The article gives the following explanation for this graph:

On that score, the big AI companies seem to think they are close to AGI. One giveaway is reflected in their own hiring practices. According to Zeki Data, the top 15 US AI companies had been frantically hiring software engineers at a rate of up to 3,000 a month, recruiting a total of 500,000 between 2011 and 2024. But lately their net monthly hiring rate has dropped to zero as these companies anticipate that AI agents can perform many of the same tasks.

I have my doubts about this, and I'm confident that we'll be able to replace at least 90% of all software engineers within the next decade. However, I can't think of any other plausible explanations. Progress in AI definitely looks more promising than it did in 2011, so a new AI winter cannot be the explanation.

Source: https://www.ft.com/content/49ad6c69-9b67-48da-a274-8408dfdaa7cb

No paywall: https://archive.is/c4lT0

The Pope actually chose the name Leo because of AI

Sources:
1. https://www.detroitcatholic.com/news/chilean-cardinal-gives-insight-to-the-conclave-that-elected-pope-leo-xiv
2. https://apnews.com/article/pope-leo-vision-papacy-artificial-intelligence-36d29e37a11620b594b9b7c0574cc358

Smarter people tend to be less violent: https://pubmed.ncbi.nlm.nih.gov/30058504/

See also:

1. Intelligence and criminal behavior: Examined a total birth cohort of Finnish males born in 1987 http://www.sciencedirect.com/science/article/pii/S016028961500077X

• Examined a total birth cohort of Finnish males born in 1987
• Lower levels of intelligence are associated with greater levels of offending.
• The IQ-offending association is mostly linear.
• Pattern is consistent across multiple measures of intelligence and offending.

2. Huge 2012 study involving longitudinal data for over 3 million individuals found that genetics explains about 50% of the variation in violent criminal behavior, while the family environment explains only about 15%. https://pubmed.ncbi.nlm.nih.gov/21761238/

Why Small Probabilities (Often) Add Up

When a risk 'p' is tiny (e.g., 1-in-a-million), the chance it happens twice independently (p^2) is minuscule (1-in-a-trillion). This "tiny-squared" p^2 is often so small we can ignore it, making simple addition a good shortcut.

🤔 If I face a risk 'p' twice, is my danger 2p?

Math time!
Prob. event never happens in 2 tries: (1-p)^2
Prob. it happens at least once: 1 - (1-p)^2
Expanding this: 1 - (1 - 2p + p^2) = 2p - p^2

Let's break 2p - p^2 down:
* The 2p part (from p+p) is what you get from simply "adding the two risks."
* The p^2 part is the chance the event happens in both tries (the "overlap").
Why is p^2 subtracted? When you add the first p (chance for try 1) and the second p (chance for try 2):
* The first p includes cases where it also happens in try 2.
* The second p also includes those same cases where it also happened in try 1.
So, this p^2 overlap gets counted twice in the simple 2p sum. Subtracting p^2 once corrects this double count, giving the true chance of it happening at least once.

✨ When 'p' is tiny (e.g., 1-in-a-million, or 10^{-6}):
p^2 = (10^{-6})^2 = 10^{-12} (1-in-a-trillion!).
This p^2 is lost in the noise of real-world uncertainties. So, 2p is a great approximation for 2p-p^2.

😬 When 'p' is large (e.g., p=1/2 for a coin flip):
p^2 = (1/2)^2 = 1/4. This is NOT negligible!
Naively adding (2p = 2 * 1/2 = 1 or 100%) is wrong. The true chance of at least one tail in two flips is 2p - p^2 = 1 - 1/4 = 3/4 (75%).

🌍 Micromorts Example (1-in-a-million risks):
Flight (1000 miles) ≈ 1 micromort.
Chest X-ray ≈ 1 micromort.
Added risk ≈ 1+1 = 2 micromorts.
The p^2 error here is (10^{-6})^2 = 10^{-12}, or 0.000001 micromorts. This tiny error is dwarfed by uncertainties in the original "1 micromort" estimates (is it really 1.0, or 0.9, or 1.1?). It's like a grain of sand on a beach.

---
Beyond Two Tries: Summing Many Probabilities (np)

What if you face a risk 'p' many times ('n' tries)?
Simply summing them (np) gives the average expected number of events.

Is np also the chance of it happening at least once?
➡️ Only if np is very small (e.g., < 0.1). Then np ≈ 1-(1-p)^n.
Why? The simple sum np is a first-order approximation. It ignores further "overlap" terms (involving p^2, p^3, etc.), similar to how 2p ignores p^2 for two tries.

Example: p=1/10^6, n=10^6 tries.
* Expected number np = 1.
* But actual chance of at least one event is 1-(1-p)^n ≈ 1 - e^{-np} ≈ 1 - e^{-1} ≈ 0.632 (or 63.2%).
The np=1 doesn't mean a 100% chance or that it will happen for certain in one set of a million tries; it means an average of 1 occurrence if you looked at many such sets. The difference (1 vs 0.632) is due to those overlap terms becoming significant when np isn't tiny.

Key Takeaway:
* For a few tiny risks (p_1, p_2), adding them (p_1+p_2) is a good estimate for the chance of at least one.
* For many tiny risks (n tries, probability p each):
* np = great for the average number of events expected.
* 1-(1-p)^n = the actual chance of at least one event.
* If np is very small (e.g., <0.1), then np is also a good estimate for 1-(1-p)^n.

This helps understand "once-in-a-lifetime" headlines: they often refer to the large expected number (np) of such rare events globally, implying a near certainty that many will occur across a large population or many locations.

Throughout 2024, France consistently emitted less CO2 than Germany. Even at its worst, France outperformed Germany at its best. This stark comparison serves as a reminder that the world could have largely mitigated climate change as early as the 1980s if all industrialized nations had followed France's lead and adopted nuclear power.

Taking action against climate change decades ago would have been far less costly than the measures nations are now pledging to implement. More importantly, it would have provided a sustainable solution that supported continued economic growth. Unfortunately, environmental activists opposed this technological pathway, hindering progress.

Some additional notes:

1. Nuclear waste isn't a problem.

- High-level nuclear waste, primarily spent fuel rods, is solid and compact. It is stored safely in secure facilities. Compared to waste from other energy sources, its volume is small, and it has a perfect safety record. Dry cask storage is safe, cost-effective, and proven, and it is used globally. Expensive projects like Yucca Mountain address a political fear rather than a safety concern and waste billions with no health benefit.

- While nuclear waste is radioactive, its radioactivity diminishes significantly over time. It decreases by over 99% in just 40 years. In contrast, many industrial wastes (e.g., mercury and arsenic) remain toxic indefinitely.

- Spent nuclear fuel still contains over 90% of its potential energy and can be recycled into new fuel, a common practice in some countries.

2. Nuclear fuel isn't limited.

- Current reactors use only a small fraction (about 1%) of the potential energy in uranium, which leads to inefficient fuel use. Breeder reactors can utilize nearly 100% of uranium by converting U-238 into Pu-239, which significantly extends fuel reserves.

- Known uranium reserves (6.15 million tons) and thorium reserves can power humanity for hundreds of years using breeder technology. Additional resources exist in unconventional sources like granite, phosphate deposits, and seawater. Seawater holds about 4.2 billion tons of uranium, replenished naturally, offering fuel for millions of years. The Earth's crust and oceans contain enough fissionable material to sustain nuclear power for billions of years.

3. Nuclear power is safe.

- With 440 operating reactors globally, many of which have been in operation for over 40 years, as well as numerous others powering ships and submarines, nuclear energy has an exceptional safety record.

- There have only been three significant accidents in its history, all with minimal casualties. For example, the Fukushima incident caused no direct deaths, and the United Nations confirmed no significant health impacts from the event.

4. Nuclear power is efficient and affordable.

- The real obstacle is overregulation, not the technology itself. On average, it takes six to eight years to build a nuclear reactor worldwide, though some are completed in as little as three to five years.

Many people still keep coping that AI is about to hit a wall. But all the leading AI forgers keep disagreeing. Some statements from the last few days:

1. Google Chief Scientist, Jeff Dean: We will have AI systems operating at the level of junior engineers within a year.

Source: https://youtu.be/dq8MhTFCs80?si=HmvJS_HS4EVkxkou&t=1545

2. Sam Altman predicts 2025 will be the year AI agents do real work, especially in coding. By 2026, AI's will help make major scientific discoveries, driving the next wave of economic growth. In 2027, these breakthroughs will move into the physical world, with robots shifting from novelty to serious creators of economic value.

Source: https://youtu.be/ctcMA6chfDY?si=Nka_jmdXrR86_yXK&t=1691

3. Anthropic Lead Engineer, Boris Cherny: "about 80–90% of the code we use is written by Claude"

Source: https://youtu.be/zDmW5hJPsvQ?si=fv-wDIM1t5pRfmw3&t=1092

And no, Google doesn't need to say this to get funding. Indeed, AI is the biggest threat to their main business model, and it would be great for them if it failed.

Mathematician Terence Tao, a recipient of the Fields Medal, now has a YouTube channel: https://www.youtube.com/@TerenceTao27

In his latest video, he tests Claude 3.7 Sonnet and o4-mini-high's ability to formalize a proof in Lean. He concludes that LLMs like Claude and o4 can accelerate the process of formalizing proofs in Lean. They can successfully convert informal proof steps into Lean code.

THE “NAMES IN BOXES” (100 PRISONERS) PUZZLE

INTRODUCTION: FROM HOPELESS TO HOPEFUL

Imagine a life‑or‑death puzzle where the odds are so astronomically stacked against you that success seems impossible. Yet a simple, coordinated strategy can transform this near‑certain failure into a significant, very real chance of success. This walk‑through explains the puzzle, the clever plan, and the mathematics that turn despair into opportunity.

1. THE RULES IN ONE BREATH

- 100 prisoners are numbered 1–100.
- 100 boxes are numbered 1–100. Each box hides exactly one prisoner’s number (all numbers appear once).
- One prisoner at a time may enter the room and open up to 50 boxes.
- If every prisoner finds their own number, everyone goes free; if even one fails, they all lose.
- The prisoners may agree on a strategy beforehand but cannot communicate once the process starts.

2. WHY “JUST GUESSING” IS HOPELESS

If a prisoner opens 50 random boxes, their chance of success is 50⁄100 = 1⁄2.

For all 100 prisoners to succeed this way the probability is (1⁄2)^100 ≈ 7.9 × 10^-31.

3. A SIMPLE, STARTLINGLY GOOD PLAN

Start at your own label. Prisoner k first opens Box k.

Follow the breadcrumb.

- If Box k contains number m (where m ≠ k), next open Box m.
- If Box k contains k, you’re done — success!

Continue until:

- you find your own number (success), or
- you have opened 50 boxes (failure).

That’s the whole strategy.

4. WHY “FOLLOWING THE ARROWS” WORKS (AN INFORMAL PROOF)

1) Arrow‑paths must form cycles. Picture each box label (1–100) with an outgoing arrow to the label found inside. After more than 100 steps you must revisit a label. Since each label has only one incoming arrow (from the unique slip pointing to it), the first label revisited in your path must be your starting label, closing a loop. Every box lies on exactly one such closed loop (cycle).
2) Your starting box is on the cycle you trace. Beginning at Box k and following arrows keeps you on that loop.
3) Your number is on the same cycle. If slip k were outside this loop it would send a second arrow into Box k, contradicting the “one slip each” rule.

Consequence: If the loop containing Box k has length L, Prisoner k will open L boxes and succeed as long as L ≤ 50. Therefore everyone succeeds iff every loop has length ≤ 50.

5. CALCULATING THE ODDS OF SUCCESS

5.1 The deciding factor: They win iff no cycle is longer than 50.

5.2 How likely are long cycles? For a random shuffle of 100 items, the probability that it contains a cycle of length K (> 50) is 1⁄K.

5.3 Only one long cycle is possible. Two cycles longer than 50 would need more than 100 boxes.

5.4 Chance of failure: Pr(failure) = (1⁄51) + (1⁄52) + … + (1⁄100) ≈ 0.688 (68.8 %).

5.5 Chance of success: Pr(success) = 1 − Pr(failure) ≈ 1 − 0.688 = 0.312 → about 31.2 %.

6. WHY THE CHANCE OF ONE LONG LOOP OF LENGTH K IS 1⁄K (K > 50)

How many of these shuffles contain a cycle of a particular length K (where K > 50)?

Step ①: Choose K numbers for the cycle. There are "N choose K" ways (or C(N,K)) to select which K numbers will form this cycle.

Step ②: Arrange these K numbers into a cycle. To form a cycle with K chosen numbers, fix one number (say, the smallest) in place. The remaining K-1 numbers can be arranged in (K-1)! ways to complete the cycle. (Fixing one prevents overcounting rotations as distinct cycles).

Step ③: Arrange the other N-K numbers. These remaining numbers can be arranged in the remaining N-K boxes in (N-K)! ways. These will form other, shorter cycles.

The number of shuffles containing a specific K-length cycle is the product of these choices:

Number of shuffles = C(N, K) * (K-1)! * (N-K)!

Since C(N, K) = N! / ( K! * (N-K)! ), this simplifies to:

C(N, K) * (K-1)! * (N-K)! = (N! / ( K! * (N-K)! )) * (K-1)! * (N-K)! = N! / K.

Step ④: Calculate Probability

The total number of possible shuffles is N!.

So, Pr(K-cycle) = (Shuffles with a K-cycle) / (Total shuffles)

= (N! / K) / N!

= 1/K.

US vs. Houthis

The Houthis nearly shot down several F-16s and an F-35 (despite its stealth capability).

So the US decided to end the campaign.

In those first 30 days, the Houthis shot down seven American MQ-9 drones (around $30 million each), hampering Central Command's ability to track and strike the militant group. Several American F-16s and an F-35 fighter jet were nearly struck by Houthi air defences, making real the possibility of American casualties, multiple US officials said.

That possibility became reality when two pilots and a flight deck crew member were injured in the two episodes involving the F/A-18 Super Hornets, which fell into the Red Sea from the aircraft carrier Harry S. Truman within 10 days of each other.

Source: https://www.nytimes.com/2025/05/12/us/politics/trump-houthis-bombing.html [no paywall: https://archive.is/0aTiI]

VDT: a solution to decision theory https://www.lesswrong.com/posts/LcjuHNxubQqCry9tT/vdt-a-solution-to-decision-theory

...no comprehensive decision theory that resolves all decision theory dilemmas has yet been formalized. This paper at long last resolves this dilemma, by introducing a new decision theory: VDT.

He went from being imprisoned by American forces to defeating Iran and Russia, and even meeting the President of the United States!

The guy might be good or bad. Either way, this demands recognition. An incredible success story!

Links for 2025-05-14

AI

1. Sakana AI presents Continuous Thought Machines, mimicking brain-like neural timing for dynamic, process-based reasoning. https://sakana.ai/ctm/

2. Seed-Coder: Let the Code Model Curate Data for Itself https://github.com/ByteDance-Seed/Seed-Coder

3. “In order to break past the "pre-training ceiling," we will need to continually collect and invent new tasks and environments, likely based on systems grounded to real-world applications with both humans and models in the loop.” https://x.com/MinqiJiang/status/1921176396228952253

4. How far can reasoning models scale? It appears that the rapid scaling of reasoning training, like the jump from o1 to o3, will likely slow down in a year or so. https://epochai.substack.com/p/how-far-can-reasoning-models-scale

5. Slow corporations as an intuition pump for AI R&D automation https://www.lesswrong.com/posts/hMSuXTsEHvk4NG6pm/slow-corporations-as-an-intuition-pump-for-ai-r-and-d

6. 9 Years to AGI? OpenAI’s Dan Roberts Reasons About Emulating Einstein https://www.youtube.com/watch?v=_rjD_2zn2JU

7. Tool-using LLMs can learn to reason—without reasoning traces. https://arxiv.org/abs/2505.00024

8. Person Recognition at Altitude and Range: Fusion of Face, Body Shape and Gait https://arxiv.org/abs/2505.04616

9. UniVLA: Learning to Act Anywhere with Task-centric Latent Actions https://arxiv.org/abs/2505.06111

10. Multi-agent Embodied AI: Advances and Future Directions https://arxiv.org/abs/2505.05108

11. DanceGRPO: Unleashing GRPO on Visual Generation https://arxiv.org/abs/2505.07818

12. MLE-Dojo: Interactive Environments for Empowering LLM Agents in Machine Learning Engineering https://arxiv.org/abs/2505.07782

13. Learning from Peers in Reasoning Models https://arxiv.org/abs/2505.07787

14. Dynamic Byte Latent Transformer: An alternative to traditional tokenization https://ai.meta.com/blog/meta-fair-updates-perception-localization-reasoning/

15. Cognitive Behaviors that Enable Self-Improving Reasoners, or, Four Habits of Highly Effective STaRs https://arxiv.org/abs/2503.01307

16. “Muscle Mem is a cache system for AI agents, allowing them to learn and efficiently replay complex behaviors.” https://www.youtube.com/watch?v=hToIl9PRyRk

17. OpenAI enables PDF export for detailed research reports with sources and rich formatting. https://x.com/OpenAI/status/1921998278628901322

18. Norway’s giant sovereign wealth fund said it would scale back hiring thanks to AI improvements. https://www.bloomberg.com/news/articles/2025-05-13/norway-wealth-fund-s-use-of-ai-triggers-hiring-freeze-ceo-says [no paywall: https://archive.is/HDb4t]

19. From 12 weeks to 10 minutes: How Novo Nordisk accelerates time to value with GenAI and MongoDB https://www.mongodb.com/solutions/customer-case-studies/novo-nordisk

20. Saudi Arabia and NVIDIA to Build AI Factories to Power Next Wave of Intelligence for the Age of Reasoning https://nvidianews.nvidia.com/news/saudi-arabia-and-nvidia-to-build-ai-factories-to-power-next-wave-of-intelligence-for-the-age-of-reasoning

21. Microsoft is offering to give up some of its equity stake in OpenAI's new for-profit in exchange for continued access to new models developed beyond 2030 https://www.ft.com/content/8d9e5149-7e4f-4886-a035-9d200204972a [no paywall: https://archive.is/1NIP8]

22. Republicans push for a decadelong ban on states regulating AI https://www.theverge.com/news/666288/republican-ai-state-regulation-ban-10-years

Miscellaneous

1. A protein from tiny tardigrades may help cancer patients tolerate radiation therapy https://news.mit.edu/2025/tiny-tardigrades-protein-may-help-cancer-patients-tolerate-radiation-therapy-0226

2. Apple to Support Brain-Implant Control of Its Devices https://www.wsj.com/tech/apple-brain-computer-interface-9ec69919 [no paywall: https://archive.is/Xw3Wo]

3. Mass spectrometry method identifies pathogens within minutes instead of days https://phys.org/news/2025-05-mass-spectrometry-method-pathogens-minutes.html

Google AlphaEvolve: a Gemini-powered coding agent for algorithm discovery.

It’s able to:

🔘 Design faster matrix multiplication algorithms
🔘 Find new solutions to open math problems
🔘 Make data centers, chip design and AI training more efficient across Google.

The system uses:

🔵 LLMs: To synthesize information about problems as well as previous attempts to solve them - and to propose new versions of algorithms
🔵 Automated evaluation: To address the broad class of problems where progress can be clearly and systematically measured.
🔵 Evolution: Iteratively improving the best algorithms found, and re-combining ideas from different solutions to find even better ones.

Over the past year, Google has deployed algorithms discovered by AlphaEvolve across Google’s computing ecosystem, including data centers, software and hardware.

It’s been able to:

🔧 Optimize data center scheduling
🔧 Assist in hardware design
🔧 Enhance AI training and inference

They applied AlphaEvolve to a fundamental problem in computer science: discovering algorithms for matrix multiplication. It managed to identify multiple new algorithms.

This significantly advances their previous model AlphaTensor, which AlphaEvolve outperforms using its better and more generalist approach.

Google also applied AlphaEvolve to over 50 open problems in analysis ✍️, geometry 📐, combinatorics ➕ and number theory 🔂, including the kissing number problem.

🔵 In 75% of cases, it rediscovered the best solution known so far.
🔵 In 20% of cases, it improved upon the previously best known solutions, thus yielding new discoveries.

Read more: https://deepmind.google/discover/blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/

Links for 2025-05-15

AI

1. Whole-Body Pose Control for Legged Manipulation https://www.youtube.com/watch?app=desktop&v=D0bvsit_jqE

2. AI headphones translate multiple speakers at once, cloning their voices in 3D sound https://www.washington.edu/news/2025/05/09/ai-headphones-translate-multiple-speakers-at-once-cloning-their-voices-in-3d-sound/

3. CrowdStrike CEO announces 5% of workforce to be slashed globally, citing artificial intelligence efficiencies created in the business https://www.theguardian.com/technology/2025/may/09/crowdstrike-to-cut-jobs-and-use-ai

4. Early evidence on human + Ai in accounting: “We document significant productivity gains among AI adopters, including a 55% increase in weekly client support and a reallocation of approximately 8.5% of accountant time from routine data entry toward high-value tasks such as business communication and quality assurance. AI usage further corresponds to improved financial reporting quality, evidenced by a 12% increase in general ledger granularity and a 7.5-day reduction in monthly close time.” https://marginalrevolution.com/marginalrevolution/2025/05/early-evidence-on-human-ai-in-accounting.html

5. LLMs Get Lost In Multi-Turn Conversation https://arxiv.org/abs/2505.06120

6. Revealing economic facts: LLMs know more than they say https://arxiv.org/abs/2505.08662

7. Why OpenAI projects only $174B of revenue by 2030? https://www.lesswrong.com/posts/Yhfgygybmkyfgs64k/untitled-draft-fqpt

8. Yudkowsky and Soares Announce Major New Book: “If Anyone Builds It, Everyone Dies” https://intelligence.org/2025/05/15/yudkowsky-and-soares-announce-major-new-book-if-anyone-builds-it-everyone-dies/

Miscellaneous

1. Ozempic is a miracle drug. This looks like another: SGLT2 inhibitors may be a miracle cure for kidney disease – but also for liver disease, dementia, respiratory diseases, and even, very possibly, for old age. https://www.worksinprogress.news/p/everything-drugs

Scientists have been publishing climate models since ~1970.

A good way to evaluate their skill is to compare what they expected to happen in the years after the model was published to observed climate changes.

It turns out most models were pretty spot-on

Source: https://x.com/hausfath/status/1922794856054702160

How to Spot and Lock-In “Free-Money” Gaps Between Betting Sites

---

1. What’s an arbitrage in prediction markets?

Think of buying a product cheaply in one shop and instantly reselling it at a higher price next door—guaranteed profit, zero risk.

In betting, the “product” is a bet on each possible outcome. The prices are given as decimal odds (for example, odds of 3.0 mean a €1 bet returns €3 if it wins, for a €2 profit). If the odds across different bookmakers for all possible outcomes don’t add up correctly, you can place bets on every outcome in the right proportions and finish with more money than you started—no matter which outcome occurs.

---

Lemma (Constant-payout stakes)

This lemma provides the formula for calculating stakes that ensure a constant profit across all outcomes, a core principle for achieving risk-free arbitrage.

Let O_i be the decimal odds on n mutually exclusive, exhaustive outcomes labeled 1 through n. The implied probability of outcome i according to the bookmaker is 1 divided by O_i. In a fair market without a bookmaker’s margin, the sum over i from 1 to n of (1/O_i) would equal 1.

An arbitrage opportunity arises when:

S = sum over i from 1 to n of (1/O_i) < 1.

If this holds, then for any total capital T > 0, staking:

B_i = T / (S * O_i) for i = 1, ..., n

yields the same payoff, equal to T/S, regardless of which outcome occurs.

Proof.

* Payoff for outcome i: B_i * O_i = T/S.
* Total stakes: sum over i=1 to n of B_i = (T/S) * sum over i of (1/O_i) = (T/S) * S = T.

This shows:

1. The payoff T/S is the same for any outcome.
2. The total amount staked equals the initial capital T.

---

Corollary (Arbitrage)

Profit = (T/S) - T = T * (1/S - 1) > 0 because 0 < S < 1.

---

Full proof that price discrepancies across markets generate arbitrage

Theorem (Cross-market prediction-market arbitrage)

Let M_1 through M_k be k betting venues on the same event with outcomes labeled 1 through n. For each outcome i, let O_bar_i be the best available odds—that is, the maximum odds for outcome i across all k markets. Define:

S_bar = sum over i from 1 to n of (1/O_bar_i).

If S_bar < 1, then risk-free profit is attainable:

1. Market aggregation. By construction, no other bookmaker offers better odds for outcome i than O_bar_i.
2. Stake selection. With budget T > 0, wager:
B_i = T / (S_bar * O_bar_i).
3. State-independent payoff. Whichever outcome occurs, return = B_i * O_bar_i = T/S_bar.
4. Positive profit. Initial outlay = T; payoff = T/S_bar. Profit = T * (1/S_bar - 1) > 0 if 0 < S_bar < 1.
5. Risk-free. Payoff is identical across all outcomes, so variance is zero and no outcome can eliminate the gain.

End of proof.

Month-by-month question volume for Mathematics Stack Exchange and Stack Overflow, with vertical dashed lines marking the public announcements of GPT-3 (June 2020), ChatGPT (November 2022), and o1-preview (Sep 2024).

(Note: I used o3 to both extract the data from Stack Exchange Data Explorer and to generate the graph.)

Google presents LightLab: https://nadmag.github.io/LightLab/

Controlling Light Sources in Images with Diffusion Models