For a strictly increasing $G:\mathbb{N}\to\mathbb{N}$ with $G(0)\geq 1$, we study minimal-basis function algebras generated by recursion along generalized inverses of its iterates. Bounded recursion requires an earlier pointwise majorant; unbounded recursion has no such requirement. If the orbit $G^{[t]}(0)$ has no polynomial upper bound, we obtain exact fixed-function membership spectra, reverse-divisibility inclusion, and indexed intersections that need not be single-stride algebras. Every finite divisibility-downset is realized in both regimes, and mixed strides admit fixed functions expressing any finite monotone Boolean condition on their additive stride monoid. For every generator, the realized bounded spectra form a bounded meet-semilattice: a construction sharing a bounding input coordinate realizes finite intersections without an unrestricted selector. Polynomial orbit growth is equivalent to definability of each of ordinary predecessor, unary doubling, and addition in any fixed unbounded stride algebra. In this case the algebra is ordinary primitive recursion relative to its inverse map, giving full collapse for primitive-recursive $G$. Ordinary predecessor belongs to a bounded stride algebra exactly when the orbit is polynomially bounded and $G$ has bounded successive differences; equivalently, ordinary bounded recursion embeds. A locality obstruction explains the additional bounded requirement, and two primitive-recursive generators with quadratic orbits have opposite bounded collapse behavior.
We study finite-terminating, address-only step recursion on the nonnegative reals, from zero, unit successor and projections. For reduced positive rational strides $s=a/q$ and $t=b/v$, we prove $\mathcal{C}s\subseteq\mathcal{C}t$ if and only if $q$ divides $v$, for both the bounded and unbounded algebras $\mathcal{C}\in{\mathcal{B},\mathcal{U}}$. The bounded compiler preserves earlier global majorants on every auxiliary input. Mixed rational stride sets are classified by least common denominators and supernatural denominators. Primitive-recursive translated-coordinate grid codes characterize the unbounded algebras and yield $\mathcal{U}{1/q}\cap\mathcal{U}{1/v}=\mathcal{U}{1/\gcd(q,v)}$. For bounded intersections, we prove an exact reduction: all corresponding identities hold if and only if the single identity $\mathcal{B}{1/2}\cap\mathcal{B}_{1/3}=\mathcal{B}_1$ holds, equivalently if the Boolean parts of the integer successor and addition algebras under scalar bounded recursion coincide. A Boolean reconstruction reduces general functions to predicates; a clipped fine-grid simulation with a frozen global resource bound identifies the union of predicate precision classes with the addition level. These reductions do not decide the resulting predicate equality, so literal bounded intersections remain unresolved. We also prove that unbounded-program equality is $Π^0_1$-complete and extensional bounded-algebra membership is $Σ^0_2$-complete. The model has no free real arithmetic or Zeno limits; semantic majorant validity is distinguished from syntax transformation.