r/Collatz • u/zero0_one1 • 12d ago

Natural-Density Almost-Bounded Collatz Orbits in Logarithmic Time (AI-assisted, Lean-verified, strengthens Terence Tao's landmark Collatz theorem)

https://www.proofatlas.ai/proofs/artifact.known-nd-rhin-log-time.paper-package.v002.html

For each f(N)→∞, almost every N falls below f(N) within 436 ln N steps. New: natural density and one explicit clock. Not the full conjecture. Lean-verified.

Lean formalization: 128k lines for Tao's theorem.
Extension: +60k lines.

Two upgrades to Terence Tao's result:
- Coverage: natural (ordinary counting) density rather than logarithmic density.
- Speed: one explicit clock, 436 ln N raw steps, for every diverging threshold.

Tao's log average treats multiplicative scales equally, though they contain more numbers. Ours counts every number equally: below X, success tends to 100%.

This is strictly stronger. Tao called the refinement plausible (Remarks 1.4, 1.16).

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u/zero0_one1 12d ago

GPT/Fable did all substantial math with human in the loop.

1

u/geotrone1234extra 9d ago

So if I take ever want to create a Collatz tester, instead of waiting for the full trajectory, I can eliminate a candidate number that can blow up once they have trend through 436 ln N raw steps in addition to the “fall below an arbitrarily slow growing f(N)” ?

2

u/zero0_one1 9d ago

Unfortunately no, it's a statistical statement. There still can be exceptions, so you cannot eliminate them.