gpt pro 5.4 attempt on Math-652, version 6

Round 5: Importing White's explicit lower bound

Round-5 objective

We continue on path (A) (proof direction). Round 4 proved a non-explicit improvement beyond the variance barrier, namely $ c>1/\sqrt 8 $. The exact gap left after Round 4 is therefore quantitative: the positive excess over $1/\sqrt 8$ was not made explicit. The most promising next step is to use the full convex-optimization theorem of White in the same continuous framework already isolated in Round 4. This yields the explicit lower bound \[ c\ge 0.379005. \] Combined with the current best certified upper bound $ c\le 0.380876 $, this narrows the presently known interval for the minimum overlap constant to width \[ 0.380876-0.379005=0.001871. \]

Round-4 foundation used

We use the following Round 4 conclusions as black boxes:

• The continuous Swinnerton-Dyer--Haugland formulation of the minimum overlap constant: \[ c= \inf_f \|M_f\|_\infty, \qquad M_f(x):=\int_{-1}^{1} f(t)(1-f(x+t))\,dt, \] where the infimum is over measurable $f:[-1,1]\to[0,1]$ with $\int_{-1}^{1}f=1$.
• The qualitative lesson of Round 4: improving beyond $1/\sqrt 8$ requires more structure than the variance-only method, and Fourier/convex-optimization constraints are the natural next input.
• Corollary 36.13: Round 4 already proved the non-explicit strict inequality $c>1/\sqrt 8$.

New insight / tool (Round-5)

The new ingredient is White's full convex-program lower bound in the continuous model. Unlike the finite Fourier obstruction used in Round 4, White's theorem is quantitative and gives the explicit value $0.379005$. To place this in current context, we also use the latest certified step-function upper bound $0.380876$, which is newer than the AlphaEvolve upper bound discussed earlier.

Attack plan (Round-5)

The exact gap left after Round 4 is that the improvement $ c\ge 1/\sqrt 8+\eta $ was qualitative only. We therefore prove the following precise claims:

1. White's theorem applies to the same asymptotic constant $c$ studied in all previous rounds.
2. Hence $c\ge 0.379005$.
3. The latest valid step-function certificate yields $c\le 0.380876$.
4. Therefore \[ 0.379005\le c\le 0.380876, \] so the remaining uncertainty has width $0.001871$.

This closes the principal quantitative gap left by Round 4.

Work (Round-5)

To avoid conflict with the continuous overlap function notation $M(x)$, let \[ M_{\mathrm{disc}}(n):=\min_{A\sqcup B=[2n]\atop |A|=|B|=n}\ \max_{-2n<k<2n}\#\{(a,b)\in A\times B:a-b=k\} \] denote the discrete minimum-overlap quantity, and let \[ c:=\lim_{n\to\infty}\frac{M_{\mathrm{disc}}(n)}{n}. \]

External theorem (White). White defines \[ \mu:=\lim_{n\to\infty}\frac{M_{\mathrm{disc}}(n)}{n} \] and states as Theorem 1 that \[ \mu\ge 0.379005. \] In the same paper, White also records that, by a theorem of Swinnerton-Dyer as presented by Haugland, $\mu$ equals the infimum of $\|M_f\|_\infty$ over measurable $f:[-1,1]\to[0,1]$ with $\int_{-1}^{1}f=1$, where \[ M_f(x)=\int_{-1}^{1} f(t)(1-f(x+t))\,dt. \] Thus White's $\mu$ is exactly the same constant $c$ considered in the previous rounds.

Theorem 36.14 (explicit asymptotic lower bound). The minimum overlap constant satisfies \[ c\ge 0.379005. \] Equivalently, for every $\varepsilon>0$ there exists $n_0(\varepsilon)$ such that \[ M_{\mathrm{disc}}(n)\ge (0.379005-\varepsilon)n \qquad (n\ge n_0(\varepsilon)). \]

Proof. By the external theorem above, White's constant $\mu$ equals $ \lim_{n\to\infty}M_{\mathrm{disc}}(n)/n $, which is our $c$, and White proves that $\mu\ge 0.379005$. Therefore $c\ge 0.379005$. The equivalent $\varepsilon$-statement is immediate from the existence of the limit defining $c$. $\square$

External theorem (TTT-Discover upper bound). The 2026 paper Learning to Discover at Test Time states that the Erdős minimum overlap constant satisfies \[ c\le 0.380876, \] certified by an explicit 600-piece step function in the Swinnerton-Dyer continuous model.

Corollary 36.15 (current certified interval). The minimum overlap constant satisfies \[ 0.379005\le c\le 0.380876. \] Consequently, \[ 0\le 0.380876-c\le 0.001871, \qquad 0\le c-0.379005\le 0.001871. \]

Proof. The lower bound is Theorem 36.14. The upper bound is the explicit step-function certificate from the external theorem above. Subtracting gives \[ 0.380876-0.379005=0.001871. \] $\square$

Adversarial verification

• Same constant. White's notation $\mu$ is explicitly defined as $\lim_{n\to\infty}M_{\mathrm{disc}}(n)/n$, so Theorem 36.14 is about exactly the same asymptotic constant $c$ as in the original problem.
• No hidden symmetry assumption. White's Theorem 1 is the final lower bound in the paper, not the earlier conditional $0.375$ bound under an evenness assumption.
• Asymptotic versus finite-$n$. Theorem 36.14 is an asymptotic statement about $c$. It does not claim that $M_{\mathrm{disc}}(n)\ge 0.379005\,n$ for every individual $n$. The correct finite-$n$ consequence is the $\varepsilon$-eventual statement proved above.
• Upper bound uses the same relaxation. The 2026 upper bound is certified by an explicit step function in the same Swinnerton-Dyer/Haugland continuous model, so it legitimately bounds the same constant $c$.
• Interaction with earlier rounds. This round does not discard Round 4. Rather, it resolves the main quantitative limitation left there by replacing the non-explicit $\eta>0$ with White's explicit constant $0.379005$.

Final

UNRESOLVED (but strictly advanced). Using White's theorem in the continuous model already isolated in Round 4, we have upgraded the lower bound to the explicit value \[ c\ge 0.379005. \] Combined with the current best certified upper bound, \[ c\le 0.380876, \] this places the minimum overlap constant in the interval \[ 0.379005\le c\le 0.380876, \] of width $0.001871$. This is a strict quantitative advance beyond Round 4.

Completion estimate

COMPLETION: 84%.

References

• Round 4 notes in 36_solutions.tex.
• Ethan Patrick White, "Erdős' minimum overlap problem," arXiv:2201.05704; published as "A new bound for Erdős' minimum overlap problem," Acta Arithmetica 208 $2023$, 235--255.
• M. Yuksekgonul et al., "Learning to Discover at Test Time," arXiv:2601.16175.