Compression Spectra of Collatz Orbits: Sharp Residual Ordering, Exact Valuation Cylinders, and Near-Extremal Sparsity

The accelerated Collatz (Syracuse) map separates each odd event into multiplication-addition followed by complete dyadic contraction. We develop a finite-time compression framework in which orbit displacement is decomposed exactly into multiplicative compression and an additive arithmetic residual. The principal new result is a sharp ordering theorem for positive valuation words with fixed length m and fixed total dyadic valuation K: front-loading all excess valuation uniquely maximizes the residual kernel, whereas back-loading it uniquely minimizes it. Closed-form extrema yield explicit start-size thresholds separating universal non-descent, an ordering-sensitive strip, and universal descent at fixed terminal compression. We then connect this combinatorial theorem back to arithmetic Collatz trajectories. Using the classical parity-vector bijection, we prove that every finite positive valuation word is realized by exactly one odd residue class modulo 2ᴷ⁺¹ and give its explicit representative; consequently, the sharp extremal words occur infinitely often and have exact density 2⁻ᴷ among odd integers. We further derive an exact arithmetic witness formula: when 2ᴷ > 3ᵐ, the starts realizing a fixed word but failing to descend by time m form a finite initial segment of that residue progression, with a closed-form count. This converts the ordering-sensitive strip into a residue-aware criterion rather than a purely combinatorial envelope. We then prove a quantitative near-extremal concentration theorem. Along every regime K/m → ρ > 1 and for each fixed relative tolerance ε < 1/3, only a bounded amount of excess valuation can be delayed beyond the first coordinate; consequently, near-extremal valuation words form an exponentially vanishing fraction of all fixed-(m,K) words, and the union of their odd-start cylinders has exponentially vanishing relative density. The usual geometric/negative-binomial “random valuation” law then becomes an exact finite-word residue-density identity, while long-orbit dependence remains a separate issue. A reproducible computation on 200,000 consecutive odd starts near 10³⁰ follows blocks through 128 accelerated odd steps and measures finite-scale rate spectra, finite-window occupancy of extremal residue classes, and exceptional residual kernels. At m = 128, 16 anti-compression blocks are observed versus 17.72 predicted by the exact residue-density law. A separate census of 1 ≤ n ≤ 200,000 gives Spearman ρ = 0.6698 between maximum compression deficit and logarithmic excursion ratio; deterministic holdout and independent windows at 10⁶, 10⁹, and 10¹² retain stable associations. The results do not prove the Collatz conjecture. They provide arithmetically attainable sharp residual bounds, a quantitative asymptotic sparsity law for near-extremal words, exact finite-word densities, and reproducible rare-event diagnostics for exceptional finite Collatz structure.