The Tower Is Coordinate-Free: What Survives When You Change the Vertical Axis of a Prime Walk

A companion construction walks each integer's prime factorization as a set of compass directions: one leg per prime, a turn of 137.50776° between legs, and a climb at each new prime, giving every integer a unique address in space. The vertical axis of that picture was an authorial choice. This paper asks which of the measured regularities depend on that choice and which would hold under any vertical. Eight of ten never touch the vertical — they are computed entirely from sideways motion, and climbs do not move the walk sideways — so they reproduce identically, not merely similarly, under any vertical. One regularity, the octave law, reads the vertical and survives only under the original choice. The result is negative in the useful sense: most of what was found is not an artifact of how it was drawn. Three further results follow. The closure deficit is shown to be isolated with an effective lower bound by Baker's theorem on linear forms in logarithms, making this the one substrate in the program where a noise floor is discharged by citation rather than by measurement. The tower's phase is measured to be Benford-distributed rather than uniform — a registered prediction that failed informatively — and Benford is precisely the invariant measure of the multiplicative action the same construction exhibits, so the two results are one result. And the oldest measured instance of the same object is musical: twelve perfect fifths do not close into seven octaves, missing by 23.46 cents. No new theorems are claimed. Every core object is classical and is cited where the work rederived it independently, including the arithmetic taxicab metric and the factorization-torus. A credentialed zbMATH/MathSciNet prior-art pass has not been run; novelty language is restricted to "not located."