Research proof index

Coordinate-wise special soundness of SendWitness. The verifier has no challenge rounds, so CWSS collapses (via the no-challenge bridge coordinateWiseSpecialSound_of_isEmpty_challengeIdx) to a transcript-level extraction obligation. The extractor is e := fun _ tr => tr 0: the witness is the (single) prover message. Since the verifier is pure wi...

The Serre derivative of a bounded holomorphic function is bounded at infinity. serre_D k f = D f - (k/12)·E₂·f. Both terms are bounded: - D f is bounded by D_isBoundedAtImInfty_of_bounded - (k/12)·E₂·f is bounded since E₂ and f are bounded

Serre derivative is equivariant under the slash action. More precisely, if F is invariant under the slash action of weight k, then serre_D k F is invariant under the slash action of weight k + 2.

General limit: if f → c at i∞ and f is holomorphic and bounded, then serre_D k f → -k*c/12. This is the continuous mapping theorem applied to serre_D k f = D f - (k/12) * E₂ * f: - D f → 0 (Cauchy estimate from boundedness) - E₂ → 1 - f → c Therefore serre_D k f → 0 - (k/12) * 1 * c = -k*c/12.

The Serre derivative of E₂ is weight-4 slash-invariant. This requires explicit computation since E₂ is not modular. Proof strategy: Write serre_D 1 E₂ = serre_D 2 E₂ + (1/12) E₂². Then: - (serre_D 2 E₂) ∣[4] γ = serre_D 2 (E₂ ∣[2] γ) by serre_D_slash_equivariant - E₂ ∣[2] γ = E₂ - α D₂ γ where α = 1/(2ζ(2)) = 3/π² - (E₂²) ∣[4] γ = (E₂ ∣[2] γ)² After e...

A more precise version of sfHomotopy_in.