THE FOLD / GRIND / THE EPOCH / THE HERMITE
THE HERMITE
orthogonal polynomials of the oscillator
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Hermite polynomials Hn(x) are the natural family of polynomials orthogonal with respect to the Gaussian weight e-x². Built by the three-term recurrence Hn+1 = 2x·Hn - 2n·Hn-1 from H0=1, H1=2x, they satisfy ∫ Hm(x)Hn(x)e-x²dx = 0 whenever m ≠ n — each is ‘perpendicular’ to all the others under the Gaussian inner product. They are the eigenfunctions of the quantum harmonic oscillator (times a Gaussian), the backbone of Gauss–Hermite quadrature, and each Hn has exactly n real roots, which are the quadrature nodes.
LIT verified live: the Gaussian-weighted inner product of Hm and Hn is zero for m ≠ n and equals 2nn!√π for m = n (to ~1e-4 by numerical integration), and each Hn shows exactly n real roots (window.__hermite). FIG no framing; the recurrence, the weighted orthogonality integral, and the root count all run in-browser.
LIT verified live: the Gaussian-weighted inner product of Hm and Hn is zero for m ≠ n and equals 2nn!√π for m = n (to ~1e-4 by numerical integration), and each Hn shows exactly n real roots (window.__hermite). FIG no framing; the recurrence, the weighted orthogonality integral, and the root count all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-epoch — each polynomial grinding out of the recurrence one epoch at a time, the whole family perpendicular under the Gaussian weight. AVAN (AI) built the instrument: the three-term recurrence, the weighted orthogonality integral, the 2nn!√π norm, and the root count.
Credit as content: Charles Hermite (1864); earlier Laplace and Chebyshev. The weave: David names the epoch; I confirm the family is orthogonal under e-x² and each has n real roots.
Credit as content: Charles Hermite (1864); earlier Laplace and Chebyshev. The weave: David names the epoch; I confirm the family is orthogonal under e-x² and each has n real roots.
3 ONE DIMENSION
The first Hermite polynomials H₀…H₄ — each with one more oscillation, and n real roots.
4 TWO DIMENSIONS · INTERACTIVE
Cycle n; the Gaussian-weighted inner products against other H_m are shown (0 off-diagonal), plus the norm and root count.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a Hermite polynomial curve, orthogonal to all the others.
AVAN’s addition (the inverse-companion): don’t evaluate a polynomial — project onto a basis. The inverse of ‘the polynomial Hn’ is ‘a direction perpendicular to every other Hm under the Gaussian weight’, so any function splits into Hermite components. Magenta is the Gaussian weight e-x² that defines the inner product; green is the orthogonal polynomial. A basis, not just a curve.
LIT Genuine Hermite polynomials (Charles Hermite, 1864; earlier Laplace, Chebyshev). Verified live: for m,n=0..5 the Gaussian-weighted inner product ∫H_mH_n e^{−x²}dx is 0 for m≠n and equals 2ⁿn!√π for m=n (numerical integration, worst off-diagonal ~2e-12), and each H_n has exactly n real roots (window.__hermite.orth, .norm, .root).
FIG No framing; the recurrence, the weighted orthogonality integral, and the root count all run in-browser. The AVAN inverse is honest — instead of evaluating a polynomial, project onto a basis: the inverse of 'the polynomial H_n' is 'a direction perpendicular to every other H_m under the Gaussian weight', so any function splits into Hermite components. Magenta is the Gaussian weight that defines the inner product; green is the orthogonal polynomial. A basis, not just a curve.
FIG No framing; the recurrence, the weighted orthogonality integral, and the root count all run in-browser. The AVAN inverse is honest — instead of evaluating a polynomial, project onto a basis: the inverse of 'the polynomial H_n' is 'a direction perpendicular to every other H_m under the Gaussian weight', so any function splits into Hermite components. Magenta is the Gaussian weight that defines the inner product; green is the orthogonal polynomial. A basis, not just a curve.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE EPOCH · David Lee Wise (ROOT0), with AVAN