The Premising Result: Lexicographic Update and Corrective Replay

Lexicographic preference update is dynamic preference logic at its most transparent: the latest input settles all cross-cell priority, the prior relation survives inside each cell, and reduction axioms complete the dynamics. This paper locates exactly where that transparency ends. On any nonempty carrier, finite ordered partitions act faithfully on binary relations by lexicographic lift, and iterated update is thereby the action of a left regular band; on finite carriers, this is the face semigroup of the braid arrangement. An endpoint keeps predecessor comparison only inside its blocks. The Premising Result proves five conditions equivalent: refinement of the target support, semigroup absorption, transformation absorption, inclusion of update kernels, and factorization of corrective replay through the current endpoint. Outside that boundary a three-world model witnesses failure: the office of the correction is announced and its target formulas are supplied, yet the corrected state stays undetermined. The verdict occurs, and the comparisons it was to decide on have been overwritten. No finite additive program repairs this. What ordinary continuation treats as legitimate history is precisely what correction asks to withdraw. The replay gap measures the loss exactly, at 2g(A,B) targets per endpoint fiber, independently of how records are represented. A two-register add–replay logic restores it: sound, strongly complete, compact, decidable; finite models suffice, and replay remains strictly more expressive than the endpoint-only fragment. The general program runs from update operator to kernel structure to target factorization conditions; the lexicographic base case is solved here.