Step Recursion: A Three-Parameter Refinement of the Grzegorczyk Hierarchy

Abstract

We ask whether asymptotic recursion depth determines the expressive strength of a bounded recursive algebra, and prove that it does not. We replace ordinary predecessor recursion by generalized-inverse descent along a fixed iterate $g_n^{[l]}$ and obtain classes $H^m_{n,l}$, where $m$ measures initial-function strength, $n$ the growth row, and $l$ the traversal stride. For all rows $n,n'\ge2$ we prove an exact inclusion criterion. At a fixed row $n\ge2$ three regimes occur: below the critical basis ($m<n$), equal-row inclusion is exactly reverse divisibility $l'\mid l$; at $m=n$ every stride collapses to one class; and from $m=n+1$ this class is ordinary bounded recursion $E^m$. Hence pairwise $\Theta$-equivalent descent depths can induce infinite descending chains, infinite antichains, and copies of every finite partial order. The separation is therefore controlled by traversal alignment rather than by growth rate or recursion depth alone. The proof combines exact-depth simulation, trace sparsity, and selected dependency chains. The exceptional doubling row has the same reverse-divisibility order at basis zero, but all strides collapse from basis one onward; from basis three it equals ordinary bounded recursion, while at basis two $H^2_{1,l}\subsetneq FP$. At basis zero the doubling-row classes are proper subclasses of deterministic functional logspace, so the same dual-divisibility order already occurs inside $FL$.