Central content sectors are zero or primitive cyclotomic Hecke blocks

Base cases. At rank $0$ the algebra is $F$, the empty-word projector is $1$, and this is its unique primitive central idempotent. At positive rank and level $0$ the defining polynomial is $1$, so the algebra is $0$ and every content sector is $0$, with no nonzero blocks. Hence assume rank $d\ge 1$ and level $\mathrm{\ell }\ge 1$.

The degenerate branch. In the branch $q_H=1$ necessarily $\mathrm{char}F=e>0$. The center is the full symmetric-polynomial image over this $F$ by [F3], and [F4] applies to the early CRT projectors and proves that their content sums are zero or primitive. This uses neither a field extension nor a power-sum generation assertion.

The nondegenerate branch: labels of sectors. In the branch $q_H\mathrm{\ne }1$ choose a charge list $c_1,\dots ,c_{\mathrm{\ell }}$ repeating each $i$ exactly ${\lambda }_i$ times, so $Q_a=q_H^{c_a}$ and the first polynomial is unchanged. The arbitrary-level Murphy cellular basis and the finite cellular head lemma [F5] supply all simple $H_d^{\mathrm{\Lambda }}$-modules as nonzero Specht quotients $D^{\lambda }$, and the content-triangular JM action [F6] says every word on $S^{\lambda }$, hence on $D^{\lambda }$, has the node content $C(\lambda )$. Thus a simple module lying in a nonzero sector $e_{\alpha }$ has a label $\lambda$ with $C(\lambda )=\alpha$.

Primitivity of the nonzero sectors. Any two simple modules in the sector $e_{\alpha }$ have labels with equal content, so [F7] connects the labels by integer Jantzen edges and [F8] transfers this to a common Hecke block. If $e_{\alpha }$ decomposed into two nonzero orthogonal central idempotents $u,v$, the nonzero finite algebras $u{H}_d^{\mathrm{\Lambda }}$ and $v{H}_d^{\mathrm{\Lambda }}$ would each have a simple module, obtained from a proper right ideal of maximal dimension. Those simple modules cannot lie in one block, because the central idempotent $u$ acts as the identity on the first and as zero on the second, and all modules and extensions split by $u$. This contradicts the one-block assertion, so $e_{\alpha }$ is primitive.

Every block is a content sector. In either branch let $b$ be a primitive central idempotent and write $b={\sum }_{\alpha }b{e}_{\alpha }$. The terms are central orthogonal idempotents, so precisely one is nonzero and $b=b{e}_{\alpha }$; primitivity of the nonzero $e_{\alpha }$ from step 1.2 or step 2.1 gives $b=e_{\alpha }$. Hence the nonzero content sums are exactly all primitive central idempotents and the decomposition $H_d^{\mathrm{\Lambda }}={⨁}_{\alpha }{e}_{\alpha }{H}_d^{\mathrm{\Lambda }}$ is precisely the block decomposition [F2]. All sums are finite, the field characteristic never reduces an integer $K_0$ adjacency, and no affine-block classification or citation-only center theorem is consumed.