Suffix Diffusion and Pressure Recovery in the Collatz Map: A CPD Technical Framework and Empirical Test Battery

This technical note develops a CPD (complexity-pressure-diffusion) framework for studyingthe Collatz map through binary suffix dynamics. The note does not claim a proof of theCollatz conjecture. Its purpose is to formulate a proof-relevant mechanism and to document areproducible empirical battery supporting that mechanism. We work with the odd acceleratedmap U(x) = 3x+1/2v2(3x+1)for odd positive integers. The valuation k(x) = v2(3x + 1) is interpreted as a pressuredepth, while the binary shift-add identity 3x + 1 = x + 2x +1 is interpreted as accelerationplus deterministic carry-driven suffix transport. We define escape windows as finite oddmacro blocks with negative pressure surplus, and suffix diffusion observables based on suffixcurvature and Hamming transport under U. Numerical scans over 2 ≤ n ≤ 50000 show thatlonger escape windows strongly correlate with accumulated suffix diffusion, and that suffixdiffusion strongly correlates with pressure recovery. A key rigorous correction is also recorded:low-pressure valuation words are abundant and can be extended symbolically, so the correctproof target is not universal bounded recovery for all symbolic branches, but instability ormeasure collapse of pressure-gate-avoiding continuations in positive integer orbits.