Research Compendium From Topological Rupture to a Falsifiable Proof Skeleton: A Logbook of an Experiment in the Collatz Dynamics

This research note documents the complete trajectory of an experimental program aimed at generating a novel approach to the Collatz Conjecture. The inquiry began with a forced modal rupture, reframing the arithmetic problem as one of topological dynamics on the 2-adic integers (the "Geometric Gesture"). This led to a multi-stage effort to translate the topological intuition into a falsifiable computational protocol. Early attempts revealed a conceptual gap, which was ultimately bridged by a "Finite-Determinacy Theorem," the proof of which is provided herein. The technical complexity of this proof then forced a final ascent to a simpler, base-invariant principle: a "Valuation-Cocycle Invariance" that governs a wide class of p-adic affine dynamics. The program culminates in a precise, conditional proof skeleton for the Collatz Conjecture, hinging on a single, falsifiable inequality derived from this principle. This document provides a human mathematician with the full context, from philosophical motivation to the final, concrete mathematical target.