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Abstract
We build on established modular analyses of the Collatz map, which show that odd iterates occupy only the residue classes 1, 3, 5 (mod 6) while even iterates are confined to {2, 4} (mod 6). From these constraints, the module–LCM iteration equation naturally emerges, demonstrating that all trajectories evolve strictly within this three-class modular subspace. Within this structure, the affine–dyadic boundary equation Qn + x = 2^t and the invariant ray 5·2^t identify the only admissible intersection capable of neutralizing affine expansion. Consequently, every valid orbit of the 3n + 1 map enters the Collatz even-division chain and ultimately terminates at the absorbing state n = 1.
Supplementary materials
Title
Manuscript LaTeX Source (ZIP)
Description
In addition to the Modular‑LCM Equation, I provide a compact implementation of the iteration mechanism to illustrate the underlying dynamical structure and to facilitate reproducibility.
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Supplementary weblinks
Title
Deterministic Algebraic Framework for the Collatz Dynamical System: Modular-LCM Confinement and Spectral Rank Contraction
Description
The module–LCM equation arises from a structural need to describe the Collatz
map within its true modular constraints. Classical formulations of the 3n+1
iteration implicitly generate unresolved mod‑2 and mod‑3 cycles, which obscure
the behavior of odd and even iterates and introduce non‑terminating residue
patterns. To avoid these artificial loops, we reconstruct the iteration using
only the admissible modular classes dictated by the dynamics themselves.
By restricting the evolution to the invariant subspace {1,3,5} mod 6 for odd
states and {2,4} mod 6 for even states, the module–LCM equation provides a
closed algebraic description of all valid transitions. This formulation removes
the spurious residues created by the classical expression and reveals the
underlying affine–dyadic structure that governs every legitimate Collatz orbit.
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Deep Dive into the Collatz Dynamics: Why Divergent Trajectories Cannot Exist
Description
The classical stopping-time formula for the Collatz map is not applicable within
the modular–LCM framework. Its expression relies on the raw iteration 3n+1,
which generates non-admissible residue patterns in mod 2 and mod 3, producing
virtual cycles and artificial “singular numbers” that do not correspond to any
valid modular trajectory. These artifacts arise from the formula itself rather
than from the dynamics of the map.
By contrast, the modular–LCM formulation restricts the iteration to the
admissible residue classes determined by the Collatz dynamics, ensuring that
all transitions occur within the invariant modular subspace. Under this structure,
the classical stopping-time expression no longer reflects the true evolution of
integer iterates and therefore cannot serve as a valid descriptor of the system’s
behavior.
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