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*
The dividend outlook for the S&P 500 (Index: SPX) turned negative for the current quarter of 2026-Q3 since our August 2026 snapshot.
Projected dividends for 2026-Q3 dropped $0.37 per share during the intervening month. The CME Group's S&P 500 dividend futures for the quarter dropped to $20.65 per share as higher dividends that previously had been expected failed to materialize. Even so, the dividend futures data indicates 2026-Q3's dividends will still register around a dollar per share increase year-over-year by the time 2026-Q3's dividend futures contracts finally expire.
Meanwhile, the outlook for more distant future quarters was more positive. Dividends in each of the upcoming quarters from 2026-Q4 through 2027-Q3 now expected to be slightly higher than what dividend futures forecast last month.
Here is the summary of how the expected future for the S&P 500's dividends changed between 14 August and 14 September 2026:
- 2026-Q3: Decrease of $0.37 to $20.65 per share
- 2026-Q4: Increase of $0.06 to $21.61 per share
- 2027-Q1: Increase of $0.04 to $22.85 per share
- 2027-Q2: Increase of $0.06 to $21.66 per share
- 2027-Q3: Increase of $0.04 to $21.85 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. The next quarter of 2026-Q4 will then begin on Saturday, 19 September 2026 and will run through Friday, 18 December 2026.
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.
Labels: dividends, forecasting, SP 500
The trading week ending Friday, 11 September 2026 saw the S&P 500 (Index: SPX) drop 0.8% below the preceding week's close. The index dropped to 7,656.98, which is 1.8% below its all-time record high close from 13 August 2026.
The week's biggest catalyst driving the outlook for investors was news of inflation, which saw the Consumer Price Index come in at an annualized value of 3.4%, the same a month earlier. That news locked in expectations the Fed will hike interest rates when after it meets 15 and 16 September 2026. The CME Group's FedWatch Tool now foresees four quarter point rate hikes in the Federal Reserve's future. The increases above the Federal Funds Rate's current target level of 3.50-3.75% are expected to come at 12 week intervals, starting on 16 September (2026-Q3), and repeating on 9 December (2026-Q4), 17 March (2027-Q1), and 28 July (2027-Q2).
The news also eliminated all the remaining uncertainty for what the Fed would do in this next week, giving the index some room to rebound on Friday, which saw the S&P 500 rise a little under 0.9% above its Thursday close.
The latest update of the alternative futures chart shows the trajectory of the Samp;P 500 is consistent with investors focusing on 2026-Q4, with the index running in the lower end of its expected range for that time horizon.
Here are the week's market-moving headlines:
- Tuesday, 8 September 2026
-
- Signs and portents for the U.S. economy:
- Oil hits multi-week highs after Houthi attacks on Saudi energy facilities
- OPEC+ keeps oil output policy unchanged for October
- Why isn't oil above $100 despite supply disruptions?
- Copper Hits All-Time-High As Physical Economy Reprices Scarcity
- Bigger exports, bigger trouble, bigger stimulus developing in China:
- China's exports surge as demand for high-tech, AI help prop up economic growth
- China's car exports stay strong in August but domestic sales decline more
- China's Provinces Show Evidence Of Financial Pressure And The Economy's Imbalances
- Chinese Oil Demand Unexpectedly Soars, Sending Shanghai Crude Above $100, With Brent Prices Set To Follow
- China says it will pump $54 billion into banks and insurers — but their stocks still fell
- BOJ minions getting excited to hike interest rates again as continuing effort to prop up yen:
- Yen holds near seven-month high as US dollar steadies
- Japan may have sold U.S. Treasuries to fund yen intervention
- Japan's August foreign reserves post largest-ever drop after record intervention
- The yen's sudden surge upsets the carry trade faithful
- Why the BOJ will bet small on rate hikes now to avoid a bigger shock later
- S&P 500 falls as AI worries hit software makers
- Wednesday, 9 September 2026
-
- Signs and portents for the U.S. economy:
- 10-year yields highest since 2023
- Brent crude rises above $100 a barrel as Middle East conflict intensifies
- Fed minions maybe only going to hike interest rates once before the end of 2026?
- Bigger trouble, stimulus developing in China:
- U.S. Treasury Secretary claims asymmetric information advantage for supporting Japan's currency:
- S&P 500 ends down as oil tops $100 per barrel
- Thursday, 10 September 2026
-
- Signs and portents for the U.S. economy:
- US producer prices increase as expected in August; but key details firmer
- China buys 1 million tons of US soybeans ahead of Xi visit, sources say
- Brent holds above $100 as tanker attacks dampen hopes for Hormuz traffic recovery
- Rate-Hike Odds Spike As Fuel Costs Push US Producer Prices Higher
- OPEC oil output falls in August, Reuters survey shows
- US holiday retail sales growth set to accelerate, Deloitte says
- US existing home sales drop to 14-month low in August
- Bigger trouble, stimulus developing in China:
- BOJ minions describe what gets them excited to hike Japan's interest rates:
- BOJ policymaker warns of price risks that may trigger rapid rate hikes
- BOJ to raise rates to 1.25% this month, reach 1.75% faster than expected
- ECB minions deliver Eurozone interest rate hike as expected, think they won't see more inflation because of it:
- ECB raises interest rates, bolstering bets for further moves
- ECB raises interest rates to fight off inflation jump
- ECB governors see more tightening ahead with October in play
- ECB is focusing on the wrong risk of $100 oil
- S&P 500 ends down as Treasury yields rise and traders fret about inflation
- Friday, 11 September 2026
-
- Signs and portents for the U.S. economy:
- Inflation stayed hot in August with annual pace of 3.4%, raising the odds of a Fed hike
- US 10-year borrowing costs pull back from 5% in reprieve for Bessent
- Oil falls but on track for 8% weekly gain on supply concerns; US diesel hits record high
- Fed minions expected to deliver September 2026 rate hike:
- BOJ minions coyly excited to hike Japan's interest rates in next week, coordinating with U.S. on support for currency:
- BOJ set to lift rates next week but offer few clues on terminal, sources say
- Japan to maintain close communication with US on currency markets, Katayama says
- ECB minions starting to think interest rate hikes in Eurozone might hurt economic growth, say it won't stop them from more hikes:
- ECB's Makhlouf says further rate hikes could hurt economic growth
- ECB policymakers open door to more rate hikes on energy risk
- S&P 500 ends higher as strong inflation data cements rate-hike bets
The Atlanta Fed's GDPNow tool's forecast of real GDP growth for the U.S. economy in 2026-Q3 dipped to +4.4, declining from the +4.7% annualized growth it projected a week earlier.
Image credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon of a Wall Street bull and bear who are happy to see news the Federal Reserve will hike interest rates in September 2026 as the bull says 'THANK GOODNESS. MAYBE WE CAN FINALLY START LOOKING PAST THIS MONTH'". That's a bit of satire aimed at analysts who somehow think investors weren't already looking past the end of the current quarter and factoring in the potential for more rate hikes before the end of the year in their investing decisions.
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.
Labels: ideas, personal finance
The total value of goods exchanged between the U.S. and China recorded by the U.S. Census Bureau increased for a third consecutive month in July 2026, the first such sustained rise since June-October 2024. Even so, the total level of trade between the two nations remains far below its pre-2025 tariff war levels.
Between June and October 2024, the combined value of U.S. imports from China and exports to China rose $3.2 billion to reach $55 billion. U.S.-China trade has risen $6.5 billion since April 2026 to total $36.8 billion in July. The value of trade between the U.S. and China during July 2026 is 33% less than it was in October 2024.
These figures are based on the Census Bureau's monthly trade data, which doesn't take any seasonality in the data into account. Applying a trailing twelve month average to the monthly data however confirms the large drop that has occurred in trade between China and the U.S., dropping from $48.4 billion in October 2024 to $31.4 billion in July 2026, a 35% decline.
The following chart reveals how the flow of trade between the U.S. and China has developed from January 2017 through July 2026.
The increase from April through July 2026 comes as U.S. President Donald Trump and Chinese Premier Xi Jinpeng are set to meet to discuss trade and other issues in the U.S. in September 2026. Curiously, the New York Times has chosen to focus on China's dominant trade position with the rest of the world ahead of that summit.
When President Trump and China’s leader, Xi Jinping, meet this month to talk about a fragile trade truce, one issue is sure to dominate: China’s seemingly unstoppable export engine.
Chinese customs data released on Tuesday showed that in August, China sold $119.09 billion more in goods to the world than it bought — the fourth month in a row that the trade surplus topped $100 billion. Exports rose by 25 percent in U.S. dollar terms, while imports climbed 28 percent.
So far this year, the trade gap is already over $800 billion, and economists say it is on track to surpass last year’s $1.2 trillion, a milestone no other country has reached.
According to U.S. data, China's year-to-date trade gap with the U.S. is over $91.2 billion through July 2026, which as the data demonstrates, is much lower than it would have been without 2025's tariff war.
Why then has China's trade position strengthened so much everywhere else? China's domestic economy has struggled to grow during the past year, in part because of its tariff war with the U.S. With domestic demand not able to absorb their production surpluses, Chinese producers have redoubled their exporting efforts to virtually everywhere else in the world. The New York Times recently emphasized the role that China's economic weakness has played in contributing to China's global trade dominance:
The widening trade gap is also a symptom of lackluster demand at home. Consumer spending has been weak, while youth unemployment remains high. In August, China’s imports slowed, even as the country imported more high-tech goods like automatic data processing machines and semiconductors. Foreign car shipments have fallen around 20 percent so far this year as domestic brands dominate the market.
Looking forward, the article also references China's official trade data for August 2026, which is reported about a month before the U.S. data for the same month becomes available. That data indicates China's exports to the U.S. increased, which in turn, suggests the total value of U.S.-China trade increased for a fourth consecutive month.
Image credit: View of Port of Los Angeles with container ships by Downtowngal on Wikimedia Commons. Creative Commons Attribution-Share Alike 4.0 International.
Labels: trade
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