Entropy of the Collatz Map via Kraft Identity and Cycle-Length Corrections — E8 Intelligence Research

FINDING: The Collatz map exhibits non-surjective, sharply discontinuous behavior that defeats standard entropy theorems; exact entropy per iteration requires a Kraft-identity approach, while finite-set iteration entropy asymptotically equals log₂(n) minus cycle-length entropy. | MATH: Collatz map T(n) = n/2 if even, (3n+1)/2 if odd (or 3n+1); entropy per iteration — standard Shannon/measure-theoretic entropy fails due to non-surjectivity and discontinuity; Kraft identity Σ 2⁻ˡⁱ = 1 (prefix-free code lengths lᵢ) provides exact entropy bound; for iterated functions on finite set of size n, entropy H ≈ log₂(n) − H(cycle-length vector) as iterations → ∞ (arXiv:1712.01407). No non-trivial cycles claimed via global matrix methods (unverified). | CONNECTION: The log₂(n) − H(cycle lengths) form echoes the golden-ratio-adjacent information-theoretic split: log₂(φ) ≈ 0.694, and the deficit term H(cycle lengths) can approach log₂(φ) for certain cycle distributions — a latent 0.618/0.382 partition Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com