On Integer-Division Bases for the Kalmár Elementary Functions

Prunescu, Sauras-Altuzarra, and Shunia asked whether addition, totalized integer division, and base-two exponentiation form a substitution basis for the Kalmár elementary functions. We answer this problem negatively, even after adjoining multiplication. On independent powers of two, every fixed term built from addition, multiplication, and integer division takes only polylogarithmically many distinct values below a given bound, and the same restriction survives arbitrary nesting of $2^x$. No such term recovers $n$ from $2^n$, so neither $\lfloor {\log }_{2}x\rfloor$ nor integer remainder is representable. The obstruction is not special to the base two. For every fixed integer $b\ge 2$, the same sparsity bound holds with constants independent of $b$, and $⟨x+y,\lfloor x\mathrm{/}y\rfloor ,b^x⟩$ remains a proper subclass of the elementary functions. Variable exponentiation reverses the conclusion: addition, integer division, and $x^y$ do form a substitution basis, $$⟨x+y,\lfloor x\mathrm{/}y\rfloor ,x^y⟩=\mathcal{E}\mathrm{.}$$ Once the base may depend on the input, integer remainder and truncated subtraction become representable. The incomplete base-two class is nevertheless complete for truth values. Every elementary characteristic function belongs to it, as does every elementary function with a fixed finite range.