Sharp Accessibility Thresholds and Structural Limits for Fitness Landscapes

This research manuscript develops a mathematical theory of evolutionary accessibility under random fitness variation, incomplete observations, environmental switching, and neutral mutation dynamics. It combines sharp probabilistic thresholds with structural classification theorems, exact optimization results, and constructive obstructions. Throughout, the paper distinguishes statements about the existence of uphill paths from statements about their discovery times, evolutionary destinations, and robustness to missing information. The principal probabilistic result determines the high-dimensional critical fitness gradient for Rough Mount Fuji accessibility on directed mutation lattices. For independent noise with any continuous probability density on the unit interval, the dimension multiplied by the critical gradient converges to the reciprocal of the product of Euler’s number and the maximum noise density. Uniform noise therefore gives the sharp constant one over Euler’s number. The result applies to coordinate-increasing paths and, equivalently, to nearest-neighbor paths on the integer lattice whose graph distance from the origin increases at every step. The central difficulty is that lattice paths can separate and later meet again, sharing the same random fitness label. The proof retains these dependencies through an exact collision-renewal operator. A label-strip construction then restricts accessible paths to a short interval of noise values while preserving a sufficient branching advantage. An explicit estimate shows that genuine spatial reunions become sufficiently rare in high dimension. Together, these ingredients establish the sharp threshold without assuming that different lattice paths are independent. The lattice analysis also gives a finite-dimensional sufficient condition for percolation and an explicit positive lower bound on survival probability. Under local Hölder regularity near a density maximum, the relative threshold error is bounded by a constant times the inverse square of the logarithm of dimension. This quantitative result includes uniform noise, endpoint maxima, and density plateaus. The convergence bound is not asserted to be the optimal lattice correction. A separate obstruction shows why uniformly bounding all reunion costs by a polynomial in inverse drift fails in the natural full-interval Perron norm. For branching-tree Rough Mount Fuji models, the manuscript establishes explicit two-sided spectral bounds and sharp small-gradient asymptotics. For continuous densities with finitely many interior nondegenerate global maxima, it determines the exact next threshold coefficient. Maximum density controls the leading term, while the smallest curvature among the highest peaks controls the correction. Thus, among equal-height density peaks, the flattest determines the first correction. A finite-type branching construction transfers the spectral result to accessibility percolation using only a finite offspring mean. For compactly supported noise distributions with atoms, a separate theorem characterizes exactly when the critical gradient vanishes, including the distinction between isolated and non-isolated critical atoms. A second major contribution concerns neutral exploration and the time required to discover beneficial innovations. With mutation and opportunity rates specified, partial phenotype rankings yield attained bounds on discovery-time distributions and competing destinations. Sharp stationary survival and mean-time envelopes are supplemented by an exact worst-case mean discovery-time formula for a single innovation genotype on every connected mutation graph. The value depends on the stationary masses of the connected components remaining after deletion of that genotype, while finite attainment requires additional structural conditions. Under the stated additive-symmetrization gap condition, allowing nonreversible exploration does not increase this single-site supremum. For arbitrarily many innovation genotypes on a mutation tree, the paper gives one exact semidefinite program for every fixed nonzero hazard profile, including finite mutation-rate caps. It also identifies a precise structural boundary: the spectral-gap feasible region is convex in positive edge-resistance coordinates exactly when the connected graph is a tree. Cycles can change the optimum essentially. An alternating four-cycle has an exact unit-parameter optimum of five halves, strictly exceeding the supremum of seventeen sevenths for every spanning tree. Further examples show that scalar embeddings and spanning-tree reductions do not generally recover the required graph geometry. Successive-discovery envelopes are established under stationary entry and independent routing, while a counterexample shows that one phenotype ranking need not attain optimal bounds at every time horizon. The deterministic theory addresses what incomplete fitness comparisons establish across all compatible landscapes. It characterizes the nonlocal observation orders for which universal accessibility always has a common uphill-path certificate, uniformly across mutation graphs and compatible local comparisons: these are precisely the interval orders. A parameterized algorithm handles arbitrary comparison data through a vertex-cover parameter. Sharp examples establish minimum separations between universal accessibility and common-path certification, together with families requiring arbitrarily many alternative paths. For environmental switching, the manuscript characterizes universal target capture under arbitrary dependence within a specified family of admissible environments. Minimum target selection reduces to weighted vertex cover, with tractable specializations for affine uncertainty and explicit bipartite genotype graphs. Target capture and target invariance receive separate characterizations. Additional structural results include a universal face-extension theorem excluding a forbidden-unmarked-face characterization of subset–superset accessibility; complete local criteria for hereditary accessibility; and prescribed-peak constructions on ordered multiallelic spaces and prerequisite-constrained genotype lattices. A finite-jet nonidentifiability theorem states a separate limitation on recovering mutant curvature from resident dynamics. The deposit contains the 87-page manuscript, a matching Overleaf-ready source bundle, reproducible verification programs, machine-readable results, and research-status and source-provenance notes. Verification combines exact rational calculations, finite enumeration, and clearly identified numerical diagnostics. All earlier mathematical sections are retained. The results concern explicitly defined mathematical models. They do not establish unrestricted backtracking thresholds, finite-hypercube probability-one results, population-level evolutionary forecasts, or empirical biological validation. Internal proof audits and computational checks do not replace external peer review or establish historical priority.