THE FOLD / SPAWN / CHECKPOINT ZERO / THE REMAINDER
THE REMAINDER
CRT — a number as its coprime remainders
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Chinese Remainder Theorem. An ancient puzzle: a number leaves remainder 2 when divided by 3, 3 by 5, 2 by 7 — what is it? (23.) The theorem says that as long as the divisors are coprime, the remainders pin the number down uniquely up to their product, and you can always reconstruct it.
That makes a number equivalent to its tuple of remainders — a residue number system. And arithmetic splits perfectly across the channels: add or multiply the remainders independently, modulus by modulus, with no carries between them, and reconstruct at the end. Big multiplications become several tiny parallel ones.
LIT verified live: over 20,000 random coprime-modulus sets the reconstructed x satisfies every congruence, and a brute-force search confirms it is the only solution below the product (window.__crt.allCongruences && unique). mod [3,5,7] = [2,3,2] → 23. FIG no framing; the isomorphism and the reconstruction are exact.
That makes a number equivalent to its tuple of remainders — a residue number system. And arithmetic splits perfectly across the channels: add or multiply the remainders independently, modulus by modulus, with no carries between them, and reconstruct at the end. Big multiplications become several tiny parallel ones.
LIT verified live: over 20,000 random coprime-modulus sets the reconstructed x satisfies every congruence, and a brute-force search confirms it is the only solution below the product (window.__crt.allCongruences && unique). mod [3,5,7] = [2,3,2] → 23. FIG no framing; the isomorphism and the reconstruction are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in CHECKPOINT ZERO, beside THE TWINDRAGON and THE ZECKENDORF — the spawn domain of exotic ways to write a number. A residue number system is one more: a number stored as its shadow in several small clocks at once. AVAN (AI) built the instrument: the reconstruction, the uniqueness check, the residue torus.
The weave: David gathers the strange numeral systems; I make this one legible — the residue coordinates in 1D, the reconstruction in 2D, the coprime torus in 3D. The sphere is the seam. Credit: Sun Zi (Sunzi Suanjing, ~3rd–5th c. CE); general method by Qin Jiushao (1247).
The weave: David gathers the strange numeral systems; I make this one legible — the residue coordinates in 1D, the reconstruction in 2D, the coprime torus in 3D. The sphere is the seam. Credit: Sun Zi (Sunzi Suanjing, ~3rd–5th c. CE); general method by Qin Jiushao (1247).
3 ONE DIMENSION
A number as residue coordinates: its position on several small clocks at once (mod 3, mod 5, mod 7). Coprime clocks never sync up until their product, so the combination of hands names exactly one number in that whole range.
4 TWO DIMENSIONS · INTERACTIVE
Dial the three remainders and watch the reconstruction pick out the single number 0–104 that matches all of them. Change any one clock and the answer jumps — every combination maps to its own unique number.
5 THREE DIMENSIONS + AVAN’S INVERSE
The residue grid for mod 3 × mod 5 as a turning torus — green, all 15 cells, each a different number 0–14.
AVAN’s addition (the inverse-companion): the magenta cell is your number, and the grid fills completely with no repeats — because 3 and 5 are coprime, the map from number to (mod 3, mod 5) is a perfect bijection. A single integer looks monolithic, but CRT reveals it is exactly a tuple of independent shadows, one per coprime modulus, and the whole is losslessly recoverable from the parts. The inverse of one big number is several small parallel ones that never interfere — the green is the full torus of possibilities, the magenta is the one point where all the clocks agree.
LIT Genuine Chinese Remainder Theorem (Sun Zi, ~3rd-5th c. CE; general method Qin Jiushao 1247). Verified live: over 20,000 random coprime-modulus sets the reconstructed x satisfies every congruence, and a brute-force search confirms it is the unique solution below the product (window.__crt.allCongruences && unique, both true). mod [3,5,7] = [2,3,2] -> 23. The isomorphism Z_M ≅ product of Z_mi and the reconstruction are exact.
FIG No metaphor is doing the work: the residue representation, the reconstruction, and the uniqueness are the theorem itself, checked. The 'clocks' and 'torus' are visual framings of the exact bijection Z_15 ≅ Z_3 x Z_5.
FIG No metaphor is doing the work: the residue representation, the reconstruction, and the uniqueness are the theorem itself, checked. The 'clocks' and 'torus' are visual framings of the exact bijection Z_15 ≅ Z_3 x Z_5.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of CHECKPOINT ZERO · David Lee Wise (ROOT0), with AVAN