OD-34: Alpha-Driven Sieve-Scale Extrapolation Model (OD-33 Companion)

OD-34 is an alpha-driven sieve-scale extrapolation model designed as a conceptual and methodological companion to OD-33. While OD-33 estimates prime-related quantities through static global fitting of correction parameters, OD-34 adopts a fundamentally different approach: it models the evolution of growth itself. The core quantity α(k), defined as the logarithmic growth rate between consecutive sieve scales, is shown to follow an exact linear law:α(k) = U + V·k,with constant increment Δα independent of scale. This property is empirically validated over k = 19…29 with machine-level stability. Curvature diagnostics reject 1/k or higher-order terms, confirming that the growth acceleration is constant rather than asymptotically curved. As a consequence, OD-34 achieves scale-invariant predictability: instead of fitting values, it extrapolates dynamics. Predictions remain stable across many orders of magnitude, with deviations limited to a few integer units even at extreme scales (10^29). This repository contains:• the OD-34 reference implementation,• full diagnostic tables,• comparative data versus OD-33,• graphical evidence of alpha linearity and stability. OD-34 represents a paradigm shift from static approximation to dynamic evolution modeling and provides a robust foundation for future density estimators and factorization-guided heuristics. OD-34 nasce come evoluzione concettuale di OD-33. Mentre OD-33 cerca di approssimare la distribuzione dei primi tramite un fitting statico di costanti globali, OD-34 cambia completamente prospettiva: non stima direttamente i valori, ma studia come cresce il sistema. La scoperta chiave è che il parametro di crescita α(k) evolve in modo perfettamente lineare. Questo significa che la “accelerazione” del sistema è costante e non presenta curvature residue. In pratica:OD-33 è una fotografia.OD-34 è un film. Questo rende OD-34 estremamente stabile e predittivo anche su scale molto lontane da quelle di calibrazione.