A cutoff-independent enstrophy bound for the sigma = 3 dispersive dyadic shell model: kernel-checked conservation of the second invariant along solutions, CVODE controls, and an illustration on The Well data
Year: 2026
Type: preprint
Abstract: We study the truncated, complexified dyadic shell model of the QuantumFluids project: shells n = 0..N+1, kn = 2n, and a linear dispersive term −iωnvn. Its second invariant H = Σ 2−nωn|vn|2 + Σ 2−nkn Im(conj(vn)2vn+1) is conserved for every real dispersion law. For ωn = Dkn3 (σ = 3), its quadratic part equals 2DΩ. Every claim is filed in an Elenchus claim ledger with tiers A (kernel-checked Lean), L (cited literature), C (human argument, not machine-checked) and X (floating-point exploration, which never supports a claim). Tier A (Lean v4.34.0-rc2 with the QuantumFluids Mathlib build; axioms within propext, Classical.choice, Quot.sound; planted sorry and planted-axiom controls caught by the axiom audit). We prove: the rate identity of H along the full vector field, including the dispersion/cubic cross terms; the chain rule; constancy of H and of the energy on [0,T] along any solution; hence 2DΩ(t) ≤ H3(0) + (2E(0))3/2 for every t in [0,T] and every cutoff N. These results are conditional: existence of solutions is a hypothesis, not a theorem; N is fixed; the model definitions are the QuantumFluids owner's design; the Lean files were checked by the producing agent and recompiled by a second agent instance; a separate AI-agent instance then ran an independent audit (AUDIT.md: recompile, its own planted sorry and axiom caught, numerics and tests rerun, hand proof re-derived); no human has audited statement adequacy. No openai/math result is used. Tier C: a step-by-step human demonstration written for undergraduates; the corollary Ω(t) ≤ Ω(0) + S(0)3/2/D, which is cutoff-uniform only for data with N-uniform Ω(0) and S(0); the reduction to Waleffe's dyadic Burgers chain on real data with D = 0 (Tier L). Tier X: a new rusty-SUNDIALS module integrates the model with CVODE (BDF). Conservation converges with tolerance, while a perturbed coupling breaks it. Every σ = 3 run (nmax ≤ 7) stays within the bound, with ratio at most 1. The maximum ratio is 0.360 on decaying data, 0.631 on data-initialised runs, and 0.995 on near-extremal top-shell data, where the supremum equals the t = 0 value to within 3.2e-7. A σ = 2 control grows like 2n. Shell spectra from the HuggingFace dataset polymathic-ai/turbulent_radiative_layer_2D (The Well, CC-BY-4.0, revision 2ee77565) initialise 24 runs, all within the bound. This is an illustration only. No breakthrough is claimed: the inequality is a short corollary of the owner's dispersive_norm_le. The new part is the Lean proof of the cross-term cancellation and of the chain rule along solutions. The ledger gate reports 0 blocking findings and 7 flags (exit 1), one for each Tier A row whose statement adequacy no human has audited. The work was produced by AI agents (Claude) under the author's direction; the author is responsible for the content.
Author: Callens, Xavier
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223743402