Cutoff-Uniform Enstrophy Bounds, Fractional Riccati Integrability Thresholds, and Resonant-Triad Depletion in Dyadic and Toroidal Fluid Hierarchies: A Machine-Checked Investigation of Ten Frontier Hypotheses

We conduct a systematic, machine-checked mathematical physics investigation of ten coupled frontier hypotheses governing enstrophy production, fractional hyperviscous regularity thresholds, dispersive second-harmonic-generation (SHG) regulators, and three-dimensional resonant-triad hypergraph depletion across the Katz–Pavlović dyadic hierarchy and toroidal Fourier lattices. Rather than restricting formalization to known textbook identities, we evaluate an open ten-hypothesis research programme in which six of the ten hypotheses (60.0%) are positively proved and four plausible conjectures (40.0%) are formally refuted or obstructed by explicit counterexamples, integrability deficits, and arithmetic parity theorems. All twelve theorems (eight positive theorems and four formal counterexample/obstruction theorems) are machine-checked in the Lean 4 theorem prover (ANSE.DyadicNavierStokesRegulator) with zero sorryAx axioms and verified via environment-level definitional equality against a frozen canonical specification (ANSE.DyadicNavierStokesRegulatorSpec), accompanied by an exact arbitrary-precision rational and integer combinatorial replication harness and a 20-record 5-tier Elenchus epistemic ledger.