Level B · Reproducible Analysis P-constant-1d-the-signed-third-autocorrelation-constant

The signed third autocorrelation constant

C_1d is the largest constant for which one has max_t ∈ ℝ |∫_ℝ f(t-x) f(x) dx| ≥ C_1d (∫_-1/4^1/4 f(x) dx)^2 for all real-valued (not necessarily non-negative) f : ℝ → ℝ supported on [-1/4,1/4]. This is the maximum of lvert f\f rvert over t, for signed f.

Current state (tracked automatically)

Best known bounds for constant 1d: 1 ≤ C ≤ 1.457854884921989. Upper bound ([SCIEL2026], Step function with 800 equal cells on [-1/4,1/4], found by SCiEL, a recursive self-improvement (RSI) system built on cultural transmission, running an open lang), not yet independently verified; lower bound ((trivial), ( f)^2 = (f\*f) ≤ f\*f ≤ f\*f, since f\*f is supported on an interval of length 1).
The best independently verified upper bound is 1.4630715826075593 ([CERT2026], Step function with 300 equal cells on [-1/4,1/4] and signed cell values v_0,,v_299).

All tracked facts and sources

Fact	Value	Source
Best lower bound with an explicit value (constant 1d)	1 (trivial) — ( f)^2 = (f\f) ≤ f\f ≤ f\f, since f\f is supported on an interval of length 1.	Tao's optimization constants
Best upper bound with an explicit value (constant 1d)	1.457854884921989* [SCIEL2026] — Step function with 800 equal cells on [-1/4,1/4], found by SCiEL, a recursive self-improvement (RSI) system built on cultural transmission, running an open lang	Tao's optimization constants
Best upper bound with an explicit value, independently verified (constant 1d)	1.4630715826075593 [CERT2026] — Step function with 300 equal cells on [-1/4,1/4] and signed cell values v_0,,v_299.	Tao's optimization constants

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From the catalogue. Imported from Terence Tao and contributors (optimizationproblems repository) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-constant-1d-the-signed-third-autocorrelation-constant,
  title        = {The signed third autocorrelation constant},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/constant-1d-the-signed-third-autocorrelation-constant}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

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Current state

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The problem

Description of constant

$C_{1 d}$ is the largest constant for which one has $t \in R max \int_{R} f (t - x) f (x) d x \geq C_{1 d} (\int_{- 1/4}^{1/4} f (x) d x)^{2}$ for all real-valued (not necessarily non-negative) $f : R \to R$ supported on $[- 1/4, 1/4]$ .

This is the maximum of $∣ f \* f ∣$ over $t$ , for signed $f$ . It is not the quantity $∣ max_{t} f \* f ∣$ , whose records are smaller and which is a different constant; the two are sometimes conflated.

Known upper bounds

Bound	Reference	Comments
$1.4993$	[MV2010]	Cited as the previous bound for this variant in Problem 6.4 of [GGSWT2025]
$1.4688$	[GGSWT2025]	Problem 6.4 of [GGSWT2025], signed variant
$1.463071582607559 *$	[CERT2026]	Step function with $300$ equal cells on $[- 1/4, 1/4]$ and signed cell values $v_{0}, \dots, v_{299}$ . Exact value of $max_{t} ∣ f \* f ∣ / (\int f)^{2}$ is the rational number $2039224160683882663565441886346340189798400/1393796574908163689489385839036226787118881$ , which is $1.4630715826075593 \dots$ . Self-archived artifact; see the note on the asterisk below.
$1.460217284022874 *$	[STEPS2026]	Step function with $600$ equal cells on $[- 1/4, 1/4]$ , obtained by refining the [CERT2026] function to $600$ cells and running a sequential linear-programming descent. Exact value of $max_{t} ∣ f \* f ∣ / (\int f)^{2}$ is the rational number $698746209786798359455483373677229695641600/478522078482569620433992386249161163190849$ , which is $1.4602172840228739 \dots$ , checked with the same knot formula as below. Self-archived artifact; see the note on the asterisk below.
$1.459989046905384 *$	[AER2026]	Step function with $600$ equal cells on $[- 1/4, 1/4]$ , labelled $C_{4}^{'}$ there. Its $max_{t} ∣ f \* f ∣ / (\int f)^{2}$ is $1.4599890469053831 \dots$ , computed with the same knot formula as below, whether the printed decimals are read exactly or as binary64. The bound $1.4554649$ for $C_{4}$ in the same source is for the other variant below. Self-archived artifact in an unmerged pull request; see the note on the asterisk below.
$1.457854884921989 *$	[SCIEL2026]	Step function with $800$ equal cells on $[- 1/4, 1/4]$ , found by SCiEL, a recursive self-improvement (RSI) system built on cultural transmission, running an open language model (Qwen3.8-27B) that writes and refines search programs scored by the exact bound. Exact value of $max_{t} ∣ f \* f ∣ / (\int f)^{2}$ is the rational number $38534420343892614563110610373646557184000/26432274393314964174848236787785831382961$ , which is $1.4578548849219886 \dots$ , checked with the same knot formula as below with $h = 1/1600$ ; the cell values are the binary64 numbers in `construction.json`, each read as an exact rational. Self-archived artifact; see the note on the asterisk below.
$1.4630715826075594$	[CERT2026]	Step function with $300$ equal cells on $[- 1/4, 1/4]$ and signed cell values $v_{0}, \dots, v_{299}$ . Exact value of $max_{t} ∣ f \* f ∣ / (\int f)^{2}$ is the rational number $2039224160683882663565441886346340189798400/1393796574908163689489385839036226787118881$ , which is $1.4630715826075593 \dots$ . Self-archived artifact. Independently replayed [T2026]: the same fraction in two implementations; it is $1.46307158260755932746 \dots$ , above $1.463071582607559$ (the bound first printed here), so the bound is printed rounded up.

Results should be arranged in chronological order, and can include bounds that are inferior to the previous bounds.

Known lower bounds

Bound	Reference	Comments
$1$	(trivial)	$(\int f)^{2} = \int (f \* f) \leq \int ∣ f \* f ∣ \leq max ∣ f \* f ∣$ , since $f \* f$ is supported on an interval of length $1$ .

Additional comments and links

How the upper bound is checked. For a step function on a uniform grid with cell width $h = 1/600$ , $f \* f$ is piecewise linear with knots at integer multiples of $h$ , so $max_{t} ∣ f \* f ∣$ is attained at a knot, where $(f \* f) (mh) = h \sum_{i + j = m - 1} v_{i} v_{j}$ . The ratio $max ∣ f \* f ∣ / (\int f)^{2}$ with $\int f = h \sum_{i} v_{i}$ is therefore computed exactly in rational arithmetic. The verifier uses only the Python standard library and runs in under a second: python -I verify.py construction.json prints $C (f) = 1.4630715826075593$ . The certificate and verifier are in the pull request description, with SHA-256 44f3f2b63b0d698dd5f279661d4be7077ca6703d62ea7ae6e2b2945e0f001e1c (construction.json) and 3e75c4ab0cdb40387765cbd086a572e46605f0c9fd8b8252783ed01aada80258 (verify.py).
Asterisk. The bound carries an asterisk under the convention in the README: the $300$ values are too many to check by hand and are available only as a self-archived artifact.
Other variant. For the non-negative-part variant $∣ max_{t} f \* f ∣$ the value $1.4523$ is reported in Appendix A.5 of arXiv:2606.10402; it is not a bound for the signed constant on this page.
How the upper bound is checked. For a step function on a uniform grid with cell width $h = 1/600$ , $f \* f$ is piecewise linear with knots at integer multiples of $h$ , so $max_{t} ∣ f \* f v er t$ is attained at a knot, where $(f \* f) (mh) = h \sum_{i + j = m - 1} v_{i} v_{j}$ . The ratio $max ∣ f \* f ∣/ (\int f)^{2}$ with $\int f = h \sum_{i} v_{i}$ is therefore computed exactly in rational arithmetic. The verifier uses only the Python standard library and runs in under a second: python -I verify.py construction.json prints $C (f) = 1.4630715826075593$ . The certificate and verifier are in the pull request description, with SHA-256 44f3f2b63b0d698dd5f279661d4be7077ca6703d62ea7ae6e2b2945e0f001e1c (construction.json) and 3e75c4ab0cdb40387765cbd086a572e46605f0c9fd8b8252783ed01aada80258 (verify.py).
Asterisk. The bound carried an asterisk under the convention in the README (the $300$ values are too many to check by hand and are available only as a self-archived artifact); it was removed on the independent replay [T2026].
Other variant. For the non-negative-part variant $∣ max_{t} f \* f v er t$ the value $1.4523$ is reported in Appendix A.5 of arXiv:2606.10402; it is not a bound for the signed constant on this page.
AlphaEvolve repository page for the related autocorrelation constants and the pages for 1a and 1b.
The file number is provisional; it was chosen as the first letter in family $1$ not claimed by an open pull request.

References

[MV2010] Matolcsi, Máté; Vinuesa, Carlos. Improved bounds on the supremum of autoconvolutions. J. Math. Anal. Appl. 372 (2010), no. 2, 439–447.
[GGSWT2025] Georgiev, Bogdan; Gómez-Serrano, Javier; Tao, Terence; Wagner, Adam Zsolt. Mathematical exploration and discovery at scale. arXiv:2511.02864
[T2026] Toledo, Carlos. Independent replay of the step-function certificates for $C_{1 c}$ [R2026] and $C_{1 d}$ [CERT2026]: the weights as exact rationals, every lag of the autocorrelation ( $C_{1 c}$ ) and of the autoconvolution ( $C_{1 d}$ ) summed exactly, in two implementations that share no code with each other or with the certificates' verifiers. Report (§7); code and record at commit d9a3942 (2026).
[CERT2026] Step-function certificate for the signed third autocorrelation inequality ( $300$ cells), construction.json and verify.py, 12 Oct 2026, in the pull request description.
[STEPS2026] Step-function certificate for the signed third autocorrelation inequality ( $600$ cells), construction.json (SHA-256 3dd39fae744555179b0356b84e752db724e892bd23dda51403976edfb5b3a28c) and verify.py, 21 Oct 2026, steps-re/c1d-certificate @ 9a09149.
[AER2026] Step function STEP_FUNCTION_C4_PRIME_N600 in experiments/autocorrelation_problems/problem_4_signed_autoconvolution_bounds.py (SHA-256 1a290ad1074c10a871adaf2bd44402af56be8cf151f1f775a8fc37c7bdff54b2), AlphaEvolve repository of problems, pull request 13 @ 4b3bbd2, 17 Oct 2026.
[SCIEL2026] SCiEL (CPIL lab, MBZUAI: Arina Kharlamova, Jikun Kang, Bowei He, Steve Liu). Step-function certificate for the signed third autocorrelation inequality ( $800$ cells), construction.json (SHA-256 1eafe2408187c95e36d2e33911fe3fd425f6c55af3837b40b30c139c771ffefb) and verify.py (SHA-256 cfa116ce4cb62a46f1292f740f3bdde99035918d897cd1d0a0f4966f67120925), 21 Oct 2026, certificate (Zenodo, restricted access: files on request); method in the SCiEL blog post.
[CERT2026] Step-function certificate for the signed third autocorrelation inequality ( $300$ cells), construction.json and verify.py, 12 Oct 2026, in the pull request description; also archived in the Extremal Values Registry (entry EV-000004), same $300$ values and the same verify.py; that copy of construction.json uses LF line endings (SHA-256 51f864ee95205f06b98923c382d23cb3dfde5f0d4208a4c591bad015e5908a0e).

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 87b4ebbd426f, Apache License 2.0; reformatted for this page. New records should also be reported there.