THE FOLD / CHEAT / THE ROOT KIT / THE AUTOMORPHIC
THE AUTOMORPHIC
a number whose square ends in itself
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Automorphic numbers are numbers whose square ends in the number itself. 5² = 25, 6² = 36, 25² = 625, 76² = 5776, 376² = 141376, 625² = 390625, 9376² = 87909376. For each number of digits d there are exactly two nontrivial ones — one ending in 5, one ending in 6 — and they always add up to 10d + 1 (25 + 76 = 101; 625 + 376 = 1001). They are the nontrivial idempotents of arithmetic mod 10d (solutions of x² ≡ x), built by the Chinese Remainder Theorem from the split 10d = 2d·5d. Extended leftward forever they become the two nonzero 10-adic idempotents.
LIT verified live: the two nontrivial idempotents mod 10d (for d = 1 to 12) each satisfy x² ≡ x, end in 5 and 6, and sum to 10d + 1; the known 5, 6, 25, 76, 376, 625, 9376, 90625 are all confirmed automorphic (window.__automorphic). FIG no framing; the idempotents are constructed by CRT and squared, all in-browser with exact BigInt.
LIT verified live: the two nontrivial idempotents mod 10d (for d = 1 to 12) each satisfy x² ≡ x, end in 5 and 6, and sum to 10d + 1; the known 5, 6, 25, 76, 376, 625, 9376, 90625 are all confirmed automorphic (window.__automorphic). FIG no framing; the idempotents are constructed by CRT and squared, all in-browser with exact BigInt.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-root-kit — the cheat: a number that reproduces itself in the tail of its own square, a self-installing fixed point. AVAN (AI) built the instrument: the CRT idempotents, the square-ends-in-itself check, and the 10d+1 pairing.
Credit as content: the classical theory of idempotents in Z/10d and the 10-adic integers. The weave: David names the self-reproducing cheat; I confirm x² ≡ x mod 10d for the two nontrivial idempotents.
Credit as content: the classical theory of idempotents in Z/10d and the 10-adic integers. The weave: David names the self-reproducing cheat; I confirm x² ≡ x mod 10d for the two nontrivial idempotents.
3 ONE DIMENSION
A number and its square, with the shared trailing digits highlighted — the square ends in the number.
4 TWO DIMENSIONS · INTERACTIVE
Grow the digit-length d; the two automorphic numbers are shown, each x²≡x, summing to 10^d+1.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the number reappearing in the tail of its own square.
AVAN’s addition (the inverse-companion): don’t square and read forward — look for the fixed point of squaring. The inverse of ‘x’ is ‘the idempotent x² ≡ x that grows one digit at a time’, and its partner completing it to 10d+1. Magenta are the two idempotents; green is the tail where the square reproduces the number. A number that is its own square’s ending.
LIT Genuine automorphic-number / idempotent theory in Z/10^d and the 10-adic integers. Verified live with exact BigInt: the two nontrivial idempotents mod 10^d (d=1..12) satisfy x²≡x, end in 5 and 6, and sum to 10^d+1; known automorphics 5,6,25,76,376,625,9376,90625 all confirmed (window.__automorphic.ok, .kok).
FIG No framing; the idempotents are constructed by CRT and squared, all in-browser with exact BigInt. The AVAN inverse is honest — instead of squaring and reading forward, look for the fixed point of squaring: the inverse of 'x' is 'the idempotent x²≡x that grows one digit at a time', and its partner completing it to 10^d+1. Magenta are the two idempotents; green is the tail where the square reproduces the number. A number that is its own square's ending.
FIG No framing; the idempotents are constructed by CRT and squared, all in-browser with exact BigInt. The AVAN inverse is honest — instead of squaring and reading forward, look for the fixed point of squaring: the inverse of 'x' is 'the idempotent x²≡x that grows one digit at a time', and its partner completing it to 10^d+1. Magenta are the two idempotents; green is the tail where the square reproduces the number. A number that is its own square's ending.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE ROOT KIT · David Lee Wise (ROOT0), with AVAN