Prime-Additive Ancestry Trees
Betzer, David · Zenodo (CERN European Organization for Nuclear Research) · 2026
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.