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The system of difference inequalities introduced by Krasikov for counting functions of integers below x whose 3x+1 (Collatz) trajectories satisfy given parity constraints, and its residue-class refinements (e.g. restricting to classes modulo 3^k), used to prove lower bounds on the density of Collatz preimage sets. Density claims about other integer sequences stay under their own tags.

For all sufficiently large x, more than x^0.84 integers up to x have Collatz orbits reaching 1.

VerifiedThe evidence, examined directly, establishes the claim.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Collatz conjecture Number theoryimportance · settledImportance 0.18, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

The 3x+1 counting functions restricted to residue classes modulo 3^k satisfy Krasikov's system of difference inequalities.

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Number theory Collatz conjectureimportance · settledImportance 0.10, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →

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