THE RECOGNITION ALPHABET A PHYSICAL LANGUAGE OF HISTORIES, ITS FINITE QUOTIENT, AND ITS COMPOSITION FRONTIER

A finite recognition window supports an alphabet of motions once its transport class is stated. Its letters are closed histories on the three-cube, its one-loop meanings form a finite holonomy vocabulary of 2520 elements, and its grammar is generated by operations whose algebra matches the operations on content. We assemble these parts as one mathematical object and separate what is forced from what is attached. State readings telescope. Edge readings factor through signed traversal counts. A fourteen-act closed history is the shortest one invisible to both yet visible to ordered degree two. The first two terms of the defined formal connection expansion are precisely net winding and ordered area paired with commutators. A selected discrete transport yields the alternating group on seven letters in one window, while a fixed tie convention on shortest representativesgivesacompletefinitenormalform. Namingthesixcontentatomsleavesexactly15 relabelling orbits, whose fixed-length identification count is log2 15 bits; in dimension d the residue is log2(2d− 1)!! bits. The cube has one invariant pairing of its faces; the bare atoms have none. On the content side, a repaired binder naming rule removes the known collision and preserves the proved finite fragment. A separate frozen corpus measurement is reported only as a measurement of logical shape, not as a faithfulness claim for the weave. The selected finite quotient is completely enumerated for one window. Its extension remains open at one precise joint: the local observations force postings but do not select how postings from successive windows compose. We prove the exact underdetermination, exhibit an order-sensitive phase-clocked model, and state the finite list of theorems that would turn the alphabet into an unbounded physical language. Core algebraic identities, semantic repair, settlement laws, and finite coupling claims have machine-checked counterparts in Lean 4; the public subset and the pending release modules are distinguished in Appendix B, while the group-theoretic paper proof and corpus measurements remain separate.