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

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 \(B_1=\{1\}\), we define \(Bᵢ₊₁=(B_i⁺)^{}=\{(a+1)^k:a∈ B_i,\ k^+\}\). The construction is complete, since every \(n^+\) belongs to \(B_n\). The sets \(B_i\) 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 \(B_i\) is at least \(i\), hence every prime occurring in layer \(i\) is at least \(i\), and the least prime occurring in the tail \(j≥ tB_j\) 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 \(∑_j log k_j\). 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 \(S_m\), and derive recursive formulas for first and second derivatives. A key geometric observation is that the continuous tower is not globally convex: already \(z=(2^x+1)^y\) 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 \((1-ε)log N/loglog N\). 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 \(Y^2=X^3+1\); 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