Research proof index

Subtraction bound. The squared ℓ₂ norm of a difference of two vectors, each within boundSq, is within subL2NormSqBound boundSq = 4·boundSq.

ℓ∞ subtraction bound. The ℓ∞ norm of a difference of two vectors, each within bound, is within subLInftyNormBound bound = 2·bound.

Splitting the degree-2^α sum into low/high halves and factoring ζ^{2^{α-1}} = s collects the coefficient of ζ^j as a_j + s·a_{2^{α-1}+j}.

If multiplication by g permutes Hexp (Hexp_generator_smul), then σ_g fixes Tr_H. The per-term step uses σ_g(σ_i a) = σ_{g·i} a = σ_{(g·i) mod 2^{α+1}} a (composition + exponent-periodicity, both proven); the sum then reindexes along the permutation.

The kernel Tr_H(X^{e_p + e_q·σ_{-1}}) = (d/k)·[p = q] (Hachi [NOZ26, §3, Theorem 2]). With e_v = packExp v and n = d/2k: on the diagonal the exponent is e_p·2d, so X^{…} = 1 and the trace is d/k; off the diagonal, n ∣ (e_p + e_q·σ_{-1}) ↔ p ≡ q mod n, so when p ≢ q mod n Claim 2 applies, and when p ≡ q mod n (but p ≠ q) the exponen...

Hachi [NOZ26, §3, Theorem 2]: Tr_H(ψ(a)·σ_{-1}(ψ(b))) = (d/k)·⟨a,b⟩, where ⟨a,b⟩ = Σ_i a_i b_i is the inner product over R_q^H. Expand the product as a double sum, pull the R_q^H-coefficients out of Tr_H (traceH_smul_fixed), evaluate the monomial kernel (traceH_kernel, which is (d/k)·[p=q]), and collapse the diagonal.