Chebyshev Rigidity from a Punctured Germ: What the Excluded Point Costs, in Mathematics and in a Proof Assistant

Let H : ℝ → ℝ satisfy the Chebyshev doubling equation H(2t) = 2H(t)² − 1 at every real t, with no regularity hypothesis of any kind, and suppose the single limit κ = lim (2(H(t) − 1)/t²) as t → 0 along t ≠ 0 exists. That one number determines H off the origin: it is cosh(√κ t) when κ ≥ 0 and cos(√−κ t) when κ ≤ 0, the two agreeing at κ = 0, where H ≡ 1. At the origin the hypothesis determines nothing beyond H(0) ∈ {1, −1/2}, and both values occur: cosh with its value at the origin replaced by −1/2 satisfies the equation everywhere and has the same punctured germ. So the hypothesis determines the function exactly off the origin and cannot reach the origin at all.

The hypothesis is strictly weaker than d'Alembert's functional equation, and the distinction carries the paper. The doubling relation only ever relates t to 2t, so distinct dyadic orbits are uncoupled and none of the classical structure theory for the cosine equation applies; the dented cosh is a solution of the doubling equation that solves no d'Alembert equation. The proof is a descent, and it needs no sign hypothesis because the Chebyshev map T(z) = 2z² − 1 preserves exactly two sets, [1, ∞) and [−1, 1], with membership in the first inherited upward along a dyadic orbit and membership in the second inherited downward. Every orbit is therefore eventually of one type, and on each type iterating the relation outward from a small dyadic point gives an exact identity at every scale, H(2ⁿs) = cosh(2ⁿ arcosh H(s)) or H(2ⁿs) = cos(2ⁿ arccos H(s)), in which the germ reads off the constant. Equivalently, in the Schröder coordinate L(y) = (arcosh y)² that linearizes T at its repelling fixed point 1, the equation says L∘H scales by 4 under doubling and the germ says L(H(t)) = κt² + o(t²), and those two facts force L(H(t)) = κt².

We then apply this to the uniqueness theorem for the reciprocal cost J(x) = ½(x + x⁻¹) − 1, which is usually stated with four hypotheses: nonnegativity, normalization F(1) = 0, a composition law, and a calibration. Two of the four are conclusions, and the composition law is consumed at exactly one of its instances, y = x.

The one-variable setting turns out not to be a restriction. On a real vector space, if every line through the origin carries a germ, the germ assignment is forced to be a fixed sign times the square of a single ℝ-linear functional, and the solution is cosh of that functional or cos of it, with the constant 1 the degenerate member of either family. Nothing is assumed of the germ assignment beyond its existence, not the parallelogram law, not that its polarization is anything in particular, not even a constant sign, and each of those is a conclusion. The proof is two identities, obtained by reading d'Alembert's equation and its squared-difference consequence at a single order in the scale, followed by the algebraic half of the Jordan and von Neumann argument. The analytic half, which is where the classical treatment spends its continuity hypothesis, is not needed: rescaling the argument of a germ is a change of variable inside the defining limit, so real homogeneity is free. In particular a germ positive in every direction is impossible as soon as two independent directions exist, so the componentwise several-variable cost has no isotropic calibration, and every such cost is the one-variable cost read along a linear functional of the logarithms.

Finally we report the same phenomenon in mechanical form. The calibration is a limit. In a proof assistant whose division is total, with x/0 = 0, stating it over the full neighbourhood filter rather than the punctured one makes the hypothesis reach the origin, where it finds the junk value; the hypothesis then forces the curvature to be 0 and is satisfied by no function at all at the calibrated value 1. Two results in our library consumed it, including one whose entire purpose was to weaken a regularity premise, and no build failure or axiom audit could have said so. Everything here is machine-checked in Lean 4 with no axioms beyond propext, Classical.choice, and Quot.sound.