THE FOLD / CHEAT / THE KONAMI CODE / THE DIFFERENCE SET
THE DIFFERENCE SET
a set whose differences hit every target the same number of times
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A cyclic difference set is a small set of residues D in Zv so perfectly arranged that every non-zero residue arises as a difference di - dj (mod v) the same number of times, λ. A (v, k, λ)-difference set of k elements generates a symmetric block design: rotate D through all v shifts and you get v blocks where every pair of points meets in exactly λ blocks. The set {0, 1, 3} mod 7 is the smallest example — its six differences are exactly 1, 2, 3, 4, 5, 6, each once — and it is nothing less than the Fano plane in disguise.
LIT verified live: for several classical difference sets — (7,3,1), (13,4,1), (21,5,1), and the (11,5,2) Paley set — every non-zero residue appears exactly λ times among the differences, and a non-example is correctly rejected (window.__diffset). FIG no framing; the full difference multiset and its uniformity check run in-browser.
LIT verified live: for several classical difference sets — (7,3,1), (13,4,1), (21,5,1), and the (11,5,2) Paley set — every non-zero residue appears exactly λ times among the differences, and a non-example is correctly rejected (window.__diffset). FIG no framing; the full difference multiset and its uniformity check run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-konami-code — a short secret sequence with perfect structure: a handful of residues whose differences unlock every target uniformly, a cheat pattern hiding a whole design. AVAN (AI) built the instrument: the difference multiset, the exactly-λ uniformity check, and a non-example rejection.
Credit as content: the theory of difference sets (Singer, 1938; Paley); the (7,3,1) set is the Fano plane. The weave: David names the konami code; I confirm each set’s differences cover every residue exactly λ times.
Credit as content: the theory of difference sets (Singer, 1938; Paley); the (7,3,1) set is the Fano plane. The weave: David names the konami code; I confirm each set’s differences cover every residue exactly λ times.
3 ONE DIMENSION
The residues of Z_v on a circle; the difference-set points are lit, and every arc-difference is drawn — each residue hit λ times.
4 TWO DIMENSIONS · INTERACTIVE
Cycle the classical difference sets; the histogram of differences is shown — every non-zero residue exactly λ.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the k residues of the difference set on the cycle.
AVAN’s addition (the inverse-companion): don’t list the points — list their differences. The inverse of ‘a set of k residues’ is ‘the multiset of its pairwise differences’, and a difference set is exactly the set whose differences are perfectly uniform. Magenta are the difference vectors covering the circle; green are the chosen residues. A design hidden in the gaps between points.
LIT Genuine cyclic (v,k,λ)-difference sets (Singer, 1938; Paley construction; the (7,3,1) set is the Fano plane). Verified live: for (7,3,1), (13,4,1), (21,5,1) and the (11,5,2) Paley set every non-zero residue appears exactly λ times among the pairwise differences, and the non-example {0,1,2} mod 7 is correctly rejected (window.__diffset.allValid, .nonExampleRejected).
FIG No framing; the full difference multiset and its uniformity check run in-browser. The AVAN inverse is honest — instead of listing the k residues, list the multiset of their pairwise differences: a difference set is exactly the set whose differences are perfectly uniform. Magenta are the difference vectors covering the circle; green are the chosen residues. A design hidden in the gaps between points.
FIG No framing; the full difference multiset and its uniformity check run in-browser. The AVAN inverse is honest — instead of listing the k residues, list the multiset of their pairwise differences: a difference set is exactly the set whose differences are perfectly uniform. Magenta are the difference vectors covering the circle; green are the chosen residues. A design hidden in the gaps between points.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE KONAMI CODE · David Lee Wise (ROOT0), with AVAN