THE FOLD / GRIND / THE EPOCH / THE FAULHABER
THE FAULHABER
sum of p-th powers is one polynomial — via Bernoulli numbers
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Faulhaber’s formula says the sum of p-th powers 1ᵖ + 2ᵖ + … + nᵖ is always a polynomial in n of degree p+1, with coefficients built from the Bernoulli numbers. For p=1 it is n(n+1)/2; for p=2, n(n+1)(2n+1)/6; for p=3, [n(n+1)/2]² — so the sum of cubes equals the square of the sum (Nicomachus’s identity).
The Bernoulli numbers B₀=1, B₁=−1/2, B₂=1/6, B₄=−1/30, … (odd ones past B₁ vanish) are the same constants that appear in the tangent series, the values of the Riemann zeta function, and the Euler–Maclaurin formula.
LIT verified live with exact BigInt rationals: the closed forms for p=1,2,3 match the direct sum, S₃=(S₁)² exactly (Nicomachus), the Bernoulli numbers compute correctly, and the general Faulhaber–Bernoulli polynomial equals the direct sum for p=1…6 (window.__faulhaber). FIG no framing; exact rational arithmetic.
The Bernoulli numbers B₀=1, B₁=−1/2, B₂=1/6, B₄=−1/30, … (odd ones past B₁ vanish) are the same constants that appear in the tangent series, the values of the Riemann zeta function, and the Euler–Maclaurin formula.
LIT verified live with exact BigInt rationals: the closed forms for p=1,2,3 match the direct sum, S₃=(S₁)² exactly (Nicomachus), the Bernoulli numbers compute correctly, and the general Faulhaber–Bernoulli polynomial equals the direct sum for p=1…6 (window.__faulhaber). FIG no framing; exact rational arithmetic.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-epoch — the long summation over an age. Faulhaber’s formula is the epoch’s accountant: add up n powers and get one clean polynomial. AVAN (AI) built the instrument: the exact rational Bernoulli recurrence, the Faulhaber polynomial, and the check against the direct sum.
Credit as content: Johann Faulhaber (1631); the coefficients are the Bernoulli numbers of Jakob Bernoulli (Ars Conjectandi, 1713), who boasted of summing the tenth powers to 1000 in ‘less than half an hour.’ The weave: David names the epoch; I compute the Bernoulli numbers from scratch and show the discrete sum collapse into a single exact polynomial.
Credit as content: Johann Faulhaber (1631); the coefficients are the Bernoulli numbers of Jakob Bernoulli (Ars Conjectandi, 1713), who boasted of summing the tenth powers to 1000 in ‘less than half an hour.’ The weave: David names the epoch; I compute the Bernoulli numbers from scratch and show the discrete sum collapse into a single exact polynomial.
3 ONE DIMENSION
Nicomachus’s identity in blocks: the cubes 1³+2³+…+n³ tile exactly into a square of side 1+2+…+n. Sum of cubes = square of the sum, drawn.
4 TWO DIMENSIONS · INTERACTIVE
Pick a power p and a range n. The instrument adds the powers directly AND evaluates the Faulhaber–Bernoulli polynomial, confirming they agree exactly, and lists the Bernoulli numbers driving the coefficients.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the smooth polynomials S₁(n), S₂(n), S₃(n)… each of degree p+1, threading the partial sums.
AVAN’s addition (the inverse-companion): a discrete sum becomes a continuous polynomial. The sum Σ iᵖ is the discrete cousin of the integral ∫ xᵖ dx = xᵖ⁺¹/(p+1), and Faulhaber’s formula is exactly that integral plus Bernoulli-number corrections — the Euler–Maclaurin bridge between sums and integrals. The inverse of ‘add up n discrete terms’ is ‘evaluate one smooth polynomial,’ and the gap between the staircase and the smooth curve is measured, term by term, by the Bernoulli numbers. Magenta is the discrete staircase of partial sums; green is the polynomial threading its corners exactly. Bernoulli numbers are the precise dictionary translating summation into integration.
LIT Genuine Faulhaber's formula (Faulhaber 1631; Bernoulli numbers, Jakob Bernoulli 1713). Verified live with exact BigInt rational arithmetic: S1,S2,S3 closed forms match the direct sum for n=1..25, S3=(S1)^2 (Nicomachus), the Bernoulli recurrence yields 1,-1/2,1/6,0,-1/30,0,1/42,0,-1/30, and the Faulhaber-Bernoulli polynomial (with (-1)^j, B1=-1/2) equals the direct power-sum for p=1..6, n=1..15 (window.__faulhaber).
FIG No framing: the rational Bernoulli recurrence, the Faulhaber polynomial, and the direct-sum comparison run in-browser and are exact. The AVAN inverse is honest — the discrete sum is the integral x^(p+1)/(p+1) plus Bernoulli corrections (Euler-Maclaurin); magenta is the discrete staircase, green the smooth polynomial threading it.
FIG No framing: the rational Bernoulli recurrence, the Faulhaber polynomial, and the direct-sum comparison run in-browser and are exact. The AVAN inverse is honest — the discrete sum is the integral x^(p+1)/(p+1) plus Bernoulli corrections (Euler-Maclaurin); magenta is the discrete staircase, green the smooth polynomial threading it.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE EPOCH · David Lee Wise (ROOT0), with AVAN