Parity vectors and paradoxical sequences in the accelerated Collatz map
Pith reviewed 2026-05-22 10:24 UTC · model grok-4.3
The pith
The accelerated Collatz map has a sharp finitary parity-vector density and zero density for bounded-length paradoxical sequences.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We prove a sharp finitary form of Terras's parity-vector density. We give a closed-form analytic count of paradoxical Ω_k(n) for each fixed length k. We establish a density-zero theorem for bounded-length paradoxical sequences with explicit constant. Numerically, every paradoxical reduced ratio q/j is a left convergent, left semiconvergent, or Stern-Brocot mediant of adjacent convergents of log_3 2.
What carries the argument
Parity vectors and the paradoxical sequences Ω_k(n) defined from the accelerated Collatz map T.
Load-bearing premise
The analysis assumes the standard definitions and properties of parity vectors and paradoxical sequences from the existing literature on the accelerated Collatz map.
What would settle it
Running the accelerated Collatz map on all n up to 10^12 and counting the number of paradoxical sequences of length at most 10, then checking if their proportion exceeds the paper's explicit constant, would test the density theorem.
read the original abstract
This note studies parity vectors and paradoxical sequences in the accelerated Collatz iteration $T(n) = (3n+1)/2$ for $n$ odd, $T(n) = n/2$ for $n$ even. Building on Rozier and Terracol (arXiv:2502.00948, 2025), Terras (1976), Lagarias (1985), and Tao (2019), we prove three theorems and add one numerical observation. The first is a sharp finitary form of Terras's parity-vector density; the second is a closed-form analytic count of paradoxical $\Omega_k(n)$ for each fixed length $k$. The third is a density-zero theorem for bounded-length paradoxical sequences with explicit constant. As for the numerical piece, among the seven $(j, q)$ pairs that show up in the Rozier-Terracol enumeration with first term $n \le 10^9$, every paradoxical reduced ratio $q/j$ turns out to be a left convergent, a left semiconvergent, or a Stern-Brocot mediant of adjacent convergents/semiconvergents of $\log_3 2$. The three theorems are unconditional. The fourth observation is verified for $n \le 10^7$ and conjectured for all $n$. We make no claim toward the Collatz conjecture or Terras's coefficient-stopping-time conjecture.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This note studies parity vectors and paradoxical sequences in the accelerated Collatz iteration T(n) = (3n+1)/2 for odd n and T(n)=n/2 for even n. Building on prior work, it proves three unconditional theorems: a sharp finitary form of Terras's parity-vector density; a closed-form analytic count of paradoxical Ω_k(n) for each fixed length k; and a density-zero theorem for bounded-length paradoxical sequences with explicit constant. It also reports a numerical observation that every paradoxical reduced ratio q/j (among seven pairs with n ≤ 10^9 from prior enumeration) is a left convergent, left semiconvergent, or Stern-Brocot mediant of adjacent convergents of log_3 2, verified for n ≤ 10^7 and conjectured in general. No claim is made toward the Collatz conjecture.
Significance. If the theorems hold, the sharp finitary density and closed-form counts would supply precise quantitative control over the distribution of parity vectors and the occurrence of paradoxical sequences under the accelerated map. The density-zero result with an explicit constant strengthens the analytic toolkit for studying the iteration's dynamics. The observed link between paradoxical ratios and continued-fraction approximants of log_3 2 points to a possible Diophantine structure that could be of independent interest in number-theoretic approaches to Collatz-type problems.
major comments (1)
- Abstract: the three theorems are asserted to be unconditional with explicit constants, yet only the abstract is available; without the derivations, hidden assumptions, or proof details it is impossible to verify the central claims or check for gaps.
Simulated Author's Rebuttal
We thank the referee for their careful reading and for highlighting the need for verification of the central claims. We address the major comment below.
read point-by-point responses
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Referee: [—] Abstract: the three theorems are asserted to be unconditional with explicit constants, yet only the abstract is available; without the derivations, hidden assumptions, or proof details it is impossible to verify the central claims or check for gaps.
Authors: The full manuscript containing the complete derivations, explicit constants, and proof details for all three unconditional theorems is available on arXiv:2605.13886. The abstract provides only a summary of the results, which build directly on the cited works of Rozier-Terracol, Terras, Lagarias, and Tao; the body supplies the finitary density argument, the closed-form count for paradoxical sequences of fixed length k, and the density-zero theorem with its explicit constant. We are prepared to supply specific sections, lemmas, or clarifications from the proofs to facilitate verification. revision: no
Circularity Check
No significant circularity detected
full rationale
The abstract states three unconditional theorems (sharp finitary Terras parity-vector density, closed-form count of paradoxical Ω_k(n), and density-zero theorem with explicit constant) as proven results building on cited prior literature (Terras 1976, Lagarias 1985, Tao 2019, Rozier-Terracol 2025) whose basic properties are taken as given. The numerical observation on reduced ratios q/j being convergents or mediants of log_3 2 is explicitly limited to n ≤ 10^7, verified numerically, and conjectured thereafter, with no claim toward the Collatz conjecture. No equations, derivations, or self-referential reductions appear in the provided abstract; the central claims are presented as independent of the inputs by construction rather than fitted or renamed quantities.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption The accelerated Collatz map T(n) and the definitions of parity vectors and paradoxical sequences Ω_k(n) behave as stated in Terras (1976), Lagarias (1985), Tao (2019), and Rozier-Terracol (2025).
discussion (0)
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