▲ 67 r/sportsanalytics+1 crossposts

Hydration Breaks Change the Flow of the Game - A statistical analysis of xG and more

I ran a statistical analysis to see the effect hydration breaks had on match momentum (via cG created). I'll go into more detail below, but summarize the findings right here:

TLDR: Match data suggests that hydration breaks create larger swings in xG advantage before versus after the hydration break in each half, comparing the 2026 world cup with 2018-2022. Total xG and shot activity around the first hydration break also swung more in 2026, and even total attacking activity, regardless of team, swung more across hydration breaks in 2026. Restricting to only group stage games or only knockout or round-of-16 games preserves the pattern, and so the new 48-team format is unlikely to blame for this shift.

Putting aside ad revenue for FIFA and hydration of the players, for teams the hydration break gave a timeout that allows for tactical adjustments / reset. We should expect the momentum and chance distribution to change as a result: Is a team that had lots of chances before the hydration break as likely to continue creating chances after the hydration break as it would have been without the hydration break? Does the hydration break help teams with no chances reorganize to create something? These would lead to swings in chance creation, meaning larger changes in team advantage in xG, shots, or goals. How can we measure such swings in chance creation? The best quantity to look at is the change in a team’s xG advantage from the “first quarter” (Q1), the period of the first half before the hydration break, to the “second quarter” (Q2), the period after it. We then make the same comparison between Q3 and Q4 in the second half. I beg forgiveness for my footballing sin of calling them quarters here.

https://preview.redd.it/yl3gcyd8d8jh1.png?width=2564&format=png&auto=webp&s=d517a6f6d5155c611a7518d5c138f7918cf5f49c

The above figure compares those xG-advantage changes with the equivalent changes in matches from previous World Cups without hydration breaks. There is a noticeable shift toward larger swings around the break timing in 2026. 2026 saw 20% of matches have an xG swing of 1 or more, which happened in only 11% of world cup matches in 2018-2022. The average absolute swing in xG advantage was 0.47 xG; in 2026 it was 0.56 xG.

Running a statistical analysis that asks, “Are these swings in the 2026 World Cup?” gives a p-value of 0.032. That tells us, roughly, that observing this kind of a signal only as a result of randomness (and not some real underlying effect) is only 3.2%; the hydration breaks seem to have a real, measurable effect. Statistical details, including assumptions, methodologies, and multiple-testing adjustment, are provided at the end of the article.

But hey! Maybe the new format is to blame, since the Round of 32 could just create matches with big chance-creation swings. We can rerun the same analysis on the group stage only (p = 0.050) or the Round of 16 onward only (p = 0.077, with far fewer matches), and the data still points in the same direction; see the figure below. The new format is unlikely to be the reason for these huge xG swings.

https://preview.redd.it/4r9ol1z9d8jh1.png?width=3519&format=png&auto=webp&s=054a55ff1c45c51d3f2c0643909f5491e57ede8c

Beyond which team holds the xG advantage, there also seems to be a change in overall attacking activity on both sides of the pitch. The plot below compares the absolute difference in total xG and total shots before and after the first hydration break, and the spread for 2026 is quite a bit wider. Average total-xG movement rises from 0.41 xG in 2018 and 2022 to 0.54 xG in 2026 (unadjusted p = 0.063). The corresponding average shot movement rises from 2.49 to 2.90 shots (unadjusted p = 0.053).

https://preview.redd.it/2layoi02e8jh1.png?width=3164&format=png&auto=webp&s=80506c49e1ff205c374f440ec4654e2b60d3f5fc

for more details, see the substack article here:

Do FIFA Hydration Breaks Change the Flow of the Game? Statistical Evidence of Chance Creation Swings

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u/pkerger — 6 days ago
▲ 689 r/OpenAI

I used 5.6 Sol Ultra to Close a 30-Year Open Gap in Mathematical Optimization Theory, following OpenAI's CDC Proof Prompt Methodology

TL;DR: In a single 148 min session, with a prompt modeled after the one OpenAI used to prove CDC, GPT 5.6 Sol PRO supplied a proof that closed a complexity gap in convex optimization that has existed since 1996. The result was formally verified in Lean. Links to everything are at the bottom of this post.

Note: I am the author of the preprint and Lean repository linked below. I have a PhD in applied mathematics and am a teaching prof in IEOR at UC Berkeley. The result has not yet been peer reviewed. I am happy to answer any questions or provide thoughts below. I've also given a slightly more technical summary and thoughts over in r/math, for those interested.

Edit: I can't edit my title, but this was Sol PRO, not Ultra. I had been working in codex before this, where the level above XHigh is Ultra. But I did this in the web interface, where the highest is Pro, which is in fact not quite the same as Ultra.

Following the recent announcement that GPT-5.6 Sol Pro had produced a proof of the Cycle Double Cover Conjecture, I adapted the prompting methodology used in that project to a problem in convex optimization. After 148 minutes of uninterrupted work, GPT-5.6 Sol Pro produced the main argument for a lower bound that I had been unable to prove myself (and a lot of my past work has been proving complexity lower bounds in different settings). My prompt is about ten pages long and attached at the end of the preprint (see collection of links below), and was also designed together with 5.6 Sol. There is a lot baked into this prompt, on approaches to try and also on how exactly the model should proceed, but it's built exactly in the style of OpenAI's CDC prompt. With it, 5.6 Sol Pro solved the problem it in one shot. After checking things myself, I formally verified the proof in Lean, and it passed the formal verification checks (for those unfamiliar, Lean is a programming language in which one can formalize and computationally verify mathematical statements). Regarding the problem, this is not some obscure problem that no one has attempted to solve or that has been forgotten over the years. I've thought about this problem on and off for about a year, lots and lots of related work exists from top researchers, the equivalent problems in related settings have been solved for a long time (some as far back as 1979), and I've heard an expert in the area say "we have no idea" about how to prove this result at an optimization conference just last year.

For mathematical and theoretical CS research, 5.6 Sol seems to be a huge improvement in its capabilities. We've seen OpenAI's proof on the CDC conjecture, and either I have been extremely lucky or the approach they used has serious potential to be successful on a lot of other open problems. I had tried using 5.4 and 5.5 on this problem after seeing folks like Ernest Ryu having success with them, but that went nowhere. I'll share here a chat for example, where I tried the approach that Sol 5.6 ended up using that worked in the end (excuse my shortness in my follow-ups there, I was trying to run deep dives on a couple different approaches in parallel, and was a bit frustrated against 5.5 at the time!).

Links:

An accessible account I wrote on Medium:

https://medium.com/@kerger.p/an-ai-assisted-breakthrough-in-convex-optimization-an-optimization-problem-dating-back-30-years-a-db5c631119de

The preprint, Lean code, complete prompts, proof map, and build instructions are available here:

https://github.com/PhillipKerger/zero-order-bounds-lean-verification

ArXiv preprint: Closing the Oracle-Complexity Gap in Derivative-Free Convex Optimization: A Near-Quadratic Lower Bound from Exact Function Values

The original uninterrupted 148-minute chat that produced the initial proof:

https://chatgpt.com/share/6a55aa50-b484-83ea-85c0-c7e7b4bda41c

The later chat that led to the d⁻¹ᐟ² refinement on accuracy requirements:

https://chatgpt.com/share/6a55ad10-7644-83ea-859e-5483d2e0dff0

OpenAI’s CDC prompt, that I structured things after:

https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_prompt.pdf

u/pkerger — 1 month ago
▲ 901 r/math+2 crossposts

After OpenAI’s CDC proof announcement, GPT-5.6 used a similar prompt to close a 30-year gap in convex optimization, verified in Lean

TL;DR: In a single 148 min session, with a prompt modeled after the one OpenAI used to prove CDC, GPT 5.6 Sol Pro supplied a proof that closed a complexity gap in convex optimization that has existed since 1996. The result was formally verified in Lean. Links to everything and thoughts on AI capabilities are at the bottom of this post.

Disclosure: I am the author of the preprint and Lean repository linked below. I have a PhD in applied mathematics and am a teaching prof in IEOR at UC Berkeley. The result has not yet been peer reviewed.

Following the recent announcement that GPT-5.6 Sol Pro had produced a proof of the Cycle Double Cover Conjecture, I adapted the prompting methodology used in that project to a problem in convex optimization. After 148 minutes of uninterrupted work, GPT-5.6 Sol Pro produced the main argument for a lower bound that I had been unable to prove myself (and a lot of my past work has been proving complexity lower bounds in different settings).

The problem concerns deterministic zeroth-order convex optimization: Let B_d be the Euclidean unit ball in ℝᵈ, and consider all convex, 1-Lipschitz functions f: B_d → ℝ. An algorithm may query any point x ∈ B_d, and receives only the exact real number f(x), no other information (but the algorithm "knows" that f is convex and Lipschitz). The algorithm is otherwise completely unrestricted, and can use unlimited computation and memory. These function-value-only problems arise naturally when an objective is evaluated through a physical experiment or simulator. One can imagine choosing d engineering parameters and observing only the cost returned by the simulation. If evaluations are expensive (think of measuring a physical system), the natural question is how many are fundamentally required. This is formalized as oracle complexity. Specifically, this is the oracle complexity of convex optimization under an exact function value oracle.

Let Q(d, ε) denote the worst-case number of queries required to find an ε-optimal point of f. An algorithm due to Protasov from 1996 shows that order d² function evaluations are sufficient, which gives Q(d, ε) = O(d²), an upper bound on the complexity. Lower bounds were practically nonexistent for this setting, and the strongest previously applicable bound was only Ω(d), inherited from the stronger first-order oracle model (where the algorithm receives both function values and gradients). That means we didn't know for certain whether gradients actually help in optimization, since the function-value only and first-order oracle models have had this same lower bound, and so there was a linear gap in d in the complexity of this fairly fundamental convex optimization setting since 1996. So, can you find an algoritm that is better than Prosatov’s, and only needs d evaluations? Or can you show that no such algorithm can exist, and we can sleep well at night knowing that Protasov’s algorithm using d² evaluations is best possible? What 5.6 Sol proved is the latter.

I had worked on this problem sporadically for about a year (I ran into needing such a bound for a different complexity paper I was working on). I had some ideas that didn't pan out, and also spent long sessions trying to solve it with GPT-5.4 and GPT-5.5 with no luck, after reading of folks like Ernest Ryu having success with these in some work on optimization bounds.

After seeing OpenAI’s CDC result, I wrote a much more elaborate prompt following the same general methodology. My prompt is about ten pages long and attached at the end of the preprint (see collection of links below). There is a lot baked into this prompt, on approaches to try and also on how exactly the model should proceed, but it's built exactly in the style of OpenAI's CDC prompt. One note is that I gave it a relatively small error requirement, to prove the quadratic lower bound under order d⁻⁴ accuracy. After 148 minutes, GPT-5.6 Sol Pro returned a proposed proof resolving the quadratic dimension dependence at accuracy of order d⁻³. After checking things myself, I formally verified the proof in Lean, and it passed the formal verification check. The construction and main invariant used also make genuine sense to me and are closely related to some other results in complexity of convex optimization (for example, Nemirovsky and Yudin's tight bound for first-order convex optimization also uses constructions that are maxes of affine functions).

Lastly, some important comments about the work relating to AI capabilities: In a lot of cases, proving lower bounds like this result relies on finding that right construction that works (in this case, family of difficult functions and a strategy for how an "adversarial" oracle should answer queries from an algorithm to reveal minimal information) and then proving things about it. There are only so many function classes which would be reasonable to look at (here, quadratics for example would have also been reasonable with order d² degrees of freedom, or any variation of maxes of some simpler families of convex functions as well), but the actual proof mechanics once the "correct" function class and correct strategy for adversarial oracle answers is found are often not so complicated, and often employ existing results from convex geometry or similar (this is also the structure of two previous but much more niche, less important results of mine). So I wouldn't really say that this result is using or creating some fundamentally new techniques in convex geometry or optimization theory. What this means from my perspective is that if a result is attainable with existing techniques, modern AI methods will be able to solve those problems. I don't think researchers in math/TCS will be made obsolete, but I think it will instead no longer make sense to work on any low-hanging, or even medium-hanging (you know what I mean) fruit. We'll be needed for problems where actual novel approaches are needed.

Links:

The preprint, Lean code, complete prompts, proof map, and build instructions are available here:

https://github.com/PhillipKerger/zero-order-bounds-lean-verification

ArXiv: Closing the Oracle-Complexity Gap in Derivative-Free Convex Optimization: A Near-Quadratic Lower Bound from Exact Function Values

The original uninterrupted 148-minute chat that produced the initial proof:

https://chatgpt.com/share/6a55aa50-b484-83ea-85c0-c7e7b4bda41c

The later chat that led to the d⁻¹ᐟ² refinement:

https://chatgpt.com/share/6a55ad10-7644-83ea-859e-5483d2e0dff0

OpenAI’s CDC prompt, that I structured things after:

https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_prompt.pdf

And a more accessible account I wrote on Medium:

https://medium.com/@kerger.p/an-ai-assisted-breakthrough-in-convex-optimization-an-optimization-problem-dating-back-30-years-a-db5c631119de

Edit: This was Sol PRO, not Ultra. I had been working in codex before this, where the level above XHigh is Ultra. But I did this in the web interface, where the highest is Pro, which is in fact not quite the same as Ultra.

u/InstitutionBuilder — 1 month ago