THE FOLD / GLITCH / HEISENBUG / THE PISOT
THE PISOT
powers that creep toward integers but never quite land
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A Pisot–Vijayaraghavan number is a real algebraic integer θ > 1 whose every Galois conjugate has absolute value strictly below 1. That single condition has a startling consequence: the powers θn creep arbitrarily close to whole numbers. The reason is exact — θn plus its conjugate powers is always an integer (a linear-recurrence term), and since the conjugates shrink, what is left over vanishes. The golden ratio φ is the classic case: φn + ψn = the Lucas number Ln, and |ψ| = 0.618, so φn races toward Ln. The smallest Pisot number of all is the plastic number ρ ≈ 1.3247.
LIT verified live: φn rounds to the Lucas number with distance exactly |ψ|n (dist(φ35) ≈ 7e-8); the silver ratio 1+√2 rounds to the Pell–Lucas number with distance |1-√2|n; and a non-Pisot algebraic integer (1+√13)/2, whose conjugate exceeds 1, keeps missing the integers (mean distance ≈ 0.26) (window.__pisot). FIG no framing; the companion recurrences (exact BigInt), the powers, and the nearest-integer distances all run in-browser.
LIT verified live: φn rounds to the Lucas number with distance exactly |ψ|n (dist(φ35) ≈ 7e-8); the silver ratio 1+√2 rounds to the Pell–Lucas number with distance |1-√2|n; and a non-Pisot algebraic integer (1+√13)/2, whose conjugate exceeds 1, keeps missing the integers (mean distance ≈ 0.26) (window.__pisot). FIG no framing; the companion recurrences (exact BigInt), the powers, and the nearest-integer distances all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at heisenbug — a bug that vanishes the closer you look: θn appears to be an integer, but the tiny discrepancy is real and only shrinks as n grows, never quite gone at any finite n. AVAN (AI) built the instrument: the companion recurrences (Lucas, Pell–Lucas), the nearest-integer distances, the |conjugate|n match, and the non-Pisot counter-example.
Credit as content: Charles Pisot & Tirukkannapuram Vijayaraghavan (1930s); Axel Thue and G. H. Hardy earlier. The weave: David names the heisenbug; I confirm the powers approach integers exactly when the conjugates lie inside the unit circle — and fail when one does not.
Credit as content: Charles Pisot & Tirukkannapuram Vijayaraghavan (1930s); Axel Thue and G. H. Hardy earlier. The weave: David names the heisenbug; I confirm the powers approach integers exactly when the conjugates lie inside the unit circle — and fail when one does not.
3 ONE DIMENSION
Distance of θⁿ to the nearest integer vs n: the Pisot numbers (green, cyan) collapse to 0; the non-Pisot (magenta) scatters.
4 TWO DIMENSIONS · INTERACTIVE
Cycle through θ; see θⁿ, its nearest integer (a companion recurrence), and the distance shrinking — or not, for the non-Pisot.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: θⁿ snapping onto the integer ladder.
AVAN’s addition (the inverse-companion): don’t watch θn — watch what it hides. The inverse of ‘θn approaches an integer’ is ‘its conjugate power ψn is the vanishing remainder that carries it there, shrinking geometrically.’ Magenta is that shrinking conjugate remainder; green is θn landing on the integer. The gap is a heisenbug — real, but gone in the limit.
LIT Genuine Pisot-Vijayaraghavan numbers (Charles Pisot & T. Vijayaraghavan, 1930s; earlier Thue, Hardy). Verified live with exact BigInt companion recurrences: φⁿ rounds to Lucas Lₙ with distance exactly |ψ|ⁿ (dist(φ³⁵)≈7e-8), the silver ratio 1+√2 rounds to Pell-Lucas Qₙ with distance |1-√2|ⁿ, and the non-Pisot (1+√13)/2 (conjugate |·|>1) keeps missing integers with mean distance ≈0.26 (window.__pisot.goldenRounds, .goldenDist, .silverRounds, .nonPisotStaysAway).
FIG No framing; the companion recurrences (Lucas, Pell-Lucas — exact BigInt), the powers, and the nearest-integer distances all run in-browser. Honest scope: the distance is positive at every finite n and only tends to 0 in the limit — never exactly reached. The AVAN inverse is honest — instead of watching θⁿ, watch its conjugate power ψⁿ, the vanishing remainder that carries θⁿ to the integer. Magenta is that shrinking remainder; green is θⁿ landing on the integer ladder.
FIG No framing; the companion recurrences (Lucas, Pell-Lucas — exact BigInt), the powers, and the nearest-integer distances all run in-browser. Honest scope: the distance is positive at every finite n and only tends to 0 in the limit — never exactly reached. The AVAN inverse is honest — instead of watching θⁿ, watch its conjugate power ψⁿ, the vanishing remainder that carries θⁿ to the integer. Magenta is that shrinking remainder; green is θⁿ landing on the integer ladder.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HEISENBUG · David Lee Wise (ROOT0), with AVAN