Power–Successor Generation: Discrete Depth, Continuous Geometry, and Analytic Interfaces

Abstract This paper develops a discrete generation theory in which positive integers are produced from the unit by alternating two operations: additive expansion and power expansion. Starting with \ (B₁=\1\\), we define \ (B₈+₁= (Bᵢ^+) ^=\ (a+1) ᵏ: a Bᵢ, \ k^+\\). The construction is complete, since every \ (n^+\) belongs to \ (Bₙ\). The sets \ (Bᵢ\) admit an exact nested parametrisation by power–successor expressions based at \ (2\). We introduce the generation depth \ (d (n) \), prove a root recursion for it, and classify the primes of depth at most three as precisely \ (2\) together with the Fermat primes. Several subtle points are clarified. First, minimal nested representations are not unique in general; uniqueness is recovered only after imposing an explicit canonical selection rule. Second, the naive claim that the least prime in successive prime-containing layers is strictly increasing is false; the correct statement is a tail prime lower-bound theorem: every element of \ (Bᵢ\) is at least \ (i\), hence every prime occurring in layer \ (i\) is at least \ (i\), and the least prime occurring in the tail \ (₉ ₓBⱼ\) tends to infinity as \ (t\). Finally, the correct linearising transform for fixed-depth power–successor towers is the double logarithm, not an iterated logarithm whose depth grows with the tower. In double-logarithmic coordinates the discrete tower is governed, up to a controlled additive error, by the linear form \ (ⱼ kⱼ\). The second part continues this analysis to continuous hypersurfaces that approximate the discrete parameter regions. We define the real-parameter tower on \ ( (0, ) ᵐ\), study the graph hypersurfaces \ (Sₘ\), and derive recursive formulas for first and second derivatives. A key geometric observation is that the continuous tower is not globally convex: already \ (z= (2ˣ+1) ʸ\) has strictly negative Gaussian curvature. Higher mixed derivatives are governed by the multivariate Faà di Bruno formula in its set-partition form. The resulting hypersurface theory provides a disciplined bridge from lattice sums to projected surface integrals. The third part develops the analytic number-theoretic and complex-geometric interfaces. We define bounded-depth prime counting functions, prove that fixed depth contains zero proportion of primes, and show that almost all primes up to \ (N\) have depth exceeding \ ( (1-) N/ N\). We introduce depth-weighted and trace-based Dirichlet series, explain their convergence domains, and formulate exponential sums on power–successor traces as natural objects for future circle-method work. On the complex side, the tower is continued on principal logarithmic branches, its last-coordinate Riemann surface is a cylinder over \ (C^\), negative exponents become reciprocal branches, and special slices yield rank-one automorphic pullbacks from \ (Y²=X³+1\) ; the associated power–successor theta series are holomorphic on the upper half-plane but are not claimed to be classical modular forms without a separate transformation law. The fourth part places the real tower inside Banach, Sobolev, kernel, and joint tower–logarithmic spaces. It proves compact-domain density via an exponential subalgebra, gives tower-admissible weights for global integrability, and corrects several tempting but false statements about ordinary Fourier transforms, global inverses, and compact convolution operators. The joint dictionary identifies its classical and weighted-uniform completions and records the domain thresholds, Orlicz closures, and differential-algebra corrections needed to keep logarithmic linearisation analytically honest. The fifth part turns the joint dictionary into a genuine function ring, defines congruences modulo ideals, identifies point and zero-set quotient completions, and separates legitimate tower-automorphic factors and finite-field reductions from unproved modularity or pseudorandomness claims. --- Keywords additive expansion; power expansion; factor sets; generation depth; Fermat primes; prime layers; double-logarithmic linearisation; power–successor towers; hypersurface geometry; Gaussian curvature; Faà di Bruno formula; bounded-depth primes; Dirichlet series; exponential sums; modular forms; automorphic forms; elliptic curves; theta series; Riemann surfaces; complex dynamics; Banach spaces; Sobolev spaces; weighted spaces; approximation theory; reproducing kernels; iterated logarithms; Orlicz spaces; differential algebras; function rings; ideals; quotient rings; finite-field reductions; coding theory