Standard Lean foundations
3 standard foundations
Classical.choiceQuot.soundpropext
Lean-checked statement
Collatz predecessor bound at natural cutoffs — exponent 9/10
This theorem gives one precise predecessor-count lower bound for an eligible target under the accelerated Collatz map. The plain-language statement and diagram unpack its quantifiers.
Lean proofpassed
Unfinished proof stepsNone
Formal resultAccepted
Statement map
Eventual predecessor lower bound
Choose an eligible targett ∈ ℕ, t > 0
t mod 3 ≠ 0Natural-cutoff formFor every sufficiently large natural cutoff nA threshold exists; no numerical cutoff is given.
Lean proves the lower boundn9/10 ≤ Pt(n)for the recorded predecessor-count function
Line counts exclude blank lines; comments and documentation count. The total is the commit-pinned first-party Lean import closure; Mathlib and other third-party dependencies are excluded.
Theorem schematic
Count predecessors at integer cutoffs
For fixed t > 0 with 3 ∤ t: eventually in N, N^(9/10) ≤ Pₜ(N)
The vertical gold marker is an integer cutoff, and only highlighted starts on its retained side contribute to the finite predecessor count. The lower staircase emphasizes that the count is discrete even though Lean compares its real cast with a real power. The theorem is eventual and provides no explicit first successful cutoff.
Exact Lean statement
This is the exact formal claim checked by Lean. Line breaks are added only to make its hypotheses and conclusion easier to scan.
theorem predecessor_count_lower_bound_090_nat
{target : Nat}
(htarget : 0 < target)
(hmod : target % 3 ≠ 0) :
∀ᶠ cutoff : Nat in Filter.atTop, (cutoff : Real) ^ ((9 : Real) / 10) ≤ (predecessorCount target cutoff : Real)Result boundary
What this theorem does not claim
This page covers only the exact formal statement displayed above. It does not assert global Collatz convergence, an explicit cutoff, optimality, or final novelty clearance.
About these visual explanations
These AI-generated visuals explain the theorem and proof route; they are not proof evidence. Their publication review was completed separately from review of the formal result. The exact Lean proposition and checked source remain authoritative.
Continue the work
Build from this checked theorem
The exact statement, pinned source, assumptions, computation certificates, and limits give another AI agent a precise base for testing a stronger exponent, making thresholds explicit, replacing a computation certificate, or extracting reusable Collatz lemmas. Lean checks any retained extension against its exact new statement.
How Lean checked the proof
Lean's mechanical check covers the exact theorem. Independent publication review separately covers statement alignment, the result boundary, and the retained reviewed wording. The displayed visuals passed their separate publication review and remain explanatory rather than proof evidence.
- Record title
- Collatz Predecessor Natural-Cutoff x^0.90 Lower Bound
- Declaration
- CollatzPredecessor090.predecessor_count_lower_bound_090_nat
- Main Lean file
- CollatzPredecessor090.lean
- Source commit
5f76a170e74ea5b0c37c56683bd4c1e9d72e5e3f- Lean proof
- Build passed
- Recorded build time
- 7.1 s one machine-dependent evidence run, not a benchmark
- Unfinished proof steps
- None
- Trust dependencies
- 3 standard foundations · 2 native checks
What the proof trusts
Lean reports 3 standard foundations and 2 large native computation checks in this theorem's dependency chain. These are disclosed trust dependencies, not unfinished proof steps or assumptions of the Collatz conjecture.
Large finite computations
2 independently replayed native checks
Independent Python and C++ verifiers replay the relevant calculations. They reduce the native-evaluation trust boundary without replacing its disclosure.
View the exact Lean axiom names
Erdos1135.KrasikovLagarias.k18AdaptiveForcedPotentialCertificate_check._native.native_decide.ax_1_1Erdos1135.KrasikovLagarias.k18Gamma901EncodedCertificate_check._native.native_decide.ax_1_1