The climbing limo method of forecasting future GDP in the United States projects the nation's economic output in the recently finished third quarter of 2025 will be around $30.9 trillion.
This estimate assumes the momentum the U.S. economy recorded in growing between 2024-Q2 and 2024-Q4 will be sustained through the current quarter. Since the U.S. economy's momentum has been slowing however, it's likely 2025-Q3's will come in below that value, which is not adjusted for inflation.
That's the same scenario we anticipated for 2025-Q2's finalized GDP estimate, which the following chart indicates held even as the BEA revised all the GDP data used to create the climbing limo forecasts.
Because the BEA revised all its GDP data going back to the first quarter of 2020, this chart shows the climbing limo GDP forecasts as if that data had been available throughout the period it covers.
Generally speaking, the BEA revised its nominal GDP data for all quarters from 2020-Q1 through 2025-Q1 upward by varying amounts, with the largest adjustments in the period from 2023-Q3 through 2025-Q1.
Looking at the recently released estimate for 2025-Q2, the climbing limo method had forecast GDP would clock in at about $30.58 trillion, which after the BEA's revisions, is now shown as nearly $30.76 trillion. The BEA's now official estimate of GDP for this quarter is $30.49 trillion. The official estimate came in lower than both the original and revised forecasts.
The climbing limo forecasting method is a "momentum"-based projection. As such, even when recorded GDP deviates considerably from the forecast values that are projected three quarters ahead in time, it provides valuable information in confirming the economy's underlying momentum has changed. For much of the period the chart shows, it confirms the U.S. economy's growth momentum has been slowing since its initial recovery from 2020's Coronavirus Pandemic Recession.
References
U.S. Bureau of Economic Analysis. National Income and Product Accounts. Table 1.1.5. Gross Domestic Product. [Online Database]. Accessed 26 September 2025.
Image Credit: Microsoft Copilot Designer. Prompt: "Image of a long limousine driving up a bumpy area chart labeled 'GDP'".
Labels: gdp forecast
Believe it or not, the S&P 500 (Index: SPX) has mostly behaved in an orderly manner since the end of 2023.
Mostly. The index experienced a brief outbreak of chaos during 2025, which proved to be little more than an outlier in a longer period of order. That brief outbreak coincided with two major market events:
- The brief 622 point deflation of the AI bubble between 19 February 2025 and 13 March 2025 after the reveal of China's DeepSeek AI system "broke" what had been a virtual AI monopoly by U.S. firms.
- The market's reaction to President Trump's 2 April 2025 "Liberation Day" tariffs, which prompted the index to plunge by 688 points through 8 April 2025.
Altogether, these two events caused the S&P 500 to drop by 1,161.38 points between 19 February 2025 and 8 April 2025. And had the market chaos unleashed by them continued, the period of order in the stock market that began after 29 December 2023 would have quickly come to a definitive end.
Except it didn't. Stock prices rebounded so quickly that the plunge became little more than the equivalent of a statistical outlier. The following chart mapping the relationship between the value of the S&P 500 and the index' underlying trailing year dividends per share illustrates that remarkable development.
Two things combined to preserve order in the U.S. stock market. First, the Trump administration quickly adapted its global tariff policies to reduce their damaging potential. Second, the AI bubble reflated, as the valuations of firms involved in the technology behind artificial intelligence systems recovered and then reached even higher highs.
Through 14 October 2025, we find the recent drop in stock prices associated with the reignition of the U.S.-China tariff war following China's provocative attempt to assert monopoly status over rare earth material processing and the Trump administration's reaction to it would have to become much deeper to break the established period of order in the market. In the short term, the index' 20-day moving average would need to drop below about $5,500 by the end of 2025 to qualify as a definitive breakdown of order in the U.S. stock market.
Given that the S&P 500 just recorded an all time high of 6,753.72 on 8 October 2025, we'll observe that if such a breakdown in order occurs within this timeframe, there will be no mistaking it for what it is.
Image Credit: Microsoft Copilot Designer. Prompt: "Conceptual art of the S&P 500 quickly moving from a period of order to chaos and back again".
Labels: chaos, data visualization, ideas, SP 500
The market capitalization of the S&P 500 (Index: SPX) grew by 13% during the third quarter of 2025, more than double the pace it rose during the preceding quarter.
According to Standard and Poor, the index closed out 2025-Q3 with a market cap of nearly $59.32 trillion. This figure represents an increase of $6.82 trillion from the $52.50 trillion S&P reported at the end of 2025-Q2.
Over 56% of the total gain in the S&P 500's market cap was delivered by its top 10 component stocks. Together, these 10 stocks account for 38.9% of the entire valuation of the index, a record high concentration.
The overall membership of the Top 10 stocks of the S&P 500 is unchanged this quarter, but there have been some changes in the relative rankings of the bottom three stocks of the table.
The following chart shows the relative shares of the top 10 stocks in the S&P 500 at the end of the second quarter of 2025.
Here are the approximate market capitalizations of each of the S&P 500's top ten component firms at the end trading on 30 September 2025:
- Nvidia (NASDAQ: NVDA) $4,733,844,220,324 (7.98%)
- Microsoft (NASDAQ: MSFT) $4,003,323,779,663 (6.75%)
- Apple (NASDAQ: AAPL) $3,929,288,628,536 (6.62%)
- Amazon (NASDAQ: AMZN) $2,215,800,814,217 (3.74%)
- Meta Platforms (A) (NASDAQ: META) $1,656,272,653,214 (2.79%)
- Broadcom (NASDAQ: AVGO) $1,613,514,964,354 (2.72%)
- Alphabet (A) (NASDAQ: GOOGL) $1,470,425,649,566 (2.48%)
- Tesla (NYSE: TSLA) $1,297,642,736,446 (2.19%)
- Alphabet (C) (NASDAQ: GOOG) $1,182,620,660,455 (1.99%)
- Berkshire Hathaway (B) (NASDAQ: BRK.B) $958,603,483,954 (1.62%)
One important thing to note is that nine of the top ten stocks within the index have valuations in excess of $1 trillion. Only Berkshire Hathaway falls below that still exclusive club.
The index' top-most valued stock, Nvidia, saw its market cap rise enough to account for 8% of the index by itself. Microsoft saw its share of the index dip by a quarter percent to 6.75%, while Apple's share increased from 5.84% to 6.62%.
The next four firms saw small changes in their individual market capitalizations, with Amazon and Meta Platforms losing some relative share of the index' total valuation and Broadcom and Class A shares of Alphabet gaining.
That brings us to where the rankings changed. Both Tesla and Class C shares of Alphabet moved up one slot to eighth and ninth place respectively, while Berkshire Hathaway dropped two positions into the tenth position.
All ten of the S&P 500's Top 10 stocks grew their market capitalizations during 2025-Q3.
Six months ago, the AI-bubble of the preceding two years had all but deflated, but with Nvidia's surge in valuation during 2025-Q2 and 2025-Q3, it appears to have reinflated and expanded.
References
Standard and Poor. S&P Market Attributes. [Excel Spreadsheet]. 30 September 2025. Accessed 1 October 2025.
Labels: market cap, SP 500
The trading week ending on Friday, 10 October 2025 saw the S&P 500 (Index: SPX) drop over 2.4% from its previous week's close. The index closed the week at 6,552.50.
All of that loss came on Friday, 10 October 2025, which fully qualified as interesting after President Trump indicated he was considering breaking of a meeting with China's premier and was considering new, large tariffs on China. The announcement followed China's action earlier in the week to expand its export controls on rare earths and other materials that are vital to several industries in the United States.
The S&P 500 lost 2.71% of its value from the previous day's close following President Trump's comments, while the tech-heavy NASDAQ Composite (Index: COMP) lost 3.56%. By contrast, the price-weighted Dow Jones Industrial Average (Index: DJI) which has lesser exposure to the tech sector, lost 1.9%.
The latest update of the alternative futures chart finds the trajectory of the S&P 500 is still tracking along with the dividend futures-based model's projection associated with investors focusing their forward-looking attention on 2026-Q2, but went from a little below the middle to the low end of that expected range.
We'll be paying attention in the upcoming weeks to the market-moving headlines to determine whether the reignition of 2025's US-China tariff war represents a regime change for the market. The potential is there given what happened earlier this year. Through Friday, 10 October 2025 however, it's too early to make that determination as we cannot yet reject the hypothesis its a noise event and investors remain focused on 2026-Q2 whose projections are defined by the market regime that took hold on and after 28 April 2025.
Here are the market-moving headlines for the trading week ending on 10 October 2025.
- Monday, 6 October 2025
-
- Signs and portents for the U.S. economy:
- US holiday sales growth to be muted this year, forecasts show
- Oil rises after OPEC+ hikes output less than expected
- US regional bank mergers gain momentum under Trump administration
- US unemployment claims rise moderately amid labor market freeze
- Trump bailout for trade-hit US farmers expected this week
- Fed minion says US interest rates are perfectly set:
- BOJ minions wondering if they should proceed with rate hikes since Japan's government has a new boss:
- Japan's stocks zoom, yen slumps as Takaichi win dims BOJ hike bets
- BOJ keeps cautiously upbeat economic view, warns of wage uncertainty
- Growth signs developing in Eurozone:
- U.S. stocks race to record closes as AMD jumps on deal with OpenAI
- Tuesday, 7 October 2025
-
- Signs and portents for the U.S. economy:
- Oil rises as oversupply fear eases after OPEC+ restrains output increase
- As oil glut fears mount, OPEC+ restrains output rises for now
- Oil price fall turns up the heat on Big Oil's bloated payouts
- White House says no shutdown-related layoffs yet, but warns they could come
- Former Fed minion says go slow on rate cuts, current minion worries that AI is pushing up inflation:
- Fmr. Dallas Fed President Robert Kaplan warns against aggressive rate cuts amid shutdown
- Fed's Kashkari says he's skeptical AI is replacing workers, but it may push up rates
- Fed's Miran says calm bond market shows support for rate cuts
- Bigger doomsday prepping developing in China:
- BOJ minions getting inflation data to support plans to hike Japan's interest rates:
- ECB minions wants more Euros in Eurozone:
- ECB's Lagarde renews calls for beefed up role for euro
- Euro zone ministers to look at how to boost euro stablecoin issuance
- S&P 500 ends week-long winning streak amid government shutdown, gold nears $4K
- Wednesday, 8 October 2025
-
- Signs and portents for the U.S. economy:
- Oil up over 1% on OPEC+ output increase restraint
- Brazil's soybean exports to hit record as US out of market, Chinese demand strong
- Fed minions starting to worry more about job market than inflation:
- Fed last month saw rising risks to job market, but remained wary on inflation
- Fed's view of neutral, reshuffled by Miran, a key to inflation concerns
- Bigger global supply chain trouble, stimulus developing in China:
- China expands rare earth restrictions, targets defence, semiconductor users
- China’s new restrictions on rare earth exports send a stark warning to the West
- Explainer: How China's new rare earth export controls work
- China's September bank lending seen more than doubling from previous month
- BOJ minions warned by Japan's new prime minister to avoid hiking interest rates as minions' independence called into question:
- BOJ should be wary of more rate hikes, adviser to Japan's likely next PM says
- Japan Takaichi says BOJ rate decisions must align with govt goal
- ECB minion says Eurozone interest rates are just where they need to be:
- Nasdaq and S&P 500 notched record highs while gold topped $4,000
- Thursday, 9 October 2025
-
- Signs and portents for the U.S. economy:
- Oil settles 1.6% lower after Gaza ceasefire
- US 30-year fixed mortgage rate falls; prospective buyers stay on the sidelines
- US companies' profit growth seen softer, spotlight on AI spending
- Fed minions getting worried about sluggish US labor market:
- Fed's Williams backs more rate cuts this year due to labor market slowdown risks, he tells NYT
- Fed's Daly: labor market is at a 'worrisome' point
- ECB minions thinking their interest rate policy is perfect for Eurozone:
- Wall Street closes lower, pausing record-setting rally as earnings approach
- Friday, 10 October 2025
-
- Signs and portents for the U.S. economy:
- Trump says no reason to meet China's Xi, threatens tariffs in new rift
- VIEW Trump threatens 'massive increase' in China tariffs
- After the bell: US will impose additional 100% tariff on Chinese imports from November, Trump says
- Oil prices fall as risk premium fades after Gaza deal
- US to publish September CPI report this month despite government shutdown
- Trump Fires Federal Workers Amid Shutdown Fight With Democrats
- Fed minion says they will be cautious in setting US interest rates, potential new Chief Minion says regular size rate cuts will help US job market:
- Fed should 'tread with caution' on rate cuts, Musalem says
- Fed can address job market weakness with normal-sized rate cuts, Waller tells CNBC
- Bigger trouble developing in China:
- BOJ minions get data that goes against their plans to hike Japan's interest rates, Japan's new prime minister sees her political coalition break down:
- Japan wholesale inflation holds steady, yen falls add pressure on BOJ
- Political Chaos In Japan: LDP Partner Exits Ruling Coalition In Shock Blow To Takaichi, What Happens Next
- ECB minions wonder if their plan for digital Euro will cause a Eurozone bank runs:
- Wall Street suffers worst day since April tariff shock
The CME Group's FedWatch Tool projects a greater than 97% probability of two more quarter point cuts in 2025, coming on 29 October (2025-Q4) and 10 December (2025-Q4). In 2026, the FedWatch tool now forecasts a slower pace for additional rate cuts, with a 95% probability for a quarter point rate cut on 18 March (2026-Q1) and an 80% probability of another quarter point rate cut on 17 June (2026-Q2).
The Atlanta Fed's GDPNow tool projection of real GDP growth in the U.S. during the current quarter of 2025-Q3 held steady +3.8% with data reports on hold because of the Senate Democrats' ongoing refusal to fund government operations. That refusal led to the announcement of mass firings on Day 10 of the shutdown that resulted from their actions.
We've probably given it more electronic ink than it deserves, but the government shutdown once again failed to have any real market-moving impact. We are however tracking it because the announcement of mass firings of nonessential U.S. government employees on Friday, 10 October 2025 would affect U.S. employment data, which in turn, would influence how the Federal Reserve sets its monetary policies.
On a final note, sharp-eyed readers will recognize we've reused our editorial cartoon from 7 April 2025. It's one of our favorites and is, once again, timely!
Image credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon of a frightened Wall Street bull and a very excited Wall Street bear reacting to a news ticker that says 'TARIFF WAR'".
For all the press that AI and all the companies behind it get, it's rare to find any story that identifies something the Large Language Models (LLMs) behind today's Artificial Intelligence technologies have accomplished that represents a true advancement.
Most stories are about AI's teething problems, such as its problems in creating images of people with too many figures or some other form of body horror dystopia. Other news items deal with how the ability of AI to completely automate writing is negatively impacting education, publishing, music, filmmaking, and other fields.
But stories involving AI systems doing anything new and useful, that hasn't been seen or done before, are rare.
That story still hasn't been written. At least in any traditional media. But there is breaking news of an AI system that's broken new ground in mathematics. By generating a new mathematical proof that mathematicians have verified is correct
That story came to our attention through a post at X and it's probably best to let that post tell the tale:
GPT-5 just casually did new mathematics.
Sebastien Bubeck gave it an open problem from convex optimization, something humans had only partially solved. GPT-5-Pro sat down, reasoned for 17 minutes, and produced a correct proof improving the known bound from 1/L all the way to 1.5/L.
This wasn’t in the paper. It wasn’t online. It wasn’t memorized. It was new math. Verified by Bubeck himself.
Humans later closed the gap at 1.75/L, but GPT-5 independently advanced the frontier. A machine just contributed original research-level mathematics.
If you’re not completely stunned by this, you’re not paying attention.
We’ve officially entered the era where AI isn’t just learning math, it’s creating it.
Here's the X post in which Bubeck announced the accomplishment.
But as Bubeck later notes, while GPT-5 did something that was both new and novel, humans still beat AI to the punch and delivered a better proof focused on the convex optimization problem that outperforms the AI-generated proof:
Now the only reason why I won't post this as an arxiv note, is that the humans actually beat gpt-5 to the punch :-). Namely the arxiv paper has a v2 https://arxiv.org/pdf/2503.10138v2 with an additional author and they closed the gap completely, showing that 1.75/L is the tight bound.
While that better proof clearly beat it, the unexpected artificially generated proof demonstrates AI systems are becoming more capable and useful. Because they are, more things are becoming possible.
The open question however is when will AI's promise and ability to deliver on it outshine all the AI-generated slop that dominates what it has done to date?
Image Credit: Microsoft Copilot Designer. Prompt: "An image illustrating the concept of an artificial intelligence system creating an entirely new mathematical proof".
Labels: math, technology
Welcome to the blogosphere's toolchest! Here, unlike other blogs dedicated to analyzing current events, we create easy-to-use, simple tools to do the math related to them so you can get in on the action too! If you would like to learn more about these tools, or if you would like to contribute ideas to develop for this blog, please e-mail us at:
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