Mass increase and acyclic cancellation for phase flows on quiver complexes

Abstract

For two-term complexes over a two-vertex parallel-arrow quiver, with degree-one term supported at the source, trace-norm mass can increase along the Haiden–Katzarkov–Kontsevich–Pandit phase flow defined by positivity. We give an explicit noncommuting example. Nevertheless, for invertible source differential and vertex charges in the strict upper half-plane, every positive initial metric yields exponential decay of the acyclic coefficients and convergence of the normalized cohomology metric. For a balanced rank-two three-arrow family, a sufficiently large initial acyclic scale gives quantitative cancellation for arbitrary bounded mixing, uniformly through a change of cohomology stabilizer. The evolving coefficient approaches its instantaneous cohomology coefficient; the proof controls the polar phase using only the nonsingular cohomology block. As the differential scale tends to infinity, the static mass and its first derivatives in bounded arrow coordinates converge uniformly. A rescaled profile forces third derivatives to diverge. An explicit primitive has a negative Levi direction on a complex gauge orbit and identifies the positive limiting cohomology potential.