Did you know that...

• if you rent, you're paying the property owner's income and property taxes?
• if you buy food, you're paying a farmer's income and property taxes?
• if you fly, you're paying the airline's income taxes, payroll taxes, and property taxes? And also the airline employees' income taxes and payroll taxes?
• if you buy a product made outside the country, you're paying an importer's tariffs and customs duties? In addition to their income and property taxes?
• if you buy a product, you're paying the maker's income and property taxes?
• if you borrow money, you're paying your lender's income and property taxes?

It would be more accurate to say you're paying a portion of the other party's taxes for each of these items and not all their taxes, but you get the idea.

Exit question: Have you paid your fair share of their taxes?

Image credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon of a giant iceberg labeled 'Hidden Taxes.' Above the water, only a tiny tip is visible and labeled 'Sales Tax.' Below the waterline is a massive submerged iceberg labeled with: 'Business Income Tax,' 'Property Tax,' 'Payroll Tax,' 'Regulatory Costs,' 'Licensing Fees,' 'Import Duties,' and 'Fuel Taxes.' In the foreground, a family at a grocery checkout studies a receipt and notices only the small sales tax line. Arrows from the hidden underwater taxes flow upward into grocery price tags and consumer goods. Caption: The tax you see is often the smallest one you pay."

On 6 October 2026, OpenAI casually announced it was "releasing a broad range of new mathematical results produced by an internal frontier model", the same system the technology company used to produce its solution to the Navier-Stokes millennium challenge problem a month earlier.

"Broad range" is something of an understatement. The firm's still-internal AI system solved 376 open problems in several fields of mathematics, backed up by autoformalized proofs using the Lean proof assistant which it made publicly available on a GitHub repository.

On 11 May 1959, Eugene Wigner gave a lecture in which he described "the unreasonable effectiveness of mathematics in the natural sciences", pointing out mathematics was uniquely capable of describing observations in the physical world. So much so that abstract mathematical concepts developed without any connection to any physical science were proving capable of describing and predicting physical reality.

While not quite the same, what's happening now with the specially-trained artificial intelligence models making breakthroughs in solving so many open problems at such a torrid pace might be described as the unreasonable effectiveness of AI in mathematics. The rapid development is shaking up the discipline.

A Productivity Explosion

To show how much AI is rattling mathematicians, here's an updated version of a chart we first featured two months ago to illustrate AI's productivity explosion. It shows the number of math papers submitted to the arXiv repository each month from January 1992 through September 2026.

In December 2022, when OpenAI's ChatGPT was first made publicly accessible, arXiv reported some 3,122 math preprint papers had been posted in its database. In July 2026, arXiv reported the posting of 5,696 math papers during the month. For September 2026, arXiv reports 8,919 math papers were added to the archive.

That doesn't include any of the results OpenAI released in October 2026. When we next feature an updated version of this chart, we expect we may have to adjust the vertical scale because the number of newly math papers could exceed 10,000 per month. For its part, arXiv announced it would limit authors to posting no more than two papers per month, but the daily numbers we've seen suggest that is doing little to slow the pace.

Mathematicians Go Into John Henry Mode

In American folklore, John Henry is the story of a former slave who became a "steel driving man" building railroads after the Civil War. In the tale, John Henry took on new steam-powered technology in a contest to find out whether man or machine could do the job of driving steel faster. He won, but died from the exertion.

A team of mathematicians experienced their own version of John Henry's story in September 2026. Quanta Magazine sets the stage for their story:

On the morning of September 11, 2026, Dor Minzer received a text message from a friend asking whether he was close to settling one of the most famous open questions in theoretical computer science. Minzer, a professor at the Massachusetts Institute of Technology and a leading expert on the “unique games” conjecture, initially thought it was a joke. Then more messages started coming in, and the rumors began to cohere. The artificial intelligence company OpenAI, fresh from announcing a bombshell proof about the behavior of fluids that sent ripples through the math world, had allegedly discovered a proof of the unique games conjecture. They might make the result public any day. If Minzer had any results of his own to share, the messages suggested, now would be the time to do so.

The unique games conjecture is an iconic question in computational complexity theory, the study of the inherent difficulty of mathematical problems. Roughly speaking, it states that a problem about satisfying multiple constraints at the same time can be extremely hard, even if you’re willing to settle for a poor approximation of the best possible solution. A proof of the conjecture would also automatically imply that current methods for solving many seemingly unrelated problems can’t be improved. Researchers would be a big step closer to a unified theory of computational difficulty.

Minzer had not, in fact, proved the unique games conjecture. But he and his graduate students Yumou Fei and Shuo Wang had recently proved a milestone result about a closely related question. They were in the midst of writing up their findings — a process that would typically take months or longer — when the rumors began circulating. The trio rushed to finish their draft to avoid being overshadowed by an OpenAI press release. Three days later, prioritizing completeness over clarity, they posted a 95-page paper online with a disclaimer on the first page: “The current version of the manuscript is complete mathematically, but it is not in the shape we wished to share in.”...

“From section 6 onward, there are literally no connecting words,” Minzer said — just an unbroken sequence of definitions and proofs of intermediate results. “We felt the need to apologize for that.”

It's not as tragic an ending as John Henry's story, yet it is telling all the same. For mathematicians who until recently haven't had any need for urgency in doing what they do, AI is pressing them to become more productive than they've ever been before.

How will they adapt? And for that matter, how many other professions will soon find themselves in a similar situation?

References

arXiv. Math Submissions. [Online Database]. Accessed 15 October 2026.

arXiv. Monthly Submissions. [Online Article and CSV Data]. Accessed 15 October 2026.

ScriptByAI. OpenAI & ChatGPT Timeline: GPT Release Dates to GPT-5.6 (2026). [Online Database]. Accessed 20 August 2026.

Image Credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon of a dumptruck labeled 'OpenAI' depositing hundreds of math papers in front of mathematicians who look on sadly and worry about their future".

The S&P 500 (Index: SPX) continued tracking along above its mean trendline with respect to its trailing year dividends per share in September 2026. The index stayed mostly within one standard deviation of its long-established central trend during the month.

Through the end of trading on Wednesday, 30 September 2026, the index' closing value of 7,651.54 per share is half a standard deviation above its mean trend and is slightly below its trailing 20-day moving average of 7,671.31. The S&P 500's trailing year dividends per share came in at 82.89 per share during the calendar month.

The following chart visualizes the relationship between the value of the S&P 500 and its underlying trailing year dividends per share from 29 December 2023 through 30 September 2026:

The S&P 500's current period of relative order with respect to the index' underlying trailing year dividends per share has been in place since the end of 2023. While we're using the kind of analysis that applies to standard normal distribution bell curves from statistics to describe the variation of stock prices during that now long-established trend, that variation is not really normal.

You can see that in the data, with a much higher than expected number of data points within one standard deviation of the central trendline than would be predicted using a normal distribution to quantify the variation. There's also a higher than expected number of data points falling more than three standard deviations away from it.

That's a characteristic of a Lévy alpha-stable distribution, which looks more like the red curve on our featured chart illustration than it does like the green Gaussian normal distribution. Which is to say that stock prices can have stable distributions about central trends, but aren't really normal!

Image Credit: Levy distribution probability distribution functions by PAR on Wikimedia Commons. Public domain image.

It finally happened. One of 2025's Thanksgiving Leftover stocks has been removed from the S&P 500 (Index: SPX).

That stock belongs to The Trade Desk (NASDAQ: TTD), which we previously described as the worst of 2025's Thanksgiving Leftover stocks. It was ejected from the S&P 500 earlier this month because of its ongoing dismal performance. The move came a day after the company announced it would lay off 15% of its staff as it goes through a restructuring.

The writing on the wall for the company was visible as early as February of this year. In August, it was clear that action was coming and that the stock would have further to fall after it did.

Flashing forward to last week, that latter reality began taking hold, as managers of funds that track the S&P 500 began clearing out TTD stock from their holdings.

Through the close of trading on Monday, 28 September 2026, the Trade Desk finds itself at the very bottom of our spaghetti chart tracking the relative movements of 2025's ten Thanksgiving Leftover stocks during the last ten months with respect to their value on the day after 2025's Thanksgiving holiday.

Also through 28 September 2026, we find that eight of the ten Thanksgiving Leftover stocks we've been following have dropped below the values recorded for each on 28 November 2025.

While The Trade Desk is arguably having the worst year of its existence, one other member of the S&P 500's Thanksgiving Leftovers of 2025 is starting to also break away from the pack for its disappointing performance in 2026. Lululemon Athletica (NASDAQ: LULU) plunged in September thanks to the worsening outlook for its apparel business.

Compared to our August 2026 snapshot, both the market cap-weighted and equal-weighted indices composed of the ten Thanksgiving Leftover Stocks dropped lower. The market cap-weighted version of the index dropped to its lowest level of the year, having lost 18.5% of its day-after-Thanksgiving Day 2025 level.

Here is the latest version of our chart tracking our two hypothetical indices and the S&P 500.

The market cap-weighted version of the Thanksgiving Leftover stock index was pulled lower as its three most heavily weighted components, Chipotle Mexican Grill (NYSE: CMG), Fiserv (NASDAQ: FISV), and Lululemon Athletica (NASDAQ: LULU), which together account for 51.7% of its value, all dropped during the past month.

There are only two months to go to find out how many of the Thanksgiving Leftover stocks will remain underwater with respect to where they were on the day after Thanksgiving 2025. Which ones do you suppose will rise above the waterline by the time the day after Thanksgiving Day 2026 rolls around?

Saudi Arabia's NEOM project ranks among the largest megaprojects ever conceived. And cancelled, far from ever being realized.

The project to construct a 500-meter tall and 170-kilometer long city of nine million people in a straight line across deserts, mountains, and coastal plains in the westernmost corner of Saudi Arabia was both visionary and ambitious. This City of Tomorrow however was undone by the unrealistic assumptions that its planners adopted to try to keep it alive long after its true costs could no longer be sustained. The following 11-minute video from the WSJ provides a good overview of the unrealistic assumptions that ultimately led to the project's suspension on 16 September 2025.

In the year since, billions of dollars worth of contracts to build the city have been cancelled. Officially, construction on the project has been put on hold until after 2030.

But in truth, the dream of the City of Tomorrow will remain just a dream. An aspiration forever out of reach.

Even so, some parts of the project will go forward to completion. Amazingly, there are worthwhile things that can still be salvaged from NEOM, which is not a total loss. It's those parts that we find interesting because they answer the question of what becomes of a City of Tomorrow after the dream is abandoned.

We've queued the next video to start with what will continue after NEOM's story ends.

These much smaller, yet still significant projects all share one thing in common. They can generate a realistic and positive return on investment, on their own merits, without needing to be an inseparable part of the original, centrally-planned concept to survive.

We find the tale of the aftermath more interesting than the story of NEOM itself because that's how real cities come about. Not as the result of a single grand plan, but as a hodgepodge of varied endeavors by different groups of people who figure out what their City of Tomorrow should be a little at a time, keeping what works and is worthwhile and abandoning what isn't.

NEOM is just the latest case study in a very long history lesson for such discarded megacity projects. If their history tells us anything, it won't be the last.

Previously on Political Calculations

• Linear Cities
• The Future of the Linear City

In cracking one of the biggest unresolved problems in mathematics, OpenAI launched a controversy over the information it used to train the developmental artificial intelligence system behind it, which raises questions over to whom credit for the accomplishment belongs. The controversy is overshadowing AI's more important contribution to the advancement of mathematics: its role in "autoformalizing" mathematical proofs in the Lean proof assistant.

Autoformalization refers to the automated formal verification of a mathematical proof utilizing proof assistant software. That matters because developing the computer code to verify a proof is often a time consuming and often tedious task for mathematicians.

As a general rule of thumb, it takes 40 hours of labor for a mathematician to formalize the equivalent of one page of an established mathematical proof presented in a textbook with all supporting material preceding it using the popular Lean proof assistant. John D. Cook, an applied mathematician and statistician who runs a consulting firm, estimates a research paper in mathematics takes about 20 times more effort to formalize into a Lean-verified proof, mainly because time needed to collect and validate all the supporting material for it, which may or may not be presented within the paper.

OpenAI's Navier-Stokes proof runs 166 pages. By Cook's back-of-the-envelope math, formalizing all the math needed to verify it could take as much as 132,800 hours for mathematicians to execute.

OpenAI autoformalized its Navier-Stokes proof in just 17 hours.

It's also not just OpenAI's technology making these kinds of advancements toward automating the most tedious and time consuming aspect of modern mathematics. On 4 September 2026, less than a week before OpenAI unveiled its Navier-Stokes proof, Anthropic announced its Claude AI system had successfully generated Lean-verified code for Andrew Wiles' proof of Fermat's Last Theorem (FLT). In doing so, Anthropic beat the mathematicians who had been working for years to formalize it by manually coding it in Lean.

Here's how Anthropic describes what its Claude AI system accomplished:

One way to check a proof’s correctness is to ask a computer to do it. Proof assistants like Lean verify the logic of a proof algorithmically, demonstrating its correctness beyond a doubt. The difficult part for humans is rewriting the proof so Lean can understand it. While a proof written for human readers will skip many obvious steps, Lean needs to see every step, no matter how trivial. Human proofs also build on centuries of published work, while a formalization starts from the tiny fraction of math that’s been formalized already.

For FLT, the formalization process was expected to take years. Just the blueprint the mathematical community has been using to describe the initial phase of the project runs to 86 pages.

Claude completed the proof in 11 days, producing computer-verifiable proofs of 30,300 theorems along the way (using 29,500 in the final proof). Dozens of Claude agents collaborated to define concepts, prove intermediate theorems, and use those theorems to prove ever harder statements. At 13 million lines of Lean code, Claude’s proof is over 5x the size of Mathlib, the principal community library of mathematical proofs this theorem builds on.

Here's some more back of the envelope math. If mathematicians were given $20 million and told to generate the Lean code to verify the Navier-Stokes proof using 132,800 hours of labor, no more and no less, assuming they got it done, they would effectively be paid a little over $150 for each hour of labor. That may even be close to what it would actually cost a single trained mathematician or coder per hour of their labor. Although assuming they worked 40 hours a week, 50 weeks a year, it would also take them nearly 64 years to perform the task.

How many trained mathematicians do you think it might take to accomplish the same task in 17 hours? Keep in mind that level of execution would also take extraordinary planning, preparation, coordination, and execution on their part to accomplish the task in that time. Do you think that extraordinary effort would cost more or less than $20 million?

Businesses around the world are already running numbers like these. The potential to realize massive savings in time and cost is why AI technology has sparked an investing and development boom in the last few years. It might have cost OpenAI $20 million to develop its AI systems to crack the Navier-Stokes equations including generating the Lean code to verify the proof, which perhaps appears excessive next to the Clay Mathematical Institute's $1 million prize for the achievement. But how much would the alternative of having an army of trained mathematicians to do the same job in the same time have cost?

Previously on Political Calculations

We've been covering developments toward the resolving the open question of when the Navier-Stokes equations describing fluid motion works and when it doesn't for some time. Here's our coverage in chronological order:

• The Biggest Math Story of 2017
• The Biggest Math Story of 2018
• Navier-Stokes and the Emergence of Order from Disorder
• What is Emergence?
• The Biggest Math Story of 2019
• A Proof for Batchelor's Law
• The Biggest Math Story of 2022
• Breakthrough in Unifying Math to Describe How Fluids Behave at Different Scales
• The Biggest Math Story of 2025
• AI Cracks the Navier-Stokes Millennium Problem, Part 1
• AI Cracks the Navier-Stokes Millennium Problem, Part 2

On 8 September 2026, OpenAI officially announced it had determined the Navier-Stokes differential equations for describing the motion of fluids can develop a "singularity", or "blow up" or "break" to use more expressive terms, while running in a finite period of time.

Here's how they described the open question about the math involved in the Navier-Stokes equations:

The Navier–Stokes equations use Newton’s second law of motion (“F=ma”) to describe how fluids move. Importantly, they treat a fluid as a continuous medium rather than tracking individual molecules. These equations are used for aircraft design, weather forecasting, and the study of blood flow.

A fundamental open question for these dynamical equations has been whether the continuum approximation of the fluid can break down. Specifically, can the Navier–Stokes equations for a three-dimensional incompressible fluid with constant density develop a “singularity,” even when the motion starts smoothly? Here, a singularity means the dynamics lead to speeds in the fluid growing without bound within a finite amount of time. The development of a singularity would have to happen despite the presence of viscosity, which tends to smooth out motion. Because a real fluid cannot move infinitely fast, this would mark a breakdown in how the equations model the fluid. To continue modeling the system, one would then need to track the behaviour of each particle individually.

The equations date to the nineteenth-century work of Claude-Louis Navier and George Gabriel Stokes. In 1934, Jean Leray proved that solutions exist in a generalized sense, but whether they always remain smooth became a central unanswered question. In 2000, the Clay Mathematics Institute named the Navier–Stokes existence and smoothness problem one of seven Millennium Prize Problems.

And here's their result:

Our system produced an analytical proof and a Lean formalization that an initially smooth fluid at rest can develop a singularity in a finite time. The fluid has a smooth force applied to it, and its energy remains finite through the entire dynamics, from rest to the formation of the singularity. This resolves the Navier–Stokes Millennium Prize problem by establishing statement “C” (and also “D”) in the official Millennium Prize formulation⁠.

The solution is a vortex, a spinning swirl of fluid, that spirals inward and gets increasingly elongated, like spaghetti. This central region shrinks while it speeds up in such a way that its energy still stays finite, as required by the laws of physics. The technical challenge is for the equations to develop the breakdown through the motion of the fluid itself, rather than, for example, us putting in an infinite force by hand. More mathematically, the terms in the Navier–Stokes equations that describe the motion—acceleration, pressure gradients, momentum transfer, viscosity—must both become big yet cancel in a precise way. This detailed balance leaves a smooth external force even as the velocity of the fluid grows without bound.

Assuming it holds, OpenAI's solution to the Navier-Stokes Millennium Prize challenge, for which it might win $1 million, will have cost the firm $20 million.

The Clay Mathematics Institute, which established the "Millennium Prize" challenges for solving seven long-standing open problems in mathematics back in 2000, issued a statement indicating the million dollar prize for cracking the Navier-Stokes millennium problem is now pending their official verification.

Controversy soon erupted because of questions about whose mathematical research OpenAI's still-internal AI system used to develop their proof the Navier-Stokes equations will not always work and that they can break down under specific circumstances. The following Code Report from Fireship provides an entertaining rundown of the conflict:

Terry Tao's blog post announcing the accomplishment and assigning credit as best as could be done on 7 September 2026 is here.

The controversy over credit has the potential to arrest the rapid progress in mathematics that is being boosted by AI technologies. That potential exists because of the AI systems' ability to rapidly absorb new developments by mathematicians and apply them to a multitude of other problems without really understanding the work. In doing so, the AI developers are creating a strong, adverse incentive for the mathematicians to hold back their work until it has been fully verified to ensure they retain credit for it.

On 11 September 2026, Tao signed onto a declaration with other Fields Medal-winners (the math equivalent of the Nobel prize for science disciplines) who are alarmed at the major disconnect they're seeing between mathematicians and AI developers, which they argue puts advances in mathematics at risk.

That's a shame because the controversy is obscuring one of the biggest math stories of the year, which points to the massive capabilties that AI technologies are unlocking to advance mathematics. We'll cover that part of the story in Part 2.

References

Tristan Buckmaster. Announcement of three results with Levent Alpöge on finite-time blowup with smooth forcing for incompressible porous media, for Boussinesq, and for 3d incompressible Euler. [Mastodon post]. 7 September 2026.

Terence Tao. Finite time blowup with smooth forcing term for the incompressible porous medium, Boussinesq, and incompressible Euler equations. [Online article]. 7 September 2026.

OpenAI. On the Navier-Stokes Millennium Prize Problem. [Online article]. 8 September 2026.

Complexity. Visualizing the OpenAI solution to the Navier Stokes Equation. [Online video]. 10 September 2026.

Clay Mathematics Institute. Navier-Stokes Announcement. [Online article]. 11 September 2026.

Math and AI. A Severe Misalignment of AI in Mathematics. [Online article]. 11 September 2026.

Previously on Political Calculations

We've been covering developments toward resolving the open question of when the Navier-Stokes equations describing fluid motion will work and when it won't for some time. Here's our coverage in chronological order, the articles marked with an asterisk are directly relevant to the resolution of the Clay Mathematical Institute's Navier-Stokes Millennium Problem:

• The Biggest Math Story of 2017*
• Navier-Stokes and the Emergence of Order from Disorder
• The Biggest Math Story of 2019
• A Proof for Batchelor's Law
• The Biggest Math Story of 2022*
• Breakthrough in Unifying Math to Describe How Fluids Behave at Different Scales
• The Biggest Math Story of 2025
• AI Cracks the Navier-Stokes Millennium Problem, Part 1*

There's an entire cottage industry that has sprung up for Americans thinking about how long should they wait to start taking Social Security's retirement benefits.

If you only look at size of your monthly benefit, the numbers seem to suggest waiting for as long as possible to start taking the benefit is the right path forward. If you compare the amount of your benefit with what you would get if you waited until you reached what Social Security considers "normal retirement age", which has been set at Age 67 for those born in 1960 or later, your initial monthly benefit will be reduced by 8% per each year early. For example, if you start drawing Social Security as early as possible at Age 62, your benefit will be 40% less than what it would be if you had waited until Age 67 to start.

Similarly, if you hold out longer until Age 70, the longest you can hold out, your initial benefit will be 24% higher than what it would be if you had started taking benefits three years earlier.

But whether waiting like that actually makes sense depends on more than just how old you will be when you start taking Social Security benefits. If you have health problems that might keep you from living longer, for example, taking benefits earlier might make a lot more sense for your situation. If however you have a reasonable expectatation you'll live much longer, waiting to be older before tapping Social Security could be more beneficial for you.

The answer to the question of when to start taking benefits can also depend upon whether you're married or single. If you're married, the right age for you to start pulling Social Security's pension benefits may be different from the optimal age for your spouse to maximize your total household benefit. So what's the right thing to do?

That topic was recently covered on the RetirementNerds podcast, in which host Erik Soderborg ran through a number of examples with financial planner Zacc Call. The following hour-long video provides one of the better overviews we've seen of the factors that can complicate the major life event of deciding when to start taking Social Security retirement benefits:

We came across this video while researching an upcoming article that we're still developing behind the scenes, in which we'll feature another video by the pair discussing a different retirement-related topic.

August 2026 was a good month overall for the Thanksgiving Leftover Stocks of 2025.

Compared to their July 2026 snapshot, both our hypothetical indices of the ten worst stocks within the S&P 500 (Index: SPX) during 2025 saw month-over-month gains. The market cap-weighted index of these stocks increased from 87.4% to 91.4% of its value on the day after Thanksgiving 2025, while the equal-weighted index grew more, rising from 88.5% to 94.7%.

Both these indices are still lagging behind the overall S&P 500 index. The benchmark index increased from 108.2% to 112.1% of its day-after-Thanksgiving Day 2025 value in the month from the July to August snapshots.

The following chart shows the performance of all three sets of stocks, with the two Thanksgiving Leftover stock indices continuing to lag behind the S&P 500 index by a wide margin.

It's worth noting why the equal-weighted version of the ten stock index is performing better than the market cap-weighted version. The largest component of the market-cap weighted index is Chipotle Mexican Grill (NYSE: CMG), which accounts for 23.5% of its value. Shortly after the July 2026 snapshot, Chipotle's stock plunged when jalapeno peppers served at the chain's restaurants in Minnesota were linked to an outbreak of salmonella.

Although Chipotle acted quickly to pull all potentially affected jalapenos from its restaurants, investors sent its stock down sharply, losing nearly 16% of its value in a week. Since then, Chipotle's stock has largely recovered to its pre-jalapeno recall level.

That recovery however lagged behind the improvement of several other Thanksgiving Leftover stocks, which gave the edge to the equal-weighted version of the index. The stock of Gartner (NYSE: IT) led the month, rising from 63.4% to 82.5% of its value on 28 November 2026 as investors shook off some of the AI disruption discounting they had earlier imposed on it. the stock price of Factset Research Systems (NYSE: FDS) also saw outsized gains for the same reason, rising from 95.6% to 106.6% of its post-Thanksgiving Day 2025 level.

Two of the individual Thanksgiving Leftover stocks lost notable value over the past month. Deckers Outdoor (NYSE: DECK) declined from 111.0% to 101.6% of its 28 November 2025 value, while the stock price of Trade Desk (NASDAQ: TTD) continued to fall through its continually lowering floor.

The spaghetti chart tracks the relative movements of 2025's ten Thanksgiving Leftover stocks during the last nine months with respect to their value on the day after 2025's Thanksgiving holiday.

Nine months after Thanksgiving 2025, five of the S&P 500's Thanksgiving Leftover stocks have risen above their 28 November 2025 level, while the other five have dropped below it.

In cased you missed it, our extended discussion of The Trade Desk's woes as the worst of the 2025's Thanksgiving Leftover Stocks is available here. We'll check back in with the Leftover Stocks near the end of September 2026.

We recently discussed how advances in technology are contributing to boosting the productivity of mathematicians during the last thirty years. But the chart we featured in that article depicting how the number of math papers being published each month has nearly doubled in the last three years may not do full justice to how fast the pace of new papers coming out has changed.

Another way to communicate that acceleration in output is Matt Parker's approach of focusing on a small subset of the new papers being published in the last few months to show how they represent a big increase in the capability of the AI large language models used to generate their findings. In the following video, in what is becoming an annual tradition, he interrupts his vacation to cover breaking news in the world of mathematics.

Parker makes an explicit point of noting the use of the Lean proof assistant to verify their findings, which gives greater confidence the findings in these papers will stand up to serious scrutiny.

Not all the preprint papers that have recently populated the mathematics category of the arXiv database meet that standard, as Reddit math community poster Salt_Attorney recently observed.

By contrast, Lean verification was used in the ten advances spanning several disciplines in mathematics OpenAI claims its Astra AI system has made. As such, they have a much better chance they'll be found valid when the findings are reviewed.

But the question now being raised is whether verification of findings by a proof assistant is enough to accept AI-generated results as presented. For further reading on that topic, we'll recommend Gary Marcus' discussion of OpenAI's claimed accomplishment to appreciate it may not be either as grand or the Astra AI system as capable as presented in OpenAI's announcement.

It may not seem like it, but the millennia-old academic discipline of mathematics has been undergoing a productivity revolution during the last thirty years.

That revolution is being enabled by the adoption of several new technologies, each of which is following a S-shaped logistic growth curve model. Mike Roberts of Strategic Tool Kits describes the pattern of how technological advancements increase performance over time:

The S curve is a strategic concept that describes how the old ways mature and are superseded by new ways.

In the early days of new technology, it takes a long time to improve performance. People are working out the technology, and the applications, ironing out the flaws, and building the ecosystem. Slowly over time, performance accelerates.

After a certain time, the rate of improvement hits a peak and then starts to slow down. Easy wins have all been made and the learning curve has been fully ridden for continuous improvement. Some of the fundamental barriers of the technology are reached. Eventually, the improvement tapers off and a plateau with this technology is reached. This is the “S” shaped curve.

When a new productivity-enhancing technology comes along after the older technology reaches its mature phase, the process repeats with performance building on its old plateau and rising to new heights.

For mathematicians, the two great technological improvements of the twenty-first century have been the widespread adoption of the arXiv database for publishing preprint papers to more quickly communicate their discoveries and the more recent development of Artificial Intelligence (AI) systems paired with proof assistant systems like Lean that are helping automate large portions of their work.

ArXiv provides data on the number of new papers that are uploaded to its preprint paper database each month. The following chart reveals its data for math papers, not counting cross listings from other categories of academic papers, from January 1992 through July 2026:

The data shows adoption of the arXiv database for communicating advancements in maths went through the full S-curve advancement cycle, reaching a fully mature phase by the early 2020s with output holding fairly level for several years. But after ChatGPT (3.5) was launched at the end of November 2022, the Large Language Model (LLM) technology facilitiated a steady increase in the productive output of mathematicians for the first time in years. By making it possible to automate a portion of writing academic papers, the number of papers published to arXiv each month began to rise in a steady, linear trend.

But soon after OpenAI's o3-mini LLM-based reasoning model was released on 31 January 2025, the rate of output for publishing new papers to arXiv exploded in what appears to be the exponential growth phase of the S-shaped logistic growth pattern. The new cost efficient o3-mini reasoning model was specifically developed to automate analytical tasks in Science, Technology, Engineering, and Mathematics fields and also coding.

For mathematics, the new AI technologies make rapid advancements possible by drawing on the body of published work, like the thousands of math preprint papers documented in the arXiv database and other resources that catalogued large numbers of unresolved conjectures in digital-friendly formats, to test new possible ways of testing those conjectures. When promising proofs or disproofs of the conjectures are identified, the technology automates their verification using the proof assistant software technology that has also come into its own during this period. Using proof assistants also has the benefit of coding any new successful proofs into their proof libraries, which can then be mixed and matched as needed to test other conjectures.

When that's done, the remaining step for the mathematicians orchestrating what's effectively become a massive, automated collaboration exercise is to write up a new preprint paper and publish it to the arXiv database.

Mathematics isn't the only field experiencing a boom in preprint papers. Economics has likewise seen the number of papers published each month almost double since ChatGPT 3.5's public release. Unlike math however, economists don't have the equivalent of maths' proof assistants to verify their findings, which raises questions about the quality and validity of the flood of new papers in the field.

References

arXiv. Math Submissions. [Online Database]. Accessed 8 August 2026.

arXiv. Monthly Submissions. [Online Article and CSV Data]. Accessed 8 August 2026.

Ecology.net. Logistic Growth. [Online Article]. 15 December 2025.

Pablo Groisman. Math papers uploaded to arXiv per month, January 1992 to July 2026. [Online Article]. 3 August 2026. [Our article was inspired by Pablo's chart!]

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ScriptByAI. OpenAI & ChatGPT Timeline: GPT Release Dates to GPT-5.6 (2026). [Online Database]. Accessed 20 August 2026.

Image credit: S Curve Concept in Science by Oliver Tacke on Wikiversity. Creative Commons CC BY-SA 3.0 Attribution-ShareAlike 3.0 Unported Deed. 17 July 2012.

When we launched the S&P 500's Thanksgiving Leftover project the day after Thanksgiving 2025, we knew we were going to spend the next year following the stock prices of companies that weren't doing very well. After all, to even make the list, the ten companies whose stocks we would track ranked as the S&P 500's worst performing stocks of 2025.

In the nine months since then, a few of those stocks have outperformed the index, while the rest have lagged behind. Most of those stocks have fallen below their post-2025 Thanksgiving Day level, but not by anywhere near as much as they had fallen to qualify as one of the S&P 500's worst performing stocks in 2025.

But one stock in particular has gone on to plumb new depths. It has continued to fall so much more that it is on track to qualify as one of the S&P 500's worst performing stocks of 2026.

That stock is The Trade Desk (NASDAQ: TTD), the digital advertising firm analyst David Desjardins believes is facing an "existential crisis". Here's how he describes the company's now nearly two year long fall from grace:

After reporting highly disappointing financial results for the second quarter of 2026, shares of The Trade Desk, Inc. (TTD) declined by a whopping 21.9% last Friday, which came on top of a 6.8% decline on the prior day. Since the publication of my initiating coverage in early February 2026, TTD's stock price has basically been cut in half, from ~$27.00 per share at the time of publication to around $13.39 as of today's market close.

Relative to its all-time high of $141.53 reached in December 2024, The Trade Desk has now declined by a massive ~90%. As David Einhorn famously said, a stock down 90% is just a stock that was down 80% before being cut in half again, and this is exactly what happened to TTD since last February. The depth of TTD's sell-off is quite something, but what is even more impressive to me is its speed....

At this point, The Trade Desk has become one of the most hated stocks that I am aware of, and this is on top of being the worst-performing constituent in the S&P 500 (SPX) on a year-to-date basis. Pretty much everything said or written about the company is negative, and it is precisely why I decided to write an update today. In less than two years, TTD went from a market darling that could do no wrong at over 26.0x forward sales to being viewed as a melting ice cube changing hands at 2.3x forward sales today.

The following chart compares The Trade Desk's stock performance with the S&P 500, from 29 November 2024 (aka "the day after Thanksgiving Day 2024) through 18 August 2026:

Believe it or not, despite the company's continued misfortune, Desjardins views the company's low stock price as a speculative strong buy opportunity, where he makes the argument that the company has some potential for a turnaround based on its available cash balance, lack of debt, and cash flow.

We disagree, because we think The Trade Desk has further to fall before it might reach that point.

Here's why. According to SlickCharts, The Trade Desk's market cap has fallen to where the company now ranks 502 out of the 503 stocks that make up the S&P 500 index. Because it has, and because its fall is continuing, the company's stock is verging on the point where S&P will act to remove it from the index. If and when that happens, as increasingly seems likely, its stock price will experience the opposite of what happens when a company's stock is included in the index, which is to say its stock price will fall even further.

A deeper decline is almost ensured given the negative outlook CEO Jeffrey Green communicated during the company's 2026-Q2 earnings call. Gytis Zizys, who formerly held a buy rating for the company on the hope it will see a turnaround, reacted to that development:

The Trade Desk, Inc. (TTD) provided one of the worst guidances I’ve seen in recent months, which put the last nail in the coffin for many shareholders who were still clinging to the idea of a turnaround. It seems I was prematurely too bullish on the turnaround as well, and this report is forcing me to downgrade it to a hold. I don’t think there’s a point in selling at these low prices, unless you want to harvest some tax losses. If it gets to under $10 per share, I will be jumping in to see what happens over the next couple of years. It’ll either recover, or my investment will go to zero.

The only problem with this investing strategy is we can argue that the bar for being able to beat TTD's stock performance is very low. It's so low that investing almost anywhere else or just parking the money in a cash savings account would be more advantageous.

This article is a standalone feature in our ongoing Thanksgiving Leftover series, which will continue with its regular monthly installment later this month. The ongoing tragedy of the performance of The Trade Desk's stock demanded a special edition.

Something new is coming to the world of sports: real futures trading.

On 11 August 2026, the world's largest futures and options trading house, CME Group (NASDAQ: CME) announced it would partner with the National Hockey League and a startup called FutureSports to launch the world's first index-based hockey futures contracts.

Unlike the kind of futures you might find at multiple sports wagering outfits that might involve things like betting on which team will win the Stanley Cup next year or which goalie will win the Vezina Trophy, the kind of futures contracts now being developed for the NHL have a lot more to do with giving sports-related businesses and investors new tools to hedge their risks in addition to simple speculation.

The Chicago Tribune gives an overview of what the CME Group/FutureSports/NHL partnership is looking to bring about:

Futures contracts, tied to the value of indexes, allow investors to buy or sell an asset at an agreed price by a specific date. It also allows traders to hedge risks.

For decades, futures contracts have primarily been associated with agriculture products, like corn and soybeans, or energy commodities, such as oil and natural gas, as well as stock indexes like the S&P 500. Futures even got the Hollywood treatment with the 1983 film “Trading Places,” when Dan Aykroyd and Eddie Murphy’s characters cashed in on orange juice futures, and over 20 years later, inspired a federal provision nicknamed the Eddie Murphy Rule.

FutureSports and CME’s new offering means if the Blackhawks play poorly, for example, a season ticket holder could take a short position on the team’s index to try and recoup some of their ticket investment losses. Corporate sponsors, which spend millions annually on sports deals, could buy hockey futures to limit their exposure if a key athlete gets injured or the team just fails to meet expectations.

The companies highlighted other potential market participants like garage and parking lot operators, retailers and even the sports franchises.

There are real businesses and people who have real money on the line that depend on how well a team performs. For example, consider one of the Chicago Blackhawks' star players, Connor Bedard. One of the top offensive players in the NHL when healthy, Bedard's career has been repeatedly knocked by injuries, forcing him to miss playing in significant parts of several seasons.

Every time he's been sidelined, the Blackhawks' ability to score goals and win games has been notably reduced. While not the only reason for the team's bottom-of-their-division status in recent seasons, his absence from the ice when injured has certainly been a contributing factor.

With futures trading however, the businesses whose revenues rise and fall with the team's fortunes would have a way to cushion the losses they might otherwise face. For example, if Connor Bedard isn't playing, it's tougher to sell hockey sweaters with his name and number on them. Or to sell Connor Bedard Funko-pop figurines at Chicago's sporting goods stores. Or to fill restaurants near the United Center on the Blackhawks' game nights. Hedging using futures contracts could make a lot of sense for the owners of these businesses to offset their otherwise unmitigated loss of revenue and potentially even stabilize it in the face of an adverse event like a key player not being able to play.

Futures trading falls into the kind of higher risk investing we consider to be suitable mainly for well-established and well-funded operators. It's definitely not for the faint-of-heart or for those who have low tolerances for risk. The Chicago Tribune's article understates that aspect of the venture:

Trading futures is complex and its investors are savvy. Even retail traders, who buy for themselves and often make smaller trades, use similar financial analysis tools as institutional investors.

“Anyone considering trading them should understand the mechanics, costs and risks first and seek professional guidance as appropriate,” said Joseph Cusick, senior vice president and portfolio specialist at Calamos Investments in Naperville. “Futures traders can experience rapid gains or losses because of margin requirements.”

That's putting it mildly, especially since real futures trading might involve using leverage (or borrowing) to fund an inherently speculative investment.

It will be interesting to see how the proposed NHL futures market plays out. If all goes as planned, NHL futures will go live 28 September 2026. Right in time for the NHL's 2026-27 season to get underway.

Image Credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon of a Wall Street bull and bear playing hockey where the scoreboard shows 'CME Group / FutureSports / NHL Index' with value of 7500".

July 2026 saw positive changes overall for the Thanksgiving Leftover portfolio made up of the ten worst-performing stocks in the S&P 500 (Index: SPX) as of Thanksgiving 2025. At least, as compared to how they fared in June 2025.

The equal-weighted weighted version of the portfolio overtook the market cap-weighted version over the past month. Through the close of trading on 27 July 2026, the equal-weighted group of Thanksgiving Leftover stocks rise to 88.5% of their value recorded on 28 November 2025. That compares with the 87.4% valuation of the market-cap weighted version of the ten stock portfolio.

That's a change from most of the preceding seven months that had the market-cap weighted version of the 2025 Thanksgiving Leftover stock portfolio outperforming the equal-weighted version. It's also developed as the S&P 500 index itself has largely moved sideways, rising from 107.4% to 108.2% of its post-2025 Thanksgiving holiday valuation.

The following chart shows the performance of all three sets of stocks, with the two Thanksgiving Leftover stock indices continuing to lag behind the S&P 500 index by a wide margin.

Much of the gain of the equal-weighted version of the Thanksgiving Leftover stock index has come about because the three worst performing individual stocks in the portfolio, Lululemon Athletica (NASDAQ: LULU), Gartner (NYSE: IT), and The Trade Desk (NASDAQ: TTD) stopped falling and even rebounded a bit in the past month.

More significantly for the equal-weighted Leftover stocks, Factset Research Systems (NYSE: FDS) rose 23% over its level a month earlier.

At the same time, three stocks that account for 45% of the makeup of the market-cap version of the Thanksgiving Leftover stock portfolio, Chipotle Mexican Grill (NYSE: CMG), Fiserv (NASDAQ: FISV), and Alexandria Real Estate Equities (NYSE: ARE), saw positive but smaller gains over the preceding month while the Leftover stocks' highest flyers, Molina Healthcare (NYSE: MOH), Deckers Outdoor (NYSE: DECK), and Dow Inc. (NYSE: DOW) were little changed from where they were a month earlier, though they changed quite a bit in between!

The spaghetti chart tracks the relative movements of 2025's ten Thanksgiving Leftover stocks with respect to their value on the day after 2025's Thanksgiving holiday.

Will the equal-weighted continue pulling ahead of the market-cap weighted version of the Thanksgiving Leftover portfolio? Or will the market-cap weighting win out? We'll next see where things stand near the end of August 2026.