The template side conditions for large primes

The template side conditions for large primes.

The following holds (Lean statement, in the scope of module Solutions.KIL6Tok):

theorem NLogN.KIL.tok_edgeTpl {R : Type*} [CommRing R] {W : ℕ} (Qc : QCircuit R m W) (X : IMat m)
    {q e Bb : ℕ} (hq : q.Prime) (he : 1 ≤ e)
    (hg : GoodAtQ q ((Qc.den : ℚ)⁻¹ • ratM X)) (hlt : tokBnd ((Qc.den : ℚ)⁻¹ • ratM X) < q)
    (hBb : zsz ((tokBnd ((Qc.den : ℚ)⁻¹ • ratM X) * (q ^ e + 1) ^ 2 : ℕ) : ℤ) ≤ Bb) :
    ∀ T, TStep.sh T ∈ edgeTplQ Qc X → TOK T (q ^ e) Bb

Context. The lemma belongs to the part KIL6: the template side conditions for large primes.

Formalization Note. This lemma belongs to the item-language (IL) layer of the integer-multiplication-kappa campaign: tape-level programs (register programs over 64 item stacks of the NLogN multitape framework, definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8) that implement the data-movement primitives of the OpenAI preprint Integer multiplication below $n\log n$ (field moves, row split/merge, shifts, the uniform Proposition 11 interchange, Lemma 12 with fine $O(1)$-per-element costs, Gaussian record primitives, Proposition 14 pieces) with exact list denotations from the array-model bundles IntMul_Kappa_K_Defs* and explicit cost bounds. All names refer to the definition bundles up to IntMul_Kappa_K_Defs18.