Collatz Convergence through Logarithmic Descent and Modular Reduction

description: |
  This fourth version (v4) represents a structural closure of the descent-based framework proposed across earlier versions.
  Building upon the original φ(n)-based logarithmic descent, this update introduces a hybrid piecewise function F(n),
  defined over a three-way partition of ℕ⁺ into structural states S₁ (even), S₂ (odd with ν₂(3n+1) ≥ 3), and 
  S₃ (resistant odd values with ν₂ ∈ {1,2}). 

  New in this version:
    - A state-sensitive descent model that dynamically adapts φ(n), ψ(n), and log₂(n).
    - The function ψ(n) incorporates resistance-depth penalties and modular triggers (e.g., n ≡ 5 mod 8).
    - Full logical closure showing that all resistant states are eventually absorbed into guaranteed descent paths.
    - Empirical evaluation confirming descent within ≤2 steps for n ∈ [1, 200].
    - Final section asserts that while this is not a full proof, the structure now admits no known failure modes.

  This version supersedes prior drafts (v1–v3), consolidates exception handling, and prepares the descent framework for formal verification.