A false sum-check claim rarely becomes true

Let $q_i(T)$ be the true partial sum at the fixed prefix. Its degree is at most $d_i$ because specialization and addition preserve degree upper bounds, and $q_i(0)+q_i(1)=S_{i-1}$. Any well-formed consistent $h_i$ instead has $h_i(0)+h_i(1)=C_{i-1}\mathrm{\ne }{S}_{i-1}$, so $h_i-q_i$ is a nonzero formal polynomial of degree at most $d_i$. Messages failing these checks reject immediately.

A field is an integral domain, since $ab=0$ with $a\mathrm{\ne }0$ implies $b=a^{-1}ab=0$. Apply the root bound to $h_i-q_i$: it has at most $d_i$ roots in $F$. The next scalar is true precisely if $h_i(r_i)=q_i(r_i)$, because $S_i=q_i(r_i)$. The fresh $r_i$ is uniform even after conditioning on the fixed message, so this probability is at most $\min (1,d_i\mathrm{/}\mathrm{\mid }F\mathrm{\mid })$.

The bound holds for each allowed message, so averaging its conditional probability over any prover randomness preserves it. If $d_i=0$, the nonzero difference is constant and the probability is zero; if $d_i\ge \mathrm{\mid }F\mathrm{\mid }$, the cap at one is merely trivial. The argument includes $i=n$, $n=1$, characteristic two, and challenges $0,1$. A false updated claim need not be detected in this round; only its becoming true is the exceptional event bounded here.