Tao reformulated Erdős #1038 over discrete probability measures and asked for the supremum of |{U_μ < 0}|, conjecturing 2√2 — “this may be hard to prove completely.” For rational weights the question is about root-constrained polynomials, decidable degree by degree. This machine decided the first seven degrees: the odd ones fall strictly below the conjecture, and the even ones pin its witness to within 2.3×10⁻⁵ of optimal.
Machine-derived, per-degree theorems for rational-weight measures — Tao's conjecture itself (all measures, all denominators) remains open, and the even-degree equality case is localized, not characterized. Every number on this page is a certified outward enclosure recomputed at this build; nothing is decided in floating point. Not peer-reviewed.
Erdős #1038 concerns how small the sublevel set of a polynomial with constrained roots can be. Its infimum side was resolved in 2026 by Darvas–Peng–Tao — this lab independently verified the computational fragment of that manuscript. In erdosproblems#179, Tao poses the other end: over discrete probability measures μ = Σ pᵢ δ_{aᵢ} on [−1,1], with logarithmic potential U_μ(x) = Σ pᵢ log|x − aᵢ|, how LARGE can |{x : U_μ(x) < 0}| be?
His conjecture: the supremum is 2√2, attained by the uniform measure on {−1, +1} — and “this may be hard to prove completely.” The thread had no answers when this campaign began.
So the conjecture restricted to rational weights of denominator N is a statement about degree-N polynomials — a finite-dimensional, certifiable object. That is the machine's opening.
The boundary of {|q| < 1} consists of roots of the integer polynomials A ∓ dᴺ (A the root-scaled form of q). Those are isolated and refined by the BigInt Sturm machinery of the trigmin instrument — the code calibrated on Mercer's closed forms in the λ(4) campaign — and each gap between boundary roots is decided by one exact rational comparison. The measure is summed outward: a true enclosure, never a float.
| polynomial | certified measure of {|q|<1} | exact value |
|---|---|---|
| x² − 1 | [2.828427124746, 2.828427124747] | 2√2 = 2.8284271247… |
| x² | [1.999999999999, 2.000000000000] | 2 |
| x | [2.000000000000, 2.000000000001] | 2 |
| (x−1)² | [2.000000000000, 2.000000000001] | 2 |
| (x²−1)² | [2.828427124745, 2.828427124746] | 2√2 again — even powers of the witness keep it |
The last row is why even degrees are different: powers of the witness keep its measure, so (x²−1)^{N/2} attains 2√2 at every even degree, and no even degree can fall strictly below the conjecture the way the odd ones do.
| degree | certified statement | boxes | depth | time | |
|---|---|---|---|---|---|
| N = 3 | every degree-3 polynomial stays below 2.82 < 2√2 — the whole degree falls strictly under the conjecture | 127 | 16 | 0.0 s | strict |
| N = 4 | the degree supremum lies in [2√2, 2.82845]; (x²−1)^2 attains the left end | 549 | 59 | 0.1 s | localized |
| N = 5 | every degree-5 polynomial stays below 2.82 < 2√2 — the whole degree falls strictly under the conjecture | 1,583 | 37 | 0.2 s | strict |
| N = 6 | the degree supremum lies in [2√2, 2.82845]; (x²−1)^3 attains the left end | 4,219 | 89 | 0.6 s | localized |
| N = 7 | every degree-7 polynomial stays below 2.82 < 2√2 — the whole degree falls strictly under the conjecture | 28,773 | 63 | 8.8 s | strict |
| N = 8 | the degree supremum lies in [2√2, 2.82845]; (x²−1)^4 attains the left end | 31,425 | 118 | 5.7 s | localized |
Each row quantifies over EVERY root configuration in [−1,1]ᴺ — collisions and multiplicities included — through the ordered-and-mirrored branch-and-bound: a box is closed when the certified measure of {Π dist(x, Iᵢ) < 1} falls below the threshold, and that bound equals the true measure on thin boxes (a battery check). In measure language: every discrete probability measure on [−1,1] whose weights have denominator 3, 5, or 7 satisfies |{U_μ < 0}| < 2.82; for denominators 4 and 6 and 8, the two-atom witness is within 2.3×10⁻⁵ of the best possible.
deg9: bnb: box budget exhausted — recorded as attempted and open, not silently dropped.