Research &
Publications.

By Akpalu Elliot Elikplim

We formalize the structural properties of the Π̇(a,b,c) constraint manifold and its associated stochastic logistic framework. Through exact symbolic computation and high-precision numerical experiments, we establish: (i) a complete Gaussian curvature partition showing 87.80% hyperbolic dominance with power-law parabolic boundary; (ii) a fold catastrophe cusp governed by the quintic b⁵+b⁴−1=0; (iii) non-existence of saddle-node bifurcation via positive discriminant proof; (iv) closed-form Shannon entropy and exact Fisher information matrix for the stationary Fokker-Planck density; and (v) detailed balance with zero entropy production. These results bridge algebraic geometry, nonlinear dynamics, and stochastic thermodynamics within a unified mathematical ecosystem.

By Akpalu Elliot Elikplim

We formulate and investigate the arithmetic geometry and integer structures of the variety constraints underlying the Π̇(a,b,c) system. Through exact symbolic computation, series reversion, and modular analysis over finite fields, we establish three arithmetic discoveries: (i) the Integer Asymptotic Coefficients Conjecture, showing that the stable equilibrium root power series expansion consists of exact integers verified symbolically up to order 20; (ii) the Diophantine Isolation Conjecture, proving that positive integer solutions are isolated to the trivial family (1, b, 0, 0), forcing the coordinate c to a non-integer rational; and (iii) a Hasse-Weil quadratic growth fit N(p) ≈ 2p² − 2.58p − 2.92 for modular solutions mod p. These findings unveil a rich, hidden modular and algebraic structure underlying continuous SDE parameter constraints.

By Akpalu Elliot Elikplim

We formulate the foundations, dynamics, and constraints of the Π̇(a,b,c) mathematical framework. Built upon the parametrized rate equation Π̇ = (ab² - ca)/b as a primary axiom, we derive the linear autonomous dynamical system dΠ/dt = Π(b - c/b) under the identification a ≡ Π. We prove that the system's defining constraint equations (the n-law and √n-law) select a smooth, two-sheeted parameter manifold Μ. Under dynamic state projection, the restricted flow possesses exactly two positive equilibria Π₁*(b) and Π₂*(b) representing stable sink and unstable separatrix states, respectively. A transcritical bifurcation is proven at the critical boundary b² = c, and stochastic multiplicative forcing is modeled via Fokker-Planck equilibrium densities. High-precision numerical stress-tests verify manifold solvability and error bounds under extreme parameter ranges.