THE FOLD / GRIND / GRADIENT DESCENT / THE STATIONARY DISTRIBUTION
THE STATIONARY DISTRIBUTION
a chain that forgets where it started
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The stationary distribution is the long-run equilibrium of a Markov chain. A chain hops between states by a transition matrix P (each row a probability distribution). If the chain is irreducible (every state reaches every other) and aperiodic, then no matter where it starts, the distribution over states converges to a unique vector π that is fixed by the dynamics: πP = π, with Σπ = 1. The chain forgets its starting point. It is the mathematics behind PageRank, MCMC sampling, and equilibrium in queueing and physics.
LIT verified live: for thousands of random irreducible aperiodic chains, power iteration from any start converges to a π satisfying πP = π and Σπ = 1, and different starts reach the same π (window.__stationary). FIG no framing; the fixed-point equation and start-independence checked exactly.
LIT verified live: for thousands of random irreducible aperiodic chains, power iteration from any start converges to a π satisfying πP = π and Σπ = 1, and different starts reach the same π (window.__stationary). FIG no framing; the fixed-point equation and start-independence checked exactly.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at gradient-descent — keep applying the transition matrix and the distribution settles onto the equilibrium π, sliding to the same place from any start. That convergence is the mechanic. AVAN (AI) built the instrument: the random stochastic matrix, the power iteration π ← πP, the πP = π fixed-point check, and the start-independence (uniqueness) test.
Credit as content: Andrey Markov (chains, 1906); the ergodic convergence is the Perron–Frobenius theorem for stochastic matrices. The weave: David names gradient-descent; I run the distribution forward under P until it stops changing, confirm the limit is fixed by P and sums to one, and check that every starting distribution lands on the same π — the chain’s memory of its origin fades to nothing.
Credit as content: Andrey Markov (chains, 1906); the ergodic convergence is the Perron–Frobenius theorem for stochastic matrices. The weave: David names gradient-descent; I run the distribution forward under P until it stops changing, confirm the limit is fixed by P and sums to one, and check that every starting distribution lands on the same π — the chain’s memory of its origin fades to nothing.
3 ONE DIMENSION
πP = π, Σπ = 1. Start anywhere; apply P again and again; the distribution converges to the same π. The stationary π is the left eigenvector of P for eigenvalue 1.
4 TWO DIMENSIONS · INTERACTIVE
A chain’s transition graph and the distribution converging to π from two different starts; πP = π checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: an equilibrium that forgets the start.
AVAN’s addition (the inverse-companion): don’t track where the chain is — ask where it settles, the π fixed by πP = π and reached from every start. The inverse of ‘follow the random walk’ is ‘the equilibrium distribution is a fixed point, blind to the origin.’ Magenta is the starting distribution; green is the stationary π it converges to. Equilibrium forgets the beginning.
LIT Genuine Markov-chain stationary distribution (Andrey Markov, 1906; the ergodic convergence is Perron–Frobenius for stochastic matrices). Verified live: for 2000 random irreducible aperiodic chains, power iteration converges to a π with πP = π (window.__stationary.fixed) and Σπ = 1 (window.__stationary.sums), and a corner-start reaches the same π as the uniform start (window.__stationary.unique).
FIG No framing: the random stochastic matrix, the power iteration π ← πP, the πP = π fixed-point check, and the start-independence test all run in-browser. The AVAN inverse is honest — asking where the chain settles (the π fixed by πP=π, reached from every start) rather than tracking where it is, is the equilibrium view; magenta is the starting distribution, green the stationary π it converges to. Equilibrium forgets the beginning.
FIG No framing: the random stochastic matrix, the power iteration π ← πP, the πP = π fixed-point check, and the start-independence test all run in-browser. The AVAN inverse is honest — asking where the chain settles (the π fixed by πP=π, reached from every start) rather than tracking where it is, is the equilibrium view; magenta is the starting distribution, green the stationary π it converges to. Equilibrium forgets the beginning.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GRADIENT DESCENT · David Lee Wise (ROOT0), with AVAN