A Δ-Core Reduction Framework for the Collatz Conjecture

This paper develops a Δ-core reduction framework for the Collatz conjecture based on valuation-depth recursion and centered congruence displacement coordinates. The main unconditional result establishes the recurrence bound # {k ≤ T : |s_k| ≤ B} = O_B(log* T), derived deterministically from congruence rigidity and the Lifting-the-Exponent lemma, without probabilistic or ergodic assumptions. The framework further shows that any infinite bounded-strip recurrence must induce a tower-sparse 2-adic congruence structure. The remaining obstruction is thereby reduced to a realizability problem concerning nonperiodic tower-sparse 2-adic locks. A central conceptual distinction emerging from the framework is the separation between local admissibility and global realizability: congruence constraints may remain locally compatible at arbitrarily large valuation depths while still failing to arise from globally realizable positive-integer Collatz orbit sections. The paper does not claim to prove the Collatz conjecture. Its contribution is structural: reducing the remaining obstruction to a sharply isolated arithmetic-dynamical realizability problem.