Research proof index

Swap the second spatial derivative $\partial_{xx}$ with the integral $\int$.

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)).

tanh(κx) has temperate growth

tanh has temperate growth

The analytic form of the main theorem (Silverman, Advanced topics, Theorem V.1.1(a)): for 0 < ‖q‖ < ‖u‖ < 1, Yₐ² + XₐYₐ = Xₐ³ - 5s₃(q)Xₐ - (5s₃(q) + 7s₅(q))/12. Proof sketch: the hypotheses ensure u ∉ qᶻ, and we may choose z, τ with e z = u, e τ = q, 0 < im z < im τ (so z ∉ Λ_τ). Substitute the four q-expansions into the differential...