Reduced coordinates for the Collatz map: exact per-step laws, anchor dynamics, and the limits of counting arguments for cycles

We study the Collatz dynamics in a reduced coordinate system that compresses each deterministic valuation run into a single block, producing a self-map F on states (ω, d): an odd core prime to 3 and a depth. The reduction is faithful: the Collatz conjecture is equivalent to every F-orbit reaching (1, 1), with nontrivial cycles in bijection. In these coordinates the local arithmetic admits exact laws. A single 2-adic quantity, the anchor M(ω) = −2 log ω/ log 9 ∈ Z2, governs the step: the exit valuation obeys the global law s = 2 + v2(d − M(ω)) whenever 3dω ≡ 1 (mod 8) and is constant on the remaining residue classes; the depth evolution closes exactly in terms of the anchor displacement together with a stated 3-adic absorption law; and the anchor increment along one step obeys an exact law modulo any power of 2, computable from graded residues of the state. A finite window of digits consequently decides each step in an error-free trichotomy, while a digit-budget accounting indicates that no bounded window can decide infinite horizons, localizing the difficulty of the problem in the digit supply of the anchors. On the cycle side, a oneline elimination identity yields short rederivations of the classical exclusions for cycles of one, two, and three blocks. Our main new theorem is a sharp dichotomy for counting arguments: a trim uniform in the number of blocks p exists, giving effective finiteness at every period, but its constant necessarily degrades like (log2 3)−p: an explicit family of near counterexamples (staircases: geometric climbs closed by a single crash, precisely divergent orbit profiles bent into loops) shows counting arguments cannot do substantially better, so uniform cycle exclusion requires arithmetic (divisibility) input, not sharper counting. Finally we state the equidistribution hypothesis implicit in the classical heuristics as a precise conjecture about an exactly computable product law, prove its consequences conditionally, and report a calibration campaign whose four apparent anomalies all dissolved under controls, one of them via an exact routing lemma that a biased estimator had been reflecting