Restricted Full-Adjoint Gain Bound for a Fixed Fourier–Galerkin Navier–Stokes Background

This mathematical technical note presents a computer-assisted restricted full-adjoint gain theorem for a prescribed Fourier–Galerkin Navier–Stokes background over the physical-time interval [0, 4/5]. For the original 376-wavevector, radius-six Galerkin initial-value problem, the analysis establishes the quantitative bound G_F([0, 4/5])² ≤ 34609/2700, and consequently G_F([0, 4/5]) < 3.99. The certification combines continuous, non-resetting six-shell storage across nine physical-time intervals; exterior boundary coercivity; continuous-time Riccati positivity; exact-arithmetic computational witnesses; original-Galerkin trajectory enclosures; backward-adjoint passivity; and an analytic infinite-tail limiting argument. The theorem establishes a specified class of shell-two receiving targets and a restricted high-frequency observation on shells 13–18. It does not establish unrestricted three-dimensional Navier–Stokes global regularity or a universal full-adjoint gain estimate. This publication is a Structural Stability Science (SSS) mathematical technical note. Its mathematical result is independently defined and does not introduce or modify Structural Identification Method (SIM) operations, SSS field mechanics, or Dimensional Human Field Theory (DHFT) variables. The supporting computational proof records, exact-arithmetic witnesses, and source-verification materials are maintained separately from this technical note. This mathematical technical note states the certified theorem and its principal quantitative bound; the supporting computational proof records, exact-arithmetic witnesses, and verification materials are maintained separately. Source Records Tonucci, C. (2026). Order Preservation Under Real Extension: Admissibility Conditions, Counterexamples, and the Distinction Between Iteration and Input Limits. Mathematical Note, Version 1, November 5, 2026. Zenodo. https://doi.org/10.5281/zenodo.23985122 Tonucci, C. (2026). Structural Identification and the Preservation of Relations Under Extension. Mathematical Note, Version 1, November 5, 2026. Zenodo. https://doi.org/10.5281/zenodo.23985887 Scientific Method, Field, and Theory Tonucci, C. (2026). Structural Identification Method: Deriving Invariant Mechanics from Somatic Detection. Zenodo. https://doi.org/10.5281/zenodo.19337749 Tonucci, C. (2026). Structural Stability Science (SSS): Field Conditions & Definition. Zenodo. https://doi.org/10.5281/zenodo.19740956 Tonucci, C. (2026). Dimensional Human Field Theory (DHFT): Architectural Identification of a Closed Explanatory System. Foundational Preprint, February 17, 2026, Version 1. Zenodo. https://doi.org/10.5281/zenodo.19075948 Tonucci, C. (2026). Minimal Stability Geometry of Dimensional Human Physics: A Mathematical Formalization within Dimensional Human Field Theory. Zenodo. https://doi.org/10.5281/zenodo.19076075 Tonucci, C. (2026). Stability Field Dynamics in Dimensional Human Physics: Mathematical Mechanics within Dimensional Human Field Theory (DHFT). Zenodo. https://doi.org/10.5281/zenodo.19076116 Method–Field–Theory Relationship Tonucci, C. (2026). Method–Field–Theory Relationship in Dimensional Human Field Theory (DHFT). Zenodo. https://doi.org/10.5281/zenodo.19741205 Technical Orientation & Source Architecture Structural Stability Science (SSS) — Technical Orientation & Source Architecture https://github.com/Cara-Tonucci/Structural-Stability-Science/wiki GitHub Repository Structural Stability Science (SSS) https://github.com/Cara-Tonucci/Structural-Stability-Science