ÆLITH Σ - Geometric Cascades in Every Fixed Dimension: A Proposed Proof of the Higher-Dimensional Falgas-Ravry-Sarkar Propagation Theorem

ÆLITH Σ presents a self-contained proposed resolution of the higher-dimensional forced-droplet propagation problem for bootstrap percolation on Poisson random geometric graphs. The research addresses the extension explicitly proposed by Victor Falgas-Ravry and Amites Sarkar in Bootstrap percolation in random geometric graphs, published in Advances in Applied Probability in 2023. PureOne/AELITH-Sigma-Geometric-Cascades · Datasets at Hugging Face The central question is whether activating a small spatial region can initiate an irreversible cascade through almost an entire disordered network. The manuscript proposes a proof that the propagation boundary established in two dimensions remains valid in every fixed spatial dimension d ≥ 2: θ = (1 + p)/2. The theorem concerns the two strict-inequality regimes on either side of this boundary. Its main geometric construction replaces planar boundary arguments with bounded, separated enclosures around rare defects and connected routes through their exterior. Mathematical model Vertices form a homogeneous Poisson point process of intensity one on a d-dimensional flat torus of volume n. Two distinct vertices interact when their periodic Euclidean distance is less than r_n, where v_d r_n^d = a log n, and v_d denotes the volume of the Euclidean unit ball. Each vertex is initially active independently with probability p. An inactive vertex becomes permanently active once it has at least ceil(θ a log n) active neighbors. The activation threshold is normalized by the expected degree, rather than each vertex’s realized degree. The parameters d, a, p and θ remain fixed as n tends to infinity, with d ≥ 2, a > 0, and p, θ ∈ [0,1]. A forced seed activates every vertex inside a ball of radius C r_n, in addition to the independently activated background. For fixed C, this seed occupies volume proportional to log n and therefore a vanishing fraction of the system. Principal results stated in the manuscript Almost-global propagation below the boundary. When θ < (1+p)/2, there exists a sufficiently large fixed seed multiplier C such that, with probability tending to one, the forced seed activates all but O(n^(1−γ)) vertices for some γ > 0. Every connected component induced by the finally inactive vertices has diameter O(r_n). Sublinear additional activation above the boundary. When θ > (1+p)/2, every fixed seed multiplier C produces at most O(n^(1−γ)) additional activations outside the initially active set and the forced seed, with high probability. Every component induced by those additional vertices has diameter O(r_n). This permits small exceptional components far from the seed; it does not assert that all additional activation remains in its immediate neighborhood. Uniformity over seed locations. Both conclusions hold simultaneously for every seed center on the same high-probability environmental event. The center may therefore be selected after inspecting the random configuration. Quantitative propagation time. Below the boundary, almost-full activation occurs within O((n/log n)^(1/d)) synchronous rounds. In the purely seed-driven regime p = 0 and 0 < θ < 1/2, a matching lower bound gives the same asymptotic order. These bounds count updates, rather than physical time. Relaxed density assumption. The proposed argument applies to every fixed a > 0. It does not require full connectivity of the original random geometric graph, because a sublinear exceptional set is allowed. Proof method The proof combines explicit spherical-cap estimates, fine-cell approximations, Poisson concentration and finite-dependence bounds. These establish local growth below the boundary and protection of inactive regions above it. The central deterministic construction encloses irregular regions in rectangular boxes of bounded diameter, separated by several lattice cells. A controlled merging argument prevents these enclosures from becoming macroscopic. In dimensions at least two, each box has a connected exterior shell, allowing paths to detour around defects while retaining controlled length. This supplies a route from local activation estimates to global propagation without requiring planar dual cycles. The manuscript treats periodic seams, threshold rounding, dependent local events, arbitrary seed placement, exceptional vertex counts and component diameters. Supporting verification and reproducibility The archive includes executable supporting checks and their complete recorded outputs: 312 spherical configurations. 1,176 geometric inequalities, including 720 checks explicitly aligned with manuscript formulas. Eight finite periodic-grid cases containing 39,084 vertices. Fifteen exterior shells and 127,870 ordered shell endpoint pairs examined by exact breadth-first search. Seventy-nine detour paths, including 21 requiring obstacle avoidance. Zero failed assertions in the recorded execution. During release preparation, the original verification script was executed unchanged. Its regenerated JSON report and both figure files were byte-identical to the original outputs in the same recorded software environment. These computations examine geometric formulas and selected finite constructions. They do not constitute a proof of the asymptotic theorem, a Monte Carlo study of random geometric cascades, or experimental validation of a physical system. Numerical quadrature error estimates are not interval-certified bounds. Research contribution and scope The intended contribution is a proof strategy for the specified higher-dimensional propagation extension, together with quantitative exceptional-set and propagation-time bounds. The separated-enclosure construction may also support other disordered spatial models that admit suitable local transport rules and rare-defect estimates. The critical equality θ = (1+p)/2, spontaneous nucleation thresholds, minimum seed radius, optimal seed shape, activation of every vertex, growing dimension and nonperiodic boundary effects remain outside the theorem. Applications to physical materials or devices require a separate justification of the model assumptions. Archive contents The release contains the original 15-page manuscript, editable LaTeX source, verification code, original and reproduced results, vector and raster illustrations, derived CSV summaries, a proof-review guide, structured claims, citation metadata, provenance records and SHA256 integrity manifests. A guided Windows Hugging Face uploader is also included. Research status This work is released as a proposed mathematical resolution with a complete analytic argument and reproducible supporting checks. It has not undergone external peer review or proof-assistant verification. Mathematical correctness and publication priority remain open to independent scrutiny. Primary reference Falgas-Ravry, V., and Sarkar, A. (2023). Bootstrap percolation in random geometric graphs. Advances in Applied Probability, 55(4), 1254–1300.https://doi.org/10.1017/apr.2023.5 Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki