Discrete Functional Quotients and Integer Logarithmic Quantization

We study elementary asymptotic families of the form (G_f(n,a)=(f(n+a)/f(n))^n), where (f) is a positive function and (a) is an integer shift. Writing (g=log f), the local behavior is governed by the scale (Δ_f(n)=n g'(n)=n f'(n)/f(n)). Under a standard Taylor condition, if (a(n)Δ_f(n)→λ), then (G_f(n,a(n))→ e^λ). This gives a unified elementary formulation for approximating constants and positive functions by integer shifts. In the logarithmic case (f(n)=log n), constants (C>0) may be encoded, at finite resolution, by (a_C(n)=round((log C)log n)). For positive functions (φ(x)), the analogous integer field (a_φ(x,n)=round(logφ(x)/Δ_f(n))) yields pointwise reconstruction when the quantization scale is refined. We also formulate an inverse viewpoint for externally given integer sequences. For consecutive primes, if (p_n) is the (n)-th prime and (g_n=pₙ₊₁-p_n), then ((pₙ₊₁/p_n)^n≈ eg_n/log n), so the quotient transforms normalized prime gaps into an exponential scale. Its extremal limiting behavior is determined unconditionally by known results on small and large prime gaps. The framework is presented as an elementary quantization and asymptotic representation principle, not as a prime-generating or Diophantine theorem.