Additive Carries, Multiplicative Closure, and Prime Dependency Topology: Nested Generation, Homological Filtrations, and Logical Boundaries

Abstract We formalize prime generation as an interaction between additive order and multiplicative closure. For a finite prime alphabet \(S\), the least integer outside its free commutative multiplicative monoid \(\langle S\rangle_{\times}\) is the next prime; iterated least-gap completion therefore enumerates all primes in increasing order. This process is distinct from the Euclid product-plus-one recursion. Starting from the multiplicative unit, the latter has the maximal initial prime spine \[ 1;\quad 2,3,7,43, \] followed by \(1807=13\cdot139\). The spine is not closed, but its iterated prime-power carry closure contains every prime, with unbounded required depth. A typed expression language separates the pure multiplication forming \(p-1\) from the final successor operation \(+1\). Recursive predecessor factorization gives every prime a finite unit-rooted expression and a Pratt dependency tree. Dirichlet’s theorem supplies strong cross-level coupling, and the Euclid–Sylvester branch gives double-exponential growth. Uniform bounds on expression depth or additive-node count nevertheless omit infinitely many primes. These are algebraic growth statements, not a claim of dynamical chaos. Retaining every prime-factor choice produces an unfolded history tree with a compact ultrametric boundary and a state graph that identifies histories reaching the same prime set. A split into \(r\) choices creates \(r-1\) degree-zero branch classes. If \(m_n\) is the incoming merge excess at rank \(n\), then \[ \beta_1(\Gamma_{\leq N})=\sum_{n=1}^{N}m_n, \] and the persistent state-graph barcode contains \(m_n\) copies of \([n,\infty)\). Profinite, cohomological, metric, and Shannon constructions then quantify finite-resolution branch information. Their asymptotic consequences remain conditional on arithmetic recurrence of branching or recombination. The transitive prime-dependency relation also defines Alexandrov order complexes. The global complex is contractible, while finite links and open intervals have nontrivial local homology. Chain links decompose as joins, prime attachment yields a height spectral sequence, and a simultaneous-coupling nerve is Dowker-dual to a high-witness complex. These constructions give precise meanings to nested connectivity, closed chains, and high–low coupling. Finally, the results imply finite-depth, finite-generator, and literal-output limitations, but not a universal lower bound on proof length. They prove neither the Riemann hypothesis nor its independence from ZFC; such conclusions would require separate proof-complexity or independence arguments. Keywords **prime generation; additive carries; multiplicative closure; Euclid–Mullin graph; Pratt trees; prime dependency topology; Alexandrov topology; ultrametric boundary; persistent homology; cohomological multiresolution; Dowker duality; arithmetic information; proof-complexity boundaries.**