Common-null transport and viscous response from an internal periodic cell: strengthened resolvent estimates and multi-reviewer verification — interim report v0.1

Abstract

Interim progress report on the study of viscous response from a kinetic model on ℜ3+1×S1 with a common null constraint coupling the observed and internal (circle) directions. Under a two-branch time-reading hypothesis, two signed internal-momentum species (σ=±1) follow an Anderson–Witting relaxation toward local equilibrium. For a linearised shear perturbation with a sinusoidal collision rate, an internal periodic cell yields positive viscous response with error bounds uniform in particle energy. New results in this report (relative to the base manuscript v0.3): Uniform resolvent gap. For a constant collision rate, the linearised operator has a spectral gap uniform in Fourier mode k, proved via a Woodbury–Fredholm reduction to a 6×6 dispersion matrix and a slab estimate (measure-theoretic bound on momentum concentration near the transport velocity’s zero set). A Gearhart–Prüss-type theorem yields exponential decay. The Kawashima-type condition (K) is shown necessary and sufficient, with an explicit counterexample to a weaker proposal. Basis mismatch estimate. An Aubin–Nitsche type bound separates truncation error (controlled by ε and J) from frame-mismatch error (controlled by rapidity ρ). With ε=1/2, J=5, 128 unknown fields achieve 0.098% relative error ceiling for the frozen DC shear viscosity. Harmonic rank repair and stress extraction. The 55 fourth-degree harmonic labels in H4(ℜ5) have image rank 35 due to 20 linear relations; an S5-covariant quarter-turn repair restores rank 55. The unique stress-type V2 copy in the SO(3) branching is extracted via F(A)=tr(A3)A−⅓tr(A2)[A2−⅓tr(A2)I], with Gram matrix δab/135. All core claims have been checked by the author and independently reviewed by two AI systems (Codex/GPT-5.6-Sol, Devin/Gemini-4-Argon) with zero P0 (critical) findings across both work packages. This interim report has not been externally peer-reviewed by human experts. The main text is in English; a Japanese translation is included. If the two differ, the English text is authoritative. The Markdown files contain the LaTeX source for machine-readable equations. 日本語要旨: 直積空間 ℜ3+1×S1 上の運動論モデルを研究する。 共通null拘束条件のもとで、 正弦波的衝突率をもつ内部周期セルが正のせん断粘性応答を与える。 定数衝突率の場合の一様レゾルベントギャップ、 Kawashima型条件の修正と反例、 基底ミスマッチ評価、 調和ランク修復と応力抽出を報告する。 中間報告であり、人間の外部査読は未完了。