THE LAW OF HISTORIES POSTING WINDOWS IN A FIXED SETTLED OCTAVE, AN EXACT ABELIAN BOUNDARY, AND A PHASE-CLOCKED NONCOMMUTATIVE MODEL

Fix a settled Gray-octave interface and a one-tick-local readout. Within that declared class, vacuum, unit, and neutrality force the posting of every commitment: +1 at its commit phase, −1 at the next phase, and zero elsewhere. The chosen Gray loop is also uniquely determined by its directed transition tally. Neither theorem selects the Gray interface itself. Composition is not xed. An additive licensed model is order-blind. A free-stream model and a bicyclic model prove that the licensed class also contains order-sensitive readings. We dene clause-receipted as embedding the target monoid into an abelian group. This is a MODEL formalization of clause-level provenance. Under it, every clause-receipted composition is commutative. We then construct a licensed phase-clocked model. Its window component is the forced posting, its shift component is the observed one-phase advance of every strict native edge, and its action is the inverse of the proved cyclic shift. The semidirect multiplication (a,s)· (b,t) = (a+ τsb,s+ t) remains a disclosed model choice. This readout separates a transition-equal backtrack pair (the window at tick 0 is 2 in one order and −1 in the other); the trivial-action control merges the pair. The readout is proper and non-injective. These results establish a conditional posting theorem, an exact abelian-embeddability boundary, and a concrete noncommutative model. Classication of licensed compositions, receipt of the semidirect rule, and physical selection remain open. Every named theorem is kernel-checked in Lean 4. Appendix A gives the exact declarations and axiom basis.