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

The Tower Is Coordinate-Free — companion to Taking a Loop Walk Through the Integer Forest (10.5281/zenodo.21449569). v3.4 (2026-08-22): names D. Dominici on the arithmetic-metric citations; adds an origin note crediting the Sophomore's Dream identities (Bernoulli 1697) and the accident that pointed to them; strengthens the closure-floor citations with the sharp two-logarithm bounds (Rhin 1987, exponent ≈7.616; Wu 2003; Zudilin 2004); adds standard references for the Benford dictionary, the chain count, and the divides-relation graph; adds a small result — the Liouville sign as the holonomy of a flat ℤ/2 connection on the subway network; runs the boundary-holonomy measurement the first edition stated as open — the transition vertices carry exactly zero vertical vector area, so there is no boundary holonomy in the measured channel, and 99.92% of the horizontal area; adds a Threads Left in the Open section handing off the never-computed diffraction pattern of the prime-rank golden-angle point set to the aperiodic-photonics literature, the analysis of the sideways constant, and one fenced Hofstadter parallel; records that a dated catalog sweep did not locate the sideways constant; moves related identifiers to concept DOIs; updates the supplement's interactive bench to v1.3. No previously published value changed.

multiplicative number theory; prime factorization; golden angle; Liouville function; Dirichlet series; Laplace transform; Baker's theorem; linear forms in logarithms; Benford's law; musical temperament; coordinate-free; closure deficit; flat connection; Vogel spiral; aperiodic photonics; holonomy; open problems; arithmetic metric; hyperuniformity; irrationality measure; Sophomore's Dream