From First Principles

Claude did not solve the Riemann Hypothesis. But what it actually did may be one of the clearest examples yet of how rapidly AI systems are changing the way difficult mathematics can be attacked.
In Episode 53, Lester Nare and Krishna Choudhary go from first principles on arguably the most famous unsolved problem in mathematics.
We begin with Euler and the Basel problem, build the Riemann zeta function from the ground up, explain its deep connection to prime numbers, move into the complex plane and analytic continuation, unpack the famous 1 + 2 + 3 + 4 + … = -1/12 result, and finally arrive at the Riemann Hypothesis itself: the claim that every non-trivial zero of the zeta function lies on the critical line.
Then we get into Claude.
An unreleased Anthropic model was prompted to take a serious run at the problem. It orchestrated roughly 60 autonomous sub-agents, tested hundreds of mathematical approaches, executed code, searched academic literature, challenged its own strategies, created adversarial referees to attack its work, and ultimately produced a result pushing a related mathematical bound well beyond the previous state of the art.
The human behind the prompt was not a mathematician. One of his instructions was essentially: believe in yourself.
We explain what Claude actually accomplished, what it absolutely did not accomplish, why moving a bound toward two-thirds does not mean the Riemann Hypothesis is “two-thirds solved,” and what the process tells us about agentic AI, mathematical research, scientific discovery, and AI safety.
Then it’s transfer season.
For the first FFP Summer Transfer Window for Scientists, we look at prominent researchers leaving American institutions for universities and research centers abroad. Using the language of football transfers, we examine major moves in chemistry, battery research, gravitational-wave astrophysics, and neuroscience—and what they reveal about research funding, immigration, scientific infrastructure, and the global competition for talent.

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In this anniversary episode, Lester Nare and Krishna Choudhary look back at how two longtime friends turned their regular conversations about science into a show now shared by millions of people around the world, and what they hope to build with FFP Nation in Year Two.
Then we turn to a new Nature paper challenging the idea that the precolonial Amazon was sparsely populated. Airborne LiDAR revealed hundreds of geometric earthworks hidden beneath the rainforest canopy. Combining the new survey with earlier archaeological evidence, the researchers estimate that the region could contain more than 20,000 earthworks and may have supported 1.25–3 million people around AD 100–300.
Lester and Krishna explain how LiDAR sees through dense vegetation, why early European accounts of crowded Amazonian settlements were dismissed, how disease and forest regrowth could erase the visible traces of large societies, and what the findings mean for our understanding of the Amazon’s human and environmental history.
The conversation then becomes a thought experiment: if our civilization disappeared, what would future archaeologists—or extraterrestrial visitors—recognize as our pyramids? Apollo landing sites, CERN, LIGO, and the James Webb Space Telescope become candidates for the enduring signatures of a curiosity-driven civilization.
Finally, we christen the From First Principles library. Krishna shares the mathematics, physics, biology, history, and philosophy books that shaped how he thinks, including Baby Rudin, Landau–Lifshitz, Fermi, Jackson, Sakurai, Einstein, Schrödinger, Gibbs, Newton’s Principia, Plato, the Upanishads, and Adam Becker’s What Is Real?

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Research and show notes:
Over 20,000 precolonial earthworks in the Southwest Amazonia
Nature Research Briefing
FFP episode archive and research library

What happens when electronics can operate at temperatures hot enough to melt aluminum?
In this deep-dive episode, Lester Nare and Krishna Choudhary examine a new high-temperature memory device developed by researchers at USC, the Air Force Research Laboratory, Kumamoto University, and their collaborators.
Published in Science, the experimental memristor combines tungsten, hafnium oxide, and graphene. It operated reliably at 700°C—roughly 1,300°F—retained data for more than 50 hours, and survived more than one billion switching cycles.
We begin by explaining why conventional electronics and flash memory fail when temperatures rise. From deep-earth drilling and hypersonic aircraft to nuclear systems and the surface of Venus, many environments where intelligent electronics would be useful remain inaccessible to today’s hardware.
Krishna then builds the memristor from first principles. We explore the history of the “missing” fourth circuit element, how oxygen vacancies create low- and high-resistance memory states, why conventional platinum electrodes fail under extreme heat, and how graphene prevents tungsten atoms from diffusing through the device.
Finally, we examine the implications for artificial intelligence. Memristors can potentially store neural-network weights and perform matrix multiplication in the same physical location, reducing the energy wasted moving information between processors and memory.
Could that combination of heat tolerance and energy efficiency make AI data centers in space more practical? Lester and Krishna work through thermal radiation, radiator size, power consumption, radiation resilience, and the considerable engineering challenges that remain.

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Research and Show Notes
High-temperature memristors enabled by interfacial engineering
USC: A memory device that operates at 700°C
The development of carbon-neutral data centres in space
NASA Venus facts

Hosted by Lester Nare and Krishna Choudhary, this episode moves from astrobiology to science policy to the rapidly changing frontier of artificial intelligence and mathematics.
First, researchers analyzing pristine samples returned from asteroid Ryugu report all five canonical nucleobases used by DNA and RNA. We explain what that does—and does not—mean for the origin of life, how JAXA’s Hayabusa2 mission collected uncontaminated asteroid material, and why comparisons with NASA’s Bennu samples strengthen the case that prebiotic chemistry may be widespread across the Solar System.
Next, we examine the fight over who controls federal research funding. A proposed overhaul of the rules governing federal grants would give political appointees greater influence over awards, reduce the controlling role of expert peer review, and expand the government’s power to stop grants that no longer align with an administration’s priorities. We break down the roles of Congress, OMB, federal agencies, universities, and the courts—and why this dispute could reshape the American research ecosystem.
Finally, we go deep on an AI-assisted counterexample to the Jacobian conjecture, a major open problem in mathematics. Krishna explains coordinate transformations, Jacobian determinants, invertibility, special relativity, and why this result appears fundamentally different from simple brute force. We close with the growing debate over AI-generated mathematics, human verification, open science, attribution, and the future role of mathematicians.
Summary
All five canonical nucleobases found in pristine asteroid Ryugu samples
Hayabusa2, Bennu, and the possibility of widespread prebiotic chemistry
The fight over political control of federal research grants
Congress, OMB, peer review, and the American science-funding system
The Jacobian conjecture and an AI-assisted counterexample
Special relativity, coordinate transformations, and invertibility
AI-generated mathematics, open science, attribution, and verification

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Show Notes
A complete set of canonical nucleobases in asteroid Ryugu
OMB proposed federal-grant rule
Association of American Universities response
Levent Alpöge’s Jacobian counterexample announcement
Leiden Declaration on Artificial Intelligence and Mathematics
Human-verified remarks on the OpenAI-generated Erdős result

In this special interview episode, Lester Nare speaks with Juan Busso, Senior Football Data Scientist at FIFA, and Arron Ackerman, FIFA’s Team Lead for Football Performance Analysis, about the data science behind the Match Momentum visualization featured throughout the 2026 World Cup.
What does “momentum” actually mean in football—and how can it be measured without reducing the game to possession or shots? Juan and Arron explain how FIFA translates football principles into mathematical models, validates those models with coaches and technical experts, and turns complex tracking data into a graphic that fans can understand at a glance.
We break down the underlying “threat” model, including kinetic pitch control, player speed and acceleration, ball trajectories, defensive spacing, distance to goal, sight lines, and the creation of space. Match Momentum is calculated from player-tracking data captured 50 times per second, allowing the model to recognize when a team is becoming dangerous even without dominating possession.
The conversation also covers FIFA’s wider data ecosystem—including event data, skeletal tracking, and the connected match ball—why offside positioning can still create threat, whether hydration breaks alter momentum, and the next generation of football analytics focused on player energy and physical effort.
Guests
Juan Busso — Senior Football Data Scientist, FIFA
Arron Ackerman — Team Lead, Football Performance Analysis, FIFA
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