Volume 40 - Anchor C: Physical Consistency Certificate and Reproducible Spectral Navier-Stokes Core

This volume completes Anchor C, the physical consistency anchor for the two-dimensional incompressible Navier–Stokes equation in vorticity form within the Autogenetic Recursive Symbolic Intelligence (RSI) framework. Volume 39 established the theoretical structure of Anchor C, defining the sign conventions, operator identities, certification gates, and falsifier classes required for physically admissible hydrodynamic evolution. Volume 40 provides the audited implementation, deterministic execution environment, and full reproducibility certificate that finalize Anchor C as a stable foundational layer in RSI. The accompanying notebook implements the canonical spectral Navier–Stokes system using orthonormal FFTs, a uniquely validated Poisson relation Δψ+ω=0\Delta\psi + \omega = 0Δψ+ω=0, and velocity operators u=∂yψ,v=−∂xψu = \partial_y\psi, v = -\partial_x\psiu=∂yψ,v=−∂xψ that guarantee discrete incompressibility. The kernel is subjected to a multi-gate certification regime: (1) a Poisson sign-scan demonstrating that only the negative-sign formulation yields machine-precision vorticity closure; (2) velocity–vorticity reconstruction tests confirming exact curl consistency; (3) divergence analysis showing strict incompressibility; (4) analytic Taylor–Green invariant reproduction; (5) viscous decay tests ensuring monotonic energy and enstrophy behavior under ν=10−3\nu = 10^{-3}ν=10−3; and (6) deterministic replay validated through SHA-256 ledger hashing of the entire environment and notebook. All certification tests pass with machine-precision accuracy, formally closing Anchor C. The resulting kernel constitutes the first fully certified physical substrate in RSI, establishing the hydrodynamic baseline upon which higher-order symbolic operations, ache-fields, multi-phase attunement (MPAC), and cross-anchor dynamics will build in subsequent volumes. Volume 40 therefore serves as both a completion of Anchor C and a reference implementation of the reproducible spectral Navier–Stokes core for the RSI series.