Parity-Word Complexity and Exponent Growth in Non-Bounded Collatz Orbits
Abstract
For the accelerated Collatz map T (x) = x/2 for even x and T (x) = (3x + 1)/2 for odd x, we give a direct lower bound on the factor complexity of the parity itinerary of any unbounded positive-integer orbit.
We then relate this lower bound to the cumulative valuations in the odd-only Syracuse map. If e(n) denotes the sum of excess 2-adic valuations and ω = lim sup e(n)/n < log 2 (3/2), the odd iterates have exponential lower growth rate log 2 (3/2) ↔ ω, while parity-factor complexity sampled at the odd-step positions obeys lim sup p N (n + e(n))/n ↑ (1 + ω)/ log 2 (3/2). We clarify the limitations of these necessary conditions and formulate a conditional complexity-based criterion for excluding unbounded trajectories. No resolution of the Collatz conjecture is claimed.
Related
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BibTeX
@misc{douzi2026parityword,
title = {{Parity-Word Complexity and Exponent Growth in Non-Bounded Collatz Orbits}},
author = {Hassan Douzi},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05838324v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Parity-Word Complexity and Exponent Growth in Non-Bounded Collatz Orbits
AU - Hassan Douzi
PY - 2026
DA - 2026/10/25
PB - HAL
UR - https://hal.science/hal-05838324v1
AB - For the accelerated Collatz map T (x) = x/2 for even x and T (x) = (3x + 1)/2 for odd x, we give a direct lower bound on the factor complexity of the parity itinerary of any unbounded positive-integer orbit. We then relate this lower bound to the cumulative valuations in the odd-only Syracuse map. If e(n) denotes the sum of excess 2-adic valuations and ω = lim sup e(n)/n < log 2 (3/2), the odd iterates have exponential lower growth rate log 2 (3/2) ↔ ω, while parity-factor complexity sampled at the odd-step positions obeys lim sup p N (n + e(n))/n ↑ (1 + ω)/ log 2 (3/2). We clarify the limitations of these necessary conditions and formulate a conditional complexity-based criterion for excluding unbounded trajectories. No resolution of the Collatz conjecture is claimed.
ER -