Arithmetic Renormalization Dynamics for the Syracuse Map: Exact Valuation Cylinders, 3-Adic Transport, and Max-Plus Packet Certificates

Abstract

This Version 2.0 develops an arithmetic-renormalization and operator framework for the accelerated Syracuse map while keeping three logically distinct levels of the Collatz problem separate: exact finite arithmetic, averaged transport, and worst-branch deterministic obstruction. A finite valuation word determines a unique odd residue class modulo \(2^{K_m+1}\), and the normalized Haar mass of the corresponding cylinder in the odd \(2\)-adic integers is exactly \(2^{-K_m}\). The geometric valuation law underlying probabilistic Syracuse models is therefore realized as the exact cylinder law of the deterministic valuation tree. Pushing this law through the Syracuse offset recursion produces a finite Markov transfer operator modulo \(3^h\). Along every valuation word realized by an actual positive Syracuse orbit, the finite \(3\)-adic offset coordinate agrees exactly with the current Syracuse state modulo \(3^h\) once the orbit depth exceeds \(h\). For the universal branch problem, the manuscript passes from linear transfer to max-plus packet dynamics. It formulates finite Bellman certificates for exact packet paths, proves a no-go theorem for quotients covering all finite terminally bad prefixes, and introduces a normalized least-representative coordinate that separates finite \(2\)-adic compatibility from eventual positive-integer realization. The resulting bottom-cell analysis combines exact packet algebra, scale-adaptive copy collision, Diophantine mantissa holonomy, primitive multiplier recurrences, linear forms in logarithms, \(S\)-unit equations, Chinese-remainder realizability, affine resultants, bridge continuants, and reciprocal bridge factorization. Version 2.0 adds a shifted-gcd collapse audit and a corrected second Farey cycle barrier. Combining the published verification through \(2^{71}\), the corrected published cycle-length input, and an exact rational Farey certificate, the manuscript proves that every surviving nontrivial positive accelerated Syracuse cycle must satisfy\[m\geq 137{,}528{,}045{,}312\]and\[u-e_2\geq 57{,}079{,}296{,}007.\]Consequently, every such hypothetical cycle must contain at least \(57{,}079{,}296{,}007\) valuation-one entries and at least ninety-two transitions whose valuation is at least two. The paper also proves that every candidate of accelerated length at most \(2^{72}\) must lie in the minimal-excess layer\[K=\left\lceil m\log_2 3\right\rceil,\]whereas every non-minimal-excess candidate must have more than \(2^{72}\) odd states. The new reduced-denominator local-order theorem controls every proper cyclic segment below the first Farey denominator\[Q_{71}=72{,}057{,}431{,}991.\]Its two-sided refinement shows that a segment and its complement cannot both be supercritical at that rational complexity. In the minimal-excess branch, the remaining positive-split problem is reduced to an explicit mechanical-word obstruction associated with rotation by\[\log_2\!\left(\frac{4}{3}\right).\] On the unbounded terminal front, Version 2.0 proves a quantitative scale-refinement barrier and an exact scale-lift identity. These results show that arbitrarily fine recurrent positive lifts cannot remain at polynomial band scale or polynomial valuation-cylinder depth. Any surviving obstruction must escape simultaneously to stretched-exponential scale and stretched-exponential cylinder modulus. This double scale-lift escape is a substantial narrowing of the universal obstruction, but it is not yet itself contradictory. The manuscript does not claim a proof of the full Collatz conjecture. The remaining universal tasks are to obtain a candidate-independent contradiction or quantitative positive-defect theorem on the recurrent word-unbounded macroedge family and, independently, to force a strict cyclic divisibility failure excluding the remaining enormous sparse-positive accelerated cycle words.