Phase-Locked Local Grain Coarsening in an Endogenous Walker Clock

In a local integer-convolution readout, the gcd grain of the active overlap multiset can be coarser than the surrounding field would suggest. Prior static work showed two mechanisms: support-completion seams that coarsen the local gcd to the kernel mass, and uniform multiplicity scaling that grows the local gcd while leaving the normalized local distribution, local capacity, and occupied overlap count invariant. This note finds the same uniform-scaling mechanism in an autonomous self-interacting walker. In the tau = 0 flip clock, one phase of the endogenous 10-step recurrence produces mW = 2, 3,. . . , 10 at steps 19, 29,. . . , 99, with local capacity, overlap-value count, and normalized readout distribution invariant across those events. A tau = 12 control, audited identically, produces no mW > 1 events. U-turns alone are not sufficient: the mechanism requires a phase-locked recurrence that accumulates identical wake increments each period. The coarsening is a readout signature of periodic wake stacking, not a cause of the flip clock.