• Text is split into tokens; each token becomes a long vector of numbers
• Attention: each token looks back at earlier tokens and pulls in what's relevant
• Dozens of layers of this; at the end, predict the next token
• Trained on enormous amounts of text, then trained further to reason and use tools

Anthropic, "Tracing the thoughts of a large language model" (March 2025)

• Open for 87 years
• Claude Fable 5 found a counterexample in dimension 3 (posted by Levent Alpöge, Anthropic)
• Jacobian determinant \(-2\), yet three points have the same image:
  \((0,0,-\tfrac14),\ (1,-\tfrac32,\tfrac{13}2),\ (-1,\tfrac32,\tfrac{13}2)\ \mapsto\ (-\tfrac14,0,0)\)

• The loop runs thousands of times, often with many agents in parallel
• Models are also trained on problems whose answers can be checked

• A qubit: spin up/down, or a photon's polarization
• In general: a unit vector in \(\mathbb C^d\), for a system with \(d\) distinguishable outcomes
• Multiplying \(\psi\) by a phase \(e^{i\theta}\) gives the same physical state

• Qubit \(\psi=(a,b)\): outcome 0 with probability \(|a|^2\), outcome 1 with probability \(|b|^2\)
• That is why states have length 1: the probabilities add up to 1
• It is the only place probability enters quantum theory

• Measuring one disturbs the other; no state has both sharp
• Discrete version in \(\mathbb C^d\): shift \(X\) and clock \(Z\), with \(ZX=\omega\,XZ\), \(\omega=e^{2\pi i/d}\)
• Together they generate the Weyl–Heisenberg group: all the "shift and phase" moves

• Which rule applies when? What counts as a measurement?
• Schrödinger's cat (1935): is the cat alive and dead until someone looks?
• Wigner's friend (1961): the friend sees an outcome; for Wigner outside the lab, has it happened?

• Probabilities are personal (Bayesian, in de Finetti's sense)
• Collapse is just updating beliefs after a new experience: nothing physical jumps
• Wigner and his friend each have their own state assignment: no contradiction
• The Born rule is a consistency rule for beliefs, like the rules of probability

Caves, Fuchs, Schack (2002 onward); Mermin

• If a state is an agent's beliefs, it should be writable as plain probabilities
• Goal: replace the state vector by its outcome probabilities for one fixed reference measurement
• The reference measurement is still built from vectors, but it is fixed once and for all, like a ruler. Only the state becomes probabilities
• Then the Born rule becomes a rule for combining probabilities (Fuchs–Schack)

• A general state (density matrix) has \(d^2-1\) real parameters
• A measurement with \(n\) outcomes gives \(n-1\) independent probabilities
• So you need \(n\ge d^2\). The most symmetric choice: a SIC-POVM

\(p(i)\) = probability of SIC outcome \(i\);   \(r(j\,|\,i)\) = probability of outcome \(j\) of your experiment, if the system were in SIC state \(i\)

A SIC exists in every dimension \(d\)

• Built from one vector by the \(d^2\) "shift and phase" operations (Weyl–Heisenberg group)
• Numerical solutions: every \(d\le193\), some up to \(d=39604\)
• Exact solutions: every \(d\le53\), some much larger
• For twenty-five years, found one dimension at a time by computer

• Exact SICs are built from algebraic numbers that generate class fields of \(K\) (Appleby et al.)
• Their overlaps are essentially Stark units: conjectured by Gene Kopp (checked in \(d=5,11,17,23\)), extended to every \(d\) by Appleby–Flammia–Kopp
• This is Hilbert's 12th problem for real quadratic fields

• Every overlap written exactly, not just a few invariants: no guessing or optimisation step at the end
• Height control (the published bottleneck): a good basis of the number field and automatic rescaling keep the integers small
• A cheap exactness check: rebuild everything modulo primes and verify the SIC equations there

AFK, Theorem 1.4: Stark Conjecture + Twisted Convolution Conjecture ⇒ Zauner's conjecture

• The ghost is a matrix built from the cocycle values: \(\tilde\Pi=\sum_{\mathbf p}c_{\mathbf p}D_{\mathbf p}\), one coefficient for each shift-and-phase move \(D_{\mathbf p}\)
• Conjecture (AFK): it is a projector, \(\tilde\Pi^2=\tilde\Pi\)
• Moves multiply up to a phase, \(D_{\mathbf q}D_{\mathbf p-\mathbf q}=\chi(\mathbf q,\mathbf p-\mathbf q)\,D_{\mathbf p}\), so this is one identity for each label \(\mathbf p\):

• Their values: Faddeev's modular quantum dilogarithm at the real quadratic fixed point \(\tau=\gamma\tau\), one value per label \[\Phi_\gamma(z;\tau)=\frac{\big(e(z);\,e(\tau)\big)_\infty}{\big(e\big(\tfrac{z}{c\tau+d}\big);\,e(\gamma\tau)\big)_\infty}\]
• These values satisfy an overdetermined system of polynomial equations, including a pentagon identity
• The equations build fusion categories (Izumi's near-group categories); Ocneanu rigidity makes the values algebraic
• The pentagon gives AFK's Twisted Convolution Conjecture: the first obstacle, gone

• Every lemma is a node: a statement, what it depends on, and how it is justified
• Its status is computed, never typed in: proved, conditional, checked in Lean
• The proof changes only through reviewed patches; the system shows what each change affects
• Every explanation is pinned to the exact version of the statement it explains, so stale text is caught

• 18 September: Radchenko–Wheeler prove the convolution identity and algebraicity
• Left: a Galois symmetry that puts the values on the unit circle
• My route: a reciprocity law, predicting the values modulo primes \(p\)
• Model: the CM case, where Frobenius at \(p\) acts by \(p\)-th powers (Kronecker: \(j(p\tau)\equiv j(\tau)^p\)); Chebotarev then pins down the Galois action
• Everything came down to one property:

• Use it exactly: every combination re-sums the same global information, never one value
• Use only how many factors of \(p\) each term has: bad terms can cancel, and fake examples pass every test

Late September: many agents, many approaches, the same wall. "It needs a new idea."

• One vertex for each lattice in the plane, up to scaling
• An edge when one lattice is \(p\) times as dense as the other. Every vertex has \(p+1\) neighbours (picture: \(p=2\))
• Axis (crater): the most symmetric lattices. Tube: everything within distance \(R\) of it

• RW's values belong to matrices; each matrix has a lattice, so each value sits at a vertex
• Tube lemma: all the matrices we need live inside the tube, \(R=v_p(\varepsilon_0-1)\)
• Inside the tube: one neighbour points inward (toward the axis), \(p\) point outward
• On the outer sphere (distance exactly \(R\)): no outward neighbours

• RW's Fourier identity relates the values to their discrete Fourier transform. On the outer sphere it has \(p^R\) in its denominator
• That forces the values along each line into a low-degree pattern (a cyclic code)
• Result: a pole \(a\) on the outer sphere makes the inward neighbour's pole at least \(3a\)
• Since no pole exceeds \(A\): poles on the outer sphere are at most \(A/3\)

• Recall: \(A\) = the worst pole, the most factors of \(p\) in any denominator
• The Frobenius defect never shrinks moving outward (an exact average)
• At the edge, RW's Fourier identity caps it

• The real-quadratic analogue of Kronecker's congruence
• Two primes above \(p\): locally you can't tell which Frobenius you see
• A sign at the second real place decides; Chebotarev assembles all primes

The reciprocity law says what every Galois symmetry does to every value

• At the other real embedding, complex conjugation sends each value to its inverse, so it has absolute value 1
• That is Radchenko–Wheeler's remaining step, and with it, SICs in every dimension \(d\ge5\)

All four use Radchenko–Wheeler's Fourier identity (their eq. (6)).