Research proof index

This lemma proves the final algebraic step in the DG25 Theorem 3.1 proof. It shows that if R > e + 1, then e * (R / (R - 1)) < e + 1. The intuition is that the fraction R / (R - 1) is always greater than 1, but as R gets larger, it gets closer to 1. The hypothesis R > e + 1 provides a strong enough bound to ensure the product e * (fraction) do...

The k-th coefficient (k < deg φ) of a gadget-decomposition block is exactly the corresponding digit of the corresponding input coefficient.

The base-b gadget decomposition is a lawful gadget decomposition.

Each gadget-decomposition block is ℓ₂²-short: its centered squared-ℓ₂ norm is at most (deg φ)·(b-1)² (each of the deg φ coefficients contributes at most (b-1)²).

ℓ₂² shortness of G⁻¹. The full gadget decomposition has centered squared-ℓ₂ norm at most (rows·digits)·(deg φ)·(b-1)².

The gadget entry at the flattened index finProdFinEquiv (i', e) is constRq (base^e) on the diagonal block and 0 elsewhere.