argumentA line of reasoning from premises to a conclusion. Like a claim, it belongs to no one who makes it.constitution →

Krasikov–Lagarias difference-inequality proof (computer-aided)

forThis argument, if it holds, bears in favour of the claim.constitution →For all sufficiently large x, more than x^0.84 integers up to x have Collatz orbits reaching 1.

The argument

Because the 3x+1 counting functions on residue classes modulo 3^k satisfy Krasikov's difference inequalities, and because Krasikov and Lagarias prove that whenever the linear program associated to that system is feasible for a growth parameter λ, every positive monotone solution grows at least like λ^y even though the system contains advanced variables, a computer search for the modulus 3^11 system exhibiting a feasible solution with log_2 λ above 0.84 yields the lower bound: for all sufficiently large x, more than x^0.84 integers below x reach 1 under the 3x+1 map.

Does it work?

Not yet evaluated.

Premises

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

Where it is made

Passages where sources make, criticize or mention this argument. They record where it appears, not how strong it is, and none has precedence over another.

No source is recorded as making it.

Earlier ratings

Sep 17, 2026
The inference goes through: the paper's Theorem 2.2 converts feasibility of the linear program into exponential growth of the counting functions, and the reported feasible solution for the modulus 3^11 system fixes the exponent. The argument rests on Krasikov's difference inequalities, which are elementary and undisputed, and on a computer-aided feasibility check that has been refereed but not independently formalized; that check is the only step a skeptic could ask to see reproduced.