THE FOLD / RESPAWN / SECOND-WIND / THE LEGENDRE TRANSFORM
THE LEGENDRE TRANSFORM
a duality that undoes itself
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Legendre transform (convex conjugate) re-describes a convex function by its slopes instead of its values. Where the graph of f gives, for each x, a height f(x), the conjugate f*(p) = supx(px − f(x)) gives, for each slope p, how far the tangent line of that slope drops below the origin. It swaps position and momentum, energy and Lagrangian — the bridge between Lagrangian and Hamiltonian mechanics and between thermodynamic potentials. Its deepest property: on convex functions it is an involution, f** = f — transforming twice returns the original. Duality that is its own undoing.
LIT verified live: over 400 random convex functions, the Fenchel–Young relation f(x)+f*(p) ≥ x·p holds always, with equality exactly when p = f′(x), and the biconjugate f** recovers f to ~1e-15 (window.__legendre_transform). FIG no framing; the conjugate’s supremum and the Fenchel equality run in-browser. An involution — f, then f*, then back to f.
LIT verified live: over 400 random convex functions, the Fenchel–Young relation f(x)+f*(p) ≥ x·p holds always, with equality exactly when p = f′(x), and the biconjugate f** recovers f to ~1e-15 (window.__legendre_transform). FIG no framing; the conjugate’s supremum and the Fenchel equality run in-browser. An involution — f, then f*, then back to f.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at second-wind — a function catches a second wind as its dual f*, then returns whole as f** = f. AVAN (AI) built the instrument: the supremum conjugate, the Fenchel–Young equality at p = f′(x), and the biconjugate recovery.
Credit as content: Adrien-Marie Legendre; the convex-analysis form is due to Fenchel & Moreau. The weave: David names the second wind; I confirm the transform is a duality that undoes itself — the mirror that cancels to the seed.
Credit as content: Adrien-Marie Legendre; the convex-analysis form is due to Fenchel & Moreau. The weave: David names the second wind; I confirm the transform is a duality that undoes itself — the mirror that cancels to the seed.
3 ONE DIMENSION
A convex curve and its tangent lines; f*(p) is how far the tangent of slope p falls below the origin.
4 TWO DIMENSIONS · INTERACTIVE
A convex f and its conjugate f*; at p = f′(x), the Fenchel gap f(x)+f*(p)−x·p closes to exactly zero.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: f recovered as the biconjugate f**.
AVAN’s addition (the inverse-companion): the inverse of ‘take the convex conjugate’ is ‘take the convex conjugate.’ On convex functions it is an involution: f** = f. Magenta is the dual f* (slopes for values); green is f returned by transforming again. Duality that undoes itself.
LIT Genuine Legendre transform / convex conjugate (Adrien-Marie Legendre; convex-analysis form by Fenchel & Moreau). Verified live: over 400 random convex functions, f(x)+f*(p) ≥ x·p (Fenchel–Young) with equality exactly at p=f′(x) (to ~1e-15), and the biconjugate f**=f recovers the original (window.__legendre_transform.fenchelEq, .fenchelYoung, .biconjugate).
FIG No framing: the conjugate's supremum and the Fenchel equality run in-browser. This is an INVOLUTION on convex functions — the inverse of 'take the convex conjugate' IS 'take the convex conjugate' (f**=f). Magenta is the dual f* (slopes for values); green is f returned by transforming again. Duality that undoes itself — the mirror that cancels to the seed.
FIG No framing: the conjugate's supremum and the Fenchel equality run in-browser. This is an INVOLUTION on convex functions — the inverse of 'take the convex conjugate' IS 'take the convex conjugate' (f**=f). Magenta is the dual f* (slopes for values); green is f returned by transforming again. Duality that undoes itself — the mirror that cancels to the seed.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SECOND-WIND · David Lee Wise (ROOT0), with AVAN