Constant-Time 2-Adic Smoothness Screening and Modulo-4 Parity Obstructions in Arithmetic Dynamics: Achieving 96.9x Pre-Multiplication 4096-Bit Key Auditing, 50.6% Diophantine Pruning, and Transversal Euler-Manifold Escape

Iterative arithmetic dynamical systems that alternate expansion by the sum-of-divisors function on odd integers with repeated division by two on even integers offer a natural dynamical setting in which to study perfect and multiperfect numbers. Static counts of trailing zeros in the binary expansion of an odd integer yield only a trivial parity distinction. By contrast, passage to a compressed odd-to-odd operator analogous to the Syracuse map of the Collatz problem, together with the associated one-step two-adic logarithmic Lyapunov drift, reveals a far richer architecture governed by progressive two-adic cooling. Drawing on the classical Lifting-the-Exponent lemma for the two-adic valuation of prime-power divisor sums and on Euler’s well-known prime-square characterization of odd perfect numbers, we establish a universal additive lower bound on the cooling rate. This bound shows that every integer possessing at least two distinct prime factors raised to odd exponents contracts strictly in a single step. Consequently any hypothetical odd perfect number is forced to sit as a stationary fixed point on the thinnest possible non-zero valuation slice—the Euler submanifold—which itself has natural density zero among the odd integers. Our central original contribution is an exact modulo-four parity obstruction. From the arithmetic data of any point on the Euler submanifold we extract a simple parity invariant that, without any evaluation of the sum-of-divisors function, algebraically ejects more than half of all composite points from the submanifold in a single macro-step. Cyclotomic dispersion of the remaining primes then drives the great majority of the survivors out as well: nearly ninety-eight percent escape after one step and more than ninety-nine point nine percent fail to remain on the submanifold for two consecutive steps. An exhaustive computational census of every odd integer up to half a million, together with an examination of both theoretical and practical consequences—constant-time pruning inside Diophantine solvers, dramatic acceleration of smoothness screening for large cryptographic moduli, cycle-resistant whitening of pseudorandom generators, and single-cycle hardware verification of big-integer arithmetic—completes the study.