We study a deterministic finite-scale geometric index associated with the Poincaré ball. The index is defined through the volume growth of hyperbolic balls along the outward radial direction. For a fixed Euclidean radial step ε in the interval (0,1) and a point whose radial coordinate r satisfies 0 ≤ r < 2 arctanh(1 − ε), we prove that the corresponding outward radial hyperbolic step δ+(r, ε) strictly increases with r. Consequently, the composition Gε(r) = φ(δ+(r, ε)), where φ is the logarithmic derivative of the volume of a hyperbolic ball, is strictly increasing in r for dimension n ≥ 2. The quantity Gε(r) is a deterministic, direction-dependent finite-scale geometric index. It should not be identified with local intrinsic dimensionality (LID) in the sense of Houle, and the Riemannian dimension of the space equals n at every point. However, under a Lipschitz condition on a sampling density p with constant L, we establish a multiplicative envelope: the population finite-scale LID at the matched scale δ = δ+(r, ε) lies between exp(−2Lδ) Gε(r) and exp(2Lδ) Gε(r). A conjectural finite-sample extension is briefly discussed without the status of a theorem. Open problems are stated in a compact form.

We extend the finite-scale radial monotonicity result for the Poincaré ball to three settings: the inward radial direction, the tangential direction, and the case of a variable Euclidean step \(\varepsilon(r)\). In the inward case we show that the hyperbolic step \(\delta_-(r,\varepsilon)\) is strictly increasing in \(r\) on the domain \(r>2\mathrm{arctanh}(\varepsilon)\). In the tangential case, interpreting the hyperbolic step as the hyperbolic distance to the displaced point, we prove strict monotonicity of \(\delta_\perp(r,\varepsilon)\) on the domain \(a^2+\varepsilon^2<1\), where \(a=\tanh(r/2)\). We then establish a sphere-averaged analogue \(\bar{\delta}(r,\varepsilon)\) on the restricted domain \(\varepsilon<a<1-\varepsilon\) (which requires \(\varepsilon<1/2\)), where pointwise monotonicity holds for every direction of the displacement, and a volume-based characterization \(\delta_{\mathrm{vol}}(r,\varepsilon)\) on the full outward domain \(0\leq r<2\mathrm{arctanh}(1-\varepsilon)\). Finally, for a variable step \(\varepsilon(r)\) we derive a necessary and sufficient criterion for monotonicity of \(\delta_+(r,\varepsilon(r))\), show that it holds automatically whenever \(\varepsilon'(r)\geq0\), and exhibit an explicit counterexample of the form \(\varepsilon(r)=c(1-\|z\|)\) with \(c\in(0,1)\) for which monotonicity fails. The extension of the sphere-averaged result to the full domain \(a+\varepsilon<1\) remains open.

We refine the finite-scale LID bridge of [1] and extend the sphere-averaged monotonicity result of [2]. First, we replace the global Lipschitz constant L in the multiplicative envelope between G_ε(r) and the population finite-scale LID by a local constant L_z(δ) defined as the supremum of ∥∇ log p∥H over the closed hyperbolic ball of radius δ centred at z. Under a C¹ regularity assumption on the density p, we prove the local envelope e^{−2L_z (δ)δ} G_ε(r) ≤ LID_δ(z) ≤ e^{2L_z (δ)δ} G_ε(r), which refines the global bound. As an asymptotic consequence, if ∇log p is continuous at z and δ → 0⁺ at fixed z, then the relative error satisfies |LID_δ(z)/G_ε(r) − 1| ≤ 2∥∇ log p(z)∥_Hδ + o(δ), replacing the global constant L by the pointwise gradient norm. Second, we prove that the sphere-averaged hyperbolic step \( \bar{\delta} \)(r, ε) introduced in [2] remains strictly increasing in r on the full domain a + ε < 1, extending the result from the restricted domain ε < a < 1 − ε. The proof uses a symmetrization argument that reduces the integral over the sphere to a pair of pointwise inequalities. Both results preserve the multiplicative envelope structure of [1] and transfer to all five directional steps δ₊, δ₋, δ_⊥, \( \bar{\delta} \), δ_vol on their corresponding domains.

We study the geometry of the Poincaré ball equipped with a rotationally symmetric metric \(g_\eta\) conformally close to the hyperbolic metric \(g_{\mathbb{H}}\), with curvature pinched in \([-1-\eta, -1+\eta]\). We work with the matrix Jacobi field \(A_\eta\) and the shape operator \(S_\eta = A_\eta' A_\eta^{-1}\), whose trace \(\sigma_\eta = \operatorname{tr} S_\eta\) satisfies a Riccati equation. For rotationally symmetric non-homogeneous metrics and centres \(z \neq 0\), the geodesic spheres need not be isotropic: the volume element in geodesic polar coordinates centred at \(z\) is generally \(\det A_\eta(t, \omega, z)\,dt\,d\omega\), not a power of a scalar radial profile. Five results are established under the stated assumptions. First, using Günther and Bishop--Gromov comparison, we prove two-sided model-volume bounds \[ V_{-(1-\eta)}^{\mathrm{model}}(\delta) \le V_\eta(\delta, z) \le V_{-(1+\eta)}^{\mathrm{model}}(\delta) \] uniformly in \(z\). Second, we prove an exact weighted-average identity for \(\Sigma_\eta(\delta, z) = \partial_\delta\log \partial_\delta V_\eta(\delta, z)\), in which the anisotropy of the geodesic spheres enters as a positive variance term. Third, we prove a monotonicity-transfer lemma: if \(q_\eta(\delta, z) = \delta\,\Sigma_\eta(\delta, z)\) is strictly increasing in \(\delta\), then the volume indicator \(\phi_\eta(\delta, z) = \delta\,\partial_\delta V_\eta/V_\eta\) is strictly increasing. Fourth, under two explicit assumptions on the metric family — a global \(C^2\)-closeness to \(g_{\mathbb{H}}\) and a uniform \(O(\eta)\)-stability of the logarithm map along a compact family of endpoint pairs — we prove \(C^1\)-stability of the perturbed radial step \(\delta_+^\eta(r, \varepsilon)\) on compact radial intervals. Fifth, we prove an invariant LID bridge for \(g_\eta\)-Lipschitz densities. The local monotonicity of \(\phi_\eta\) on \((0, D]\) is stated as a conditional result under an explicit averaged-Riccati positivity assumption, which we do not prove; the global version is not resolved by our method.

Hemagglutinin (HA) is the primary surface protein of the influenza A virus, determining its subtype and antigenic properties. Traditional subtype classification methods rely on DNA or amino acid sequence analysis, which does not account for protein spatial folding. In this work, we propose EpitopeGNN — a graph neural network (GNN) that constructs a residue interaction network (RIN) from the 3D structure of HA and classifies the virus subtype. The model was trained on 249 structures from the Protein Data Bank (PDB), containing H1N1, H3N2, H5N1, and other subtypes. By utilizing physicochemical properties of amino acids and topological centrality measures, we achieved 100% classification accuracy on the test set and 97.6% with five-fold cross-validation. A significant correlation was found between the obtained structural embeddings and phylogenetic distances (r = 0.48, p < 0.001), confirming their biological relevance and opening opportunities for structural monitoring of virus evolution, as well as rapid analog searching for novel strains.

Background: DNA methylation is a key epigenetic mechanism whose dynamics are closely linked to ageing. Modern epigenetic clocks predict biological age based on the average methylation level. The concept of “epigenetic rejuvenation” posits that at early stages of development, the biological age of the embryo may decrease, reaching a minimum (“ground zero”) at the gastrulation stage. However, standard averaging methods may not account for important rearrangements in the internal structure of methylation. Objective: To apply the apparatus of information theory and topological data science to the analysis of scNMT-seq data and to test whether DNA methylation entropy decreases from stage E4.5 to E6.5, which would correspond to an approach towards the biological zero state. Methods: Publicly available scNMT-seq data (GSE121690) were analyzed. Five entropy measures were calculated for each cell (Shannon, Renyi, Tsallis, LZ-complexity, local gradient entropy (entropy of variations in the smoothed histogram of the methylation distribution), and persistent entropy (PE)—a topological complexity measure). For the five-dimensional entropy feature space, a Rips complex was constructed, and persistence diagrams H_0 and H_1 were computed. Results: All five entropy measures decreased significantly, with LZ complexity showing the largest relative reduction (−28.4%) and the strongest independent signal (partial r = −0.181). Among all the complexity measures studied, LZ complexity exhibited the largest relative reduction, underscoring its heightened sensitivity to the progressive ordering of the epigenetic landscape. Notably, the ternary encoding of LZ complexity showed strong correlation with Shannon entropy (r = 0.71), indicating that algorithmic complexity, when accounting for partial methylation states, aligns closely with statistical entropy while retaining sensitivity to spatial order. The consistency of results across binary and ternary encodings confirms the robustness of LZ complexity as a structural biomarker. Persistent entropy confirmed the general dynamics (decrease from 15.91 to 14.89, p = 0.01). Topological analysis of the multidimensional space revealed a qualitative reorganization: at stage E6.5, stable cyclic structures (H₁) emerge, while at E4.5 the space is dominated by a single large-scale cycle. Null model validation confirmed that the observed H₁-cycles are genuine topological features rather than random fluctuations. Comprehensive topological characterization showed that normalized persistent entropy increases from 0.846 to 0.882 (p < 0.001), while maximum persistence decreases from 0.446 to 0.218 (p < 0.001), reflecting a transition from a homogeneous state to structured diversification—multiple, evenly distributed cycles corresponding to distinct cell lineages. Consistent with this, regional disorder (RE/RD) at the single-cell level decreases from E4.5 to E6.5 (RE: −25.5%, RD: −27.4%, p < 10⁻¹³), while global entropy also decreases, together painting a picture of epigenetic rejuvenation as ordered consolidation at the whole-genome scale. An SVM model trained on 15 entropy and structural features achieved stage classification with an accuracy of 93.4% and AUC of 0.981, confirming the diagnostic potential of the approach. Conclusions: The decrease in DNA methylation entropy from E4.5 to E6.5 corresponds to an approach to “ground zero”—the point of minimum biological age in embryogenesis—and supports the hypothesis of a link between decreasing entropy and epigenetic rejuvenation. The addition of topological analysis reveals the hidden organization of epigenetic diversity, showing that ordering does not lead to homogenization but is accompanied by the formation of distinguishable cell lineages.