Sharp Linear Preimage Growth in a Modulo-Threshold Recurrence

This preprint establishes sharp finite-depth laws for synchronized preimages of a nonautonomous modulo-threshold recurrence. For every real parameter 0 < c (3,1), together with the corrected zero-loss argument. These corrections do not alter the frontier formulas. This release includes the complete manuscript, authoritative exact-predecessor code, three finite verification programs, result records, and SHA-256 checksums. Universal claims rest on analytic proofs; computational checks and external AI-assisted adversarial reviews provide supporting validation and should not be interpreted as journal peer review. This is a standalone specialist paper extending the earlier inverse-geometry work and supplying detailed component proofs summarized in the later definitive synthesis, Threshold Geometry and the Pronic Backbone.