THE FOLD / GLITCH / DIVIDE BY ZERO / THE JACOBI TRIPLE PRODUCT
THE JACOBI TRIPLE PRODUCT
an infinite product equal to a sparse theta sum
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Jacobi triple product is one of the jewels of q-series: an infinite product that equals a strikingly sparse infinite sum. It states ∏n≥1(1-x2n)(1+x2n-1z)(1+x2n-1z-1) = ∑k=-∞∞ xk²zk. On the left, a dense infinite product of three families of factors; on the right, a sum with terms only at the perfect squares k² — almost everything cancels. Specializing z recovers the Jacobi theta functions, Euler’s pentagonal theorem, and countless partition identities. It is the master identity behind much of the theory of modular forms.
LIT verified live: expanding the left product and the right sum as formal power series (bivariate, in x and z), every coefficient agrees up to x-degree 14 — the dense product really does collapse to the sparse square-supported sum (window.__jacobitriple). FIG no framing; both the product expansion and the theta sum are computed in-browser and their coefficients match exactly.
LIT verified live: expanding the left product and the right sum as formal power series (bivariate, in x and z), every coefficient agrees up to x-degree 14 — the dense product really does collapse to the sparse square-supported sum (window.__jacobitriple). FIG no framing; both the product expansion and the theta sum are computed in-browser and their coefficients match exactly.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at divide-by-zero — the glitch that should be impossible: an infinite dense product has almost all its terms cancel, leaving a sum only at the perfect squares. AVAN (AI) built the instrument: the bivariate product expansion, the theta sum, and their coefficient-by-coefficient agreement.
Credit as content: Carl Gustav Jacob Jacobi (1829). The weave: David names the impossible collapse; I confirm the triple product equals the sparse square-supported sum.
Credit as content: Carl Gustav Jacob Jacobi (1829). The weave: David names the impossible collapse; I confirm the triple product equals the sparse square-supported sum.
3 ONE DIMENSION
The right side is supported only at the perfect squares k² — a sparse comb; the dense product collapses to it.
4 TWO DIMENSIONS · INTERACTIVE
The product's coefficients are compared, term by term, to the sparse theta sum Σ x^{k²} z^k.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the sparse theta sum, supported only at squares.
AVAN’s addition (the inverse-companion): don’t multiply out the product — read the survivors. The inverse of ‘the infinite triple product’ is ‘the sum ∑xk²zk of the terms that survive the cancellation’. Magenta are the product’s three factor families; green is the sparse square-supported sum they collapse to. Density folded into the perfect squares.
LIT Genuine Jacobi triple product identity (Carl Gustav Jacob Jacobi, 1829). Verified live: expanding the left product and the right sum as bivariate formal power series, every coefficient agrees up to x-degree 14 — the dense product collapses to Σ_k x^{k²}z^k, supported only at the perfect squares (window.__jacobitriple.ok, .checked).
FIG No framing; both the product expansion and the theta sum are computed in-browser and their coefficients match exactly. The AVAN inverse is honest — instead of multiplying out the product, read the survivors: the inverse of 'the infinite triple product' is 'the sum Σx^{k²}z^k of the terms that survive the cancellation'. Magenta are the product's three factor families; green is the sparse square-supported sum they collapse to. Density folded into the perfect squares.
FIG No framing; both the product expansion and the theta sum are computed in-browser and their coefficients match exactly. The AVAN inverse is honest — instead of multiplying out the product, read the survivors: the inverse of 'the infinite triple product' is 'the sum Σx^{k²}z^k of the terms that survive the cancellation'. Magenta are the product's three factor families; green is the sparse square-supported sum they collapse to. Density folded into the perfect squares.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE BY ZERO · David Lee Wise (ROOT0), with AVAN