THE FOLD / GLITCH / UNDEFINED BEHAVIOR / THE P-ADIC
THE P-ADIC
a metric where big powers are small
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The p-adic numbers come from a strange way to measure size: a number is small when it is divisible by a high power of a prime p. The p-adic absolute value is |x|p = p−v, where v is how many times p divides x — so 1, p, p², p³… march toward zero. This metric is ultrametric: |x + y|p ≤ max(|x|p, |y|p), stronger than the ordinary triangle inequality. Numbers get infinite digit-strings running leftward, and famously …1111 = −1 in the 2-adics, because 2k−1 ≡ −1 for every k.
LIT verified live: over thousands of rationals the ultrametric inequality holds, any a/b with b coprime to p reconstructs from its p-adic digits mod pk, and the all-(p−1) digit string equals −1 p-adically (window.__padic). FIG no framing; the valuations, the digit reconstruction, and the identity all run in-browser.
LIT verified live: over thousands of rationals the ultrametric inequality holds, any a/b with b coprime to p reconstructs from its p-adic digits mod pk, and the all-(p−1) digit string equals −1 p-adically (window.__padic). FIG no framing; the valuations, the digit reconstruction, and the identity all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at undefined-behavior — a number system where “big” means small and counting up runs off to −1 is the mathematician’s undefined behaviour, perfectly consistent once you accept the rules. AVAN (AI) built the instrument: the p-adic valuation and absolute value, the digit expansion of a rational, and the ultrametric check.
Credit as content: Kurt Hensel (1897). The weave: David names the undefined behaviour; I confirm the ultrametric holds, that rationals reconstruct from their leftward digits, and that …1111 really is −1.
Credit as content: Kurt Hensel (1897). The weave: David names the undefined behaviour; I confirm the ultrametric holds, that rationals reconstruct from their leftward digits, and that …1111 really is −1.
3 ONE DIMENSION
The 2-adic string of −1: all 1s. Adding 1 carries forever and lands on 0 — so …1111 + 1 = 0, hence …1111 = −1.
4 TWO DIMENSIONS · INTERACTIVE
Pick a prime p and a rational a/b; see its p-adic digits and confirm they rebuild a/b mod pk. The p-adic size p−v shrinks as p divides more.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the p-adic sizes p−v descending toward zero.
AVAN’s addition (the inverse-companion): don’t measure distance the usual way — measure it by shared low digits. The inverse of ‘close means small difference’ is ‘close means divisible by a high power of p’, and that makes the numbers cluster into a tree. Magenta is the …1111 = −1 idea; green is the ultrametric tree of p-adic closeness. Big powers are small.
LIT Genuine p-adic numbers (Kurt Hensel, 1897): |x|_p=p^−v_p(x), ultrametric, …(p−1)(p−1)=−1. Verified live: ultrametric inequality over 3000 rationals (window.__padic.ultrametric), a/b reconstructs from its digits mod p^k (.reconstructs), and the all-(p−1) string equals −1 mod p^k (.minusOne).
FIG No framing: the p-adic valuations, the digit reconstruction (via modular inverse in BigInt), and the −1 identity all run in-browser. The AVAN inverse is honest — measuring closeness by shared low digits (divisibility by a high power of p) rather than by ordinary difference is exactly the ultrametric, and it clusters numbers into a tree; magenta is the …1111=−1 idea, green the ultrametric tree. Big powers are small.
FIG No framing: the p-adic valuations, the digit reconstruction (via modular inverse in BigInt), and the −1 identity all run in-browser. The AVAN inverse is honest — measuring closeness by shared low digits (divisibility by a high power of p) rather than by ordinary difference is exactly the ultrametric, and it clusters numbers into a tree; magenta is the …1111=−1 idea, green the ultrametric tree. Big powers are small.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of UNDEFINED BEHAVIOR · David Lee Wise (ROOT0), with AVAN