Prime-Additive Ancestry Trees

This paper supersedes the earlier universal-greedy framing of the prime alphabet 1, 2, 3, 5, 7. Instead of treating the prime-recursive word as one example of a general canonical construction for arbitrary additive bases, it identifies the full binary decomposition tree T (n) as the native object of the theory, with the frontier word E (n), terminal-count vector v (n), and complexity ℓ (n) recovered as derived invariants. The key structural result is exact subtree replay on prime blocks: T (p+r) = (p+r; T (p), T (r) ) for 0 ≤ r ≤ p′−p, and henceE (p+r) = E (p) ∥E (r), v (p+r) = v (p) + v (r), ℓ (p+r) = ℓ (p) + ℓ (r). The paper develops this tree viewpoint into results on leaf-sum conservation, injectivity, nested prime trees, ordered shape fibers, and prime-realized shapes. It should be read as a structural refinement and specialization of the prime case, not as a further universalization of the arbitrary-basis framework. What is new in this version replaces the universal-greedy framing with a prime-specific tree-theoretic one identifies T (n) as the native recursive object derives E (n), v (n), and ℓ (n) from the tree rather than treating the word alone as primary upgrades prime-block replay from word concatenation to exact subtree replay develops ordered shape fibers, left-right asymmetry, and prime-realized shape theory.