Exploring Collatz Dynamics with Human-LLM Collaboration

arXiv:2603.11066v5 Announce Type: replace-cross Abstract: We develop a structural framework for the Collatz map based on odd-to-odd dynamics, modular return structure, and burst-gap decomposition. We prove exact results for fiber-57 return dynamics, including a uniform affine channel for q = 7 (mod 8), minimum return gap five for q = 3 (mod 8), and that the chain map on the invariant core is a permutation at every depth. These yield a depth-2 partial return kernel with Perron root 129/1024. A conditional reduction via phantom-cycle gain analysis and a weak-mixing hierarchy establishes an exact geometric block law, an exponential almost-all crossing bound, and per-orbit phantom gain within a 4.65x contraction margin, reducing convergence to a single orbitwise regularity statement. New in v5: the Carry Contamination Theorem proves that at every Sturmian depth D, n_D mod 8 is exactly equidistributed over {1,3,5,7}, yielding an exact (3/4)^D survivor law. The reduction terminates at the Carry Independence Conjecture (CIC): no n_0 > 1 avoids class 5 at every depth. CIC implies Collatz via a proved Dichotomy theorem. This extends via a seven-block cross-core alphabet with spectral radius rho = 2+sqrt(2). The safe Collatz map is a 2-adic expander with measure shrinkage mu_2(T_j) <= 0.522^{jD}. No K-bit integer survives j >= 5 non-descending rounds for large K. Cycle impossibility is verified through period 13 (238,811 words). Anti-correlation is bounded (constant ~0.635, not exponential in K). This is not a proof of the Collatz conjecture but a sharpened structural reduction along two routes, each terminating at a distributional-to-pointwise gap.