Research proof index

Auxiliary lemma for proving that the polynomial sent by the honest prover is of degree at most deg

Telescoping identity (the core sum-check completeness step): summing over the (k+1)-coordinate cube equals summing coordinate 0 over its domain, then the rest over the tail cube. This is the piFinset "cons decomposition" 𝔻^{k+1} ↔ 𝔻₀ × 𝔻^k.

Core uniformity lemma: uniform keygen plus unique key per (message, ciphertext) pair implies every (message, ciphertext) conditional has probability (card K)⁻¹. Both Shannon theorems follow from this.

Constructive Shannon direction: if keygen is uniform and each (message, ciphertext) pair is realized by a unique key in support, then perfect secrecy holds. deterministicEnc asserts encryption is deterministic in distribution (singleton support for each fixed (key, message)).

The CWSS shape of the canonical ℓᵢ = 1 structure CWSSStructure.ofSpecialSound k is exactly the plain special-soundness shape distinctShape k. This is the structural heart of the equivalence between CWSS and plain special soundness: both the arity (1·(kᵢ-1)+1 = kᵢ) and the node predicate (IsSpecialSoundFamily 1 kᵢ vs. Function.Injective) agree.

Total-variation distance is convex over a shared bind: if tvDist (f a) (g a) ≤ c for every a ∈ support mx, then tvDist (mx >>= f) (mx >>= g) ≤ c. The real-valued root of the const bound, with ℝ≥0∞ companion ofReal_tvDist_bind_left_le_const.