Uniform Collatz Predecessor Counts and Conditioned Residue Laws

We study the Syracuse residue law when the total of the geometric letters is fixed. This conditioning connects the residue distribution to the height and arrival time of integer predecessors. Building on published uniform atom bounds and density regularity, we prove a uniform linear count of positive predecessors of every target prime to \(3\), within a sharp leading logarithmic arrival budget. Our central-window bound with period caps proves the conclusion of Wirsching’s Conjecture 2 without its Markov-operator hypothesis. We also quantify the dependence between the total and the residue: a projective score has finite squared norm \(\mathcal{J}\), and total–residue mutual information has expansion \(\frac{\mathcal{J}}{16n}+\frac{\mathcal{C}_2}{n^2}+O_{\varepsilon}(n^{-3+\varepsilon})\) under fair sampling. Under node counting its leading term is \(\frac{\mathcal{J}}{16n^2}\). We prove a variance-relative law for arbitrary CLT-window endpoint mixtures, finite-depth atom floors throughout \(1/\sqrt{n}\) geometric-parameter windows, and fixed-order score moment limits. Exact geometric deconvolution retains the original total and identifies the same two fair information coefficients for Sinai’s original pair. We also give finite-profile crossover estimates. Further applications include a conditioned-word transfer through every minimum in the finite Krasikov–Lagarias hierarchy, harmonic predecessor bounds, and an inverse policy checked by exact integer arithmetic yielding \(\gg X^{0.8968}\) convergent starts with shortcut stopping time at least \(7.65\log n\). Entropy and critical-renewal consequences are stated with their exact hypotheses; finite stationary overlap remains open. The earlier atom, density, integrability and timed fixed-target counting theorems are cited foundations, rather than new claims of this sequel. Reproducibility. Lean 4 checks 56 of the paper’s 60 numbered assertions. The Lean sources and build instructions are available on GitHub. The accompanying archive contains the LaTeX sources and the exact C++17 verifier, policy tables, checksums, and verification receipt for the finite K17 certificate used in the stopping-time bound.