Explorables of the intelligence theorem.
A library of interactive essays built around a single inequality — I ≥ 0, the Coherence Theorem — and what happens when you run it on different substrates. Each piece is self-contained, written in the browser, and pursues one idea to its consequence.
Phase and beats, and the audible moment two rhythms lock. The sound workshop: a digital harmonograph where locking means the drawing holds still, and Chladni plates where sand walks to the quiet lines. Rotations: what turning cannot change.
Topology and manifolds. Inside a transformer. Gauge and holonomy. Eigenvalues as a network's ways of wobbling — hear its chord, cut its bridge. The von Mises circle, where the constant 0.303 is born. Rotation groups, from a turntable to 768 dimensions.
Lohe dynamics on the unit hypersphere and tokens as paths. Living K: the learning rule as a phase portrait, and the repair demonstration. The BKT transition: why the wall sits at 2/π. Coarse-graining: squint, and the law survives. The full ladder from the essays' hardest words to their actual mathematics.
The path: beginner → intermediate → Foundations → any essay → advanced. Or skip ahead and come back when a word stops you — the essays link down to the module they lean on.
The shared physics every other essay assumes: coupled oscillators, coherence capital C = Iphase·ρ, the Coherent Learning Rule, the arrow Ĉ ≥ 0, and how a substrate remembers and thinks. Start here.
Starts from one inequality on the diamond lattice. Derives the fine structure constant to 1.5 parts per billion and the electron’s g-factor to eleven matching digits with the most precise measurement in physics. Zero free parameters.
One law from the vacuum to the mind. Intelligence is the flux of coherence; consciousness is that flux observing itself; both are sharpest at the edge of a phase transition. The companion to The Coherence Theory of Intelligence and Consciousness.
A transformer is a network of coupled oscillators. LayerNorm is reprojection, attention is coupling, the residual stream is the field. From that identity: a derived dead-head threshold, a low-dimensional coupling manifold, lossless rank-2 projection, and one rotation between any two trained models.
The next physics chapter: extending the same machinery from electrons to hadrons. Frame-sector dynamics on the diamond lattice, Skyrmion topology, and the proton’s mass as the target.