Search Results (109)

Depression is a widespread mental health disorder whose early diagnosis is often hindered by the reliance on subjective clinical assessments. This study proposes a subject-aware machine learning framework for detecting depression using wrist-worn actigraphy data, enabling continuous and objective monitoring of daily activity patterns. Unlike many previous studies that evaluate models at the day or window level, the proposed approach employs strict leave-one-subject-out (LOSO) cross-validation, ensuring that no data from the same individual appear in both the training and testing sets and thereby preventing information leakage. The framework extracts time-series features with the Time-Series Feature Extraction Library (TSFEL) from daily actigraphy signals and aggregates them into subject-level representations using distributional statistics. Classical circadian rhythm biomarkers, including interdaily stability (IS), intradaily variability (IV), relative amplitude (RA), cosinor parameters, and activity fragmentation measures, were also investigated to improve interpretability. Seven feature selection techniques and fifteen machine learning classifiers were systematically evaluated. Model selection is performed entirely inside each training partition. The feature selector, the classifier and the number of retained features are chosen together by inner cross-validation over 840 candidate configurations, so the held-out subject contributes nothing to the choice of pipeline. Averaged over five random seeds the nested protocol classifies 45.4 of 55 subjects correctly (82.5%; range 80.0–85.5% across seeds; mean Cohen’s κ = 0.637; and mean sensitivity 74.8%, mean specificity 88.1%). For the median run, 45 of 55, the Wilson 95% confidence interval is 69.7–89.8%. The estimate is above the 58.2% majority-class rate (exact McNemar, p = 0.003), and above the 73% reported by the previously published subject-disjoint evaluation of the same 55-subject cohort; per-subject predictions are not available for that study, so the latter comparison is unpaired. Recording length is a confound in this cohort—used alone as a single feature under the same protocol it reaches 72.7%—and at n = 55 the framework is not separable from that baseline (exact McNemar p = 0.09 to 0.42 across seeds; achieved power 0.40). The single best configuration, Information Gain feature selection with a Bagging classifier, reaches 92.7%. That value is the maximum over a grid of 105 selector–classifier pairs and we report it as such, not as an out-of-sample estimate. Although incorporating circadian biomarkers did not improve classification accuracy (92.7% vs. 90.9%), one bout-fragmentation biomarker, the within-subject mean of daily bout count, was retained in every cross-validation fold; it is, however, a rescaling of the daily Zero Crossing Rate already present in the signature, so the retained features summarize a single daily quantity rather than several independent ones. On this benchmark, subject-level machine learning on wearable actigraphy classifies well above chance under an evaluation protocol that matches the clinical screening setting, but this cohort is too small to establish that the learned representation adds to recording length alone. All results come from one single-site cohort of 55 subjects, and the framework has not been evaluated on an independent dataset. The findings are therefore a reference point for Depresjon, not evidence of generalization to other populations. Full article

Intensified climate change and anthropogenic activities have increased the complexity and nonlinearity of watershed runoff, challenging accurate runoff simulation and input-factor importance analysis. Physical hydrological models are constrained by parameter uncertainty and structural limitations, whereas meteorology-driven data-driven models insufficiently capture the inherent characteristics of watershed hydrological processes. To address this gap, this study develops an interpretable SWAT–Random Forest–SHapley Additive exPlanations (SWAT-RF-SHAP) coupling framework with dual inputs of hydrological process variables and meteorological forcings, using monthly natural runoff data from 13 hydrological stations in the upper Yangtze River during 1980–2016. Seven SWAT-simulated hydrological variables (surface streamflow, groundwater streamflow, percolation, soil moisture, snowmelt, and potential and actual evapotranspiration) are combined with five meteorological variables (precipitation, air temperature, relative humidity, solar radiation and wind speed) to form a 12-dimensional feature set, and a random forest (RF) model optimised by the Sparrow Search Algorithm (SSA) integrates watershed hydrological features with meteorological forcing signals. The results show that: (1) annual runoff presents a three-stage evolution pattern and is decreasing at 11 of the 13 stations; the classical rescaled-range estimate of the mean Hurst exponent is 0.70, but because that estimator is biased upward for short records, the Anis–Lloyd bias-corrected estimate is used instead (mean 0.55, with 5 stations above 0.65), indicating moderate rather than strong long-term persistence, and no trend remains significant at the 1% level once serial correlation is accounted for. (2) The SWAT-RF model outperforms both standalone SWAT and the meteorology-driven RF model: the 13-station mean Nash–Sutcliffe Efficiency (NSE) is 0.951 during calibration (meteorology-driven RF: 0.909; SWAT: 0.679) and 0.723 during validation (RF: 0.678; SWAT: 0.587), and SWAT-RF attains the highest mean validation NSE in all 10 runs, by 0.044 over the meteorology-driven RF, so that its advantage exceeds the run-to-run variability of the optimisation. The margin over the error-correction model SWAT-RF-EC, which uses only SWAT-simulated runoff as input, is larger (validation NSE 0.640) but remains station-dependent, SWAT-RF-EC attaining the higher validation NSE at TZL and YC. (3) When the observed runoff of preceding months is added as a predictor, validation performance improves at all 13 stations relative to the re-optimised baseline, and the 12-month lag window yields the highest mean validation NSE of 0.795, a 10.4% gain; because these lag-augmented schemes are one-step-ahead predictors updated with observed runoff, their skill is not directly comparable with that of the simulation models. (4) SHAP analysis identifies six meteorological-dominant, six hybrid and one hydrological-process-dominant stations, revealing a longitudinal pattern in which hydrological processes dominate the source region while meteorological factors prevail downstream. The framework improves both simulation accuracy and interpretability, offering a scientific basis for regional water resources management in large river basins. Full article

Adapting an open-vocabulary detector to work-zone imagery can improve target recognition but may reduce performance on source-domain and large-vocabulary benchmarks, represented here by COCO and LVIS. Replay and model merging can produce retention-aware checkpoints, but they do not by themselves provide a dedicated local refinement step after checkpoint selection. We propose Query Spectral-Null Adapter (QSNA), a complementary local update applied after checkpoint selection. QSNA freezes the selected checkpoint and learns an 8064-parameter low-rank residual at a single decoder FFN output projection, constrained by an estimated source-gradient complement. The residual can be rescaled after training and folded directly into the checkpoint. On the primary replay-derived checkpoint, QSNA raised ROADWork AP from 28.18 to 28.30, a +0.12 AP change, while COCO and LVIS AP decreased by 0.09 and 0.16 relative to the base. Across three fixed-base adapter seeds, the primary configuration produced similar descriptive ROADWork changes of about +0.13 AP and all three seeds remained above the adopted retention floors; this is replication evidence, not a significance test. Across nine base-specific fits, ROADWork AP repeatedly rose over a finite residual-scale range and then turned over, with the feasible range governed by the starting checkpoint. Canonical random, unconstrained, and projection-removed controls show that update orientation changes the acquisition–retention response, but they do not establish spectral projection as necessary or universally superior. Together, these results position QSNA as a base-conditioned, deployment-oriented local refinement stage after retention-aware checkpoint construction. Full article

Regional-scale gravity imaging is limited by the size of the sensitivity matrix: a direct three-dimensional formulation of the gravity operator requires on the order of ${10}^{13}$ coefficients, making explicit storage infeasible on standard hardware. We exploit two symmetry properties of the gravitational sensitivity kernel: translational invariance, giving the operator a Block Toeplitz with Toeplitz Blocks (BTTB) structure, and central symmetry, halving the number of unique kernel evaluations, to reformulate the Linear Back-Projection (LBP) adjoint operator as depth-wise two-dimensional convolutions. With finite-support truncation at cutoff radius $R_{\max }$, memory requirements drop from terabyte-scale dense arrays to compact per-layer kernels. We implement and benchmark two equivalent backends: an OpenMP spatial-domain stencil and an FFT convolution via BTTB circulant embedding. For a production-scale model ($701\times 401\times 512$ cells, $R_{\max }=150$), the stencil completed the LBP computation in 7187 s on one thread and 354 s on 32 threads ($S_{32}=20.29$), while the FFT backend required 13.8 s; both agree to within an invariant checksum ($3.3464\times {10}^{10}$). Applied to Bouguer gravity data over the Chicxulub impact crater, northern Yucatán Peninsula, the raw LBP amplitudes were calibrated to physical density contrasts through a two-parameter rescaling solved in closed form as a box-constrained convex quadratic program and cross-validated against an independent simulated-annealing search, reducing the RMS misfit from $27.3$ to $5.3$ mGal (an $81\%$ reduction). The resulting model shows density patterns consistent with the crater’s known structural organization. As a single-pass adjoint method, LBP does not resolve the non-uniqueness of the gravity inverse problem, and the model is interpreted for regional structure rather than for pointwise density values; the formulation extends directly to iterative regularized inversion. Full article

The Ewens sampling formula sits at the boundary $\alpha =0$ of the two-parameter Pitman–Yor partition, and deciding whether data lie on it is an inferential question in which the strength $\theta$ is unknown. We give tests of $\alpha =0$ that need no knowledge of $\theta$. Conditioning on the block count, which is sufficient for the strength, gives a randomized score test with exact level at every sample size, and its score agrees, to the order of the local experiment, with the efficient one. The local experiment is anisotropic: the discount and the strength are learned at the rates ${\left(\log n\right)}^{-3/2}$ and ${\left(\log n\right)}^{-1/2}$, and after each coordinate is rescaled by its own rate, the experiment is a Gaussian shift on a half-plane cone for $\alpha \ge 0$ and on the whole plane once the Dirichlet–multinomial branch $\alpha <0$ is admitted. Profiling out the strength divides the discount information by four. The asymptotic score test attains the one-sided and two-sided local power bounds; whether the exact test attains them is open. At each fixed alternative with $0<\alpha <1$ and $\theta >0$, the type II errors of both tests decay like $n^{-(\theta +\alpha )}$, which is the fastest rate available to any test whose level is controlled over all strengths. Simulations and an analysis of Williams’ Rothamsted moth abundances illustrate the tests. Full article

Network threat detection in the Internet of Things must exploit typed relations between devices: a device with unremarkable flow statistics may still be compromised relative to its ownership, co-location, and social ties. Existing graph-based intrusion detectors discard relation type, while lightweight detectors compress tabular rather than graph models. This paper presents Light-SIoT-GAD, a lightweight relation-aware graph detector that treats typed relations as first-class input. The model learns one scalar weight per relation, shares a small basis across relation transforms so that each additional relation type costs only a handful of mixing coefficients rather than a full weight matrix, and combines supervised feature selection, bounded neighbour sampling, and 8-bit post-training quantisation for gateway deployment. A dual-track evaluation isolates typed aggregation on a real 16,216-device SIoT relation graph with injected anomalies and tests attack detection on three labelled NetFlow v2 corpora against classical, deep tabular, and graph baselines, including cross-corpus transfer and ablations. Light-SIoT-GAD matches or exceeds the strongest untyped graph baselines on all three corpora, reaching an AUPRC of 0.9986 with 73,962 parameters, substantially fewer than the unshared R-GCN; on the sparsest corpus a gradient-boosted tabular ensemble still ranks better, which bounds the claim to graphs that carry usable structure. Whether the gain comes from the relation semantics or merely from having per-relation parameters is tested directly: permuting the relation labels leaves the dense corpora unchanged to four decimals and costs 0.0083 AUPRC only on the sparsest one, so the typed advantage is real there and is a capacity effect elsewhere. On the social track, where the anomalies are injected under a stated protocol rather than observed, removing message passing collapses AUPRC from 0.9990 to 0.4869 under that same injection, isolating propagation over the relation graph as the source of the gain; this track measures whether relational structure carries signal, not accuracy against real SIoT attacks. The quantised model occupies 85.4 KB at 14.38 ms per 1000 edges on CPU. The learned relation gates are shown to be identified only up to a per-relation rescaling, and a leave-one-relation-out intervention finds no relation of the SIoT taxonomy to be load-bearing under the injection protocol, so the social track supports the typed parameterisation and not a ranking of the relations. Full article

Several minimal-length constructions deform the spacetime metric conformally at leading order, ${\tilde{g}}_{\mu \nu }=C\left(x\right)g_{\mu \nu }$. We ask which observables can register such a deformation. The two structural results rest on standard properties of conformal rescalings; what is worked out here is their consequence for this class of metrics, carried to closed form on a black-hole background. For an arbitrary conformal factor the unparametrized null geodesics are unchanged, so the critical impact parameter, the shadow angular radius seen from infinity, the deflection angles and the Shapiro delay keep their undeformed values at all orders, and the photon sphere keeps its coordinate location although its areal radius changes. A minimally coupled massless test scalar on $\tilde{g}$ obeys, exactly, a wave equation on g with the single potential $V_{\omega }=\square \sqrt{C}/\sqrt{C}=-\frac{1}{6}(C\tilde{R}-R)$, into which the Weyl invariant does not enter; a conformally coupled massless scalar does not see the deformation at all. For a Reissner–Nordström background and the power-law family $\eta ={\epsilon }_0{(M/r)}^p$, with p as a model input rather than a prediction, we obtain the leading large-l scalar quasinormal shift $p(p-3){\epsilon }_0/[12·3^{p}l(l+1)]$, suppressed as $l^{-2}$, and closed forms for the ISCO radius, the disk radiative efficiency and the orbital frequency. Within this family the channels fall silent at the distinct exponents $p=3$, 2 and $5/2$. All results are checked by computer algebra. Full article

We consider the step-$\lambda$ parametrization of the type A Lauricella matrix hypergeometric function $F_{A,\lambda }$, defined by replacing the classical matrix Pochhammer symbol with ${\left(A\right)}_{m,\lambda }=A(A+\lambda I)\cdots \left(A+(m-1)\lambda I\right),\lambda >0.$ For scalar parameters, this is the step-$\lambda$ factorial used in the degenerate Gauss hypergeometric matrix function, not the Gamma-ratio convention also termed a degenerate Pochhammer symbol. For fixed $\lambda >0$, the scaling identity $F_{A,\lambda }[A,B_i;C_i;x_i]=F_A[A/\lambda ,B_i/\lambda ;C_i/\lambda ;\lambda x_i]$ shows that the function is a reparametrization of the classical Lauricella matrix function. Accordingly, the classical sufficient convergence region, differential relations, coupled matrix PDE system, Euler transformation, and Euler-type representations recorded below are consequences of the classical theory under this rescaling; they are not claimed as independent new results. The material not obtained merely by applying the scaling identity consists of the direct coefficient recursion and its path-independence criterion, the fixed-parameter zero-step analysis, and the accompanying truncation and numerical checks. Under the right-update compatibility condition, the coefficient recursion is path-independent even when A need not commute with the right-hand parameter family. With the parameters fixed as $\lambda \to 0^{+}$, the general result is coefficientwise convergence along the admissible set, together with finite-order error estimates; locally uniform convergence is proved only under the stronger simultaneous-diagonalization and positivity assumptions. The numerical section includes a genuinely noncommuting check of the coefficient recursion, a separate commuting Jordan-type comparison with Euler quadrature, and a certified Cauchy truncation bound. Full article

This study examines a non-autonomous quadratic map driven by a logarithmic cooling schedule (${\mu }_n\sim 1/{ln}^{2}n$, a phenomenological ansatz), building on our recent published result that the logistic map’s symbolic dynamics at its band-merging point is isomorphic to the prime sieve. From its trajectories, we construct an empirical, non-normal, dissipative transfer matrix and compares its complex eigenphases to the non-trivial Riemann zeros after calibrating a few free parameters against the same low-order zeros—an in-sample numerical correspondence, not an independent prediction. We quantify this gap directly: fitting on the first $M\in \{50,70,80\}$ zeros and evaluating on the rest gives a held-out MSE one to two orders of magnitude larger than the training error, with the fitted coupling drifting across M but remaining comparatively stable across ten random seeds at fixed $M=70$ (CV $\approx 4.7\%$). At low order ($N\lesssim 20$), an unselected re-computation shows a qualitative rank correlation ($\rho =0.56$, $p=0.010$) with, but no significant joint co-location ($p=0.099$) of, a residual feature reported in recent ion-trap quantum simulations of the same zeros. At larger N ($N\ge 1000$), the model matches a globally rescaled GUE surrogate’s mean counting-function trend better once conjugate eigenphases are restored, though this partly follows from the construction’s own symmetry; a separate, standard unfolded local-statistics test shows the model’s own eigenphase spacings do not match GUE, unlike the true zeros. These are numerical observations on a heuristic model, not a proof or Hilbert–Pólya-type operator construction; a dedicated table tabulates the epistemic status of every main claim. Full article

TinyStressNet is a compact neural classifier for three-class academic stress estimation (low, medium, high) from three physiological features (galvanic skin response, heart rate, and skin temperature) designed for low-cost educational sensing. We evaluate it on 1000 interval-level observations from five engineering students collected over three months, with labels derived from repeated 6-item short-form of the State-Trait Anxiety Inventory (STAI-6) self-reports aligned with contemporaneous physiological readings. For subject-independent evidence, we use leave-one-subject-out (LOSO) evaluation as the primary protocol, complemented by a per-user chronological split for temporal robustness. We study augmentation through the lens of approximate invariance to bounded, label-preserving nuisance transformations, comparing no augmentation, Gaussian jitter, a radial-rescaling family, and a bounded affine (BA) family against strong classical baselines (Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis (QDA), and Support Vector Machines (SVMs), logistic and ordered-logit models). Under LOSO, the best TinyStressNet configuration reaches 0.899 accuracy and 0.898 weighted F1; within users, it reaches 0.912 accuracy. The model uses 3139 parameters and 2800 Multiply-Accumulate operations (MACs), and post-training int8 export preserves predictions with almost the same accuracy (0.992 float–int8 agreement). Strong classical baselines remain competitive on this low-dimensional representation, which we read as evidence about the geometry of the three-feature space rather than as a limitation of the model. Repeating the full evaluation under ten seeds shows that the augmentation families are not separable on this representation: subject-independent weighted F1 spans 0.894 to 0.900 across the four conditions with a seed-to-seed standard deviation of about 0.005, and no augmentation is not measurably worse than the best family. The BA family is the most consistent condition, the strongest in five of ten seeds and never the weakest, which we report as a stability observation rather than a performance gain. TinyStressNet is presented as a compact, quantization-ready reference model and a controlled testbed for invariance-aware augmentation in low-dimensional physiological sensing. Full article

The Weibull distribution is one of the most versatile and widely applied continuous probability distributions in reliability engineering, survival analysis, wind-energy modelling, and extreme-value theory. Classical parameter estimation relies on maximum likelihood estimation or method-of-moments using integer-order moments, both of which may suffer from instability or high variance in small to moderate samples. This paper develops a rigorous framework for estimating Weibull parameters by combining the maximum-entropy principle with fractional-order moment constraints, i.e., constraints of the form $E\left[X^r\right]$ for non-integer $r>0$. A central theme of the paper is symmetry: we show that the maximum-entropy density subject to a finite set of fractional-moment constraints uniquely recovers the Weibull family, and that the underlying moment-matching system, while not symmetric in every sense considered in an earlier draft (see Remark 3), possesses a precisely characterised duality under joint rescaling and relabelling of the exponent pair, together with a log-moment map whose local curvature is strictly positive and increasing with exponent spacing rather than symmetric about a fixed midpoint. We derive closed-form expressions relating the Lagrange multipliers to the shape and scale parameters, establish new sound theoretical results on the symmetric behaviour of the moment-ratio function and its sensitivity, and propose a numerically stable algorithm for solving the resulting moment-matching system. Extensive Monte Carlo experiments demonstrate that the proposed maximum-entropy fractional-moment estimator achieves a bias and root-mean-square error that are comparable to, and for small-to-moderate samples somewhat better than, maximum likelihood estimation, with the size of the advantage depending on how closely the chosen exponent pair tracks the true shape parameter. Applications to real wind-speed data and composite-material fatigue life data illustrate the practical utility of the method. The paper contributes both to the information-theoretic foundations of distribution fitting and to applied statistical methodology, with symmetry serving as both a diagnostic tool and a unifying structural principle throughout. Full article

Let $\mathcal{A}$ denote the class of functions $g\left(z\right)=z+{\sum }_{k=2}^{\infty }{a}_k{z}^k$ analytic and normalized in $\mathbb{U}=\{z:|z|<1\}$, and let $D=\{g\in \mathcal{A}:|g^{″}\left(z\right)|\le 1,z\in \mathbb{U}\}$. We give a short, self-contained proof that D consists entirely of univalent functions by a direct line-integral estimate combined with the classical Noshiro–Warschawski theorem. We then solve, in closed form and with explicit extremal functions, two classical extremal coefficient problems for D: the Fekete–Szegő problem and the second Hankel determinant, using the exact Schur parametrization of bounded analytic functions applied to $g^{″}$. We also establish a sharp bound on a single coefficient $|a_n|$ for every $n\ge 2$, obtained by an elementary Parseval argument that does not require the full joint Schur parametrization needed for the Fekete–Szegő and Hankel problems. We then compose D with the Owa–Srivastava fractional derivative operator ${\mathsf{\Omega }}_z^{\delta }$ ($\delta \in [0,1)$), used to build fractional bi-univalent subclasses, to define the one-parameter family $D\left(\delta \right)=\{g\in \mathcal{A}:|{\left({\mathsf{\Omega }}_z^{\delta }g\right)}^{″}\left(z\right)|\le 1\}$, and extend all three results to $D\left(\delta \right)$, again with explicit extremal functions. Because the underlying weights grow at different rates in the fractional parameter $\delta$, the Fekete–Szegő problem exhibits a genuine, monotone bifurcation as $\delta$ ranges over $[0,1)$, while the Hankel and general-coefficient bounds decrease monotonically—effects that are structural consequences of the fractional operator rather than a simple rescaling of the classical case. All results are verified in two independent numerical ways: the Schur parametrization lemma itself is checked against an explicit, independently constructed rational Schur function whose Taylor coefficients are extracted numerically via the Cauchy integral formula, and the sharp bounds are checked by Monte Carlo maximization over the admissible Schur parameters. Both checks are explained in detail and agree with the closed forms to at least four decimal places. Unlike the Carathéodory class of functions with positive real part, whose coefficient body is fixed at the origin (value 1) and is classically parametrized without an extra recursive step, the Schur class $\mathcal{B}$ used here does not pin down $\phi \left(0\right)$, so its exact coefficient body genuinely requires the two-step recursive Schur parametrization applied below to $g^{″}$; identifying and exploiting this distinction is part of the technical contribution of the paper. Full article

Software-defined Networking (SDN) is a promising networking architecture that separates the control and data planes to allow flexible network management. However, the SDN architecture makes networks vulnerable to various security threats, such as Distributed Denial-of-Service (DDoS) attacks. A DDoS attack is one of the most common SDN threats, aiming to exhaust a network’s computational and bandwidth resources. Self-similarity is a statistical property of time series in which data patterns repeat at different time scales. Several studies have shown that network traffic exhibits increased self-similarity during DDoS attacks, making it a promising tool for DDoS detection. Despite the effectiveness of statistical methods for detecting DDoS, some methods, such as self-similarity, are discarded due to their high computational cost, leading to detection delays. This paper proposes a lightweight Rescaled Range ($R/S$)-based scheme for effective real-time DDoS attack detection in SDN. The scheme employs the Welford online algorithm to compute statistical parameters of the $R/S$ scheme. Experimental results demonstrate that the proposed scheme efficiently captures changes in self-similarity and detects TCP/UDP DDoS attacks in real time. Moreover, it achieves high detection performance compared to other $R/S$ methods, with a False Positive Rate (FPR) below 0.5% and an average computation time of 0.047 ms. Full article

Quantum relativistic solutions of a “square-root” version of the Klein–Gordon equation, for a particle in a one-dimensional Morse potential, are presented using methods previously proposed and applied to a particle in a harmonic oscillator. The methods lead to both numerical and analytical solutions, with the latter allowing smooth variation of the system parameters from non-relativistic to ultra-relativistic limits. Analytical expressions for the energy levels and wavefunctions are obtained, as solutions to a Schrödinger-type equation, including relativistic effects through a state-dependent rescaled mass. The eigenstates of the Morse potential exhibit suitable and smooth behavior and approach the corresponding harmonic oscillator solutions as the depth of the Morse potential well increases, as expected. A comparison is also presented between the relativistic harmonic oscillator obtained with this method and the so-called “Klein–Gordon oscillator”. Full article

Continuous-variable quantum secret sharing (CVQSS) distributes a secret key among multiple players while requiring their cooperation for reconstruction. Its performance deteriorates rapidly in sequential multiparty links because optical loss, excess noise, and receiver noise accumulate with the number of players. We investigate a dealer-side heralded hybrid linear amplifier (HHLA), formed by measurement-based noiseless linear amplification and trusted deterministic preamplification, for mitigating the relative receiver-noise penalty. The accepted data are described by an equivalent Gaussian channel and evaluated with an asymptotic reverse-reconciliation key-rate model against collective Gaussian attacks. We explicitly condition parameter estimation and Eve’s Holevo information on successful heralding, include a cutoff-dependent Gaussian-input estimate of the heralding probability in the rate per transmitted pulse, apply a single calibrated output-quadrature rescaling, and provide parallel heterodyne and homodyne calculations. The homodyne simulation includes detector noise consistently in the mutual information and Holevo bound and enforces the Gaussian-equivalent NLA feasibility constraint. For an excess-noise variance of $0.001$ shot-noise units per player, the constrained homodyne optimization gives maximum reported distances of approximately 108, 42, and $25.5$ km for 5, 50, and 100 players, respectively, at a reporting floor of ${10}^{-6}$ bit/pulse. Composable finite-size security, finite-cutoff non-Gaussian corrections, and active-insider verifiability are left for future work. Full article