The Turan variance constant falls from 5855 to 275, a factor of 21.3, by tightening the Mertens band it inherits quadratically: mertens_first_theorem from log 4 + 16 to log 4 + 3 and mertens_second_theorem from 76 to 16, with every statement's shape preserved, no sorry, and an unchanged axiom audit.
hunts/r_4218d4 (Record 27 of 98 in chronological sequence)
Guiding Question
Can the Turan variance constant 5855 be lowered by tightening the Mertens band it inherits quadratically, without weakening any statement?
Method & Verification
Three local slack steps were tightened in Mertensstheorems.lean using log t <= t/e, the two halves of Mertens I were kept apart, then mertens_second_theorem and sum_sq_dev_le were re-derived as 16^2 + 16 + 3, with lake build green against Mathlib v4.33.0-rc2.
Lineage & Relationships
Primary Sources (at pin 8fa46e134)
Editorial Notes
Case-log status is settled. Disposition maps settled to completed. Evidence is kernel-checked for the three edited modules; the remaining-slack arithmetic in probe.py is float bookkeeping and is not the grade. Classical constants 2 and 4 are not claimed to be close.
Date Provenance
commit f3b755d37f701c6a950e65a6a0371def42def21e, hunts/r_4218d4/RESULTS.md, author 2026-08-16T21:38:04Z