Abstract
Canonical positive linear recurrence numeration systems may admit multiple coefficient presentations that generate the same place-value sequence. Paper I established that every such equivalence class has a unique canonical reduction of minimal order. Paper II showed that every presentation determines the same infinite periodic maximal boundary word and asked whether this boundary word is sufficient to reconstruct the canonical reduction. In this paper, we identify the intrinsic object underlying both theories. For every place-value sequence (Hₙ), we define an infinite sequence e = (eᵢ)ᵢ≥1 through the triangular valuation identity Hₙ₊₁ − 1 = ∑ᵢ₌₁ⁿ eᵢHₙ₊₁₋ᵢ. We prove that every coefficient presentation generating (Hₙ) is uniquely determined by a period of this intrinsic sequence, while the canonical reduction is recovered by taking its primitive period. This resolves the reconstruction problem posed in Paper II. The identification also yields a complete classification of coefficient presentations: every period of the intrinsic sequence gives a valid presentation, and every valid presentation arises in this way. Thus, the periods of e are in exact bijection with the presentations generating (Hₙ). We then establish a general lift-invariance theorem showing that every iterated lift of the canonical reduction generates exactly the same legality language. Consequently, any two coefficient presentations producing the same place-value sequence define identical languages of legal representations. Although their recursive legality decompositions may differ, their accepted languages coincide. The trilogy therefore culminates in a complete hierarchy of intrinsic structure: the coefficient vector, maximal boundary word, and legality language are all determined by the place-value sequence itself.
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