= Solution
Put
$$
q_r=|TQ_p(\lambda)_r|,
\qquad
c_r=|TC_p(\lambda)_r|.
$$
The defining relation between the <quotient tower of a partition> and the <core tower of a partition> is
$$
q_r=c_r+pq_{r+1},
\qquad q_0=n.
$$
Summing the resulting telescoping identities gives
$$
\sum_{r\geq0}c_r=n-(p-1)\sum_{r\geq1}q_r.
$$
By the <Hook-length formula>,
$$
v_p(\chi^\lambda(1))=v_p(n!)-\sum_{x\in Y(\lambda)}v_p(h_x).
$$
The <abacus divisible-hook correspondence> says that the number of hooks divisible by $p^r$ is $q_r$, so
$$
\sum_xv_p(h_x)=\sum_{r\geq1}q_r.
$$
If $d_p(n)=\sum_r\alpha_r$, the digit-sum form of the <Legendre formula> is
$$
v_p(n!)=\frac{n-d_p(n)}{p-1}.
$$
Combining the three displayed identities proves the <P-adic valuation of a symmetric-group character degree from the core tower> formula
$$
\boxed{
v_p(\chi^\lambda(1))
=\frac{\sum_{r\geq0}|TC_p(\lambda)_r|-\sum_{r\geq0}\alpha_r}{p-1}.}
$$
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