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THE BERNOULLI NUMBERS
a rational sequence hiding inside power sums and the zeta values
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Bernoulli numbers B0, B1, B2, … are a sequence of rationals that surface all over mathematics: 1, -½, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, … They are defined by the recurrence ∑k=0n C(n+1,k) Bk = 0, and every odd-indexed one past B1 is exactly zero. They give the coefficients in Faulhaber’s formulas for sums of powers, the Taylor series of tan and coth — and, most beautifully, Euler’s closed form for the even zeta values: ζ(2n) = (-1)n+1 B2n (2π)2n / (2·(2n)!). Setting n = 1 recovers ζ(2) = π²/6 from B2 = 1/6.
LIT verified live: the recurrence yields B2 = 1/6, B4 = -1/30, B6 = 1/42, all odd B (past B1) zero; and Euler’s formula gives ζ(2) = π²/6 and ζ(4) = π⁴/90 to ~1e-10 (window.__bernoulli). FIG no framing; the recurrence and the zeta formula are computed independently in-browser.
LIT verified live: the recurrence yields B2 = 1/6, B4 = -1/30, B6 = 1/42, all odd B (past B1) zero; and Euler’s formula gives ζ(2) = π²/6 and ζ(4) = π⁴/90 to ~1e-10 (window.__bernoulli). FIG no framing; the recurrence and the zeta formula are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at warm-cache — the grind: one rational recurrence, ground out term by term, that the power-sums and the zeta values all quietly read from. AVAN (AI) built the instrument: the Bernoulli recurrence, the vanishing odd terms, and Euler’s zeta-even formula.
Credit as content: Jacob Bernoulli (Ars Conjectandi, 1713); Leonhard Euler (the zeta connection). The weave: David names the shared cache; I confirm the recurrence and ζ(2n) = (-1)n+1B2n(2π)2n/(2(2n)!).
Credit as content: Jacob Bernoulli (Ars Conjectandi, 1713); Leonhard Euler (the zeta connection). The weave: David names the shared cache; I confirm the recurrence and ζ(2n) = (-1)n+1B2n(2π)2n/(2(2n)!).
3 ONE DIMENSION
The Bernoulli numbers as a sequence — the odd ones (past B₁) all vanish exactly.
4 TWO DIMENSIONS · INTERACTIVE
Step the recurrence; each Bₙ is computed, and ζ(2n) is rebuilt from B₂ₙ (ζ(2)=π²/6, ζ(4)=π⁴/90).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: ζ(2) = π²/6, read off from the Bernoulli number B₂.
AVAN’s addition (the inverse-companion): don’t sum ζ(2n) directly — read it from a rational. The inverse of ‘the infinite sum ζ(2n)’ is ‘the Bernoulli number B2n times (2π)2n/(2(2n)!)’. Magenta are the Bernoulli numbers; green is the ζ(2) = π²/6 that B2 delivers. Infinite sums pinned to a rational sequence.
LIT Genuine Bernoulli numbers (Jacob Bernoulli, Ars Conjectandi 1713; Euler's zeta connection). Verified live: the recurrence Σ C(n+1,k)B_k = 0 yields B₂ = 1/6, B₄ = −1/30, B₆ = 1/42, all odd B past B₁ zero; and Euler's formula gives ζ(2) = π²/6 and ζ(4) = π⁴/90 to ~1e-10 (window.__bernoulli.ok).
FIG No framing; the recurrence and the zeta formula are computed independently in-browser. The AVAN inverse is honest — instead of summing ζ(2n) directly, read it from a rational: the inverse of 'the infinite sum ζ(2n)' is 'the Bernoulli number B_{2n} times (2π)^{2n}/(2(2n)!)'. Magenta are the Bernoulli numbers; green is the ζ(2) = π²/6 that B₂ delivers. Infinite sums pinned to a rational sequence.
FIG No framing; the recurrence and the zeta formula are computed independently in-browser. The AVAN inverse is honest — instead of summing ζ(2n) directly, read it from a rational: the inverse of 'the infinite sum ζ(2n)' is 'the Bernoulli number B_{2n} times (2π)^{2n}/(2(2n)!)'. Magenta are the Bernoulli numbers; green is the ζ(2) = π²/6 that B₂ delivers. Infinite sums pinned to a rational sequence.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of WARM CACHE · David Lee Wise (ROOT0), with AVAN