RESULTS: hunt `overlap_lower`

The ceiling procedure on the lower side of Erdos's minimum overlap constant. Run 2026-08-23.

Grades on every figure. VERIFIED means exact arithmetic with every irrational replaced by a directed-rounded rational, no floating point in the value. MEASURED means one floating-point route. INFERRED means taken from the literature or argued rather than computed here.

Status: partly settled, and the settled part is a negative

Settled: on White's simplified program the published parameter choice is already at the method's ceiling, and this hunt has the saturation curve that shows it. R = 20 is what White used; the value stops moving at R = 40 and the total remaining gain from R = 20 to R = 320 is 2.9e-6. The same shape appears in his full program, where R = 10 is what he used and the gain from R = 10 to R = 20 is 2.0e-5.

Not settled: the full program was not reproduced to its published value. The reconstruction lands 0.375 where White reports 0.379, and the reason is identified and stated below rather than absorbed: the Parseval cone cannot be enforced by cutting planes in the environment available, so what was swept is a strictly weaker program than his. Every number this hunt reports is still a valid lower bound; none of them is a good one.

No bound on C moves. The best exactly-accepted value obtained here is 0.37399241331, which is 0.00501 below the peer-reviewed lower bound 0.379005 and 0.00513 below the preprint record 0.37912: and which, being a section 4 value, bounds ||M||_inf for even M only and is not a bound on C at all. Everything here is a reproduction falling short.

A. The state of the art (INFERRED, every source read 2026-08-23)

Two independent literature passes, both reading the primary sources rather than a summary. The brief this hunt was opened from contained three errors and they are corrected here rather than quietly worked around.

Lower bounds on C

Upper bounds on C

A public leaderboard at einsteinarena.com/problems/erdos-min-overlap shows 0.3808586 from an agent submission with no paper behind it. That is a leaderboard entry, not literature, and it is not used as a reference value.

The catalogue, and one correction about it

github.com/teorth/optimizationproblems (Damek Davis, Paata Ivanisvili, Terence Tao). The entry is constants/1b.md, "Erdos minimum overlap constant", label C_1b, and it states the equivalence explicitly: "The problem of determining $C_{1b}$ is known to be equivalent to Erdős problem #36." Its README row reads

| 1b | Erdős minimum overlap constant | 0.379005 | 0.380868 |

The repository-wide disclaimer, verbatim except for one elision. [C-WORD] stands for the reserved word that zeta/rigor.py and the Lean arm own; this tree forbids that byte anywhere under hunts/, and the rule does not carve out quotations, so it is elided here and below rather than reproduced.

"IMPORTANT NOTE: while submissions to this site are reviewed to meet minimal standards of plausibility and replicability, they are not [C-WORD] by this site for correctness, and may be subject to future revision, for instance due to errors in the associated preprint or paper. Thus, readers should exercise their own judgement when assessing the validity of the bounds reported on this site, particularly if their source is not yet published by a peer-reviewed journal."

and the asterisk convention, verbatim:

"Bounds for which the level of available verification is currently at minimal levels will be marked with an asterisk in the table below."

Correction 1. The brief asked this hunt to verify that the minimum overlap problem is one of the asterisk-marked entries and to quote the marking. It is not marked. Neither 0.379005 nor 0.380868 carries an asterisk in the README table, and neither carries one on 1b.md. The general disclaimer above applies to every entry in the repository, including this one; the asterisk does not. Reporting the asterisk on this constant would have been wrong.

Correction 2. The brief states the lower bound "has not moved since 2022." It moved on 30 June 2026: Kim and Pilanci, arXiv:2606.31182, 0.379005 -> 0.37912. Their abstract, verbatim: "we improve the [C-WORD] lower bounds from 1.28 to 1.2937 and from 0.379005 to 0.37912, respectively." The catalogue has not absorbed it, the last content commit to 1b.md is 2026-04-30, two months before the preprint, which is why the catalogue still reads 0.379005 and why a reader trusting the catalogue alone would repeat the brief's error.

Correction 3. The 0.000059 figure is arithmetically exact but misattributed: it is 0.380927 - 0.380868, the 2016 human record minus the 2026 best published AI value, not twelve months of AI progress. The AI-only span is AlphaEvolve 0.380924 to the leaderboard's 0.3808586, which is 0.0000654 over about nine months.

Also worth stating. The brief's implied attribution of 0.379005 to Haugland is wrong: Haugland has only ever worked the upper side. And the 0.37912 improvement did not come from a bigger computation: Kim and Pilanci take White's program unchanged and add two Bochner semidefiniteness constraints on the Toeplitz moment matrices of f and g = 1 - f. New constraints, not a bigger N.

White's own answer to this hunt's question

Section 6, concluding remarks, verbatim:

"We expect that by computing more feasible solutions, using larger values of N, T, R, and using more accurate floating point calculations, that our lower bound of 0.379005 can be improved. Using the input N, T, R = 25000, 7000, 10 and h_1 = h_2 = 0.015, p_1 = p_2 = 0.381, -q_1 = q_2 = 0.02, we were able to find a feasible point with objective 0.37905, but not with objective 0.3791. This seems to indicate that the limit of this method is not much larger than 0.379."

and, for the simplified program of his section 4:

"By increasing N and R this bound can be improved a little, but it seems that the limit of this approach is less than 0.3755."

Both sentences are the author's inference from a single pair of solves, and neither is a sweep. That is precisely the gap this hunt exists to fill.

B. What hunt #85 settled about each side, precisely

hunts/r_828c8b/ (hunt #85, same day, same constant) settled:

• The upper side, reproduced. The m-piece step-function minimax falls 0.400000 / 0.385072 / 0.381833 / 0.381084 at m = 4/8/16/32 and stalls; the stall is the solver's, not the family's, because an m-piece step function upsamples to a feasible 2m-piece one and Haugland's object sits strictly below. Best exactly-evaluated object C <= 9990167/26214400, weaker than Haugland's.
• Three lower-side routes killed, with witnesses. Constructions bound only from above; the naive Fourier handling of the quadratic term is vacuous for every admissible weight because what(0) = 1 forces sup what = 1; and single-weight averaging is capped at 0.2526 by explicit witnesses.

It did not settle:

• It did not reproduce 0.379005, did not read White's discretisation, and did not check whether the published dual point is verified or merely reported. Its own words: "this hunt did not reproduce 0.379005."
• Its first loose thread asked whether the discretisation is sound at all, since restricting f to step functions bounds the infimum from above, in the unsafe direction.

That thread is now closed, and the answer is that the worry does not apply. White discretises M, never f. His variables w_j, v_j are cell averages of the true M, and every constraint is a necessary condition those averages satisfy, so each admissible f maps to a feasible point with Omega = ||M||_inf. The program is a relaxation, its optimum is at most ||M||_inf, and the direction is the safe one. His Proposition 9 says so in one line: "the average of M(x) on an interval cannot exceed the maximum." The discretisation error is absorbed by the envelopes alpha^-_{j,m} = cos(pi m L (j-1/2)/2) - pi m L/4 and its three siblings, and the Fourier-tail error by the slack variables of (5.8), (5.9), which are defined to be the discarded tail and then bounded by his Lemma 4.

Hunt #85 also recorded a route this hunt can now mark dead with a reason: relaxing f to its cell masses alone. Given only F_i = int_{cell i} f, the tight upper bound on the quadratic term at grid shift k is sum_i min(F_i, F_{i-k}), which is attained at f == 1/2 and returns a bound of exactly 0. Cell masses are not enough information; that is why the real method carries Fourier coefficients.

C. The free parameter

White's section 5 program has three integer parameters and a six-number box: N (grid cells per half, L = 2/N), T (Fourier truncation index), R (number of Fourier modes used as constraints, 1 <= m <= 2R), and (h_1, h_2, p_1, p_2, q_1, q_2) bounding E(M), c_1, d_1. His headline run is N, T, R = 25000, 7000, 10, solved in CPLEX. The section 4 program has two, N and R, and his run is N, R = 80000, 20.

R is the one he never varied. It is 10 in every table of the full program and 20 in the simplified one. R is therefore the parameter this hunt sweeps.

D. The section 4 ceiling (MEASURED, and it saturates)

White's (4.1)-(4.4), reimplemented from the displayed constraints in program.py:simplified_lp. This program bounds ||M||_inf for even M only; it is not by itself a bound on C, and is not reported as one.

D1. Reproduction, sweeping N at R = 20

The successive gains are 6.41e-4 then 3.24e-4, a ratio of 0.505, so the discretisation error decays as C/N to the precision available. Richardson extrapolation on the last two rows puts the N -> infinity limit at 0.37528 (MEASURED, one float route, extrapolated). White's own value at N = 80000 is 0.375169, which sits between the measured N = 20000 row and that limit exactly as it should. The reimplementation is faithful.

D2. The ceiling in R, at N = 5000

The value stops moving at R = 40. Rows 80, 160 and 320 agree with row 40 to nine decimals, and the residual wobble at the tenth is solver noise, not signal, the optimum is provably non-decreasing in R, so a decrease of 1e-9 is a floating-point artefact and is reported as one.

The measurement that answers the question: the total gain available from R = 20, which is what White used, to R = 320 is 2.9e-6. His choice was already at the ceiling of the parameter, and the sentence "it seems that the limit of this approach is less than 0.3755" is confirmed, with a number: the limit in R is reached at R = 40, and what remains is entirely the 1/N discretisation, whose extrapolated value 0.37528 is indeed below 0.3755.

D3. Exact rational dual certificate (VERIFIED)

The dual of (4.1)-(4.4) is

maximize (N/4) lam - z/3 subject to sum_j y_j <= 1 lam <= y_j + sum_m a_{m,j} u_m + s_j z for all j y, u, z >= 0, lam free.

The float solver is used only to propose (u, z). Everything after that is fractions.Fraction: alpha^-_{j,2m} is replaced by a rational lower bound (float cosine, floor at denominator 10^7, minus one unit, minus the Lipschitz term as an exact rational), which enlarges the primal feasible set and so can only weaken the bound; theta_j is formed exactly; and the largest feasible lam is found exactly by sorting theta.

At N, R = 5000, 20:

So 0.37399241331 is a VERIFIED value for the section 4 program at those parameters, with no floating point in the number. It is 0.0050 below the published lower bound on C, and, because section 4 assumes M even, it is not a bound on C at all. It is here to show that the acceptance step on this side is cheap and complete, which is the thing hunt #85 could not check.

E. The section 5 ceiling (MEASURED, and the reproduction falls short)

White's (5.1)-(5.13), reimplemented in program.py:WhiteProgram. Two departures, both stated before the numbers, both in the direction that can only lower the value:

1. The two quadratic families are outer-approximated. White's program is a second-order cone program; the environment has numpy and scipy and no conic solver, so (5.5) and (5.11) are approximated from outside by supporting hyperplanes added at the current optimum. An outer approximation contains the true feasible set, so the LP optimum is at most the SOCP optimum and every value below remains a valid lower bound.
2. (5.6) and (5.7) use the factor 4, not the 8 printed in the paper. White's Lemma 3 (3.6) gives B_m = -(4/(m pi)) sin(m pi/2) b_m and Lemma 5 sandwiches that same B_m, so 4 is what the sandwich needs and the printed 8 is a factor-of-two typo. Using 8 makes the constraint tighter than the mathematics licenses, which is the unsafe direction. Measured cost of the choice at N, T, R = 1000, 300, 10: 0.37212920 with 4 against 0.37212751 with 8, a difference of 1.7e-6. The typo is real and numerically almost irrelevant, which is worth saying in both halves.

E1. Sweeping N at T = 300, R = 10, at White's binding box cell

Box h_1 = h_2 = 0.015, p_1 = p_2 = 0.381, q_1 = -q_2 = -0.02, which is the cell his section 6 names.

Gains 2.81e-3, 2.50e-3, 2.94e-3, not decaying, so N = 2000 is nowhere near the N limit and no extrapolation is offered. N = 4000 and above did not fit the compute budget; see "What was not done".

E2. The ceiling in R, at N = 1000

The same shape as section 4, at the same place White stopped. The gain from R = 10, his choice in every table, to R = 20 is 2.0e-5. For scale: the entire improvement Kim and Pilanci obtained over White is 1.15e-4, about six times larger, and they got it by adding constraints rather than by raising a parameter. The R = 40 row is below the R = 20 row, which is impossible for the real program, adding constraints cannot lower a minimum, so it is reported as an unconverged loop, not as a measurement.

This non-monotonicity is worth one sentence because it was the bug that mattered. The cut loop originally stopped when the objective stalled, and that produced R = 5 -> 0.37433, R = 10 -> 0.37246, R = 20 -> 0.37504 at N = 2000: a value that fell when a constraint was added. Monotonicity in R is provable, so the stall was the loop's and not the program's, and the stopping test was changed to the residual of the true quadratic constraint. A ceiling procedure that cannot tell "the parameterisation ran out" from "the search ran out" measures the search, and this is the second time in two days that this family has produced that exact confusion (hunt #85, front B).

E3. T does not matter here, and that is a defect not a finding

T = 100, 300, 1000, 3000, 7000 at N = 1000 all return 0.3711470809943306 to sixteen digits. The tail slacks (5.8), (5.9) never bind, so T is inert in this reconstruction.

That is a symptom of the real problem. The Parseval cone (5.11) is not enforced. Cut-planing sum_k (c_k^2 + d_k^2) <= 1/2 one hyperplane per round in 2T = 600 dimensions leaves the ball violated by more than 200 after thirty rounds, and the program silently degenerates into one where a_m and b_m are free. Three reformulations were tried:

The last form is what the numbers above use. It is the only one in which the cosine constraints converge at all, and the ball is still open. So the figures in E1 and E2 are the ceiling of a program strictly weaker than White's, and the 0.0040 gap between the N = 2000 row and his 0.37905 is mostly this, not discretisation. They remain valid lower bounds, a supporting hyperplane never removes a feasible point, and they remain bad ones.

E4. The box grid (MEASURED)

White's residual region (5.16) is 0 <= E(M) <= 0.06, 0.35 <= c_1 <= 0.45, -0.02 <= d_1 <= 0.02. His tables cover it with degenerate points plus his Lemma 10, which spreads one dual certificate over an ellipse in (h, p). A grid of genuine cells needs no reuse argument, so that is what was run, at N, T, R = 500, 300, 10:

Minimum over the grid 0.36762389, at p in [0.383, 0.417], adjacent to, not identical with, the p = 0.381 ... 0.385 where White concentrated. The spread across the whole region is 2.3e-3, so at this N the box is a modest effect and the binding cell is not sharply localised.

F. The guard, and what it misses (MEASURED)

Every value in fronts D and E rests on the envelope alpha^-_{j,m} = cos(pi m L (j-1/2)/2) - pi m L/4 being a genuine lower bound on the trig weight over the cell. Drop it and the program stops being a relaxation, which is the failure that manufactures a bound that is not there. Faults were planted in the section 4 builder and checked against monotonicity in N and in R, both of which are provable properties of the real program. At N, R = 2000, 10:

One of two unsafe faults caught, and the miss is the important one. Monotonicity in N and R is blind to dropping the envelope, because the defective program is still monotone in both, it is simply a different, wrong program. The fault is worth 2.84e-3, which is larger than the entire gap between Moser's 1959 bound and nothing, and it moves the value the flattering way.

What would catch it is the exact certificate of front D3, because the directed rounding there is applied to alpha^- itself and a builder that omitted the envelope would produce a rational lower bound that no longer bounds the real coefficient. Stated plainly: on this program the guard is the exact acceptance step, and monotonicity is not a substitute for it.

G. Does the published lower bound move? No.

The gap from this hunt's best to the peer-reviewed record is 3.9e-3, and the reason is front E3, not the parameter sweep. No new bound on C is claimed and none is offered.

The finding that is offered is the one the brief said would be publishable if true, and it appears to be true in the parameter this hunt could actually sweep:

White's choice of R was already at his method's ceiling in both of his programs. In the simplified program, R = 20 is his and the total gain to R = 320 is 2.9e-6. In the full program, R = 10 is his and the gain to R = 20 is 2.0e-5, against the 1.15e-4 that the next real improvement obtained by adding constraints instead.

That is consistent with his own sentence, with Kim and Pilanci's route, and with the shape this lab has now seen six times: a published constant sits at the ceiling of its parameterisation, and the next move is a different parameterisation, not a bigger one. The caveat that keeps this a probe and not a result: it was measured on a weakened form of his full program, at N = 1000 rather than 25000, and the R-saturation of the faithful SOCP was not observed.

What was not done, and why

• The full program was not run at White's parameters. N, T, R = 25000, 7000, 10 needs a conic solver and roughly a hundredfold more compute than was available. The machine this hunt ran on is a 13-inch laptop that had already crashed under this lab's own compute that day, so nothing over about a minute of CPU was run on it.
• The sweep did not run on CI, though it was written for it. The workflow is hunts/overlap_lower/ci-sweep.yml, complete and shardable at 25 minutes per job. It is not installed under .github/workflows/ because the credential available refuses to create or update a workflow file without the workflow scope. Moving that one file into place and pushing is all that is needed; the sweep it defines reaches N = 25000 for the full program and N = 80000 for the simplified one.
• The Parseval cone was not enforced. Four formulations were tried (front E3); the best of them still leaves the ball violated. Closing it needs a second-order cone solver, which means a dependency this repository does not carry.
• Kim and Pilanci's two Bochner constraints were not implemented. They are semidefinite constraints on a Toeplitz moment matrix, which the LP route here cannot express at all. This is the single highest-value next step and it is recorded as a thread rather than attempted.
• No exact certificate was produced for the section 5 program. The dual of a program with box-constrained variables on both sides is materially messier than the section 4 dual, and with the primal value 0.004 below the published bound there was nothing worth certifying. The section 4 certificate exists to show the acceptance step works, not to claim anything.
• M(n) was not recomputed. Hunt #85 has it exactly for n <= 10 and this hunt has no reason to repeat it.

What this is not

Nothing here is evidence for or against RH (docs/08). No new bound on C is claimed. The reserved word that zeta/rigor.py and the Lean arm own is not used anywhere in this hunt.

Loose threads

• The Bochner constraints are the live route and they are two lines of mathematics. T_f >= 0 and I - T_f >= 0 where (T_f)_{kl} = (1/4) delta_{kl} + (1/2)(a_{|k-l|} - i sgn(k-l) b_{|k-l|}), valid because z* T_f z = int f |sum z_k e^{i k pi x}|^2 >= 0 and T_g = I - T_f. Why it might matter: it is the whole of the 0.000115 that separates the current record from the peer-reviewed one, and it is a constraint on objects this reconstruction already carries. First step: get a conic solver into the environment, since neither this constraint nor (5.11) survives a cutting-plane treatment.
• R saturation was measured on a weakened program. Why it might matter: the whole headline rests on it, and the Parseval cone is exactly the constraint that ties a_m for large m back to a finite budget, so it is the constraint most likely to change where R stops paying. First step: rerun front E2 with (5.11) enforced and see whether the R = 10 -> 20 gain is still 2e-5.
• The 1/N decay in section 4 is the envelope, not the grid. The gains halve exactly when N doubles, which is the signature of the pi m L/4 Lipschitz term rather than of the Riemann sum. Why it might matter: a tighter envelope, the exact extremum of cos(pi m x/2) over the cell rather than midpoint-plus-Lipschitz, costs nothing to compute and would move the whole N curve, which is a change to the method and therefore out of this hunt's scope but squarely inside the next one's. First step: replace alpha^- by the exact cell minimum and re-run front D1; if the N = 5000 row reaches what N = 20000 reaches now, the published N = 80000 was buying discretisation that better bookkeeping gives for free.
• The catalogue is two months stale on this constant. Why it might matter: this lab reads that catalogue as a frontier reference, and it silently disagreed with the literature by 0.000115 on the exact number this hunt was opened to move. First step: check the other constants this lab has hunted against their primary sources rather than against 1b.md's neighbours.