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The S curve is a strategic concept that describes how the old ways mature and are superseded by new ways.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

Mike Roberts. S Curve – What is it? [Online Article]. 13 July 2024.

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".

The dividend outlook of the S&P 500 (Index: SPX) once again delivered a mixed outlook for investors in August 2026.

Expected dividends for the index dipped in the current quarter of 2026-Q3, but the amount of cash dividends expected to be paid out in the upcoming future quarters of 2026-Q4 and 2027-Q1 increased. The more distant future quarters of 2027-Q2 and 2027-Q3 however both declined with respect to the July 2026 snapshot of the S&P 500's dividend outlook.

All these changes however are modest, with the magnitude of the changes for each quarter coming in at less than one percent. Here is the summary of how the expected future for the S&P 500's dividends changed between 14 July and 14 August 2026:

• 2026-Q3: Decrease of $0.11 to $21.02 per share
• 2026-Q4: Increase of $0.10 to $21.55 per share
• 2027-Q1: Increase of $0.06 to $22.81 per share
• 2027-Q2: Decrease of $0.15 to $21.60 per share
• 2027-Q3: Decrease of $0.11 to $21.81 per share

The following chart illustrates the expectations for the S&P 500's quarterly dividends per share as of 14 August 2026 and how they compare with the preceding quarters going back to 2024-Q3.

More About Dividend Futures

Dividend futures represent the quantified expectations investors have for the future income they will realize from owning shares of stocks, which in turn, affects how investors set current day stock prices. How changes in the outlook for dividends at specific points of time in the future contribute to changes in current day stock prices as represented by the value of the S&P 500 index is described by this math.

Dividend futures for the index indicate the market capitalization-weighted amount of dividends per share for all these dividend-paying stocks that are expected to be paid out over the period covered by each quarter's dividend futures contracts. These contracts start on the day after the preceding quarter's dividend futures contracts expire and end on the third Friday of the month ending the indicated quarter. For example, as determined by dividend futures contracts, the now "current" quarter of 2026-Q3 began on Saturday, 20 June 2026 and will officially end on Friday, 18 September 2026. Since the expectations for this quarter's dividend payouts can change all the way up to that final date, it counts as a future quarter all the way up through that future point in time.

Because dividend futures are tied to options contracts that run on this schedule, that makes these figures different from the quarterly dividends per share figures that are reported by Standard and Poor. S&P reports the amount of dividends per share paid out during regular calendar quarters after the end of each quarter. This term mismatch accounts for the differences in dividends reported by both sources, with the biggest differences between the two typically seen in the first and fourth quarters of each year.

Image Credit: Microsoft Copilot Designer. Prompt: "A crystal ball with the word 'SP 500' written inside it". And 'Dividends' written above it, which we added.

The S&P 500 (Index: SPX) reached a new record high close of 7,798.99 on Thursday, 13 August 2026 before falling back on Friday to end the trading week at 7,785.76.

Two developments helped to spur the index to its new heights. First, investors received good inflation news on Wednesday, 12 August 2026 when the Consumer Price Index came in lower than expected. The second came on Thursday, when the Producer Price Index was likewise reported to be lower than expected.

Because of these benign consumer and producer price inflation reports, the CME Group's FedWatch Tool no longer anticipates the Fed will act to hike the Federal Funds Rate from its current target range of 3.50-3.75% in September 2026, but will instead delay a quarter point rate hike to take place sometime in the fourth quarter of 2026. In its latest snapshot, the FedWatch Tool sees a falling 53% chance the Fed will act to hike this core interest rate to a target range of 3.75-4.00% on 28 October (2026-Q4) but gives a more solid 93% chance this rate will be in effect on 9 December (2026-Q4).

With the next rate hike likely delayed and fewer rate hikes on the table as a result of the better than expected inflation reports, the resulting lower interest rates kept investors focused on the first quarter of 2027 as they set current day stock prices. The latest update of the alternative futures chart shows the level of stock prices is right where they dividend futures-based model forecasts it would be provided investors are focused on 2027-Q1.

While those were the biggest headlines, other things happened that contributed to the random onset of new information investors absorbed and reacted to during the week that was. Here is our summary of the week's market moving headlines:

Monday, 10 August 2026

• Signs and portents for the U.S. economy:
• July US container imports hit fourth-highest on record, Descartes says
• Oil climbs 5% as Iran, US both demand compensation and Hormuz hopes fade
• Oil in U.S. Strategic Petroleum Reserve falls below 300 million barrels, lowest since 1983
• Lower inflation, global ambitions developing in China:
• China July factory-gate inflation eases to 3-month low, CPI slows
• China's central bank vows to expand yuan's international use in five-year plan
• China bypasses shipping chokepoints with ‘Ice Silk Road’ through Arctic
• Bigger trouble developing in Japan as BOJ minions plan next rate hikes:
• BOJ's rate-hike path runs into Takaichi's bond market problems
• BOJ's debate on faster hikes bolsters September rate move odds
• Bigger trouble developing in Eurozone:
• German trade deficit with China grows as Beijing relies less on European industry
• Wall Street concluded moderately lower as hopes for Hormuz reopening fade
• Wall Street ends down as expectations of Hormuz deal fade
• Market breadth strengthens: Equal-weight S&P 500 pulls ahead of market-weight index
• Meta to open source its most powerful AI model as it takes swipe at OpenAI, Anthropic
• Microsoft plans to unveil next-generation AI chip in September, The Information reports

Tuesday, 11 August 2026

• Signs and portents for the U.S. economy:
• US court rules Meta, other tech firms must face thousands of lawsuits over social media addiction
• US existing home sales fall further; small business sentiment at 11-month high
• Oil prices settle up as Iran says Strait of Hormuz to stay shut
• U.S. oil tops $84 as Hormuz deal hopes dwindle and deadlock deepens
• US auto loans hit record high in second quarter, New York Fed says
• US officials work to close supply chain gaps as foreign investment grows
• Fed minions say inflation is their biggest problem:
• Inflation is the biggest problem, Fed's Goolsbee says
• Atlanta Fed's Venable: Inflation too high, with prospect of easing dependent on events in Middle East
• Bigger trouble, stimulus developing in China:
• China's car sales extend slide as shift accelerates to overseas markets
• Surge in China's coal imports may start to taper off
• Bigger trouble developing in Japan:
• Yen sinks as effect of US-Japan intervention fades
• Wall Street ended lower as eyes remain on the Middle East and upcoming inflation data
• Wall Street ends down as US-Iran peace optimism fades

Wednesday, 12 August 2026

• Signs and portents for the U.S. economy:
• US annual inflation cooled to 3.4% in July
• Headline CPI rises 3.4% in July, slowing from June's pace
• Rate-Hike Odds Tumble As US Consumer Price Inflation Prints Slowest Since Pre-War
• Oil prices edge up as investors weigh US-Iran talks deadlock against lower demand
• OPEC further lowers 2026 global oil demand growth forecast
• Mortgage rates finally stop rising, causing demand to trickle back
• Before the CPI inflation report: Fed minions say poor Americans are struggling to make ends meet, want to hike their interest rates:
• Boston Fed's Collins open to interest rate increase in September, FT says
• Poorer Americans are struggling to make ‘ends meet’, top Fed official says
• Investors worry leaner Fed guidance may come at a price
• After the CPI inflation report:
• Fed expected to leave rates unchanged next month after soft inflation data
• Bigger trouble, stimulus developing in China:
• China's central bank pledges timely new policy rollout
• Wall Street finished up in the green after the latest inflation report
• S&P 500 ends higher as CoreWeave results fuel AI optimism
• CoreWeave, Super Micro surge on signs of sustained AI buildout
• 23 dividend safety stocks beating the S&P 500 this year

Thursday, 13 August 2026

• Signs and portents for the U.S. economy:
• US producer prices unchanged in July, further dimming rate hike odds
• Oil settles down 2% on weak demand outlook, hefty US crude build
• One Fed minion notices inflation data doesn't support rate hikes, other Fed minion is slow on the uptake, while third is flapping in the wind:
• Fed's Goolsbee says latest inflation data is better
• Fed should raise rates to restrain growth and inflation, Hammack says
• Fed's Barkin: Still an "open question" if rate hike will be needed to meet inflation target
• BOJ minions excited to get inflation data to support their plans to keep hiking Japan's interest rates:
• Japan's wholesale inflation stays hot, bolstering odds of September BOJ hike
• Rate hike bets leave yen's post-intervention gains at BOJ's mercy
• Japan may see more yen intervention, faster BOJ rate hikes, ex-top FX diplomat says
• Wall Street ended on top as the S&P 500 notched a new record high
• S&P 500 notches record-high close as rate-hike worries ease
• S&P 500 hits 27th record high of 2026 — History says more likely ahead

Friday, 14 August 2026

• Signs and portents for the U.S. economy:
• US retail sales post first decline in nine months in July
• Oil set for weekly gains after US threatens indefinite blockade of Iran
• Fed rate hike odds sink further as retail sales, consumer sentiment fall
• US retail sales post first decline in nine months in July
• Fed minions expected to sit on hands rather than hike rates in September 2026:
• Cooler inflation data may force Warsh's divided Fed to hold the line on rates
• Bigger trouble developing in and out of China:
• China July bank loans post record contraction as credit demand falters
• Chinese car sales are booming almost everywhere - just not at home
• BOJ minions getting super excited to hike Japan's interest rates again:
• BOJ eyeing September rate hike, faster pace of tightening, sources say
• Central bank minions around globe betting big on gold:
• Central banks spearhead renewed gold rush
• Wall Street ended lower a day after the S&P 500 notched a record high
• S&P 500 ends lower as investors weigh data, Middle East tensions
• S&P 500 posts weekly gain as strength spreads beyond technology

The Atlanta Fed's GDPNow tool anticipates real GDP growth for the U.S. economy of +4.3% in 2026-Q3, down from the +5.8% annualized growth it forecast a week earlier.

Image credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon of a Wall Street bull and bear who give each other a high-five after news breaks that inflation came in lower than expected while Federal Reserve officials are sad they won't be able to hike U.S. interest rates when they wanted".