^2 ≥ τ_{ms} σ^2; the authors should state this feasibility condition explicitly.","section":"Section III-A, Eq. (5)"},{"comment":"The notation θ_0 ≠ θ_c ≠ θ_k is ambiguous; pairwise distinctness of θ_0, θ_c, θ_1, and θ_2 should be written explicitly.","section":"Lemma 1 statement"}],"recommendation":"major_revision","confidential_remarks":"The two gaps in Lemma 1 are localized and likely repairable (by correcting r_{i,k} and by restricting the theorem to even N, or by finding a genuinely different construction). The rank-bound gap in Section IV-A is more serious because it concerns the claimed optimality guarantee of the main algorithmic component; if the rank-one recovery cannot be repaired, the inner SDR should be presented as a heuristic and the numerical claims adjusted accordingly. The paper's novelty relative to the cited MA beamforming/position-optimization tools is incremental but acceptable for a letter if the theoretical statements are corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is the first to apply movable antennas to physical-layer service integration, and it has one result that actually made me stop: in a LoS channel, with an arbitrarily large movement region and even N, there is an antenna placement that nulls the confidential signal at user 2 while giving full gain to user 1 and full multicast gain to both users. That is a neat extension of the null-steering idea in the MA literature, and it goes beyond the existing secrecy-only or multicast-only MA papers.\n\nWhat works well: the single-MA power allocation derivation is clean and correct as far as I checked. The point that a single moving antenna can make PHY-SI worse than time sharing, because the optimal position differs for the two messages and you can't share positions, is a genuinely new observation. The SDR-plus-discrete-sampling algorithm is reasonable for a letter, and the numerical comparisons to FPA and PSO are consistent.\n\nNow the soft spots. The main theorem is stated for any N and any LoS geometry, but the proof does not cover odd N. Condition (14) puts 2π r_c,2 d near π modulo 2π, and simultaneously requires 2π N r_c,2 d near 0 modulo 2π. These are compatible only when N is even; for odd N the confidential gain at user 2 tends to 1/N, not 0. Since the simulations include N=3 and N=5, this is not an edge case for the numerics. The theorem also depends on the r_i,k being rationally independent, which the authors mention in the proof as 'with probability one' but omit from the statement. And Lemma 1 calls d a positive integer when it is a physical spacing in meters; the Diophantine approximation presumably applies to normalized frequency differences, so the statement as written has a unit inconsistency. None of those are fatal if the theorem is restated with the right hypotheses, but as written Theorem 1 is too strong.\n\nMinor issues: the rank-count argument for the SDR says rank^2(Z*)+rank^2(Γ*) ≤ 3 and concludes both are rank one. That conclusion isn't forced by the bound alone; you need to handle the possibility of a rank-zero solution. Also no code or error bars, which is common for a letter but worth noting.\n\nBottom line: this is a worthwhile contribution to the MA/PHY-SI subfield, and the central idea deserves to see the light after a careful revision. A referee should demand that the odd-N case be either proven, fixed with a different construction, or explicitly excluded from the theorem. I would send it to review.","headline":"First MA + PHY-SI paper with a genuinely thought-provoking LoS dominance result, but the main theorem only holds for even N — stated for all N and simulated for odd N.","tokens_in":11049,"tokens_out":4286,"would_cite":true,"duration_ms":47836,"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":"Movable antennas let one base station run secrecy and multicast at full array gain, which fixed-position antennas cannot match.","keywords":["movable antennas","physical-layer service integration","secrecy rate region","multicast beamforming","semidefinite relaxation","discrete sampling","line-of-sight channels","antenna position optimization"],"falsifier":"Take an odd number of movable antennas ($N=3$) and apply the equal-spacing construction of Lemma 1 with $2\\pi r_{c,2}d \\approx \\pi \\pmod{2\\pi}$: the confidential beam gain at user 2 becomes $1/N$, not $0$, so the claimed simultaneous full gain and perfect null is not achieved by this construction, and Theorem 1's universal dominance over all antenna positions would need a different geometry to survive. Alternatively, deliberately choose user directions that make the $r_{i,k}$ values rationally dependent and search exhaustively over a finite grid: if any antenna layout yields a higher secrecy rate at the same multicast rate than the Lemma-1 layout, the asymptotic dominance claim fails for that finite region.","tokens_in":10059,"feed_emoji":"📡","tokens_out":16011,"duration_ms":155735,"temperature":0.7,"pith_summary":"Physical-layer service integration sends a public (multicast) message and a confidential message at the same time from one transmitter; conventionally, fixed-position antennas limit how well these two services can coexist. This paper asks whether movable antennas, antennas that can be repositioned inside a small square region, can expand the achievable set of (multicast rate, secrecy rate) pairs. It establishes that under line-of-sight channels with an arbitrarily large movement region, there is an antenna layout that gives the confidential beam full array gain at the intended user, zero gain at the other user, and the multicast beam full array gain at both users simultaneously. From that, it proves that this layout dominates every other antenna position in both rates. The paper also develops a two-layer optimization algorithm, semidefinite relaxation for beamforming and discrete sampling for antenna positions, and shows numerically that the secrecy rate region is much larger with movable antennas than with fixed-position antennas.","feed_headline":"Movable antennas unlock full-gain secrecy plus multicast","feed_subtitle":"Repositioning antennas cancels the eavesdropper while keeping full beam gain for both users.","key_machinery":"The load-bearing identity is the beam-gain formula for a uniform linear array, $g_{i,k}(T) = \\frac{1}{N}\\frac{1-\\cos(2\\pi N r_{i,k}d)}{1-\\cos(2\\pi r_{i,k}d)}$, where $r_{i,k}$ is the difference of inter-antenna phase increments toward user $k$ for beam $i$. The whole dominance argument reduces to choosing one integer spacing $d$ such that, modulo $2\\pi$, each $2\\pi r_{i,k}d$ takes the required value: near $0$ for the three beams that should add constructively, and near $\\pi$ for the confidential beam at the non-intended user so the numerator vanishes when $N$ is even. Lemma 1 obtains such a $d$ from a Diophantine approximation result, using the fact that the $r_{i,k}$ values are, with probability one, irrational and not rationally related. On the algorithmic side, the inner layer uses semidefinite relaxation with a Charnes-Cooper transformation, a change of variables that convexifies the fractional objective, to solve the beamforming problem for a fixed antenna layout, and the outer layer performs a sequential discrete sampling over candidate positions.","core_discovery":"The paper's central claim is Theorem 1: in a line-of-sight channel with an arbitrarily large movement region, there exists an antenna position vector $T^\\star$ such that the achievable multicast rate and secrecy rate both satisfy $R_0(T^\\star) \\ge R_0(T)$ and $R_c(T^\\star) \\ge R_c(T)$ for every other position vector $T$. The construction behind it places the $N$ movable antennas in a uniform line and chooses the spacing $d$ so that the confidential beam adds constructively at user 1 and destructively at user 2, while the multicast beam adds constructively at both users; for an even number of antennas this yields beam gains $g_{c,1}=g_{0,1}=g_{0,2}=N$ and $g_{c,2}=0$. Thus one antenna geometry simultaneously serves two service types at the maximum possible array gain, something a fixed array cannot generically do. The paper further shows that a single movable antenna can lose to simple time-sharing because one position cannot serve both services well, whereas multiple movable antennas restore the advantage of simultaneous transmission.","pith_inferences":["The one-spacing-fits-all construction is essentially spatial interference alignment: a single degree of freedom, the inter-antenna spacing, simultaneously aligns three beams constructively and one beam destructively. This suggests the same Diophantine machinery could extend to more than two users or multiple eavesdroppers by allocating more antennas to satisfy more simultaneous phase conditions.","The single-MA-versus-time-sharing failure points to a threshold effect: for each channel realization there may be a minimum number of antennas beyond which movable-antenna PHY-SI always beats time-sharing; finding that threshold as a function of channel statistics would be a natural next step, but the paper does not compute it.","Because the theory needs an arbitrarily large region while the numerical gains saturate at a few wavelengths, a practical conjecture is that a finite region of about $8\\lambda$ already captures most of the asymptotic benefit; verifying this with an exhaustive search over all feasible layouts would test how much the asymptotic assumption matters.","The reliance on direction-dependent phase constants being irrational and not rationally related means special geometries, such as users placed symmetrically so that two $r_{i,k}$ values are rational multiples of each other, may break the construction; one could deliberately construct such a geometry and check whether a different antenna layout still dominates fixed positions."],"forward_implications":["In line-of-sight scenarios with a large enough movement region, movable antennas can achieve full array gain for multicast at both users and, at the same time, a perfect spatial null for the confidential message at the non-intended user; no fixed-position array can match both.","The secrecy rate region of the movable-antenna system contains the fixed-position-antenna region and also beats the time-sharing benchmark, reversing the single-antenna case where time-sharing can win.","The proposed two-layer algorithm reaches these gains with worst-case complexity $O(N^{7.5}M^2)$, where $M$ is the number of sampling points per dimension; the inner SDR step has the same complexity order as conventional fixed-position-antenna PHY-SI.","Increasing the number of antennas $N$ or the movement region size $A$ enlarges the secrecy-multicast rate region, with the gain saturating once the region is large enough.","Coarse sampling (e.g., 25 points per dimension) already outperforms the fixed-position benchmark, so the theoretical full-array-gain result does not require fine-grained continuous positioning."],"supporting_citations":[{"why":"Supplies the Diophantine approximation fact that guarantees an integer spacing $d$ satisfying the required simultaneous modular conditions in Lemma 1.","marker":"[16]"},{"why":"Defines the physical-layer service integration rate region and superposition-coding model that the optimization problem is built on.","marker":"[1]"},{"why":"Provides the field-response channel model for movable antennas used in the simulations and the channel-acquisition techniques assumed for problem (P1).","marker":"[5]"},{"why":"Establishes the movable-antenna array's ability to achieve full array gain with a null, the precursor effect that Lemma 1 extends to two service types.","marker":"[6]"},{"why":"Provides the semidefinite relaxation and rank-one recovery argument for integrated multicast/secrecy beamforming used in the inner-layer solver.","marker":"[2]"},{"why":"Supplies the discrete sampling and sequential search method the outer layer uses for antenna position optimization.","marker":"[17]"},{"why":"Justifies the asymptotic 'arbitrarily large movement region' analysis by analogy to massive MIMO analysis.","marker":"[15]"}],"fun_headline_variants":["Antenna movement wins full gain for secrecy and multicast","Movable antennas max both secrecy and multicast rates","One antenna geometry serves secrecy and multicast at peak","Repositioning antennas beats fixed for dual services","Movable antenna trick boosts secrecy and multicast together"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The result stands on the assumption that a single antenna spacing can be chosen to make the confidential beam add up at user 1, cancel at user 2, and the multicast beam add up at both users at the same time; the proof requires the relevant direction-dependent constants to be irrational and not rational multiples of each other, the movement region to be large enough to contain the required spacing, and, in the printed construction, an even number of antennas to make the cancellation exact.","fun_headline_variants_meta":{"raw":{"variants":["Antenna movement wins full gain for secrecy and multicast","Movable antennas max both secrecy and multicast rates","One antenna geometry serves secrecy and multicast at peak","Repositioning antennas beats fixed for dual services","Movable antenna trick boosts secrecy and multicast together"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00016,"raw_usage":{"total_tokens":1253,"prompt_tokens":988,"completion_tokens":265,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":604,"completion_tokens_details":{"reasoning_tokens":193}},"tokens_in":604,"tokens_out":265,"duration_ms":3586,"temperature":1.0,"reasoning_tokens":193,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:11:39.062810+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an odd number of movable antennas ($N=3$) and apply the equal-spacing construction of Lemma 1 with $2\\pi r_{c,2}d \\approx \\pi \\pmod{2\\pi}$: the confidential beam gain at user 2 becomes $1/N$, not $0$, so the claimed simultaneous full gain and perfect null is not achieved by this construction, and Theorem 1's universal dominance over all antenna positions would need a different geometry to survive. Alternatively, deliberately choose user directions that make the $r_{i,k}$ values rationally dependent and search exhaustively over a finite grid: if any antenna layout yields a higher secrecy rate at the same multicast rate than the Lemma-1 layout, the asymptotic dominance claim fails for that finite region.","supporting_citations":[{"cited_title":"The interference channel revisi ted: Aligning interference by adjusting antenna separation,","cited_arxiv_id":null,"evidence_quote":"Supplies the Diophantine approximation fact that guarantees an integer spacing $d$ satisfying the required simultaneous modular conditions in Lemma 1."},{"cited_title":"Physical layer service integr ation in wireless networks: Signal processing challenges,","cited_arxiv_id":null,"evidence_quote":"Defines the physical-layer service integration rate region and superposition-coding model that the optimization problem is built on."},{"cited_title":"A tutorial on movable antennas for wirele ss networks,","cited_arxiv_id":null,"evidence_quote":"Provides the field-response channel model for movable antennas used in the simulations and the channel-acquisition techniques assumed for problem (P1)."},{"cited_title":"Movable-antenna array enhanc ed beam- forming: Achieving full array gain with null steering,","cited_arxiv_id":null,"evidence_quote":"Establishes the movable-antenna array's ability to achieve full array gain with a null, the precursor effect that Lemma 1 extends to two service types."},{"cited_title":"On artificial-no ise-aided transmit design for multiuser MISO systems with integrated services,","cited_arxiv_id":null,"evidence_quote":"Provides the semidefinite relaxation and rank-one recovery argument for integrated multicast/secrecy beamforming used in the inner-layer solver."},{"cited_title":"Movable-an tenna position optimization: A graph-based approach,","cited_arxiv_id":null,"evidence_quote":"Supplies the discrete sampling and sequential search method the outer layer uses for antenna position optimization."},{"cited_title":"Massive MI MO networks: Spectral, energy, and hardware efficiency","cited_arxiv_id":null,"evidence_quote":"Justifies the asymptotic 'arbitrarily large movement region' analysis by analogy to massive MIMO analysis."}],"review_version":1}