Binary Scaling of ρ = 400/11: Prime Hierarchy Under Doubling

The gap scaling ratio ρ = 400/11 − 1/2500 − 1/939939, connecting the geometric gap G = 1 − √2×ln(2) to Feigenbaum's period-doubling constant, is examined under repeated multiplication by 2 — the framework's native operation. The drift sequence produces framework primes in order (5, 11, 23) at tower landmarks (2⁴, 2⁵, 2⁶). The prime 7 is the only framework prime where 2 is not a primitive root, explaining its role as the boundary prime. At k = 14 (the tower's breath span), the drift factorizes as 7 × 23 × 37 — boundary × mirror × completion. All claims are exact integer arithmetic, verifiable by any calculator. Update — 13 April 2026: §2 corrected. The earlier text described step 36 as "the Landauer step in the binary tower (step 36, where 36G ~ 1/√2)," which conflated two distinct thresholds. Step 36 is adjacent to the geometric damping threshold (n ≈ 35.82); the Landauer crossing proper is at n ≈ 35.11 (between steps 35 and 36); the Shannon bound is at n ≈ 36.54. All §2 arithmetic content (the self-encoding decimal, the 99-closure, the 36.00° pentagonal angle, and the 7/11 gap to 37) is unchanged. Typography upgrade applied. The prior published version remains on OSF, renamed with the OLD- prefix and a date.