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Autor principal:	Moon, KyungUp
Formato:	Recurso digital
Lenguaje:	inglés
Publicado:	Zenodo 2025
Materias:	Collatz conjecture odd-only Collatz finite-state obstruction modular projection residue dynamics deterministic dynamics proof topology
Acceso en línea:	https://doi.org/10.5281/zenodo.18095567
Etiquetas:	Agregar Etiqueta Sin Etiquetas, Sea el primero en etiquetar este registro!

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This upload introduces Modular Backwash as a finite-state obstruction in the odd-only Collatz dynamics.   The work does not claim convergence, divergence, or a proof of the Collatz conjecture. It does not propose a decision procedure, stopping-time bound, or probabilistic estimate. Instead, its purpose is to fix a structural object that isolates a specific and previously implicit constraint in single-orbit Collatz behavior.   The central question addressed is the following:   Even if low 2-adic valuation steps occur frequently on average, why can a single orbit not sustain arbitrarily long sequences of shallow folding?   Rather than approaching this question through average drift, density arguments, or stochastic heuristics, the paper adopts a finite observational perspective. Odd-only Collatz trajectories are analyzed under modular projection to powers of two, together with short windows of valuation data. This induces a finite augmented residue–valuation trace space in which orbit segments are observed.   Within this framework, the paper identifies a deterministic obstruction termed Modular Backwash:   Extended shallow folding (low valuation steps) along a single orbit cannot persist indefinitely without forcing repetition inside a finite augmented state space. By finiteness and determinism alone, any sufficiently long low-band episode must either (i) revisit an already realized augmented block (producing a closed walk in the finite observational model), or (ii) exit the low-band via the appearance of a deeper valuation step.   Backwash is therefore topological and combinatorial in nature. It arises from finite-state recurrence and the pigeonhole principle, not from probabilistic behavior, ensemble averages, or assumptions about typical orbits. In particular, it addresses a limitation of “almost all” or average-case results by isolating a constraint that applies to individual orbit segments.   Importantly, the paper carefully distinguishes between recurrence in an observational factor system and genuine periodicity in the integer dynamics. The appearance of a closed walk in the finite augmented model does not, by itself, imply the existence of a true integer cycle. To promote such a recurrence to an actual cycle would require additional lift or consistency conditions, including realizability and overlap determinism.   These lift and closure questions are explicitly left open. They are identified as the next structural barrier rather than being implicitly assumed or informally bypassed. The present work therefore functions as a concept-defining anchor: it fixes the obstruction itself, while separating it cleanly from the unresolved problem of integer-level realization.   The contribution of this upload is thus not a result about termination, but a clarification of where and why certain naive or probabilistic intuitions about indefinite shallow behavior must fail. By formalizing Modular Backwash as a finite-state obstruction, the paper establishes a reference point for subsequent work on odd-only Collatz dynamics, automaton models, lift conditions, and structural closure.   Future empirical or computational studies may investigate signatures or frequencies of backwash, but such studies do not redefine the concept introduced here. This upload serves to fix the definition, scope, and logical boundaries of Modular Backwash as a structural object in Collatz dynamics.

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