The Hybrid Solver: A Composite Integer-Exact Solver Across the Input-Substrate Boundary: the Processor Period Theorem, an Integer-Reducibility Discriminator for Long-Time Solvability, and Six Reductions to Practice

The Hybrid Solver — a composite, integer-exact solver across the input–substrate boundary. A solver is usually one tool for one regime: a root-finder, an integrator, a stability test. This paper reports a different object — a composite solver that binds several heterogeneous integer-exact primitives (a finite-place arithmetic stage, a bounded-orbit stage, and a spectral-containment stage) into one pipeline that solves multi-regime problems no single primitive solves alone. The stages compose without representational translation because they share one invariant, the multiplicative order ord_p (the "Unity Clock"), and the composition is certified irreducible by an operational test: ablate any stage and the answer is withheld or corrupted — a genuine hybrid, not a multiplexer. Part I (theory) presents the original contributions: The Processor Period Theorem (proven). The power-type invariant kappa(A) of an affine processor's multiplier — computable in O(log a), before any prime is touched — sets a permanent, prime-independent "frustration floor" on the orbit period: it divides (p−1)/gcd(kappa, p−1). A kappa-saturation generalized-Artin conjecture (with a falsifier) accompanies it. Two integer-exact engines: the Frustration Transform (the discrete logarithm as the substrate-exact FFT for the affine map, realizing a pseudo-random orbit as a single Fourier tone) and the Positivity Latch (Sylvester's law of inertia with Jacobi's minor-sign criterion, deciding contained-vs-escaped from the signs of leading principal minors — containing a transcendental eigenvalue exactly, and correct where floating point fails). An integer-reducibility discriminator that sorts long-time solvability by whether a flow reduces to integers at all, with three agreeing reads (a proven prime-block embedding behind floor-invariance, PSLQ-algebraicity, and the inertia latch). Applied end-to-end to the Collatz problem, it reduces the conjecture to a complete partition — two cells proven, two named as the irreducible open residues. (No Clay problem is resolved, and Collatz is not claimed settled.) Part II reduces the solver to practice across six regimes: chaotic mixing, Navier–Stokes regularity and cascade steering, additive-manufacturing solidification, predictive deep-turbulence adaptive optics, flat-band superconductivity (a magic-angle "Totient Law"), and a productized battery-containment line that matches the open-source industry tortuosity standard (TauFactor) to 0.121% on a real micro-CT at 3.47x the speed and certifies a 3.23-billion-element electrode on a commodity workstation. Every accelerated or exact result is byte-gated against a ground-truth computation. The mathematics is dedicated to the public under the MIT license; the engineering apparatus is held in confidence and is the subject of U.S. Provisional Patent Application No. 64/098,689 (filed 25 June 2026).