#1131: the four-node minimum, exactly
The target (erdosproblems.com/1131, OPEN). Given distinct nodes x1,...,xn in [-1,1], let lk be their Lagrange fundamental polynomials and put
I(x1,...,xn) = ∫-11∑k|lk(x)|2 dx.
Erdos asked for the minimum and, in particular, whether it is 2-(1+o(1))/n. We do not resolve that asymptotic question.
The exact four-node theorem. Let y0 be the unique root in (0,1) of
21y^3 + 9y^2 + 3y - 1 = 0.Then the unique minimizing four-node set, up to reordering, is
{-1, -√y0, √y0, 1}.
Numerically, y0=0.185472093653579..., √y0=0.430664711409676..., and the minimum is 1.710758107401145.... The minimum itself is the unique real root of
165375L^3 - 519750L^2 + 491040L - 146912 = 0.Why the certificate is global. In an orthonormal polynomial basis, let Q be the four-by-four evaluation matrix. The objective is exactly tr((QTQ)-1), so it diverges when nodes collide and a minimizer exists. The Fejer-Legendre nodes give 12/7<2.
For a fixed node set write S(x)=∑lk(x)2. Expanding all nodes about a center c gives the exact first variation
I'_c(1) = I - (1-c)S(1) - (1+c)S(-1).Outside the node hull, the fundamental polynomial belonging to the nearest extreme node has absolute value greater than one. Since the minimum is below two, the displayed variation lowers I unless both endpoints -1 and 1 are already nodes.
Write the remaining nodes as h-a and h+a. Exact integration factors the difference from the centered configuration as
I(h,a)-I(0,a) = 8h^2 P(a,h)/D(a,h).Here D is positive on 0<a<1, |h|<1-a. After setting z=h2/(1-a)2, all four Bernstein coefficients of P are strictly positive; two of the non-obvious factors become polynomials with strictly positive coefficients after the substitution t=a/(1-a). Thus h=0 is forced.
The remaining one-variable objective and derivative are
I(a) = 4(35a^6-28a^4+23a^2+2)
/ (105a^2(a-1)^2(a+1)^2),
I'(a) = -16(21a^6+9a^4+3a^2-1)
/ (105a^3(a-1)^3(a+1)^3).The critical cubic is strictly increasing in y=a2, changes sign on (0,1), and the objective diverges at both endpoints. This proves the claimed unique minimum. Elimination gives the cubic for L. Its discriminant, -19439210456064000000, is negative, so L is its only real root.
Prior work and claim boundary. Brutman and Toledano (1997) numerically computed high-precision minima for n=3,...,100. The four-node optimizer and minimum were therefore already known numerically. This entry contributes an independently checkable exact global certificate and algebraic characterization; it makes no claim of priority for the exact formulas. The publisher marks the article as open archive, but its complete body remained inaccessible during this audit, so absence of an earlier exact derivation has not been established.
Verification. The public exact checker starts from the Lagrange basis, performs every integration symbolically, verifies the endpoint variation, Bernstein identities, derivative, discriminants, root isolations, and resultant. Its release is closed-world and fail-closed: optimized Python, import shadowing, extra or missing files, symlinks, FIFOs, and a deliberately false formula with regenerated local hashes are all rejected. A separate frozen-byte audit rebuilt the deterministic archive and reproduced the exact and high-precision numerical transcripts.
Scope. This proves one finite case only. It does not decide the first-order asymptotic, validate the conditional 2026 preprint, or resolve Problem 1131. On 11 August 2026 the official page had no formal proof record or current-worker tag, but its comments already contained the 26 April asymptotic claimed disproof, the 27 April standard-check reply noting the earlier numerical evidence, and the pointer to the 1997 paper.