The full native beam action of a literal word and time
Abstract
Literal positive-word decompositions determine the full native beam-group action.
Fix a finite essential Presentation P with the explicit discrete topology on P.V, h : FullAut P.graph, w : Word P P and s : Int. The only comparison assumption is the literal ALL-history equation h.val x = translate P.graph s (w.eval x). It does not assume a beam, matrix, colimit or representation comparison. In the preserved original application the actual finite-window tables and OriginalPositiveWordAssembly.positive_word construct that equality for every arbitrary h : ResetFullAut 4.
full_action_coordinate says F(fullBeamRepresentation h x) = cutShift s (rowEquiv w (F x)) for every x in the WHOLE existing BeamGroup. The proof first identifies literal image subsets of each beam, then sums over every tagged object, then uses native direct-limit hom_ext and all vertex basis generators. The existing JF identity transports the result to every formal difference in the Grothendieck group. Neither positive generators alone nor selected periodic points replace that full domain.
Action order matters. Word.comp first second evaluates second after first. The native group representation has representation(g*h)=representation(g) after representation(h), as its existing monoid-hom law states. For a decomposition translate s after w, the coordinate action applies rowEquiv w before cutShift s. No newly added action-composition wrapper is needed.
OriginalRootKernelCheck uses this theorem to obtain hrho in the native reset row limit. rowEquiv_intStage and cutShift_intStage give hstage at N=0. Through the existing integer/native equivalences it derives rho(h)=T^(-N) U with N=-s, identifies BOTH the original unit and its inverse factor, and uses the same localized cochain and corrected history word at every positive 3-divisible period. The full arithmetic application also uses the primitive-orbit counts 2 and 7, literal Fix(T^6), and the same quotient character for every smaller positive period. The full arithmetic application also uses the primitive-orbit counts 2 and 7, literal Fix(T^6), and the same quotient character for every smaller positive period.
Theorem 1.1 (Derive the complete native beam action from the literal all-history word-and-time equality.).
Lean statement: D5/S3/ConceptDynamics/Coding/BeamColimits/PositiveWordFullAction.full_action_coordinate
Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/Coding/BeamColimits/PositiveWordFullAction.full_action_coordinate (✓ std3). ∎
Source. Repository-derived.
Commentary.
Derive the complete native beam action from the literal all-history word-and-time equality.