Touch and Crossing: Dual Primitive Modalities of Threshold Monitors

Threshold monitors of real-valued signals fire on one of two operational semantics: the crossing modality, with events at upward transits of a threshold, or the touch modality, with events at local extrema. The runtime-verification and signal-processing literatures treat these as competing encoder families, usually presenting one as primitive and the other as a degenerate case. We argue that this framing is wrong: touch and crossing are dual primitives with incomparable minimal resource requirements - crossing is causal and carries one bit of memory, whereas touch is memoryless but needs a three-sample window and one step of lookahead - so neither monitor can simulate the other without genuinely new structure, and bridging them costs exactly one added bit of structure in each direction: persistent state one way, window or lookahead the other. We develop three consequences. First, the two modalities sit on a ladder indexed by the number of distinct verdicts a monitor may emit: a verdict-free baseline, a degenerate binary rung, a ternary rung that hosts the touch/crossing duality, and a four-valued rung that organises both along orthogonal touch/crossing and weak/strong axes. The two-, three-, and four-valued rungs themselves are the known runtime-verification verdict progression (the two-/three-/four-valued LTL semantics); our contribution at this layer is not the ladder but the strict tuning hierarchy L1 < L2 < L3 of tuning powers that realise the rungs. Second, the touch-crossing join is a literal parameter coincidence at the four-valued rung: a single augmented monitor realises both modalities as exact parameter choices, and the three classical engineering encoders - edge detector, Schmitt trigger, and extremum detector - appear as named special cases of its ternary subfamily. Third, the duality is operational: a pointwise count inequality (upward crossings never outnumber touches, up to a trace-boundary tick) separates the two on every signal; a linear-time algorithm synthesises tuning parameters realising any feasible target finite event pattern, with an asymmetric feasibility criterion; and a safety / cosafety / liveness / coliveness classification places both modalities inside the existing temporal-property landscape without new logical machinery. --- Version note (this version): this version incorporates a citation-integrity review of the manuscript. Unsupported or inaccurate attributions were corrected, including two mischaracterized results from cited sources. Proposition 13.11, whose justification previously appealed to a result attributed to a companion paper that does not contain it, is now justified by a short direct argument from the safety/cosafety/liveness/coliveness class definitions; the proposition's statement is unchanged. No new theorem or result is introduced.