Research proof index

A ring element with zero centered ℓ₁ norm is 0: every centered coefficient representative below deg φ vanishes (ZMod.valMinAbs_eq_zero), and the representative is reduced (degree < deg φ), so the underlying polynomial is 0.

The centered ℓ₁ norm of 1 : Rq Φ is 1 (when 1 ≤ deg φ): the trivial challenge c = 1 used by the honest committer is nonzero and ℓ₁-short.

ℓ∞ → ℓ₂² bridge (ring element): ‖x‖₂² ≤ deg φ · ‖x‖∞² , each of the deg φ centered coefficients contributes at most ‖x‖∞².

Per-element subtraction bound: ‖a - b‖₂² ≤ 2·(‖a‖₂² + ‖b‖₂²).

Per-element ℓ∞ triangle inequality: ‖a - b‖∞ ≤ ‖a‖∞ + ‖b‖∞.

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