Threshold Geometry and the Pronic Backbone: Forward Dynamics, Exact Inverse Geometry, and Sharp Preimage Growth in a Pseudo-Recamán Recurrence

This preprint presents a unified theory of the forward dynamics, exact inverse geometry, and sharp synchronized-preimage growth of a pseudo-Recamán modulo-threshold recurrence. For the canonical seed, the threshold c = 1/2 separates three regimes: finite triangular-block dynamics followed by an absorbing fold rail when 0 < c < 1/2; an infinite neutral endpoint ladder at c = 1/2; and an all-add triangular rail when c > 1/2. The corresponding inverse geometry exhibits a branching band, a single midpoint pivot, or an extinction gap. A noncontraction theorem for synchronized branch origins yields the universal frontier bound U_d ≤ d + 1 below the critical threshold and a path-or-death law above it. The paper also establishes sharp subcritical deficit formulas and exact critical frontier maxima. At critical balance, the forward and inverse structures meet in a pronic backbone. This backbone indexes canonical pivot times, cube peaks of OEIS A392243, an exact dual-square certificate, full-depth frontier growth of order Θ(sqrt(N)), and cumulative maximum growth of order Θ(N^(3/2)). This definitive synthesis integrates the theorem-level content, informative exceptions, corrections, and provenance boundaries of three preceding papers together with the later Critical Balance and Pronic Backbone theorem. A source-coverage appendix documents the disposition of the component results. The manuscript has undergone extensive exact-integer verification and external AI-assisted adversarial review. These checks are supporting validation and should not be interpreted as journal peer review. Translated residue gates and the separate rotation-density devil’s-staircase experiment are outside the scope of this work.