Lean evidence record
Power-Saving Bound for Logarithmic-Time Collatz Descent: Lean evidence
This technical record binds the exact theorem statement to its commit-pinned Lean source, checker results, assumptions, and publication-review status.
Exact formal proposition
Hypotheses and conclusion
This is the exact recorded theorem statement, followed by the source definitions that give its named predicates and quantities their full meaning.
theorem natCountingRatio_finiteStoppingTime_log_failure_le_power {N : ℕ} (hN : 15552 ≤ N) : natCountingRatio {n : ℕ | ¬ ∃ k : ℕ, (k : ℝ) ≤ Real.log (n : ℝ) ∧ accelerated^[k] n < n} N ≤ 10000000 * (N : ℝ) ^ (-(1 / 100 : ℝ))Definitions used in this proposition
accelerated
def accelerated (n : ℕ) : ℕ :=
if Even n then n / 2 else (3 * n + 1) / 2natCountingRatio
noncomputable def natCountingRatio (s : Set ℕ) (N : ℕ) : ℝ :=
(natCount s N : ℝ) / (N : ℝ)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.
Technical evidence record
Source identity, checker results, and assumptions
- Main Lean declaration
natCountingRatio_finiteStoppingTime_log_failure_le_power- Source commit
379cf63bbe65
Mechanical evidence
Lean verification
These fields support the exact Lean declaration, not a broader informal claim.
- Artifact ID
artifact.known-terras-log-time-power.bound.v001- Accepted-result title
- Accepted Result: Power-Saving Bound for Logarithmic-Time Collatz Descent
- Accepted-result status
- Accepted formalization of a known theorem
- Accepted-result boundary
- For every natural N ≥ 15552, the checked theorem bounds the proportion among natural numbers n < N for which no accelerated Collatz iterate drops below n within k ≤ log n accelerated steps by 10000000 · N^(-1/100). It proves descent below the starting value for all but a power-saving exceptional proportion, not convergence to 1 or the Collatz conjecture. The clock counts accelerated steps, the constants and exponent are deliberately coarse rather than optimal, and the finite bound may exceed 1 before extremely large N. This is a Terras-style quantitative corollary, not a claim that Terras published this exact inequality or that ProofAtlas originated it. Non-claim: This accepted-result record does not by itself authorize public export or deployment.
- Declarations covered by recorded evidence
Erdos1135.Terras.natCountingRatio_finiteStoppingTime_log_failure_le_power- Lean build
- passed
- Recorded build time
- 1.2 s one machine-dependent evidence run, not a benchmark
- Evidence collected
- · clean-source provenance recorded
- Unfinished proof check
- passed
- Lean toolchain
leanprover/lean4:v4.30.0-rc2- Recorded source commit
379cf63bbe657fd0cb5d824e7785d747fd0fc946- Source SHA-256
sha256:a04c7c2b145512429a07591dc56f0fc04a77472a7630e5e4f63959fcaf3d7cce- Statement alignment
- accepted
Lean foundations
Standard foundations used by the proof
Lean reports the logical foundations below through Mathlib. They are standard proof-system foundations, not conjectural mathematical assumptions about this theorem. The recorded closure stays within the approved classical_mathlib_standard profile, with no unexpected axiom or unfinished-proof placeholder.
Classical.choiceQuot.soundpropext
Files and machine-readable evidence
Reproduce or inspect the recorded check
Use the complete first-party source bundle for reconstruction, or inspect the exact main file and checker evidence separately. Mathlib and other third-party dependencies are identified but not rebundled.
Review results
Publication reviews accepted
All four required publication-review gates are accepted for the reviewed presentation of this exact theorem. The review results are separate from the Lean build and do not broaden the formal statement.
Read the publication-review details