For linear-rate master equations the generating function admits an exact composition-multiplier representation whose Taylor coefficients on any finite window are obtained from a closed lower-triangular ODE of size 2(N+1), independent of the truncation cap N; the same closure is combined with Strang–
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Saddle-point asymptotics for chromatic and Tutte polynomials of complete multipartite graphs prove Kotesovec's fixed-column conjecture and yield fixed-part and all-order expansions for OEIS sequences.
Tropicalized massive scalar QFT is exactly solvable via a non-linear recursion for effective action coefficients that computes graph moduli space volumes, enabling a polynomial-time sampling algorithm for high-order perturbative contributions.
Pin classes exhibit a phase transition at μ ≈ 3.28277 with countably many below the threshold and uncountably many at it; all growth rates below μ are classified via periodic pin permutations.
Implicit formula for critical curve α_c(κ) of loop percolation on d-regular trees, positive for κ>κ_c=2√(d-1)/d-1, with mean-field exponents for κ>κ_c and quadratic percolation probability at κ=κ_c.
Derives closed-form critical parameters, equation of state, binodal and spinodal curves for the cell fluid model in the J2 ≫ J1 limit matching van der Waals lattice gas, plus analytical extension to marginal J1=J2 stability using deformed exponential asymptotics.
Local goal support in policy functional graphs predicts goal failure in sparse GCRL with F1 0.925, and a taxonomy explains residual failures from competing attractors.
The mixed Hessian develops exactly one logarithmically diverging eigenvalue per symmetry block at the analytic threshold zeta_c, with the remaining spectrum bounded, and scalar Gram functions continue regularly past the geometric threshold.
Uniform random d-regular bipartite planar maps converge locally to an almost surely one-ended recurrent infinite map.
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