Piecewise Accumulators, Diophantine Constraints, and Logarithmic Tail Pinpointing in Collatz 3x + d Dynamics

We establish an algebraic, continuous, and Diophantine framework for analyzing candidate periodic orbits in generalized Collatz 3x+d dynamical systems. We prove the Master Piecewise Accumulator Identity and the Non-Uniformity Remainder Identity, providing exact, non-recursive closed-form expressions for trajectory accumulators. By introducing a continuous relaxation of the candidate seed function q^* (), we establish the two-sided Seed Range Sandwich and derive a Scale Worthiness Screening Criterion, demonstrating that for convergent scales satisfying Mₙ (a₍+₁+2) < 2. 05 10^20, continuous candidate seeds remain strictly bounded below the empirical lower bound 2^68 (and 2^71). For scales crossing this threshold, we prove the Prefix Depth Collapse Theorem: a prefix of depth kₙ = O (Mₙ) contracts the local candidate interval width W < 1, restricting each prefix sub-tree to at most one integer candidate seed. We evaluate the dual behavior of prefix-locking across the scale spectrum: non-convergent scales collapse instantly due to large denominator gaps, whereas convergent scales (1) necessitate transcendental methods such as Baker's linear forms in logarithms. We present a memory-efficient Modular Short-Circuit Screening Pipeline to filter candidates modulo small primes and 2-adic bit windows without exabyte-scale BigInt arithmetic, alongside the Logarithmic Tail Pinpointing Theorem for O (d) deterministic tail extraction. Finally, we integrate these structural components into a methodological framework for searching and bounding non-trivial Collatz cycles.