{"id":"fe68f045-0953-41be-a9f0-ff013ce761e8","arxiv_id":"2506.15338","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Analytical coverage and capacity expressions for HAP networks with RIS and urban blockages, based on approximating the signal-to-interference ratio as a generalized Beta prime distribution.","lead":"This paper derives closed-form formulas for coverage and capacity in a city network where high-altitude flying base stations connect to phones through smart reflecting surfaces. The value is a fast analytical tool for network designers, but it relies on several strong modeling simplifications that need scrutiny.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (21) uses a deterministic-count second-moment identity for a PPP sum; the Poisson identity requires Mvis^2, not Mvis(Mvis-1), so alpha_D/beta_D and all closed forms are mis-specified.","rationale":"The reader's undefined-omega_h point is real but secondary; the more decisive problem is Eq. (21). The central claim is a moment-matched Beta prime distribution for the SIR, and that requires alpha_D and beta_D to be the correct second moments of the PPP interference sum. Eq. (21) uses Mvis(Mvis-1), which is the second-factorial-moment identity for a fixed number of terms, but the number of visible HAPs in the stated nonhomogeneous PPP is Poisson, for which E[N(N-1)] = Mvis^2. The resulting variance formula undercounts interference fluctuations; for the default parameters Mvis is about 2.9 and the subtractive term is of order Mvis E[X]^2, so alpha_D can change by roughly a factor of two. That error propagates into the Beta prime parameters and therefore into the coverage probability and ergodic capacity expressions. This is an internal inconsistency in the derivation, not a disagreement with an external consensus. The fix is local, so CONDITIONAL remains the appropriate verdict rather than REJECT: correcting Eq. (21) and re-validating the figures should resolve the issue. I do not think the undefined omega_h alone would force a different verdict, since it is an ambiguity that could disappear if omega_h is explicitly set or taken to infinity. Because the reader identified a parameter in the interference-moment chain but not this exact algebraic error, my agreement with the reader's weakest assumption is partial.","tokens_in":9605,"tokens_out":11883,"duration_ms":125660,"concrete_test":"Implement an independent Monte Carlo simulation of the nonhomogeneous PPP Phi_LOS and empirically measure the variance of D for the default parameters; compare it with Eq. (21) and with the compound-Poisson expression Var[D] = Mvis E[|h_i|^4] E[R_h^{-2 epsilon_h}]. If Eq. (21) disagrees appreciably, replace Mvis(Mvis-1) by Mvis^2 in Eq. (21), recompute alpha_D and beta_D from Eqs. (16) and (19), and regenerate Fig. 3. If the corrected coverage curve shifts by more than a small tolerance (e.g., 0.05 in probability or 0.5 dB in threshold), the published closed forms are invalid as written.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The SIR approximation in Sec. III-B is the load-bearing step, and its validity requires the second moments of the interference denominator D = sum_{i in Phi_LOS} |h_i|^2 R_h^{-epsilon_h} to be correct. Eq. (21) states E[A_D^2] = Mvis E[|h_i|^4] E[R_h^{-2 epsilon_h}] + Mvis(Mvis - 1) E[|h_i|^2]^2 E[R_h^{-epsilon_h}]^2. This is the second moment of a sum with a fixed, deterministic number of terms, not of a sum over a Poisson point process. For the nonhomogeneous PPP Phi_LOS stated in Sec. II-C, the number of visible HAPs is Poisson with mean Mvis, so E[N(N-1)] = Mvis^2, not Mvis(Mvis - 1). Consequently, via Eq. (19), the paper's variance becomes Var[D] = Mvis E[|h_i|^4] E[R_h^{-2 epsilon_h}] - Mvis E[|h_i|^2]^2 E[R_h^{-epsilon_h}]^2, whereas the correct compound-Poisson variance is simply the first term. With the default parameters, Mvis = 2 pi lambda_HAP exp(-rho)/zeta^2 is about 2.9, and the subtracted term is of the same order as the first term, so alpha_D is substantially overestimated. This is a concrete internal inconsistency, independent of the undefined omega_h in Lemma 1: even if omega_h is set to infinity, Eq. (21) remains wrong, and the error propagates into Theorem 1 and Theorem 2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a stochastic-geometry model of an urban RIS-assisted HAP network in which a ground user is served by the nearest HAP through the nearest visible RIS, and interference comes from other visible HAPs. Building blockages are modeled via a Boolean scheme. To avoid the Laplace transform, the numerator and denominator of the SIR are approximated as Gamma random variables using second-moment matching, which leads to a generalized Beta prime approximation for the SIR. The paper derives closed-form expressions for coverage probability (Theorem 1) and ergodic capacity (Theorem 2), and reports Monte Carlo validation for a range of system parameters.","tokens_in":9963,"tokens_out":10907,"duration_ms":105770,"significance":"The paper targets a relevant and timely scenario, and the idea of bypassing Laplace transforms via moment-matched Gamma approximations is potentially useful. I credit the authors for a physically parameterized model with no curve fitting: the Beta prime parameters are computed from the system's statistical moments, and the qualitative conclusions (blockages can help by suppressing interference, HAP densification hurts, RIS densification eventually saturates) are plausible and worth reporting. However, the mathematical core has correctness issues that affect the closed forms, so the contribution as currently presented cannot be accepted; the path to a publishable version is clear.","major_comments":[{"comment":"The second moment of the interference power A_D is computed in Eq. (21) using E[A_D^2] = M_vis E[|h_i|^4] E[R_h^{-2*epsilon_h}] + M_vis(M_vis-1) (E[|h_i|^2])^2 (E[R_h^{-epsilon_h}])^2. This is the second moment of a sum with a fixed number of terms. For the nonhomogeneous PPP Phi_LOS in Section II-C, the number of visible HAPs is Poisson with mean M_vis, so the correct identity is E[A_D^2] = M_vis E[|h_i|^4] E[R_h^{-2*epsilon_h}] + M_vis^2 (E[|h_i|^2])^2 (E[R_h^{-epsilon_h}])^2. Consequently, the variance in Eq. (19) is underestimated (the term M_vis (E[X])^2 is subtracted instead of zero), alpha_D is overestimated, and the error propagates through Eq. (16) into Theorem 1 and Theorem 2. The authors should replace M_vis(M_vis-1) with M_vis^2 and re-derive the numerical results.","section":"III-B, Eq. (21)"},{"comment":"The PDF of the visible-HAP horizontal distance is normalized over the finite interval [0, omega_h] in Eq. (6), but omega_h is not defined in terms of the system parameters; the only statement is the vague inequality 'omega_h >= sqrt(2 exp(-rho))/zeta' in Lemma 1. Meanwhile, Eq. (2) gives M_vis as the mean number of visible HAPs over the infinite plane. These two choices are inconsistent unless the finite truncation is reflected in M_vis. Since the denominator moments in Eq. (10) depend on omega_h, all subsequent alpha_D, beta_D, Theorem 1, and Theorem 2 inherit this ambiguity. The authors must define omega_h (e.g., set omega_h = infinity, or derive it from the visibility model) and make M_vis consistent.","section":"III-A, Lemma 1 and Eq. (10)"},{"comment":"The interference moments use the unconditional visible-HAP distance distribution from Lemma 1, but the serving HAP is the nearest HAP of the PPP Phi. Thus all interfering HAPs are at horizontal distances no smaller than the serving HAP's horizontal distance, and the interference point process is Phi with the nearest point removed (a Palm distribution). The distance PDF in Lemma 1 is not conditioned on this exclusion, and the independence of N and D assumed in Section III-B is not justified. This is a load-bearing simplification: the reported numerical agreement should be re-examined once the moments are computed under the correct conditioning.","section":"III-A and III-B, Eq. (18)"}],"minor_comments":[{"comment":"In the parameter list, 'mu_RIS' appears to be a typo; it should be 'lambda_RIS' to match the notation used elsewhere in the paper.","section":"Section II-A"},{"comment":"The support of f_{omega_h,any}(w_h) is written as '0 <= omega_h <= infinity'; it should be '0 <= w_h <= omega_h' to be consistent with the normalization in Eq. (9).","section":"Eq. (8)"},{"comment":"The statement 'omega_h >= sqrt(2 exp(-rho))/zeta is the horizontal length of the link [14, Theorem 7]' is unclear; reference [14] concerns the ratio of gamma variates and does not define a physical link length. Please provide an explicit definition of omega_h.","section":"Lemma 1"},{"comment":"The term '(zeta)2' should be 'zeta^2'.","section":"Eq. (13)"},{"comment":"The regularized hypergeometric functions 2F1_tilde and 3F2_tilde are used without definition; a definition or a standard reference should be provided.","section":"Theorem 2"},{"comment":"The Monte Carlo simulation procedure is not described in detail (e.g., simulation region, number of trials, realization of the Boolean blockage model). A precise description is needed to reproduce the validation, especially given the theoretical concerns above.","section":"Section V"}],"recommendation":"major_revision","confidential_remarks":"The validation figures show excellent agreement between the approximation and simulation, which is surprising given the moment error in Eq. (21). I recommend requiring the authors to provide reproduction details or code. Also, the paper leans heavily on the authors' own prior work ([10] and [18]) for the distance distribution and the capacity integral; an independent derivation or validation of these components would strengthen the revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a readable, legitimate extension of the authors' own HAP-RIS conference paper, but there is a real bug in the load-bearing second-moment calculation. As published, the closed forms in Theorems 1 and 2 are mis-specified; the fix is small, but the paper should not appear without it.\n\nWhat is actually new: interference from visible non-serving HAPs is added to their earlier blockage-aware HAP-RIS model, the numerator and denominator are approximated by Gamma variables, and the SIR becomes a generalized Beta prime whose parameters come from physical moments rather than simulation fitting. The blockage-mitigates-interference result is a plausible, useful consequence of the model, and the Monte Carlo agreement in the figures suggests the moment-matching idea deserves attention. The paper is clearly written, the distance distributions are mostly standard, and there are no fitted parameters.\n\nSoft spots, in order of severity:\n\n1. Eq. (21) uses the second-moment identity for a fixed number of terms. The visible HAPs form a thinned PPP, so the count is Poisson with mean M_vis; for a Poisson count, E[N(N-1)] = M_vis^2, not M_vis(M_vis-1). Consequently Var[D] should be just M_vis E[|h|^4] E[R^{-2ε}], with no subtracted cross term. With their parameters M_vis ≈ 2.9, the term they subtract is the same order as what remains, so α_D is off by a factor of roughly 2–3, and Theorems 1 and 2 inherit that error. This is a concrete internal error, independent of the ω_h question: even if ω_h → ∞, the equation is wrong. Because the figures claim analytical and simulated curves agree, a referee should ask how the interference was generated — a fixed-count simulation would reproduce the wrong formula.\n\n2. Lemma 1 and Eq. (10) normalize the visible-HAP distance PDF over [0, ω_h], but ω_h is never defined in Section V, and M_vis in Eq. (2) is the infinite-plane value. That is a genuine gap, though it reads like an unreported truncation choice rather than a conceptual mistake.\n\n3. Theorem 2 imports the capacity formula from the authors' own [18] without proof. Self-citation is fine when the cited result is real and published, but as written a referee cannot verify the integration locally. Minor.\n\nWho this is for: researchers doing stochastic-geometry analysis of non-terrestrial RIS networks who want a tractable coverage/capacity estimator. The concept deserves referee time; the right verdict is major revision — fix Eq. (21), define ω_h (or compute all moments over the same domain as M_vis), re-run the validations, and show the corrected curves.\n\nI would send it to review. I would not cite it until the moment identity is fixed.","headline":"A readable extension of the authors' own HAP-RIS work, but Eq. (21) mis-specifies the second moment of the interference for a Poisson count, so the closed forms as published are wrong until fixed.","tokens_in":10478,"tokens_out":11568,"would_cite":false,"duration_ms":97256,"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":"An urban HAP-RIS network's SIR can be approximated by a generalized Beta prime distribution, yielding closed-form coverage and capacity expressions.","keywords":["high-altitude platforms","reconfigurable intelligent surfaces","stochastic geometry","Poisson point processes","Boolean blockage model","generalized Beta prime distribution","coverage probability","ergodic capacity"],"falsifier":"Run the Section V Monte Carlo setup with lambda_HAP = 5e-6 per square meter, lambda_RIS = 50e-6 per square meter, lambda_B = 100e-6 per square meter, H_HAP = 50 km, H_RIS = 50 m, L = 256, and Rician K = 1, then compute the empirical SIR distribution from many independent spatial realizations. If the Kolmogorov-Smirnov distance between that empirical CDF and the Beta-prime CDF in Eq. (22) is large, or if the fitted parameters alpha_D and beta_D change when the visible-HAP cutoff omega_h is chosen differently, then the closed-form approximation is not self-contained. The same experiment with low blockage density would test whether the blockage-mitigates-interference conclusion reverses when the desired link is more frequently obstructed.","tokens_in":9388,"feed_emoji":"📡","tokens_out":8502,"duration_ms":78959,"temperature":0.7,"pith_summary":"High-altitude platform (HAP) networks serving urban users through rooftop reconfigurable intelligent surfaces (RISs) are usually analyzed by simulation because interference from other visible HAPs makes exact signal-to-interference statistics intractable. The paper argues that in an interference-limited setting the SIR can be accurately approximated as the ratio of two Gamma-distributed random variables, which is precisely a generalized Beta prime distribution. With the four parameters of that distribution fixed by matching the first two moments of signal power and interference power, coverage probability and ergodic capacity become closed-form expressions rather than Monte Carlo estimates. If the approximation holds, a network designer can sweep HAP density, RIS density, building density, RIS height, and number of reflecting elements directly from formulas. The paper validates the approximation against Monte Carlo simulations and uses it to show that buildings can help by blocking interfering HAPs, and that denser HAP deployment hurts while denser RIS deployment helps.","feed_headline":"One distribution predicts HAP-RIS coverage and capacity in cities","feed_subtitle":"Stochastic-geometry analysis gives designers formulas for coverage and capacity, replacing Monte Carlo simulation.","key_machinery":"The central object is the generalized Beta prime distribution, the distribution of the ratio of two independent Gamma variables. The machinery is moment matching: for the desired signal term and the interference term, the paper sets the Gamma shape and scale parameters from the mean and variance of each term. The moments of the fading amplitudes come from the Rician moment formula, while the moments of the distances come from the visible-HAP distance PDF, the nearest-visible-RIS distance PDF, and the nearest-HAP distance PDF. These four parameters feed the Beta-prime CDF and PDF, which in turn yield closed-form coverage probability and ergodic capacity. The identifying maneuver is to avoid the Laplace transform of the interference and to let the ratio distribution carry all the spatial randomness of the network.","core_discovery":"The paper's central discovery is that the SIR of a user served by the nearest high-altitude platform through the nearest visible reconfigurable intelligent surface, with interference from the other visible HAPs, can be treated as a generalized Beta prime random variable. Writing the desired signal power as N and the interference power as D, the paper approximates each by a Gamma distribution with shape and scale parameters obtained by matching the first two moments, so that SIR follows the generalized Beta prime distribution. All moment ingredients, including Rician fading moments, visible-HAP distance moments, nearest-visible-RIS distance moments, and nearest-HAP distance moments, are expressed in closed form, and the resulting Beta-prime CDF and PDF are integrated in Theorem 1 for coverage probability and Theorem 2 for ergodic capacity. This replaces the usual Laplace-transform route, which has no closed form, with a four-parameter fit that the paper validates against Monte Carlo simulation.","pith_inferences":["The argument's structure does not depend on HAPs specifically: any interference-limited network whose desired and interfering powers can be moment-matched as Gamma variables would inherit the same Beta-prime CDF and capacity formulas, so the method should transfer to other aerial or terrestrial RIS deployments.","Because the blockage model enters only through the mean number of visible HAPs and the visibility probability, the 'blockages are helpful' conclusion is strongest in interference-limited regimes and could reverse when the desired link itself is frequently blocked; that conditional reading is not spelled out in the paper.","A testable extension is to compare the Beta-prime fit not only on the bulk of the SIR distribution but on tail outage probabilities, since second-moment matching protects the center of the distribution more than the extreme tail; if the tail matters, a higher-order moment match or a different ratio distribution may be needed."],"forward_implications":["Coverage probability and ergodic capacity can be evaluated as closed-form expressions for any parameter set, so system-level sweeps over densities, heights, and reflector counts require no simulation.","Denser HAP deployment degrades performance because it raises the mean number of visible interferers, a direct prediction of the model feeding into the interference moments.","Denser RIS deployment improves performance only up to a saturation point, because the benefit comes from shortening the nearest-visible-RIS distance and additional RISs add no interference of their own.","Buildings, while they block the direct HAP-user path, also block interfering HAPs; in the regimes studied, higher blockage density and larger buildings increase coverage.","Raising the RIS height lengthens the RIS-user path and lowers ergodic capacity, so low RIS placement is preferred whenever blockage permits."],"supporting_citations":[{"why":"Supplies the nearest-HAP distance PDF and the moment expressions for the HAP-RIS and RIS-user links.","marker":"[10]"},{"why":"Gives the blockage visibility probability, the mean number of visible HAPs, and the nearest-visible-RIS distance distribution.","marker":"[11]"},{"why":"Provides the nonhomogeneous Poisson process model for the set of visible interfering HAPs.","marker":"[12]"},{"why":"Provides the Rician fading moment formula used for all link amplitudes.","marker":"[13]"},{"why":"Establishes that the ratio of Gamma variates follows the generalized Beta prime distribution.","marker":"[14]"},{"why":"Supplies the CDF and PDF formulas for the generalized Beta prime distribution used in Theorem 1.","marker":"[17]"},{"why":"Provides the logarithmic-integral identity used to evaluate the ergodic capacity in Theorem 2.","marker":"[18]"}],"fun_headline_variants":["Beta prime distribution predicts HAP-RIS coverage in cities","Closed-form coverage and capacity for urban RIS-HAP networks","Stochastic geometry yields HAP-RIS performance formulas","One Beta prime distribution fits SIR in HAP-RIS","Urban HAP-RIS: closed-form coverage and capacity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the horizontal distance to visible interfering HAPs is known and bounded by a finite cutoff omega_h in Lemma 1; if that cutoff is not actually specified or is inconsistent with the infinite-plane mean number of visible HAPs used elsewhere, the interference-moment calculations and every closed-form performance expression inherit that ambiguity.","fun_headline_variants_meta":{"raw":{"variants":["Beta prime distribution predicts HAP-RIS coverage in cities","Closed-form coverage and capacity for urban RIS-HAP networks","Stochastic geometry yields HAP-RIS performance formulas","One Beta prime distribution fits SIR in HAP-RIS","Urban HAP-RIS: closed-form coverage and capacity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000886,"raw_usage":{"total_tokens":3777,"prompt_tokens":849,"completion_tokens":2928,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":465,"completion_tokens_details":{"reasoning_tokens":2847}},"tokens_in":465,"tokens_out":2928,"duration_ms":22667,"temperature":1.0,"reasoning_tokens":2847,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:36:39.601550+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the Section V Monte Carlo setup with lambda_HAP = 5e-6 per square meter, lambda_RIS = 50e-6 per square meter, lambda_B = 100e-6 per square meter, H_HAP = 50 km, H_RIS = 50 m, L = 256, and Rician K = 1, then compute the empirical SIR distribution from many independent spatial realizations. If the Kolmogorov-Smirnov distance between that empirical CDF and the Beta-prime CDF in Eq. (22) is large, or if the fitted parameters alpha_D and beta_D change when the visible-HAP cutoff omega_h is chosen differently, then the closed-form approximation is not self-contained. The same experiment with low blockage density would test whether the blockage-mitigates-interference conclusion reverses when the desired link is more frequently obstructed.","supporting_citations":[{"cited_title":"Enhancing HAP networks with reconfigurable intelligent surfaces,","cited_arxiv_id":null,"evidence_quote":"Supplies the nearest-HAP distance PDF and the moment expressions for the HAP-RIS and RIS-user links."},{"cited_title":"On the product of two κ-µ random variables and its application to double and composite fadin g channels,","cited_arxiv_id":null,"evidence_quote":"Provides the Rician fading moment formula used for all link amplitudes."},{"cited_title":"Distribut ion of the ratio of Gamma variates,","cited_arxiv_id":null,"evidence_quote":"Establishes that the ratio of Gamma variates follows the generalized Beta prime distribution."},{"cited_title":"A generalized beta prime distribution as the ratio probabi lity density function for change detection between two SAR intensity ima ges with different number of looks,","cited_arxiv_id":null,"evidence_quote":"Supplies the CDF and PDF formulas for the generalized Beta prime distribution used in Theorem 1."},{"cited_title":"Tight logarithmic approx imations and bounds for generic capacity integrals and their applicatio ns to statistical analysis of wireless systems,","cited_arxiv_id":null,"evidence_quote":"Provides the logarithmic-integral identity used to evaluate the ergodic capacity in Theorem 2."}],"review_version":2}