Research proof index

Fix $1<p<2$, an error threshold $\delta>0$, and a measure tolerance $\varepsilon\ge0$. There is an explicit bound $C(\delta,\varepsilon,p)$ such that, whenever a measurable $2\pi$-periodic function $g$ satisfies $\lVert g\rVert_{L^p((0,2\pi])}\le C(\delta,\varepsilon,p)$, the set where the maximal partial Fourier sum $\sup_N\lVert S_Ng(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$.

For every $\delta,\varepsilon>0$, there is an explicit positive uniform bound $C(\delta,\varepsilon)$ such that, if a measurable $2\pi$-periodic function $g$ satisfies $\lVert g(x)\rVert\le C(\delta,\varepsilon)$ for every $x$, then the set where $\sup_N\lVert S_Ng(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$.

The convex roof extension of g : KetUpToPhase d → ℝ≥0 applied to a pure state ψ is g (KetUpToPhase.mk ψ).

For ℓ = 1, coordinate-wise special soundness is ordinary k-special soundness: the challenge values are distinct, and there is a central vector together with k - 1 siblings differing in the single coordinate , i.e. k pairwise-distinct challenge values.

For a single coordinate (ℓ = 1), membership in the special-sound family SS(S, 1, k) is exactly injectivity of the k challenge vectors: a central vector together with k-1 siblings differing in the unique coordinate is just k pairwise-distinct values. This is the cleaner ℓ = 1 characterization underlying the bridge to plain k-special soundness.

Generic single-round CWSS assembly. Any pure statement-extending verifier of the two-round pSpec is coordinate-wise special sound for foldStructure, provided a witness assembler mkWitness that turns per-branch relOut-witnesses at star-shaped challenge families into a relIn-witness. This discharges all tree/extractor plumbing once; the protoc...