One Register: A Common Origin for the Granularity of Hilbert Space and of Space

Palmer's Rational Quantum Mechanics writes a qubit as L bits: the count m of +1 bits gives the Born probability m/L and a cyclic shift the phase. Previously we made L universal, identified one phase step with one Planck length on the horizon, and derived Newton's constant and Einstein's equations at L ≈ 6×10⁶¹. The two axioms become one: a dial of L notches on the horizon, each a Planck length; every qubit is a pattern on it; one notch is a Planck-length translation, under which a wave of κ wavelengths gains κ phase steps, Palmer's phase step being the slowest wave's. From it: light moves one notch per tick; velocity is the excess of right- over left-movers, v/c = 2m/L − 1; in matter a right-mover becomes a left-mover every quarter Compton period, its Compton clock being Palmer's hidden global phase; the circumference is the longest wavelength, the previous paper's second axiom; and Born's rule as a frequency bounds a state spanning its Hilbert space at 2^N ≤ L, N_max = 205, for any physical realization. Free dynamics is exact mode by mode in the plane of any elementary vertex, whose momenta the horizon's pixels label; whole-cell position holds along one direction only; several particles in several planes need three dimensions, not constructed here. The predictions are inherited: the qubit ceiling, now a random circuit and its inverse failing to return past 205 logical qubits, and constant Λ; it entails no energy-dependent light speed and no intrinsic matter-wave decoherence, unlike collapse models.