dyas · a CONVEX / PHYSICS duality · measured
THE LEGENDRE DYAD
energy(velocity) ⟺ H(momentum) · f(x) ⟺ f*(p) = max_x [ px − f(x) ]
A convex curve can be given two ways: by its points, or by its tangent lines. The Legendre transform trades one for the other — and it is its own inverse. In mechanics it is the bridge from energy as a function of velocity to the same energy as a function of momentum: one physics, two coordinates.
▬ energy f(v)=½·m·v²▬ dual H(p)=p²/2m
—
‖ f*(p) − p²/2m ‖ (dual = closed form)
one physics, two coordinates
The dual is computed live — f*(p) = max over v of (p·v − f(v)) — and it lands exactly on the closed form p²/2m (error ~1e−3, grid-limited). Steepen the energy (raise m) and its dual flattens by exactly 1/m: the two are locked, inverse faces of one convex object. At m = 1 the energy IS its own dual — the self-Legendre parabola, this domain's fixed point again ([[the-monad]]).
green · measured live
The Legendre dual is evaluated numerically in-browser and matches the analytic p²/2m to grid precision; the self-dual gap at m=1 falls to the floor. The energy⟺momentum bridge (Lagrangian → Hamiltonian) is exact for the quadratic, node-verified before shipping.
amber · the shown case
Only the quadratic (½mv²) is drawn — chosen because its dual is exact and clean. The Legendre transform is involutive for all convex functions, but that general round-trip is grid-sensitive to render; the picture is the cleanest instance, not the whole theorem.
red · where it breaks
Legendre duality needs convexity — for non-convex f the transform loses information (it returns the convex hull; f** ≠ f). A real, provable duality with a stated domain, not a universal one.