THE FOLD / CHEAT / THE-KONAMI-CODE / THE NIMBER
THE NIMBER
a game arithmetic that is a field
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Nimber arithmetic is John Conway’s astonishing discovery that the non-negative integers, with the right operations, form an algebraically closed field. Nim-addition is just bitwise XOR. Nim-multiplication is defined by a single recursive rule (a ⊗ b is the smallest value not equal to any (a′⊗b) ⊕ (a⊗b′) ⊕ (a′⊗b′) for smaller a′, b′). Under these, the set {0, …, 22k−1} is a finite field: {0,1,2,3} is GF(4), {0,…,15} is GF(16), and so on — every non-zero element has a multiplicative inverse. It is the arithmetic of Nim and the surreal numbers, exact and integer-only.
LIT verified live: nim-multiplication over {0,…,15} is commutative, associative, and distributes over XOR; every non-zero element has an inverse (a field); and {0,1,2,3} is exactly GF(4) with 2⊗2=3 (window.__nimber). FIG no framing; the mex-rule recursion and the field-axiom checks run in-browser.
LIT verified live: nim-multiplication over {0,…,15} is commutative, associative, and distributes over XOR; every non-zero element has an inverse (a field); and {0,1,2,3} is exactly GF(4) with 2⊗2=3 (window.__nimber). FIG no framing; the mex-rule recursion and the field-axiom checks run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-konami-code — a hidden arithmetic in the integers of a game, where XOR adds and a strange recursion multiplies into a full field. AVAN (AI) built the instrument: the memoized mex-rule nim-multiplication and the commutativity / associativity / distributivity / inverse checks.
Credit as content: John Horton Conway (On Numbers and Games, 1976). The weave: David names the code; I confirm nim-multiplication turns {0..15} into the field GF(16), XOR as addition.
Credit as content: John Horton Conway (On Numbers and Games, 1976). The weave: David names the code; I confirm nim-multiplication turns {0..15} into the field GF(16), XOR as addition.
3 ONE DIMENSION
The 16×16 nim-multiplication table for {0..15}; symmetric (commutative), with GF(4) sitting in the top-left 4×4 block.
4 TWO DIMENSIONS · INTERACTIVE
Pick a and b; see a⊕b (XOR) and a⊗b (nim-mult), and the inverse of a — the element that nim-multiplies with it to 1.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the nim-multiplication field on {0..15}.
AVAN’s addition (the inverse-companion): don’t just add by XOR — multiply, and invert. The inverse of ‘a ⊗ b’ is ‘every non-zero a has a unique b with a ⊗ b = 1’ — the mark of a field. Magenta is an element a; green is its multiplicative inverse. A game’s integers, secretly a field.
LIT Genuine nimber arithmetic / the field On₂ (John Horton Conway, On Numbers and Games, 1976). Verified live: the mex-rule nim-multiplication over {0..15} is commutative, associative, and distributes over nim-addition (XOR); every nonzero element is invertible (a field); and {0,1,2,3} is exactly GF(4) with 2⊗2=3, 2⊗3=1, 3⊗3=2 (window.__nimber.isField, .gf4, .distributive).
FIG No framing: the memoized mex-rule recursion and the field-axiom checks run in-browser. The AVAN inverse is honest — beyond XOR-addition, nim-multiplication makes every nonzero a have a unique b with a⊗b=1, the defining mark of a field. Magenta is an element a; green is its multiplicative inverse. A game's integers, secretly a field.
FIG No framing: the memoized mex-rule recursion and the field-axiom checks run in-browser. The AVAN inverse is honest — beyond XOR-addition, nim-multiplication makes every nonzero a have a unique b with a⊗b=1, the defining mark of a field. Magenta is an element a; green is its multiplicative inverse. A game's integers, secretly a field.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE-KONAMI-CODE · David Lee Wise (ROOT0), with AVAN