THE FOLD / GLITCH / SEGFAULT / THE ENESTRÖM-KAKEYA
THE ENESTRÖM-KAKEYA
roots caged in the unit disk by rising coefficients
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Eneström–Kakeya theorem cages a polynomial’s roots using only the order of its coefficients. If p(z) = a0 + a1z + … + anzn has coefficients that are positive and non-decreasing, 0 < a0 ≤ a1 ≤ … ≤ an, then all of its roots lie in the closed unit disk |z| ≤ 1. No root can escape to modulus greater than 1. The proof multiplies by (z-1) to telescope the coefficients, and the same idea run in reverse bounds the roots from below. It is a favourite tool for stability questions, where you need every root inside the disk.
LIT verified live: for thousands of random polynomials with strictly increasing positive coefficients, every root — found by a Durand–Kerner solver — has |z| ≤ 1; and with the monotonicity broken (random positive coefficients), a root with |z| > 1 appears in about 80% of cases, showing the hypothesis is necessary (window.__enestromkakeya). FIG no framing; the root-finding and the |z| test both run in-browser.
LIT verified live: for thousands of random polynomials with strictly increasing positive coefficients, every root — found by a Durand–Kerner solver — has |z| ≤ 1; and with the monotonicity broken (random positive coefficients), a root with |z| > 1 appears in about 80% of cases, showing the hypothesis is necessary (window.__enestromkakeya). FIG no framing; the root-finding and the |z| test both run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at segfault — the glitch where merely sorting the coefficients upward slams every root inside the unit disk, no root allowed past the boundary. AVAN (AI) built the instrument: the increasing-coefficient polynomial, the Durand–Kerner root solve, the |z| ≤ 1 test, and the non-monotone control.
Credit as content: Gustav Eneström (1893) and Sōichi Kakeya (1912). The weave: David names the cage; I confirm rising positive coefficients force all roots into |z| ≤ 1.
Credit as content: Gustav Eneström (1893) and Sōichi Kakeya (1912). The weave: David names the cage; I confirm rising positive coefficients force all roots into |z| ≤ 1.
3 ONE DIMENSION
The increasing coefficients (bars) and the roots (green) — all inside the unit circle.
4 TWO DIMENSIONS · INTERACTIVE
New polynomials; every root's modulus is checked ≤ 1 — and breaking monotonicity lets one escape.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the roots, all inside the unit disk.
AVAN’s addition (the inverse-companion): don’t solve then check — the coefficient order already bounds the roots. The inverse of ‘where are the roots?’ is ‘inside |z| ≤ 1, guaranteed by 0 < a0 ≤ … ≤ an’. Magenta is the unit circle boundary; green are the roots caged within it. Root location read from coefficient order.
LIT Genuine Eneström–Kakeya theorem (Gustav Eneström 1893; Sōichi Kakeya 1912). Verified live: for ~1500 random polynomials with strictly increasing positive coefficients, every Durand–Kerner root has |z| ≤ 1 (worst ~0.99); a non-monotone-coefficient control produces a root with |z| > 1 in ~80% of cases (window.__enestromkakeya.ok, .worst, .ctrlPct).
FIG No framing; the root-finding and the |z| test both run in-browser. The AVAN inverse is honest — instead of solving then checking, the coefficient order already bounds the roots: the inverse of 'where are the roots?' is 'inside |z| ≤ 1, guaranteed by 0 < a₀ ≤ … ≤ aₙ'. Magenta is the unit circle boundary; green are the roots caged within it. Root location read from coefficient order.
FIG No framing; the root-finding and the |z| test both run in-browser. The AVAN inverse is honest — instead of solving then checking, the coefficient order already bounds the roots: the inverse of 'where are the roots?' is 'inside |z| ≤ 1, guaranteed by 0 < a₀ ≤ … ≤ aₙ'. Magenta is the unit circle boundary; green are the roots caged within it. Root location read from coefficient order.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SEGFAULT · David Lee Wise (ROOT0), with AVAN