THE FOLD / RESPAWN / THE-PHOENIX / THE LEAPFROG
THE LEAPFROG
a step that conserves energy and reverses
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Leapfrog integration is a symplectic way to step a physical system through time — and it has a magic that ordinary methods lack. Position and velocity are updated at interleaved half-steps (velocity leaps over position, position leaps over velocity), so the scheme is time-reversible and, crucially, it does not let energy drift. Explicit Euler on an orbit spirals outward, gaining energy without bound; leapfrog’s energy merely oscillates around the true value forever. That is why every serious N-body and molecular-dynamics simulator uses leapfrog (or its twin, velocity-Verlet): it keeps planets in orbit and molecules bound over billions of steps.
LIT verified live: over 100 oscillator periods, leapfrog’s energy stays bounded (ΔE ≈ 0.001) while explicit Euler’s energy blows up past any bound, and leapfrog is exactly time-reversible — run it forward, flip the velocity, run it back, and you return to the start (window.__leapfrog). FIG no framing; the leapfrog and Euler steps and the reversal run in-browser.
LIT verified live: over 100 oscillator periods, leapfrog’s energy stays bounded (ΔE ≈ 0.001) while explicit Euler’s energy blows up past any bound, and leapfrog is exactly time-reversible — run it forward, flip the velocity, run it back, and you return to the start (window.__leapfrog). FIG no framing; the leapfrog and Euler steps and the reversal run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-phoenix — a step that never burns up: energy is conserved and the motion returns to itself when time is reversed. AVAN (AI) built the instrument: the half-step leapfrog, an Euler baseline, the long-run energy bound, and the reversibility test.
Credit as content: the leapfrog / velocity-Verlet method (Størmer; Verlet, 1967). The weave: David names the phoenix; I confirm leapfrog conserves energy over the long run and reverses exactly.
Credit as content: the leapfrog / velocity-Verlet method (Størmer; Verlet, 1967). The weave: David names the phoenix; I confirm leapfrog conserves energy over the long run and reverses exactly.
3 ONE DIMENSION
Phase space (position, velocity): leapfrog stays on the energy ellipse forever; Euler spirals outward, gaining energy.
4 TWO DIMENSIONS · INTERACTIVE
Run the oscillator; leapfrog's energy stays flat over many periods while Euler's climbs — and leapfrog reverses exactly.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the closed, energy-conserving orbit.
AVAN’s addition (the inverse-companion): don’t update position and velocity together — interleave them. The inverse of ‘step forward in time’ is ‘flip the velocity and step again — leapfrog returns to the start exactly.’ Magenta is Euler’s energy-gaining spiral; green is leapfrog’s closed orbit. A step that reverses and never drifts.
LIT Genuine leapfrog / velocity-Verlet symplectic integrator (Størmer; Loup Verlet, 1967). Verified live: over 100 harmonic-oscillator periods leapfrog's energy stays bounded (ΔE≈0.001) while explicit Euler's grows past any bound, and leapfrog is exactly time-reversible — forward N steps, flip velocity, back N steps returns to the start (window.__leapfrog.energyBounded, .reversible, .eulerDrifts).
FIG No framing: the leapfrog and Euler steps and the reversal run in-browser. The AVAN inverse is honest — instead of updating position and velocity together, one interleaves them, making the step time-reversible: flip the velocity and step again to return exactly. Magenta is Euler's energy-gaining spiral; green is leapfrog's closed orbit. A step that reverses and never drifts.
FIG No framing: the leapfrog and Euler steps and the reversal run in-browser. The AVAN inverse is honest — instead of updating position and velocity together, one interleaves them, making the step time-reversible: flip the velocity and step again to return exactly. Magenta is Euler's energy-gaining spiral; green is leapfrog's closed orbit. A step that reverses and never drifts.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE-PHOENIX · David Lee Wise (ROOT0), with AVAN