Grounded Operators and Finite Certificates: Quantitative accessibility, signed extinction, and evolutionary stability

This working paper develops mathematical connections between accessibility in random fitness landscapes, extinction and coexistence in ecological systems, and evolutionary stability in population games. Its organizing tools are grounded operators, positive quadratic forms, and finite certificates: explicitly checkable conditions that establish specified properties of systems whose long-term behavior may otherwise be difficult to determine. The paper identifies both the reach and the limitations of these connections, stating the assumptions under which an operator, response coefficient, or stability conclusion transfers between models. A central contribution concerns accessibility percolation on the coordinate-outward lattice. Independent random labels represent the irregular component of a fitness landscape, while a deterministic gradient favors movement in prescribed coordinate directions. The principal question is whether an accessible path can continue indefinitely. Paths that intersect share vertex labels, creating dependencies that must be retained when studying survival. The paper develops methods that preserve these dependencies and distinguish the actual survival threshold from the threshold at which a particular proof technique succeeds. For uniform labels, the paper determines the first two terms of the high-dimensional critical-gradient expansion. The leading term is 1/(e d), and the next term is 1/(e squared times d squared), where d is the lattice dimension and e is the base of the natural logarithm. The corresponding correction for a constant-density strip collision certificate is different. This establishes a separation between actual survival and the range of that certificate. Related separations are constructed for smooth positive label densities in every dimension at least four. Exact rational certificates also establish survival at gradient 1/24 in dimension ten and at gradient 1/50 in dimension twenty, alongside complementary extinction bounds. The planar theory addresses behavior exactly at the critical threshold. The paper proves critical extinction for the two-dimensional coordinate-outward lattice and extends this conclusion to every atomless label distribution with full support on the unit interval, including distributions that have no density. At the critical gradient, the probability of indefinite propagation is zero. The paper also establishes continuity of the survival probability as the gradient varies. The key step converts positive survival into a finite spatial reproduction certificate. Causal boundary sealing controls paths that leave a restricted region, including paths that subsequently return. Population estimates and reset constructions produce new seeds, a stopping-time argument synchronizes their appearance at a deterministic time, and a finite steering procedure places offspring in prescribed spatial regions. The resulting reproduction event persists when the gradient is decreased slightly. This robustness rules out positive survival exactly at the critical threshold. Additional fixed-dimension results provide an exact innovation representation of the reachable-label field, positive association, susceptibility estimates, tree inequalities, and extinction criteria. Signed renewal coefficients are represented through expected Möbius functions of the reachability order. Lexicographic discrete Morse theory supplies cancellations that reduce the combinatorial expansion. Finite-state queue approximations preserve endpoint mergers and provide parameter enclosures uniform in depth. For uniform labels, time reversal connects stationary workload distributions with survival from a single root. A second major contribution classifies extinction rates in finite signed Lotka–Volterra systems under diagonal stability. Rather than determining only which species disappear, the paper identifies the leading asymptotic equivalent of each extinct species. A terminating sequence of linear complementarity problems and Schur complements resolves neutral degeneracies into combinations of exponential decay, powers of time, and powers of iterated logarithms. The maximum logarithmic depth is bounded by the number of initially neutral species, and an explicit family attains the bound. For rational inputs, the extinction exponents admit finite certificates. The finite-system theory also connects killed Markov chains, ecological equilibria, and evolutionary games through specified grounded energies. Under an explicit comparison-matrix hypothesis, a signed absorption process gives an exact majority-probability criterion for coexistence in a prescribed ecological system. Further results relate equilibrium sensitivity, critical extinction amplitudes, and absorption probabilities. Weighted projective embeddings transfer suitable ecological stability certificates to evolutionary stability while accounting for the change of state variables and time scale. The contribution is therefore both structural and quantitative. It supplies precise correspondences between models, calculates thresholds and extinction laws, and separates properties of a system from limitations of the certificates used to analyze it. These distinctions are relevant to researchers in probability, random fitness landscapes, mathematical ecology, evolutionary game theory, dynamical systems, and computational certification. The accompanying materials include the manuscript, compilable LaTeX source, verification programs, exact-arithmetic results, and reproducibility documentation, with records for 41 verification blocks. Computational checks support finite identities, inequalities, and constructions; the infinite-volume conclusions rely on the analytic arguments in the paper. Higher-dimensional critical extinction and exact critical survival asymptotics remain open. The ecological classification assumes diagonal stability, and the correspondences between ecological and game models apply under their stated hypotheses.