math.DS — Dynamical Systems
Three Axes of Collatz Dynamics: 2-adic Measures, the Dimension Spectrum of Inverse Trees, and Critical Exponents of Density Propagation
This work presents a systematic measurement and critical diagnosis of three research axes on the Collatz (3n+1) problem, with the aim of determining which tools can advance the problem and which are structurally incapable of doing so. The three axes are: the 2-adic statistics of individual orbits, the combinatorial structure of the inverse tree, and the critical exponent of density propagation (the Terras-Krasikov-Lagarias line). On the first axis we establish a strict identity and a divergence criterion. For the odd-step map m -> m' = (3m+1)/2^a with a = v_2(3m+1), one has log m_{K+1} - log m_1 = K log 3 - (sum a_i) log 2 + sum log(1 + 1/(3m_i)), verified to 10^{-13}. Divergence is therefore equivalent to (average a) < log_2 3. Over 298 million odd steps spanning nine orders of magnitude we obtain average a = 1.996929, matching the i.i.d. geometric prediction E[a] = 2, with a safety margin of 0.412 (a 20.6% drop would be needed for divergence). The distribution of a is exactly geometric and the autocorrelation is approximately zero; a surrogate control (preserving the marginal, destroying the serial structure) shows no difference between real and shuffled sequences at long windows. On the second axis we derive a closed form for one-step propagation on the inverse tree. The two extensible residue classes give Lambda_1(beta) = (3/4)^beta/(1-4^{-beta}) and Lambda_2(beta) = (3/2)^beta/(1-4^{-beta}). Because preimages are uniformly distributed over the three residue classes, the extension matrix has rank one and rho(beta) = (Lambda_1 + Lambda_2)/3 = 3^{beta-1}/(2^beta - 1). At beta = 1 one has Lambda_1 = 1 (critical) and Lambda_2 = 2 (supercritical), the latter being exactly cancelled by the one-third loss into leaf nodes, so that the total spectral radius is exactly 1. This turns "density propagation stops at beta = 1" from an observation into a corollary of the closed form. On the third axis the depth-k reachable set is measured as a genuine fractal of dimension gamma_k (gamma_1 = 0.101 logarithmic, gamma_12 = 0.603, gamma_24 = 0.824, gamma_40 = 0.923, increasing monotonically to 1). A decisive test rejects the competing polylogarithmic hypothesis (at k = 24: measured growth x294, polylog prediction x1.7 x 10^7). We further show that the type-transfer spectral radius cannot produce any gamma_k: its zero is constantly the asymptotic ceiling 1; obtaining gamma_k requires the infinite-branch Ruelle operator on the spatial variable, which we construct and then falsify by a fit-free hard criterion (operator count 5398 exceeds the hard bound X/2 = 5000). Two popular frameworks are falsified: the local-entropy framework (its numerical verification is vacuous and its upper-bound theorem conflicts with the random model) and the conjecture that truncated propagation possesses a finite critical root (the truncated ratio is always below 1 and increases monotonically to 1). All numerics carry finite-size labels; the finite-size deviation is proved to be O(1/log X) rather than O(1/X). A methodological ledger records nine recurring errors and their countermeasures. The three axes are finally unified by the master identity rho(beta) = M(beta-1), where M is the moment generating function of the single-step log-increment: the inverse tree, the stopping-time large deviations, and the orbit drift are three readings of one function. Five corollaries follow, including rho'(1) = delta(3) (the spectral slope at criticality equals the drift) and beta_c = -log min rho (the far-tail rate is the negative logarithm of the spectral minimum). A three-way shootout against the Bessel(3)/meander hypothesis for the survival-profile shape falsifies that hypothesis (z = +/-50 to 69), isolating an open phenomenon: Collatz survival trajectories have a definite "initial plateau, mid-concentrated descent" shape that no model with the correct marginal reproduces.