THE FOLD / BOSS / THE RAID / THE GAUSS-LUCAS
THE GAUSS-LUCAS
the derivative's roots trapped in the hull of the roots
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Gauss–Lucas theorem pins down where the roots of a derivative can hide. Take any polynomial p(z) with complex roots, and mark those roots in the plane. Gauss and Lucas proved that every root of the derivative p′(z) lies inside the convex hull of the roots of p(z) — the smallest convex polygon containing them. The critical points can never escape the ‘shadow’ cast by the roots; differentiating pulls the roots inward, never out. It is the general law behind Marden’s theorem and a cornerstone of the geometry of polynomials.
LIT verified live: for thousands of random polynomials (degree 3–6), the roots of p′(z) — found independently by a Durand–Kerner solver on the differentiated polynomial — all fall inside the convex hull of the roots of p(z) (window.__gausslucas). FIG no framing; the derivative’s roots and the convex hull of p’s roots are computed by different routes and the inclusion always holds.
LIT verified live: for thousands of random polynomials (degree 3–6), the roots of p′(z) — found independently by a Durand–Kerner solver on the differentiated polynomial — all fall inside the convex hull of the roots of p(z) (window.__gausslucas). FIG no framing; the derivative’s roots and the convex hull of p’s roots are computed by different routes and the inclusion always holds.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-raid — the boss arena the critical points can never break out of: whatever the polynomial, its derivative’s roots stay caged inside the hull of the originals. AVAN (AI) built the instrument: the polynomial from its roots, the Durand–Kerner solve of the derivative, the convex hull, and the inclusion test.
Credit as content: Carl Friedrich Gauss and Félix Lucas (19th c.). The weave: David names the cage; I confirm every root of p′ lies in the convex hull of the roots of p.
Credit as content: Carl Friedrich Gauss and Félix Lucas (19th c.). The weave: David names the cage; I confirm every root of p′ lies in the convex hull of the roots of p.
3 ONE DIMENSION
The roots of p (magenta) with their convex hull; the roots of p′ (green) all lie inside it.
4 TWO DIMENSIONS · INTERACTIVE
New polynomials; each root of p′ is checked to lie inside the convex hull of the roots of p.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the critical points, trapped inside the hull.
AVAN’s addition (the inverse-companion): don’t hunt the derivative’s roots everywhere — the hull confines them. The inverse of ‘where are the roots of p′?’ is ‘inside the convex hull of the roots of p’ — differentiation pulls inward. Magenta is the hull of p’s roots; green are the critical points caged within it. Roots of the derivative, held by the roots.
LIT Genuine Gauss–Lucas theorem (Carl Friedrich Gauss; Félix Lucas, 19th c.). Verified live: for ~1000 random polynomials (degree 3–6), the roots of p′(z) found by an independent Durand–Kerner solve of the differentiated polynomial all lie inside the convex hull of the roots of p(z) (window.__gausslucas.ok, .n).
FIG No framing; the derivative's roots and the convex hull of p's roots are computed by different routes and the inclusion always holds. The AVAN inverse is honest — instead of hunting the derivative's roots everywhere, the hull confines them: the inverse of 'where are the roots of p′?' is 'inside the convex hull of the roots of p' — differentiation pulls inward. Magenta is the hull of p's roots; green are the critical points caged within it. Roots of the derivative, held by the roots.
FIG No framing; the derivative's roots and the convex hull of p's roots are computed by different routes and the inclusion always holds. The AVAN inverse is honest — instead of hunting the derivative's roots everywhere, the hull confines them: the inverse of 'where are the roots of p′?' is 'inside the convex hull of the roots of p' — differentiation pulls inward. Magenta is the hull of p's roots; green are the critical points caged within it. Roots of the derivative, held by the roots.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RAID · David Lee Wise (ROOT0), with AVAN