On Degree Powers in Intersecting Families

For a family $\mathcal{F}\subseteq\binom{[n]}k$ and $R\in\binom{[n]}r$, let $d_{\mathcal{F}}(R)=|\{F\in\mathcal{F}:R\subseteq F\}|$ and $\ell_{r,p}(\mathcal{F})=\sum_{R\in\binom{[n]}r}d_{\mathcal{F}}(R)^p$; at the codegree level we write $co_p(\mathcal{F})=\ell_{k-1,p}(\mathcal{F})$. We develop a discrete two-moment interpolation principle that majorizes $x^p$ on the integer degree lattice by a quadratic interpolant and reduces every real exponent $p\geq2$ to sharp bounds for the first two falling moments. We prove that a full $t$-star maximizes $co_p$ among $t$-intersecting families for every real $p\geq2$ throughout the sharp classical range $n\geq(t+1)(k-t+1)$, and we determine all equality cases. Using Bey's size-sensitive quadratic inequality, we extend the same framework to every nontrivial degree level: if $\mathcal{F}$ is intersecting, $n\geq2k$, and $1\leq r\leq k-1$, then a full point-star maximizes $\ell_{r,p}(\mathcal{F})$ for every real $p\geq2$, again with a complete equality classification. Thus the codegree theorem extends the sharp Wu--Zhang quadratic bound to every real $p\geq2$, completes the quadratic boundary equality classification, contains the Brooks--Linz conjecture as its $p=2$ special case, and, for integer exponents $p\geq2$, resolves the problem of Zhou--Yuan throughout the sharp Erd\H{o}s--Ko--Rado range.