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.

Political Calculations' initial estimate of the total value of new home sales in the United States during July 2026 is $28.41 billion. This value is slightly lower than the initial estimate of $28.62 billion we presented in our previous update that covered new home sales data through May 2026.

Since then, the number of new home sales has been trending downward thanks largely to an increase in mortgage rates in recent months, which has made new homes relatively less affordable during this time. The number of new home sales has dropped to its lowest level since the start of the year.

One surprising development however is that the average new home sale prices has also declined in recent months. Builder incentives are making new homes less expensive than existing homes.

As far back as modern records go, newly built homes have almost always cost more than previously owned homes.

It only makes sense: New homes are expected to have far fewer maintenance issues, brand new appliances, and designs suited to contemporary tastes, plus they can be customized to suit the homebuyer's needs.

After all, it is only in very rare circumstances that a used car would cost more than a similar make and model purchased brand new at the dealer's lot.

But in recent months, that trend has been upended to an extent that has never been seen, with the typical new home selling at a sharp discount to existing homes.

In June, the $407,200 median sales price of new homes was about $28,000 less than for existing homes, a 6.5% discount and by far the biggest inversion in at least 25 years.

Last month, the gap narrowed a bit to nearly $19,000, or 4%—still significantly larger than any discount seen before this year. July also marked the fourth straight month of price inversion for new homes, the longest stretch on record.

The following charts present the U.S. new home market capitalization, the number of new home sales, and their average sale prices as measured by their time-shifted, trailing twelve month averages from January 1976 through July 2026.

New home sales trending down:

New home prices also trending down:

We'll update our measure of the relative affordability of new homes sometime in the next week.

References

U.S. Census Bureau. New Residential Sales Historical Data. Houses Sold. [Excel Spreadsheet]. Accessed 25 August 2026. 

U.S. Census Bureau. New Residential Sales Historical Data. Median and Average Sale Price of Houses Sold. [Excel Spreadsheet]. Accessed 25 August 2026. 

Image Credit: An aerial view of a lot of houses photo by Modunite Ltd on Unsplash.

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.

The S&P 500 (Index: SPX) dropped almost 1.4% from its previous week's close to wrap up the trading week ending on Friday, 21 August 2026 at 7,678.76.

Rising bond yields was perhaps the biggest driver of stock prices during the week, which comes as the U.S. government is increasingly having to compete with Big Tech to borrow money as the company's seeking to build out the infrastructure to support the expansion of Artificial Intelligence (AI) systems are borrowing big to do it.

One outcome of that dynamic is expectations of higher interest rates. The CME Group's FedWatch Tool projections of the expected future for how the Fed will set the Federal Funds Rate changed little in the past week. It anticipates a 60% 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), while giving a much stronger 98% chance this rate will be in effect on 9 December (2026-Q4). Beyond that, the FedWatch Tool now anticipates another quarter point rate hike on 28 April (2027-Q2).

Meanwhile, stock prices behaved almost exactly as would be expected if investors were tightly focusing on 2027-Q1 as they set the level of the week's stock prices. The latest update of the alternative futures chart shows that outcome as the S&P 500's trajectory closely paced the dividend futures-based model's projection associated with investors fixing their attention on the distant future quarter of 2027-Q1.

Investors had quite a lot of other new information to absorb during the trading week. Here is the summary of the week's market-moving headlines:

Monday, 17 August 2026

• Signs and portents for the U.S. economy:
• 30-Year Treasury Yield hits 5.30%, its highest level since 2007
• Oil settles up over $2 as Iran war stalemate stokes supply concerns
• Brent oil rises above $90 as Iran rules out interim deal extension, threatens to escalate conflict
• Depleted strategic oil reserve nears level that raises concerns about damage to caverns, operations
• Economists predict Fed minions to leave interest rates unchanged in rest of 2026:
• Fed to hold interest rates this year, economists say, sticking to their view
• Bigger trouble, stimulus developing in China:
• China's recovery sputters as consumption, output lose steam
• China investment slump deepens as economy shows signs of weakness
• China new home prices stagnant in July as demand stays weak
• China's premier calls for stabilising external demand as growth sputters
• Bigger trouble developing in Japan, BOJ minions catch a break in effort to prop up Japan's currency:
• Japan Q2 growth misses forecasts on weaker spending, investment
• Japan second-quarter GDP grows 1.1% on an annualized basis, missing expectations
• Yen edges up as traders push back Fed rate hike bets
• Japan’s 10-year bond yield hits three-decade high
• Bigger trouble developing in Eurozone:
• Europe's heatwaves expose insurance gap as business losses mount
• China’s unplanned surpluses put EU in trade bind
• ECB minions say AI stocks are due for a fall:
• AI market correction is coming, ECB blog predicts
• Wall Street finished lower as investors look ahead to retail earnings
• Wall Street indexes slip as oil prices rise, retail results awaited

Tuesday, 18 August 2026

• Signs and portents for the U.S. economy:
• As US debt mounts, investors demand higher returns to lend
• US 30-year yields hit highest level since 2007 as war, oil worries fester
• US yields edge lower despite Iran worries, broader sell-off
• Oil closes at three-week high as hopes of US-Iran peace deal fade
• US diesel crack surpasses $100 a barrel for the first time on supply disruptions
• Fed minions say their monetary policy is "accommodative":
• One neutral rate estimate suggests Fed's policy stance is accommodative, paper says
• Bigger trouble, stimulus developing in China:
• China's easing steel output reflects divergent demand drivers
• China has one good reason for its credit slump
• China’s 10-year bond yield falls to 13-month low
• Bigger trouble developing in Japan:
• As Japanese Bond Yields Soar, Unrealized Losses At Life Insurers Hit $200 Billion
• Bigger trouble developing… everywhere:
• Global bond markets put governments on notice over fiscal, inflation risks
• Wall Street finished in the red as attention remains on the Middle East
• Tech selloff weighs down Wall Street as bond yields climb
• Explainer: Treasury yields are rising - why does it matter?

Wednesday, 19 August 2026

• Signs and portents for the U.S. economy:
• US housing market remains under pressure in July; factory output rises
• Oil extends climb on prolonged Hormuz export uncertainty
• Demand from US data-center boom radiates out through factory supply chains
• US 30-year Treasury yields drop from multi-year highs
• Treasury Secretary Bessent doubles US long-bond buybacks in the face of surging yields
• VIEW Yields fall after US Treasury says it will double some bond buybacks
• Port of Los Angeles posts high July volume, helped by retail goods, AI project parts
• Fed minions worried about inflation back in July 2026, just realized increase in productivity blunted it:
• Fed policymakers' inflation concerns increased at July meeting, minutes show
• Boston Fed paper says strong productivity blunted tariff inflation impact
• Bigger trouble developing in China:
• China seen holding loan rates steady in August despite economic weakness
• China robot makers seek to turn humanoid hype into useful work
• Bigger trouble developing in Japan:
• Japan bond yields near 3% as inflation, fiscal worries mount
• Japan has few answers as bond rout puts fiscal plans at risk
• Wall Street ended higher, and yields dipped after the latest Treasury Department announcement
• Wall St rises as yields ease, Moderna lifts healthcare stocks

Thursday, 20 August 2026

• Signs and portents for the U.S. economy:
• US Treasury buyback briefly eases bond rout, but debt worries persist
• Treasury's Bessent says upsized bond buybacks could increase further
• Oil settles up more than 2% after Trump threatens countries supporting Iran
• U.S. military says it aided passage of 660 million barrels of oil through Strait of Hormuz since May
• Fed minion says they're not sure how Fed will set U.S. interest rates in September 2026:
• Fed's Musalem says on CNBC he won't prejudge rate call view for September FOMC
• Bigger trouble developing in and out of China:
• China boosts imports of Russian crude, stymieing India's refiners
• BOJ minions / Bigger trouble developing in Japan:
• Japan's exports surge on chips demand, imports hit record on oil costs
• Bigger trouble developing in the Eurozone:
• Europe’s oil refining capacity set to shrink by 20% over next decade
• Wall Street ended lower as Walmart dragged down retail and oil climbed
• Wall Street sinks as bond yields rise, Walmart results disappoint
• Walmart sales growth slowdown tests consumer resilience, shares slide 9%

Friday, 21 August 2026

• Signs and portents for the U.S. economy:
• US service sector fuels acceleration in business activity, S&P Global says
• Oil rises as Trump threatens sanctions on Iran partners
• US Mortgage Rates Fall for Second Straight Week, to 6.65%
• Fed minions try not to upset their real boss:
• Fed officials tread carefully after Treasury's bond market intervention
• BOJ minions potentially planning rate hikes up through July 2027:
• A date on the calendar could shape the BOJ's endgame on interest rates
• Japan's core inflation accelerates in July, bolsters case for rate hike
• Bigger trouble developing in Eurozone (hawkish ECB minions):
• Traders are bracing for an increasingly hawkish ECB
• S&P 500 drops for week as Tech leads sector losses
• Dow surges 500 points Friday, but index posts back-to-back weekly losses
• Gains in AI company stakes juice second-quarter earnings for S&P 500
• US corporate AI debt surge tests investor limits as fatigue emerges
• Wall Street ended higher as gold and Bitcoin rallied

The Atlanta Fed's GDPNow tool anticipates +4.0% real GDP growth for the U.S. economy in 2026-Q3, dipping from the +4.3% annualized growth it projected a week earlier.

Image credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon of a Wall Street bull holding up a sign that says 'RISING BOND YIELDS' who is growling at scared investors"

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.