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THE HADWIGER-FINSLER
a sharpened Weitzenbock inequality
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Hadwiger–Finsler inequality is a sharpened version of Weitzenböck’s. Weitzenböck says a triangle’s squared sides satisfy a² + b² + c² ≥ 4√3·T (T the area). Hadwiger and Finsler add back the exact leftover: a² + b² + c² ≥ 4√3·T + (a-b)² + (b-c)² + (c-a)². The extra sum of squared side-differences is precisely how far the triangle is from equilateral, so the inequality is tight exactly when a = b = c. Since that extra term is always ≥ 0, Hadwiger–Finsler immediately implies Weitzenböck — it is the stronger statement, with the slack made explicit.
LIT verified live: for tens of thousands of random triangles, a²+b²+c² - 4√3·T - ((a-b)²+(b-c)²+(c-a)²) is always ≥ 0, reaching 0 only for the equilateral triangle (window.__hadwigerfinsler). FIG no framing; the sides, the area, and both sides of the inequality are computed independently in-browser.
LIT verified live: for tens of thousands of random triangles, a²+b²+c² - 4√3·T - ((a-b)²+(b-c)²+(c-a)²) is always ≥ 0, reaching 0 only for the equilateral triangle (window.__hadwigerfinsler). FIG no framing; the sides, the area, and both sides of the inequality are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-choke-point — the boss gate: no triangle passes without its squared sides clearing the area plus the full penalty for being non-equilateral. AVAN (AI) built the instrument: the sides, the area, and the sharpened bound with its explicit slack.
Credit as content: Hugo Hadwiger and Paul Finsler (1937); Roland Weitzenböck (the weaker parent inequality). The weave: David names the sharpened gate; I confirm a²+b²+c² ≥ 4√3·T + ∑(a-b)².
Credit as content: Hugo Hadwiger and Paul Finsler (1937); Roland Weitzenböck (the weaker parent inequality). The weave: David names the sharpened gate; I confirm a²+b²+c² ≥ 4√3·T + ∑(a-b)².
3 ONE DIMENSION
A triangle: a²+b²+c² against 4√3·T plus the squared side-differences — the gap closes as it nears equilateral.
4 TWO DIMENSIONS · INTERACTIVE
Cycle triangles; the slack a²+b²+c² − 4√3T − Σ(a−b)² is checked to stay ≥ 0.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the bound 4√3·T plus the squared side-differences.
AVAN’s addition (the inverse-companion): don’t stop at Weitzenböck — add back the leftover. The inverse of ‘a²+b²+c² ≥ 4√3T’ is ‘the exact surplus (a-b)²+(b-c)²+(c-a)², zero only when equilateral’. Magenta are the squared side-differences added to the area bound; green is the squared-side total that clears it. The area bound, plus the price of not being equilateral.
LIT Genuine Hadwiger–Finsler inequality (Hugo Hadwiger and Paul Finsler, 1937; sharpens Weitzenböck). Verified live: for ~60000 random triangles a²+b²+c² − 4√3·T − ((a−b)²+(b−c)²+(c−a)²) ≥ 0 always, reaching 0 only at the equilateral triangle (window.__hadwigerfinsler.ok, .minD).
FIG No framing; the sides, the area, and both sides of the inequality are computed independently in-browser. The AVAN inverse is honest — instead of stopping at Weitzenböck, add back the leftover: the inverse of 'a²+b²+c² ≥ 4√3T' is 'the exact surplus (a−b)²+(b−c)²+(c−a)², zero only when equilateral'. Magenta are the squared side-differences added to the area bound; green is the squared-side total that clears it. The area bound, plus the price of not being equilateral.
FIG No framing; the sides, the area, and both sides of the inequality are computed independently in-browser. The AVAN inverse is honest — instead of stopping at Weitzenböck, add back the leftover: the inverse of 'a²+b²+c² ≥ 4√3T' is 'the exact surplus (a−b)²+(b−c)²+(c−a)², zero only when equilateral'. Magenta are the squared side-differences added to the area bound; green is the squared-side total that clears it. The area bound, plus the price of not being equilateral.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CHOKE POINT · David Lee Wise (ROOT0), with AVAN