Paper LXXVIII: The Principle of Least Action as Auxetic Brane Shape Memory: How ν = −1 Mechanics Derives Hamilton's Principle, Fermat's Principle, and Newton's Laws as Theorems, and Extends Them to Macroscopic Distances

The principle of least action the postulate that particles take paths for which the action S = R L dt is stationary under variations, δS = 0 is one of the deepest and most mysterious foundations of physics. No mechanism has previously been oered for why nature should enforce this condition. We show that in the brane-bulk octonionic framework, the principle of least action is not a postulate but a theorem of ν = −1 auxetic brane mechanics. The argument has three parts. (1) The action is the trail deformation energy. A particle is a focal zone a deceleration pattern in the brane fabric (Paper I). When a focal zone moves, it carves a deformation trail in the auxetic brane. The action Spath is identied with the strain energy of this deformation trail: S ≡Utrail = R σij εij dV dt. (2) ν = −1 makes δUtrail = 0 identically. In any elastic medium, the variation δUpath under a transverse perturbation δq contains boundary terms proportional to |1+ν| times the transverse strain variation. At ν = −1: |1+ν| = 0 exactly, and these boundary terms vanish identically, leaving only the Euler Lagrange condition ∂L/∂q −d(∂L/∂˙q)/dt = 0. The equations of motion are a theorem of the auxetic mechanics, not an independent postulate. (3) Shape memory extends this to macroscopic distances. In a conventional medium (ν > 0), the path memory decays at rate 1/τeff= (1 + ν)/τrelax: the preferred path is forgotten beyond a nite relaxation length. At ν = −1: τeff→∞and the Weissenberg number Wi →∞at all observation timescales (Paper LXXI). The auxetic brane retains the deformation pattern of the preferred path indenitely and at all distances. This is why the principle of least action holds at macroscopic scales it is not merely a Planck-scale phenomenon that somehow survives to classical physics; it is a macroscopic property of the shape memory of the brane fabric. As a corollary, the quantum-to-classical transition is identied as the scale where shape memory dominates over Fano delocalization: the action principle holds to fractional accuracy εaction(E) = |1 + νeff(E)| = αs(E)/π, which decreases from ∼0.19 at the GeV scale to ∼0.006 at the Planck scale. Fermat's principle of least time, Hamilton's principle of stationary action, Maupertuis's principle of least action, and Newton's laws of motion are derived as special cases of the auxetic brane mechanics. Four new predictions follow (Predictions 172175). Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.