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

Abstract

This preprint develops an arithmetic-renormalization and operator framework for the accelerated Syracuse map. It separates three logically distinct levels of the Collatz problem: 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 an exact cylinder law of the deterministic valuation tree. This construction yields a finite Markov transfer operator modulo \(3^h\) and an exact identification of the finite \(3^h\) offset coordinate with the current Syracuse state modulo \(3^h\) once the orbit depth exceeds \(h\). For the universal branch problem, the paper passes from linear transfer to max-plus packet dynamics. It introduces a normalized least-representative coordinate, derives exact lift and integer-edge laws, identifies the finite local language of positive packets, and establishes quantitative restrictions on neutral cores and fixed-length positive packet windings. The resulting framework combines Bellman certificates, phase-aware graph quotients, scale-adaptive copy collisions, primitive multiplier recurrences, linear forms in logarithms, \(S\)-unit equations, Chinese-remainder realizability, affine resultants, bridge continuants, and reciprocal bridge factorization. Several natural finite-state, occupancy, resultant, modular-rank, relative-recurrence, and coarse quantization closure mechanisms are shown to be insufficient, thereby isolating the genuinely word-unbounded macroedge obstruction. On the bounded-cycle front, cyclic offset transport and shifted-defect identities give exact rotation-invariant gcd criteria. Combining the published verification of Collatz convergence through \(2^{71}\) with an exact Farey-neighbor certificate, the manuscript proves that every hypothetical nontrivial accelerated cycle must satisfy \[ m\geq 72{,}057{,}431{,}991 \] and \[ u-e_2\geq 29{,}906{,}536{,}378. \] Consequently, any such cycle must contain at least \(29{,}906{,}536{,}378\) valuation-one entries. The paper also proves all-weight discrepancy identities, bounded-return mantissa-holonomy exclusion, long-return penetration, strong-run extraction, reduced-ratio height growth, escape from every fixed finite prime support, and a non-wrapping positive suffix-ladder theorem. This is an open-problem research preprint and does not claim a proof of the full Collatz conjecture. The remaining universal targets are a candidate-independent contradiction or quantitative positive-defect theorem on the recurrent word-unbounded positive macroedge family, together with strict shifted-gcd compression excluding the remaining hypothetical accelerated cycle words.