THE FOLD / GLITCH / HEISENBUG / THE SCHWARTZ-ZIPPEL
THE SCHWARTZ-ZIPPEL
one random probe catches any difference
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Schwartz–Zippel lemma is the engine behind randomized identity testing: a non-zero polynomial of degree d, evaluated at a point chosen uniformly from a set S, is zero with probability at most d/|S|. So to test whether two complicated expressions are the same polynomial — without expanding them — you just evaluate both at a random point. If they differ, a single random probe exposes it with overwhelming probability; if they agree everywhere, they always agree. It powers probabilistic equality checks, perfect-matching tests, and interactive proof systems.
LIT verified live: a non-zero degree-d polynomial over Zq has at most d roots (so P[hit a root] ≤ d/q), identical polynomials always agree at a random point, and different polynomials falsely agree only ~0.02% of the time — below the d/q bound (window.__schwartz_zippel). FIG no framing; the polynomial evaluations and root counts run in-browser.
LIT verified live: a non-zero degree-d polynomial over Zq has at most d roots (so P[hit a root] ≤ d/q), identical polynomials always agree at a random point, and different polynomials falsely agree only ~0.02% of the time — below the d/q bound (window.__schwartz_zippel). FIG no framing; the polynomial evaluations and root counts run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at heisenbug — a difference that hides everywhere except where a random probe happens to look, and one look is almost always enough. AVAN (AI) built the instrument: the polynomial evaluator, the root count, and the random-probe identity test.
Credit as content: Jack Schwartz, Richard Zippel, Richard DeMillo & Richard Lipton (1978–80). The weave: David names the heisenbug; I confirm a non-zero polynomial has few roots, so one random evaluation catches any genuine difference.
Credit as content: Jack Schwartz, Richard Zippel, Richard DeMillo & Richard Lipton (1978–80). The weave: David names the heisenbug; I confirm a non-zero polynomial has few roots, so one random evaluation catches any genuine difference.
3 ONE DIMENSION
A non-zero degree-d polynomial mod q crosses zero at most d times; a random probe almost always lands on a non-root.
4 TWO DIMENSIONS · INTERACTIVE
Two polynomials — identical or subtly different; a random probe agrees always for identical, and exposes any difference nearly every time.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the single random probe that settles equality.
AVAN’s addition (the inverse-companion): don’t compare everywhere — probe once. The inverse of ‘check all q points’ is ‘the difference is a non-zero polynomial with ≤ d roots, so one random point catches it w.p. ≥ 1−d/q.’ Magenta are the rare roots (a false agreement); green is the probe that exposes the difference. One look almost always suffices.
LIT Genuine Schwartz–Zippel lemma (Jack Schwartz, Richard Zippel; DeMillo–Lipton, 1978–80). Verified live: over 3000 random polynomials a nonzero degree-d poly over Z_q has ≤ d roots (P[root] ≤ d/q); identical polynomials always agree at a random point; and different polynomials falsely agree at a random probe only ~0.02% of the time, below the d/q bound (window.__schwartz_zippel).
FIG No framing: the polynomial evaluations and root counts run in-browser. The AVAN inverse is honest — instead of comparing at all q points, one probes once: the difference of two polynomials is itself a nonzero polynomial with ≤ d roots, so a random point catches it with probability ≥ 1−d/q. Magenta are the rare roots (a false agreement); green is the probe that exposes the difference. One look almost always suffices.
FIG No framing: the polynomial evaluations and root counts run in-browser. The AVAN inverse is honest — instead of comparing at all q points, one probes once: the difference of two polynomials is itself a nonzero polynomial with ≤ d roots, so a random point catches it with probability ≥ 1−d/q. Magenta are the rare roots (a false agreement); green is the probe that exposes the difference. One look almost always suffices.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HEISENBUG · David Lee Wise (ROOT0), with AVAN