A Deterministic Logarithmic Divisibility Band Sieve for Prime Identification and Composite Elimination

This paper introduces the Deterministic Logarithmic Divisibility Band Sieve (DDBS), a fully deterministic framework for prime identification and composite elimination based on logarithmic scale decomposition and forward divisibility. The natural numbers are partitioned into successive logarithmic bands. Composite integers are eliminated using only divisibility by primes identified in strictly lower bands, enforcing a forward-only arithmetic causality. Every composite integer, including Carmichael numbers and other pseudoprimes, is eliminated within the unique logarithmic band in which it appears. Absolute integer positions and residue classes are preserved throughout the procedure. The paper provides a complete formalization of the sieve, including explicit definitions, an inductive algorithm, and full correctness proofs. Structural properties such as finite-band completeness, scale-localized elimination, residue-class invariance, and deterministic forward causality are established rigorously. No probabilistic input, congruence testing, analytic continuation, density estimates, or heuristic assumptions are used. This work is entirely elementary and is carried out within standard Zermelo–Fraenkel set theory with the axiom of choice (ZFC). While the paper is logically independent of conjectures concerning the zeros of zeta or LLL-functions, the scale-based and non-amplifying arithmetic organization introduced here is structurally compatible with rigidity phenomena studied in analytic number theory. Note. This paper supersedes earlier drafts circulated under the name Standing–Sitting Band Framework (SSBF). The present version replaces metaphorical terminology with canonical arithmetic language and provides a fully audited, proof-complete formulation.