PRH | Aux | 4.10 • A Blur–Knob Program for Collatz

A complete proof of the Collatz conjecture remains elusive despite strong average-case theorems and powerful computational verification. This paper organizes several established ingredients into a single closed "program" with explicit knobs (resolution, smoothing, spectral cutoffs) and an explicit optimization problem (a linear program on finite-state transfer operators). Each step is stated in a mathematically precise form, and the exact missing statements needed for the program to close are isolated as concrete lemmas. The program uses: (i) the 2-adic carrier/conjugacy of Bernstein Lagarias; (ii) the difference-inequality/LP method of Krasikov Lagarias; (iii) rigorous "almost all" descent results of Terras, Everett, Korec, and Tao; and (iv) a canonical spectral/transfer-operator formalism that packages all parity-paths at once. That is, rather than selecting a single favored attack, we treat the Collatz landscape as a situation where many partial controls coexist but do not naturally synchronize. Blur is designed to lower that coordination cost. The companion papers in this series implement the resulting "closure package" as an auditable pipeline: a finite-state proxy with exact-lift screening, a certified contraction instance (the S5 Witness Theorem), and the Lyapunov step that converts contraction into a finite verification problem. The point is not to claim a minimal hypothesis set, but to give an obstruction-complete and executable reduction: either the package closes with explicit constants, or it returns concrete next-step targets (which constants, which cycles, which scales). Reading order: The papers in this package are intended to be read in the alphabetic order of their PDF filenames, namely $0_{-} \ldots, 1_{-} \ldots, 2_{-} \ldots$, and so on through $13_{-} \ldots$.