Multiple-Swap Superposition and Ultrametric Memory Separation in Collatz Correction Coordinates

Ghost Languages and Unbounded Prefix Memory in Collatz Realizabilityestablished, via an explicit Swap Difference Formula, that no boundedsliding-window observer can determine the tower quotient q_k of avaluation word. That result concerns a single local perturbation. Thisnote extends it in three steps. First, we prove that the effect offinitely many non-overlapping swap handles on the correction coordinateB_m superposes linearly, using only the locality of Ghost's own swapformula. Second, combining this with a short, self-contained argumentfrom the strong triangle inequality of the 2-adic valuation, we showthat handles whose valuations are pairwise distinct produce 2^r pairwisedistinct correction signatures B_m(w_epsilon) for r handles. Third, weexhibit an explicit bounded-discrepancy construction -- using theSturmian (Beatty) word of slope log_2(3), for every intercept -- inwhich arbitrarily many such handles can be placed while the discrepancycorridor width remains independent of the number of handles r.Combining these results, no fixed-size deterministic streaming automaton(over the binary alphabet {1,2}) can compute the correction coordinateB_m exactly on all bounded-discrepancy words. Scope: this note proves an exact-value computational lower bound forthe correction coordinate B_m, not for the tower quotient q_m itself.Whether B_m-separation transfers to q_m-separation, and whether eithertransfers to hardness of deciding carry coherence q_m=0, are left asexplicit open problems; the latter is resolved in the companion paper"Zero-Target Decision-State Separation in Collatz Digit Towers." This is a companion paper to "Prefix Recovery from IRA CorrectionCoordinates" (DOI: 10.5281/zenodo.21486899) and "Zero-TargetDecision-State Separation in Collatz Digit Towers," together formingthe IRA-RCS (Recover-Compose-Separate) research program on the statecomplexity of Collatz correction coordinates.