harp Spectral Bound for the Jacobian J N(X): Lemma for FWA Contraction Radius R_{contr} = (1-\bar{a})/(\sqrt{2}c)

We add a rigorous Lemma that closes the gap between Section 4 (Frobenius bound $\|J_N\|_F \le 2cR$) and Section 9 (spectral bound $\|J_N\|_2 \le \sqrt{2}cR$).We prove for $J_N(X) = [[0,\kappa z,\kappa y],[ \mu z,0,\mu x],[\nu y,\nu x,0]]$, $R^2=x^2+y^2+z^2$, $c=\max(|\kappa|,|\mu|,|\nu|)$:\|J_N(X)\|_2 \le \sqrt{2}cRProof is elementary, via quadratic form $u^T J_N^T J_N u = \|J_N u\|_2^2$ and Cauchy-Bunyakovsky inequality $(au_i+bu_j)^2 \le (a^2+b^2)(u_i^2+u_j^2)$, without computing cubic roots of the characteristic polynomial of $J_N^T J_N$.This bound yields the exact contraction radius $R_{contr}=(1-\bar{a})/(\sqrt{2}c)$ used in Section 9 and improves the Frobenius estimate by a factor $\sqrt{2}$. The result supports the main paper "WE SEE LIGHT, GRAVITY, VACUUM, BUT WE DO NOT YET SEE SUPERLUMINAL SOUND: HOW INFORMATION AND FIELD SYMMETRY HOLD THE UNIVERSE IN ONE COCOON" where light is interpreted as sound of vacuum superfluid ocean W, with superluminal traces $45 m/s$ (Turukhin 2002) to $310c$ (Wang 2000), $6.3c$ (Kraft 2017 M87), etc., as $+e \to -e$ dynamics $w=f$.Keywords:spectral norm, Jacobian bound, contraction mapping, superluminal sound, vacuum as medium, FWA dynamics, w=f, c vs c_sound, field symmetry