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

This paper develops a mathematical framework connecting accessibility in random fitness landscapes, coexistence and extinction in ecological systems, and stability in evolutionary games. It identifies the operators and finite certificates shared by these problems while specifying the limits of each correspondence. A central distinction is between the actual behavior of a system and the conditions under which a particular mathematical certificate can establish that behavior. For accessibility percolation with independent uniform vertex labels on the coordinate-outward lattice, the paper derives the first correction to the critical gradient in high dimension. Writing d for dimension and e for Euler’s number, the expansion is: theta_c(d) = 1/(e d) + 1/(e^2 d^2) + o(d^-2). The proof counts distinct accessible endpoints and retains the dependence created when paths share physical vertex labels. The resulting threshold differs from the threshold of the constant-density strip collision certificate: their relative correction coefficients are 1/e and 2/e, respectively. Consequently, uniform-label accessibility can survive where that certificate no longer succeeds. Additional constructions establish separation for suitable strictly positive smooth label densities in every dimension d >= 4, and finite rational witnesses provide explicit threshold bounds in dimensions ten and twenty. At fixed dimension, the paper constructs an exact attractive representation of the common-root reachable-label field. For uniform labels, this representation takes the form of killed reflected queues driven by independent innovations. It supports positive association, a disjoint-witness tree inequality, finite-size extinction criteria, critical susceptibility divergence, and quantitative susceptibility lower bounds. These results preserve the actual endpoint mergers rather than replacing the lattice with independent edges or a branching process. The critical-threshold analysis distinguishes extinction, bounded expected endpoint population, and precise survival asymptotics. A summable convolution series of actual connection probabilities is sufficient for bounded first moments. The series admits an exact Fourier representation and diverges in dimensions at most three. A separate result shows why every fixed-block endpoint-sequence upper comparison is inadequate at criticality: its expected mass grows exponentially even though the actual lattice has only polynomially many vertices per layer. The exact signed renewal correction is represented through the Möbius function of the random reachability order. An unsigned expansion over irreducible path unions has radius zero at uniform criticality in dimensions at least five, demonstrating that cancellations must be retained. Version 15 applies lexicographic discrete Morse theory to reorganize the correction into at most one signed contribution per accessible path. An explicit finite interval algorithm identifies the contributing paths. This construction eliminates the fully accessible obstruction beyond depth d and supplies a smaller positive majorant for further critical estimates. This paper also establishes a weaker sufficient condition for extinction. A two-layer killing opportunity and convexity imply that positive infinite survival forces at least logarithmic growth of time-averaged frontier sizes, both in expectation and almost surely on surviving realizations. Accordingly, sublogarithmic averaged expected growth suffices for extinction. A corresponding real-radial operator criterion and scalar Schur-complement test allow some unbounded mean populations and do not require a positive temporal derivative. A further construction approximates the uniform-label process with finitely many workload states while preserving mergers at every depth. The critical gradients of these approximations lie within one grid spacing of the actual threshold. Finite-time probabilities converge under refinement, but interchanging refinement and infinite time remains an additional requirement. The paper expresses critical extinction as precisely this remaining limit-interchange question. For finite generalized Lotka–Volterra systems satisfying diagonal stability, the ecological analysis gives a complete classification of leading extinction rates. A terminating sequence of complementarity problems and Schur complements resolves neutral degeneracies into exponential, power, and iterated-logarithmic factors. Rational inputs yield finite certificates for the exponents, and an exactly solvable family attains every permitted logarithmic depth. Related results address signed absorption, harmful interactions, coexistence reentry, and specified transfers to evolutionary stability. The actual lattice’s critical extinction, bounded critical means, critical Fourier estimates, and exact survival laws remain unresolved. The new results provide rigorous reductions and improved sufficient conditions; they do not establish the missing model-specific critical bounds. The accompanying research bundle contains the 130-page manuscript, compilable LaTeX source, exact verification programs, mathematical and literature audits, provenance records, and checksums. All 34 integrated verification blocks pass in ordinary and optimized Python with identical reports. Finite computations verify the stated identities and examples; infinite-volume conclusions depend on the analytic proofs.