Reduction to transcendence theory
The argument
Because the absence of nontrivial Collatz cycles already implies a nontrivial lower bound on |2^n − 3^m|, any proof of the Collatz conjecture would deliver such a bound as a corollary; and since every known nontrivial lower bound of this kind rests on transcendence theory, such a proof would have to either invoke Baker-type results or supply a new method capable of replacing them. That all known exclusions of structured Collatz cycles already run through Baker's theorem corroborates that the entanglement is real rather than formal.
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 absence of nontrivial Collatz cycles implies a nontrivial lower bound on |2^n − 3^m| for all natural numbers n, m.
- 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 →Every known nontrivial lower bound for |2^n − 3^m| relies on transcendence theory such as Baker's method.
- 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 →All known proofs that the Collatz map has no m-cycles for small m rely on Baker's theorem on linear forms in logarithms.
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 16, 2026
The inference goes through for the qualified conclusion that a proof of Collatz must use or replace transcendence theory: it needs both the conditional lower bound on the gap between powers of 2 and 3, which Tao proved by an elementary argument, and the absence of any known non-transcendental route to such bounds, a claim about the literature that is well attested but would fall to a single elementary proof. It does not deliver the literal conclusion that a Collatz proof is at least as hard as Baker's theorem, because the bound it produces is much weaker than Baker's; the corroborating premise about m-cycle exclusions adds plausibility without closing that gap.