The curriculumFrom a single tone to the hypersphere — the math, by ear and eye, assuming nothing
The essays in this library are written to be followed, but they move through real physics and real geometry, and sometimes a single unfamiliar word — phase, rotation, holonomy — is the whole wall. This curriculum exists to take the walls down, in order. It climbs three tiers: beginner modules you can hear before you can name, intermediate modules that build the vocabulary the essays actually use, and advanced modules that walk you to the research frontier itself — rotation groups, effective coupling, and the dynamics of tokens on the unit hypersphere. No prior math is assumed anywhere. Not being "good at math" is exactly who this is for: the ideas are physical, and the notation is always the last step, not the first.
You do not need to climb it all before the essays. Read a module when a word stops you — every essay links back to the module it leans on. A module is one page unless its topic needs more; multi-section modules get their own small table of contents.
Beginner
Hear it first
Phase, sound, and shape. Nothing assumed; ears recommended. Start at 00 and everything else on the site becomes reachable.
00Phase, beats, and the sound of syncThe one idea under everything: what phase is, why detuned rhythms beat, and the audible moment two oscillators lock. Unlocks: every essay on the site.
workshopThe sound workshop: drawing with phasesA four-section studio: a digital harmonograph where two phases locking means the drawing holds still; Chladni plates where sand walks to the quiet lines; the wave laboratory that shows WHY — watch vibration flow, reflect, and lock into a standing pattern; and free play from clamped to Tyndall free-edge patterns, beyond the audible range, with coupled drivers and 3D bouncing sand.
01Rotations: same shape, another angleRotation preserves relationships; the dot product measures agreement; two star maps one rotation apart. Unlocks: the shared manifold of meaning, brain↔text alignment, and holonomy.
Intermediate
The language of structure
The vocabulary the essays actually use — one idea per module, notation always last. Read in any order; each names its own prerequisites.
02Shapes of spaces: topology and manifoldsWhat survives stretching; what a manifold is in plain words; draw your own winding number, thread the flat torus, mirror a flag on a Möbius band. Three sections. Unlocks: the shape-of-a-thought and strange-loop chapters.
03Inside a transformer, for people who have never opened oneAttention heads, the residual stream, and LayerNorm, one draggable diagram each — softmax as a competition you can hear, heads that specialize, the stream as a running sum, and the sphere where meaning lives. Four sections. Unlocks: Transformers as Oscillators, and half the mind essays.
04Gauge and holonomy: the arrow you carry around a loopCarry an arrow around a globe triangle and it comes back turned by exactly the area it enclosed — the loop learned something. The cone where curvature hides in a point, a Foucault pendulum, and a draw-your-own-loop holonomy meter. Three sections. Unlocks: memory, self-observation, and the consciousness chapters.
05Graphs, coupling, and spectraEigenvalues and eigenvectors as a network's ways of wobbling; hear the graph's chord, cut the bridge, and watch λ₂ hit zero — plus a pluckable spring-chain and a paint-the-nodes Laplacian. Three sections. Unlocks: the learning-rule chapter's hardest paragraph, and the routing chapters.
06The von Mises circle: where 0.303 comes fromWhy averaging angles is a trap, the resultant as honest agreement, and the birth certificate of the site's central constant R₀(KBKT) ≈ 0.303.
Where the curriculum meets the research itself. These assume the tiers below and end by handing you to the essays' actual claims.
08Lohe dynamics: oscillators on the hypersphere, and how tokens moveKuramoto's big brother: unit vectors on a sphere, coupling that pulls along the surface, effective coupling chosen by agreement itself — a token's journey through a transformer as a path on the hypersphere, and the quench experiment: a frozen coupling is a memory, a living coupling is a sense organ. Three sections.
09Living K: the coupling field as the dynamical objectQuenched vs. living wiring; the learning rule as a phase portrait you can seed by clicking; and the repair flagship — fray a memory's bonds and watch the rule heal them, bridges resurrected by the Fiedler channel. Three sections.
10The BKT transition: why the wall sits at 2/πPlace vortices and fail to comb them away; the energy–entropy ledger finding its tie at exactly 0.637; free vortices collapsing across the wall — with free energy taught in plain words on the way. Three sections.
11Coarse-graining and renormalization: squint and the law survivesBlock-average a real lattice; watch three fates flow to their fixed points; then try to tell two different engines apart after squinting — you can't, and that is the mechanism behind "one law, every scale". Three sections.
Further back in the queue: free energy & entropy · singular values & SVD · complex numbers as rotating arrows. (The running list lives in the curriculum review.)