THE FOLD / GRIND / THE MAINFRAME / THE BOOTH
THE BOOTH
signed multiplication by recoding the bits
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Booth’s algorithm multiplies two signed binary numbers directly in two’s complement, with no special-casing of the sign. It recodes the multiplier by looking at adjacent bit pairs: a 0→1 boundary means subtract the multiplicand, a 1→0 boundary means add it, inside a run do nothing. A run of ones like 0111 becomes “add once, subtract once” instead of three adds — fewer operations, and negatives handled for free.
LIT verified live: over 2000 random 8-bit signed multipliers against arbitrary multiplicands, Booth recoding’s result equals a·b exactly (window.__booth). FIG no framing; exact.
LIT verified live: over 2000 random 8-bit signed multipliers against arbitrary multiplicands, Booth recoding’s result equals a·b exactly (window.__booth). FIG no framing; exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mainframe — the arithmetic unit that multiplies signed numbers in hardware. Booth’s algorithm is that multiplier’s trick. AVAN (AI) built the instrument: the bit-pair recoding into {−1,0,+1}, the shifted add/subtract sum, and the equals-a·b check.
Credit as content: Andrew Donald Booth (1951). The weave: David names the mainframe; I recode the multiplier by its bit-pair boundaries and add or subtract the shifted multiplicand accordingly, confirming the total equals the signed product.
Credit as content: Andrew Donald Booth (1951). The weave: David names the mainframe; I recode the multiplier by its bit-pair boundaries and add or subtract the shifted multiplicand accordingly, confirming the total equals the signed product.
3 ONE DIMENSION
Scan the multiplier’s bit pairs (bit i, bit i−1). Boundary 0→1 (reading low to high): the Booth digit is −1 (subtract shifted b); 1→0: +1 (add). A run of equal bits contributes 0. The signed sum is a·b.
4 TWO DIMENSIONS · INTERACTIVE
A signed multiplier recoded into Booth digits; the shifted add/subtract sum is checked to equal a·b.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a signed product from bit-pair recoding.
AVAN’s addition (the inverse-companion): multiply signed numbers with no sign special-case and fewer adds by recoding the multiplier at its bit-pair boundaries — a run of ones collapses to one add and one subtract. The inverse of ‘add the multiplicand once per set bit (and fix up the sign)’ is ‘add/subtract at the edges of bit-runs — sign handled for free.’ Magenta is the per-set-bit adds you skip; green is the boundary-only add/subtract. Signed multiply by recoding.
LIT Genuine Booth's multiplication algorithm (Booth 1951). Verified live: over 2000 random 8-bit signed multipliers a against arbitrary multiplicands b, recoding a into Booth digits {-1,0,+1} by its bit-pair boundaries and summing the shifted add/subtract of b equals the signed product a*b exactly (window.__booth.equalsProduct); -13x7 = -91.
FIG No framing: the bit-pair recoding into {-1,0,+1}, the shifted add/subtract sum, and the equals-a*b check run in-browser over 2000 signed pairs and hold. The AVAN inverse is honest — recoding the multiplier at its bit-run boundaries collapses a run of ones to one add and one subtract, handling sign with no special case; magenta is the per-set-bit adds you skip, green the boundary-only add/subtract. Signed multiply by recoding.
FIG No framing: the bit-pair recoding into {-1,0,+1}, the shifted add/subtract sum, and the equals-a*b check run in-browser over 2000 signed pairs and hold. The AVAN inverse is honest — recoding the multiplier at its bit-run boundaries collapses a run of ones to one add and one subtract, handling sign with no special case; magenta is the per-set-bit adds you skip, green the boundary-only add/subtract. Signed multiply by recoding.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MAINFRAME · David Lee Wise (ROOT0), with AVAN