THE FOLD / CHEAT / THE-SHORTCUT / THE KRONECKER SUBSTITUTION
THE KRONECKER SUBSTITUTION
polynomials multiplied as one big integer
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Kronecker substitution turns polynomial multiplication into a single big-integer multiplication. If two polynomials have non-negative integer coefficients bounded below some 2b, evaluate each at a large power of two x = 2m — this just packs the coefficients side by side into the digits of one huge integer. Multiply the two integers (using any fast bignum routine), and the product’s base-2m digits are the coefficients of the polynomial product — provided m is chosen large enough that adjacent coefficients never carry into each other. It lets you borrow the world’s fastest integer-multiplication code to multiply polynomials, and vice versa.
LIT verified live: over 20,000 random polynomial pairs with 8-bit coefficients, packing into one integer, multiplying, and unpacking the base-2m digits reproduces the direct convolution exactly (window.__kronecker_substitution). FIG no framing; the BigInt packing, multiply, and digit-unpacking run in-browser.
LIT verified live: over 20,000 random polynomial pairs with 8-bit coefficients, packing into one integer, multiplying, and unpacking the base-2m digits reproduces the direct convolution exactly (window.__kronecker_substitution). FIG no framing; the BigInt packing, multiply, and digit-unpacking run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-shortcut — skip the convolution loop and let one integer multiply carry the whole polynomial product. AVAN (AI) built the instrument: the base-2m packing, the BigInt multiply, the digit unpack, and the direct-convolution check.
Credit as content: Kronecker substitution (Leopold Kronecker; standard in computer algebra). The weave: David names the shortcut; I confirm the packed integer product’s digits are exactly the polynomial product’s coefficients.
Credit as content: Kronecker substitution (Leopold Kronecker; standard in computer algebra). The weave: David names the shortcut; I confirm the packed integer product’s digits are exactly the polynomial product’s coefficients.
3 ONE DIMENSION
Coefficients packed side by side into the digits of one big integer at base 2^m — with room so they never carry into each other.
4 TWO DIMENSIONS · INTERACTIVE
Two polynomials; pack → one integer multiply → unpack digits — matching the direct coefficient convolution.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the product coefficients, read from the big integer’s digits.
AVAN’s addition (the inverse-companion): don’t convolve coefficient by coefficient — evaluate at a big base. The inverse of ‘sum aₖbₓ₋ₖ over pairs’ is ‘pack into one integer, multiply once, read the base-2m digits.’ Magenta is the packed big integer; green is the product coefficients its digits reveal. One multiply, a whole convolution.
LIT Genuine Kronecker substitution (Leopold Kronecker; standard in computer algebra). Verified live: over 20000 random polynomial pairs with 8-bit non-negative coefficients, packing at base 2^m (with m large enough to avoid inter-coefficient carries), one BigInt multiply, and unpacking the base-2^m digits reproduces the direct coefficient convolution exactly (window.__kronecker_substitution.matchesDirect).
FIG No framing: the BigInt packing, multiply, and digit-unpacking run in-browser. The AVAN inverse is honest — instead of convolving coefficient by coefficient, one evaluates at a big base: pack into one integer, multiply once, and read the base-2^m digits. Magenta is the single packed big integer; green is the product coefficients its digits reveal. One multiply, a whole convolution.
FIG No framing: the BigInt packing, multiply, and digit-unpacking run in-browser. The AVAN inverse is honest — instead of convolving coefficient by coefficient, one evaluates at a big base: pack into one integer, multiply once, and read the base-2^m digits. Magenta is the single packed big integer; green is the product coefficients its digits reveal. One multiply, a whole convolution.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE-SHORTCUT · David Lee Wise (ROOT0), with AVAN