The unadjusted surrogate-regret bound has the wrong scale and tie convention

Give the one-point feature space its power-set sigma-algebra and take $Y=+1$ surely. Let $\mathrm{\Phi }(t)=\max (0,1+t)$, a finite nonnegative nondecreasing convex function: each affine term is nondecreasing and their maximum satisfies the convexity inequality by bounding each affine term at a convex combination by the corresponding convex combination of the two maxima. Here $\eta =1$, $L_{\mathrm{\Phi }}(x,u)=\max (0,1-u)$, and the constant minimizer $h_{\mathrm{\Phi }}^{\ast }=1$ has risk zero. With $s=2,c=1\mathrm{/}4$, the margin premise is $1\mathrm{/}4\le (1\mathrm{/}4)(1-0)$, an equality. For $h=-1\mathrm{/}2$, classification excess is $1$ and surrogate excess is $3\mathrm{/}2$, whereas the claimed bound is $(1\mathrm{/}2)\sqrt{3\mathrm{/}2}<1$ since its square is $3\mathrm{/}8<1$. All scores and risks are measurable and finite. Thus the proposed coefficient is false even without zero scores and with a finite conditional minimizer.

Independently use the same one-point feature space and hinge loss, but let each label have probability $1\mathrm{/}2$. Then $L_{\mathrm{\Phi }}(x,u)=[\max (0,1-u)+\max (0,1+u)]\mathrm{/}2\ge 1$, with equality at $u=0$, so take $h_{\mathrm{\Phi }}^{\ast }=h=0$. For $s=1,c=1$ the premise is $0\le 0$ and surrogate excess is zero. But $R_{01}(h)=1$ while $R_{01}^{\ast }=1\mathrm{/}2$, so classification excess is $1\mathrm{/}2$. Both $2c$ and $2c^{1\mathrm{/}s}$ times the zero surrogate excess vanish. This refutes the uncorrected tie convention independently of the scale defect and completes the refutation.