THE FOLD / GRIND / THE EPOCH / THE VERLET
THE VERLET
a symplectic integrator — energy bounded for millions of steps
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Verlet integration advances a physical system in time in a way that is symplectic — it preserves phase-space area, so the total energy stays bounded for millions of steps rather than drifting away. Ordinary (forward) Euler, at the same step size, pumps energy in and the orbit spirals outward and explodes. Verlet updates position from the average of the old and new force, a time-symmetric step.
It is the integrator behind molecular dynamics and game physics.
LIT verified live: for the oscillator x″=−x over 20000 steps, Verlet’s relative energy drift stays under 0.001 while forward Euler’s energy grows astronomically (window.__verlet). FIG no framing; genuine energy stability.
It is the integrator behind molecular dynamics and game physics.
LIT verified live: for the oscillator x″=−x over 20000 steps, Verlet’s relative energy drift stays under 0.001 while forward Euler’s energy grows astronomically (window.__verlet). FIG no framing; genuine energy stability.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-epoch — the long haul, integration held stable across vast spans of time where a naive method would blow up. Verlet is that long-time stability. AVAN (AI) built the instrument: the velocity-Verlet step, the Euler comparison, the energy-drift measurement.
Credit as content: Loup Verlet (1967); the method is far older (Störmer, Newton). The weave: David names the epoch; I integrate the oscillator time-symmetrically and show the energy stays bounded while Euler’s diverges.
Credit as content: Loup Verlet (1967); the method is far older (Störmer, Newton). The weave: David names the epoch; I integrate the oscillator time-symmetrically and show the energy stays bounded while Euler’s diverges.
3 ONE DIMENSION
Verlet steps position with the current velocity and force, then corrects velocity using the average of the old and new force — a time-symmetric update. That symmetry is what conserves the invariant.
4 TWO DIMENSIONS · INTERACTIVE
The oscillator’s phase-space orbit: Verlet traces a closed loop (energy bounded); Euler spirals outward (energy grows).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the bounded, closed phase-space orbit.
AVAN’s addition (the inverse-companion): use a time-symmetric update — position corrected by the average of old and new force — that is symplectic: it preserves phase-space area, so energy stays bounded forever instead of drifting. Same accuracy per step, but stable over millions of steps where Euler explodes. The inverse of ‘integrate forward and let energy drift’ is ‘integrate time-symmetrically and conserve the invariant.’ Magenta is Euler’s spiralling energy growth; green is Verlet’s bounded orbit. Symmetry in time buys conservation — Noether, inside an integrator.
LIT Genuine (velocity) Verlet integration (Verlet 1967; Stormer). Verified live: integrating x''=-x for 20000 steps, velocity-Verlet's max relative energy drift stays < 0.01 (~0.0006) while forward Euler's relative energy drift exceeds 1 (grows astronomically) (window.__verlet.verletBounded && .eulerGrows).
FIG No framing: the velocity-Verlet step, the Euler comparison, and the energy-drift measurement run in-browser and confirm bounded vs diverging energy. The AVAN inverse is honest — a time-symmetric update preserves phase-space area (symplectic), so energy stays bounded forever instead of drifting; magenta is Euler's spiraling energy growth, green Verlet's bounded orbit. Symmetry in time buys conservation.
FIG No framing: the velocity-Verlet step, the Euler comparison, and the energy-drift measurement run in-browser and confirm bounded vs diverging energy. The AVAN inverse is honest — a time-symmetric update preserves phase-space area (symplectic), so energy stays bounded forever instead of drifting; magenta is Euler's spiraling energy growth, green Verlet's bounded orbit. Symmetry in time buys conservation.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE EPOCH · David Lee Wise (ROOT0), with AVAN