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(aₙ₊₁+2) < 2.05 × 10²⁰, continuous candidate seeds remain strictly bounded below the empirical lower bound 2⁶⁸ (and 2⁷¹). For scales crossing this threshold, we prove the Prefix Depth Collapse Theorem: a prefix of depth kₙ = O(log 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 Δ, whereas convergent scales (Δ → 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.