In the last decade, proof assistants have revolutionized how mathematicians establish whether a mathematical theory is valid. One in particular, called Lean, has risen to the forefront of the field. Its productivity-enhancing capabilities are behind several notable proofs that have been demonstrated in recent years. What's more, its integration with Artificial Intelligence technologies is contributing to a rapid pace of new advances in solving long standing but, until this year, unproven mathematical conjectures.

Much of that story is now being told in Kevin Hartnett's The Proof in the Code: How a Truth Machine Is Transforming Math and AI, which is proving to be an exceptionally well-timed book. It captures the short history of the rapid development of the Lean proof assistant, which is contributing to a revolution in how mathematicians do what they do. Hartnett's book is all the more remarkable because it doesn't require readers to have an extensive background in either mathematics or computer science to both follow the story and understand its significance.

And what a significant story it is. Here's a short summary of how Lean is aiding the advancement of mathematics:

• It can automate large parts of the process of verifying mathematical proofs.
• Its results are trustworthy. If a proof is validated in Lean, the proof will hold.
• Its library allows mathematicians to take advantage of proofs developed by their peers, using them as modular building blocks to help establish new proofs.
• It can handle very sophisticated proofs developed by the world's top mathematicians.
• It facilitates massive collaboration efforts among mathematicians and other contributors.
• It can be integrated with Artificial Intelligence (AI) technologies, such as reinforced learning techniques and Large Language Models, which can both improve its performance and the performance of the AI systems.

All of these things underscore the role of Lean as an amazing productivity enhancing software tool for its users. But it wasn't always that way. Perhaps even more remarkably, Lean didn't start out as a tool aimed at helping mathematicians.

As originally envisioned by Leo De Moura, who wrote several generations of its code, it was supposed to be a tool to help software developers find and fix bugs before they released software. A chance discussion with mathematician Jeremy Avigad in 2013 identified the potential of the code to aid mathematicians in developing their proofs, which ultimately set the direction of Lean in motion.

The rest of Harnett's story is how their efforts pulled in other key players who recognized its potential and worked through its development challenges to make it useful first, then to make it more and more capable.

For a story that involves many mathematicians and the math they were seeking to validate, Harnett keeps mathematical equations and symbols to a minimum. That's a vital requirement in making the story accessible to a general audience and showcases Hartnett's experience as a math writer. It's not until the later chapters of the book that mathematical statements begin appearing in the text, which are backed by plain language descriptions, which makes them very approachable.

Even more remarkably, the only example of Lean code that Hartnett presents is contained in the book's Appendix. The example is Lean's version of a 2,300+ year-old proof by Euclid that confirms there is always another larger prime number, which is to say that prime numbers extend into infinity.

If you only read one math book this year, this is the one to read. It's a fantastic introduction to some of the most advanced happenings going on in the world of maths and computer science and what is becoming possible today that hasn't been before. Highly recommended!

If you're interested in more discussion, Hartnett was recently interviewed by Breaking Math's Autumn Phaneuf and Noah Giansiracusa. Here's the video of the interview:

A bottom is now in for trade between the U.S. and China. It might even be *the* bottom for the global tariff war that began in April 2025 but only time will confirm that for sure.

What we do know for sure is that May 2026 saw the first increase in the rolling twelve month average of the value of goods exchanged between the U.S. and China for the first time since January 2025.

There are also indications that trade will continue to increase in the short term. A report out of China hints at what will be happening:

China-US goods trade totaled 2 trillion yuan ($294.1 billion) in the first half of the year, accounting for 7.9 percent of China's total foreign trade, with the second quarter rebounding to 13.7 percent growth after an 18.7 percent slump in the first quarter, a customs official said at a press conference on Tuesday in response to a question about the recent acceleration in Chinese exports to the US and the outlook for bilateral trade in the second half.

A Chinese expert said that the recovery of China-US goods trade in the second quarter, with bilateral trade turning positive after a decline in the first quarter, was mainly driven by improving trade relations and lower US tariff rates, which helped restore market confidence and support trade activity.

The apparent improvement is being acknowledged after President Donald Trump traveled to China to meet with Chinese leader Xi Jinping in May 2026. The relative truce between the two nations in their tariff war so far seems to be holding with Trump restraining trade hardliners in his administration for now.

Meanwhile, other news out of China this month points to a concrete reason behind why trade between the U.S. and China will continue to increase in the next few months:

Chinese exporters gained an advantage as U.S. retailers moved up their orders by four to six weeks to build inventory for Black Friday and Christmas sales ahead of anticipated tariff increases later this year.

Typically, goods shipped to support the Christmas holiday season in the U.S. peak in October 2026. Moving those shipments up by four to six weeks could move the annual seasonal peak to either August or, more likely, September 2026.

On the U.S. side of U.S.-China trade data, the following chart shows the uptick off a bottom for the trailing twelve month average of the value of goods exchanged between the U.S. and China in May 2026, with April 2026 representing a bottom (if not definitively yet *the* bottom):

That bottom also confirms our prediction from last month that "the level of trade between the U.S. and China will bottom in 2026-Q2. It may already have, but we won't get the data to confirm it for at least another month or two."

It looks like it had already happened at that writing!

References

U.S. Census Bureau. U.S. International Trade in Goods and Services (FT900). U.S. Trade in Goods with China, Not Seasonally Adjusted, Nominal Figures, Total Census Basis. [Online database]. Accessed 7 July 2026.

Image credit: A close up of Scrabble tiles spelling the words China, USA, and TariffsPhoto by Markus Winkler on Unsplash.

The dividend outlook of the S&P 500 (Index: SPX) had a mixed outcome for July 2026. Dividends expected to be paid in the now current quarter of 2026-Q3 are slightly lower than what had been anticipated in our June 2026 snapshot, but the farther future showed continued strength as expected dividends increased.

Here is our tally of how the expected future changed in the month from 15 June to 14 July 2026:

• 2026-Q2: Finalized at $20.72 per share (unchanged).
• 2026-Q3: Decrease of $0.09 to $21.13 per share
• 2026-Q4: Increase of $0.05 to $21.45 per share
• 2027-Q1: Increase of $0.18 to $22.75 per share
• 2027-Q2: Increase of $0.19 to $21.75 per share

The following chart shows how expectations for the S&P 500's quarterly dividends per share changed in the month from 15 June 2026 to 14 July 2026.

The initial estimate of dividends expected to be paid out to S&P 500 investors for the distant future quarter of 2027-Q3 is $21.92 per share.

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 to its end.

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 return of warm weather marks the end of soup season, and in 2026, it also marked the end of the big discounts that several grocery selling retailers were offering for Campbell's Tomato Soup in April.

Since then, prices for Campbell's second-best selling soup have risen at four of the ten major national and regional grocery stores and retailers that we track. Even so, the rolling twelve-month average price of a 10.75 fluid ounce can of Campbell's Condensed Tomato Soup declined in the three months since our previous snapshot, from $1.15 to $1.12 per can.

That's a consequence of the higher prices of early 2025 being replaced by the much lower sale prices of early 2026 in the trailing year average. Even though prices at some retailers have risen since our last snapshot, they are generally lower than what we recorded in early 2025.

Here are the prices we observed at our ten purveyors of Campbell's Condensed Tomato Soup and how they've changed since our April 2026 snapshot:

• Walmart: Increase of $0.28 to $1.28 (+28.0%).
• Amazon: Increase of $0.28 to $1.28 (+28.0%).
• Kroger: Increase of $0.50 to $1.50 (+50.0%).
• Walgreens: Unchanged at $1.99 (0.0%).
• Target: Unchanged at $1.29 (0.0%).
• CVS: Unchanged at $2.49 (0.0%).
• Albertsons: Unchanged at $1.49 (0.0%).
• Food Lion: Increase of $0.25 to $1.25 (+25.0%).
• H-E-B: Unchanged at $1.24 (0.0%).
• Meijer: Unchanged at $1.29 (0.0%).

The following chart presents the price history of Campbell's Condensed Tomato Soup in the 21st century, from January 2000 through July 2026:

Since 2022, the price of $1.00 has become the floor for an iconic can Campbell's Tomato Soup. Before 2022, that price level was the ceiling. For more history, you can find our collected price data extending back to January 1898 here.

Image Credit: Three Campbell's Tomato Soup Mugs on a Reflective Surface photo by Tina Morris on Unsplash.

The S&P 500 (Index: SPX) rose 1.2% over its previous week's close, ending at 7,575.39 at the close of trading on Friday, 10 July 2026.

Although in a new quarter, the next earnings season hasn't yet gotten underway, making the week one in which there was little news from companies to influence their outlook. But this week was also notable because there was also a notable lack of new information for investors to absorb from Federal Reserve officials.

That's by design because of one of the first major policy initiatives of the Fed's new boss, Kevin Warsh. On 1 July 2026, Warsh put a new policy into action of not providing much, if any, forward guidance for markets when the Fed announces how it will set the Federal Funds Rate.

That changes how the Fed has operated since the 2008-2009 recession, when it initiated its policy of providing forward guidance to reduce surprises in markets from the Fed's actions and to stabilize them.

In any case, investors responded by sending the S&P 500 higher, but well within the trajectory of the redzone forecast range added to the redzone forecast range of the alternative futures chart in the previous edition of this S&P 500 chaos series. In the latest update of the chart, we've rolled the chart forward to show the dividend futures-based model's projections for all of 2026-Q3.

The trajectory of the S&P 500 remained well within the redzone forecast range, which indicates there was little that happened in the week that was to influence the future outlook of investors. Here is what passed for the trading week's marketing moving headlines:

Monday, 6 July 2026

• Signs and portents for the U.S. economy:
• For one small business, AI was key to a quick start and expansion
• Oil prices settle at pre-Iran war levels as crude output grows
• OPEC+ approves further oil output increase as Hormuz exports start to recover
• US oil companies see big profit jump, gird for clash over pump prices with Trump
• Fed minions say U.S. has risk of high inflation, also say main contributor to recent inflation has weakened:
• Waller says risks in US tilted towards high inflation
• NY Fed says supply chain pressures eased in June
• Bigger trouble, stimulus developing in China:
• The largest destruction of household wealth in recorded history, executed deliberately by the state that owned it
• China's June services activity slows slightly, private PMI shows
• China to cut domestic retail gasoline and diesel prices to near pre-war levels
• Chinese independent refiners snap up discounted Mideast oil as supplies rise
• Saudi Arabia Sells Oil At A Discount For The First Time Since COVID Crash, As China Demand Collapses
• BOJ minions get wage hike inflation they've sought, bigger trouble developing in Japan:
• Japan wage hikes top 5% for third year, backing BOJ rate-hike path
• Wage Growth As A Leading Inflation Indicator
• Japan Bankruptcies Surge To All-Time High As A Result Of Plunging Yen
• Bigger trouble developing in Eurozone:
• Euro zone services sector contraction eased in June as inflation cooled sharply, PMI shows
• ECB's Schnabel says Iran shock is not over
• S&P 500, Nasdaq close sharply higher as Broadcom rallies
• Dow posts record close, Nasdaq leads Wall Street higher as semiconductor stocks rebound

Tuesday, 7 July 2026

• Signs and portents for the U.S. economy:
• Walmart rolls back barbecue staple prices after Trump says retailer acted at his request
• US May trade deficit widens as capital goods imports hit record high
• Oil gains after vessels attacked near Strait of Hormuz
• US service sector growth dips in June; employment rebounds after months of contraction
• Fed minions notice fall in oil prices:
• Fed's Williams tells Fox Business he's more sanguine on inflation due to energy price retreat
• Bigger trouble, stimulus developing in China:
• China's booming gig economy masks job market pain, strains welfare system
• China smartphone sales drop 13% during 618 festival as memory costs limit discounts
• Beijing is looking at curbing overseas access to China's top AI models, sources say
• China's DeepSeek developing its own AI chip, sources say
• ECB minions worried about weak Eurozone economy, say region's banks need to do more to fend of cyber attacks:
• ECB's Panetta says economic outlook remains fragile
• Europe's central banks may come under political pressure, Panetta says
• ECB tells banks to bolster AI cyber defences as peers take lighter approach
• Wall Street closes lower on oil spike, chip rout
• Nasdaq sinks as AI worries hit chipmakers

Wednesday, 8 July 2026

• Signs and portents for the U.S. economy:
• Oil rises more than a dollar per barrel as US launches fresh strikes against Iran
• US reinstates sanctions on Iranian oil sales after LNG, oil tanker attacks
• US diesel futures post biggest daily gains in four years after Russia bans exports
• US container imports jumped 8% in June ahead of higher fuel costs and tariff increases
• German exports rise unexpectedly in May on surge in US demand
• Fed minions worried about inflation last month:
• Fed policymakers' inflation concerns grew at June meeting, minutes show
• Bigger trouble, stimulus developing in China:
• China car sales fall again, exports stay strong
• China's central bank pledges to maintain accommodative policy amid weak demand, external shocks
• Some BOJ minions want to see demand-driven inflation before signing off on next rate hike:
• EXCLUSIVE BOJ dissenter Asada needs demand-driven inflation before backing rate hike
• S&P 500 ends down after Trump says Iran deal is 'over'
• Wall Street finished mixed as U.S.-Iran tensions flared back up
• Crude oil-to-S&P 500 ratio falls to levels last seen in 1998, Covid

Thursday, 9 July 2026

• Signs and portents for the U.S. economy:
• US power companies scramble to secure equipment as surging data center demand strains supplies
• US existing home sales fall as house prices hit record high
• Oil prices settle 2% lower as economic worries outweigh supply risks
• Fed minions talking about how they do business, think oil prices on track to come down:
• Fed's Logan says Fed open market operations would benefit from voluntary central clearing
• Fed's Perli reiterates flexible path of reserve management buying
• Fed's Warsh taps broad group of Fed outsiders to oversee review
• Fed's Williams expects energy prices to abate even as Iran war flares
• Bigger trouble, stimulus developing in China:
• China's producer inflation jumps to 4-year high, squeezing manufacturers
• China lifts fuel export curbs for July, sources say
• BOJ minions getting excited to hike Japan's interest rates again with inflation fears, Japan's bond market somewhat less impressed as BOJ minion independence under question:
• Bank of Japan sees growing inflation pressures from Iran war
• Japan bond market signals waning faith in inflation, government's fiscal management
• Japan benchmark bond yield extends rise after hitting 30-year high
• Japan considers policy wording change as BOJ independence fears roil bonds
• ECB minions learn their rate hikes didn't constrain Eurozone inflation:
• ECB saw inflation rising despite nearly three expected hikes, accounts show
• Nasdaq ends sharply higher; chip surge offsets Iran worries
• Wall Street finished higher despite Middle East concerns
• Higher bar for corporate profit growth poses challenge for US stock market rally

Friday, 10 July 2026

• Signs and portents for the U.S. economy:
• American household exposure to stocks sees record high
• Fed minions worrying about inflation:
• Fed officials fret over inflation risk, weigh rate hikes
• Fed report cites 'stepped-up' inflation due to tariffs, Iran war, AI buildout
• Bigger trouble developing in Japan, BOJ minions licking at their chops to hike interest rates again:
• Japan's wholesale inflation spikes as fuel costs, weak yen bite
• Bigger trouble developing in Eurozone, ECB minions thinking about repeating what they've been doing since the Iran war started:
• German automakers hit by sharp China sales drop in second quarter
• ECB is back to square one as US-Iran war resumes
• S&P 500 logs weekly gains as Energy and Technology stocks lead
• Wall Street ends higher as investors turn to earnings season
• American household exposure to stocks sees record high

The CME Group's FedWatch Tool still projects the Fed will hike the Federal Funds rate by a quarter point to a target range of 3.75-4.00% after the Fed meets on 16 September (2026-Q3). Beyond that date, the FedWatch tool forecasts another quarter point rate hike on 27 January (2027-Q1).

The Atlanta Fed's GDPNow tool's estimate of real GDP growth for the U.S. economy in the current quarter of 2026-Q2 ticked up to +1.3% from the previous week's real growth estimate of +1.2%.

Image credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon that shows a Wall Street bull and bear looking at the new chief of the Federal Reserve holding a folder that says CHANGES IN FORWARD GUIDANCE POLICY with the bull asking 'WHY AREN'T THEY SAYING WHERE THEY'RE GOING?'", which we had to follow up with a second prompt: "Make the chief of the Federal Reserve look more like Kevin Warsh".