An Open-Loop Endpoint Memory Model for Incompressible Vorticity with a Null-Distance Channel and a Two-Branch Time Plane: Endpoint Response, Non-identifiability, and Numerical Checks

Abstract

We study a candidate passive extension of the local rotation of an incompressible viscous fluid. The vorticity \(\boldsymbol\omega=\nabla\times\boldsymbol v\) is passed, as a boundary value, to an auxiliary half-line ("null-distance channel") \(y\ge0\) on which a diffusion field \(\boldsymbol U\) evolves, and the endpoint flux \(\boldsymbol j=-\kappa\,\partial_y\boldsymbol U|_{0}\) is returned to the fluid as \(-\beta\nabla\times\boldsymbol j\). The time variable is then lifted to a two-branch time plane \(T_{\rm time}=t_oI+t_nT_u\) with \(T_u^2=I\), using a generator that is chosen, not derived. We record the following conditional results. (1) For \(\beta=0\) the observed-fluid equation coincides with the ordinary incompressible Navier–Stokes equation and the null field decouples. (2) With zero initial channel field, eliminating the half-line gives the known square-root (Dirichlet-to-Neumann) endpoint kernel; in the incompressible case the reaction is a known fractional memory viscosity \((\kappa\beta^2/\sqrt{\nu})\,\partial_t^{1/2}\Delta\boldsymbol v\). (3) For the single-branch model a formal energy identity holds in which the sum of fluid kinetic energy and channel energy is non-increasing. (4) Input that is uniform in \(t_n\) reproduces the single-branch endpoint response; real event bands that vary in \(t_n\) change the observed endpoint response. (5) For untilted reflection-symmetric bands, the readout-line position \(\tau\) and the band weights cannot be identified separately from the observed endpoint response; tilted bands are algebraically identifiable but not quantitatively under 1% noise. Finite-difference computations are compared with erfc analytic solutions. Here the transform-domain response ratio means a component ratio of the endpoint response to a specified test sample in the two-branch transform domain; it does not denote a delayed waveform. What this paper does not show. (i) The model is open-loop: the event bands are prescribed test inputs, and the self-consistent closed loop including the two-branch time plane and the endpoint reaction is not constructed. (ii) It does not address existence, uniqueness, smoothness, or finite-time singularity formation for either the ordinary Navier–Stokes equations or the coupled system with \(\beta\neq0\). (iii) This paper neither asserts nor rules out a metric meaning or physical reality of the null direction, and it does not present any particle-theory interpretation or physics near black holes. (iv) Numerical agreement means agreement with the stated analytic solutions, grids and time steps; it is not a probability that the model or any physical theory is correct. Disclosure. Developed with AI assistance (lemma/counterexample cycles and Python checks); checked through multiple rounds of AI-assisted review but not human peer reviewed. Files: English paper (paper_en.pdf), Japanese paper (paper_ja.pdf), and a reproduction package (scripts, JSON outputs, lemma documents, SHA256SUMS). Reproduce with pip install -r requirements.txt; cd code; python run_all.py (expected: 266 checks pass).