The Physical Cutoff Does Not Restore Homogenization: Phase-Dependent Burning in the Strain G-Equation

arXiv:2608.15337v1 Announce Type: cross Abstract: We disprove the expectation stated by Xin, Yu, and Ronney that the physical positive part strain G-equation should possess an effective burning velocity in cellular flows. For the standard cellular flow in dimension two V_A(x_1,x_2)=A(-sin x_1os x_2,os x_1sin x_2), if 00 universal. The same conclusions hold for arbitrary continuous periodic perturbations of planar initial data. Under the physical scaling V_A(x/arepsilon) and d_arepsilon=arepsilon d, an order one value gap persists between points at distance O(arepsilon) at every positive macroscopic time, so the rescaled solutions have no locally uniformly convergent subsequence. The proof uses the Hamiltonian sandwich H_{unc}le H_+lewidehat H. The upper comparator widehat H is a rectangular support function, equivalently an upper expectation over a state-dependent credal set, whose reversed control dynamics possess an invariant comparison channel. We also prove that for any C^2 incompressible periodic flow, every arepsilon-outward barrier certificate has covering radius at most 2darepsilon for all sufficiently small arepsilon. We further discuss implications for statistics and machine learning: rectangular, time-consistent local uncertainty need not imply forgetting of the initial state in the long run, so additional global stability or ergodicity conditions are needed in robust sequential decision making. Two Lean 4 appendices record conditional formalizations of a sufficient p=e_1 subregime and of the logical assembly of the rigidity theorem for barrier certificates.

Source: arXiv cs.LG | 2026-08-18