Proof. By (OpenAI 2026a, Theorem 1.1), the reductions of the blocks in (i) have finitely many \(k\)-Morita classes. Their Cartan matrices, up to simultaneous permutation, consequently form a finite list. Integral blocks and their reductions have the same Cartan matrix, which proves (i).

Assertion (ii) is (Eaton et al. 2020, Theorem 3.10), applied to the finitely many possible defect exponents. The coefficient conventions of that theorem include the absolutely unramified complete discrete valuation ring \(W(k)\) and its coefficient Frobenius, which fixes the uniformizer \(p\). Its defect-zero case can also be separated: a defect-zero block over \(W(k)\) is a matrix algebra over \(W(k)\) and has basic order \(\mathcal O\). Indeed, its reduction is a full matrix algebra over \(k\); lift its matrix units over the complete ring \(\mathcal O\), whose resulting primitive corner has rank one. ◻

Source of the assertions. For a reduced pair, the structural conclusions follow from (OpenAI 2026a, Lemma 8.1). Its opening Morita reduction is stated over \(k\); we use only its conclusions about the already reduced pair, obtained here by the integral An–Eaton reduction. The compatible partial extension and its actual factors are (OpenAI 2026a, Lemma 8.2). The integral dual recovery construction and full corners are (OpenAI 2026a, Lemma 8.3).

For clarity, the dual-recovery idempotent is \(|\Xi|^{-1}\sum_{\chi\in\Xi}u_\chi\). The characters are trivial on every \(n_{x,y}\in N\), so the pure \(X\)-units preserve this idempotent and retain exactly these factors. This explains why the construction is integral and why it retains the multiplication data needed later. Finally (iv) is the integral assertion of (OpenAI 2026a, Lemma 8.5), using the trivial-subgroup case of its Proposition 9.1. The finite Picard assertion is also (Eisele 2022, Theorem B and Corollary 1.2). ◻

Proof. We bound the special-fibre orbits and then cut each one by a fixed integral slice. The latter has only finitely many possible generic points arising from the sections under consideration.

The integral tangent and horizontal components. Embed \(V\) as a closed subscheme of \(\mathbb A^r_{\mathcal O}\), with finitely many defining equations. The module \(T_xV\) is the kernel of their Jacobian map on \(\mathcal O^r\). It is therefore saturated in \(\mathcal O^r\). Write \(u=\operatorname{rank}_{\mathcal O}T_xV\). Surjectivity of the orbit derivative and saturation imply that its coordinate matrix has a unit minor of size \(u\). This follows, for example, from Smith normal form over the discrete valuation ring \(\mathcal O\).

After extension to \(K_0\), this derivative has rank \(u\), which is also \(\dim_{K_0}T_{x_{K_0}}V_{K_0}\). Every irreducible component of \(V_{K_0}\) through \(x_{K_0}\) consequently has dimension at most \(u\). Let \(V_u\) be the reduced closure in \(V\) of the union of the generic-fibre irreducible components of dimension at most \(u\). There are finitely many such components. Each of their horizontal closures has special fibre of dimension at most its generic-fibre dimension. This is the dimension theorem for an irreducible finite-type scheme over a valuation ring (The Stacks Project Authors 2026b, Lemma 33.19.2, Tag 0B2J). Thus \[ \dim (V_u)_k\leq u. \tag{6}\] The section \(x\) factors through \(V_u\), because its generic point does and \(\mathcal O\hookrightarrow K_0\) is injective.

The special-fibre orbits. Let \(x_0\) be the reduction of \(x\). The unit minor remains nonzero modulo \(p\), so the special-fibre orbit of \(x_0\) has dimension at least \(u\). This orbit lies in \((V_u)_k\). One way to verify that last assertion is to lift every element of \(J(k)\) to \(J(\mathcal O)\), using smoothness and completeness, and then apply it to the section \(x\). The transformed generic point still lies on components of dimension at most \(u\). Equation (6) now shows that the orbit has dimension exactly \(u\).

A locally closed orbit of dimension \(u\) in a scheme of dimension at most \(u\) contains an open subset of an irreducible component of dimension \(u\). Two distinct orbits cannot both contain dense open subsets of the same component. There are therefore finitely many possible special-fibre orbits for this \(u\).

A fixed Hensel slice. Fix one such orbit and a representative \(x_0\). By lifting elements of \(J(k)\), move all integral points under consideration so that their reduction equals \(x_0\). Choose \(u\) coordinate functions whose orbit derivative has a nonzero \(u\)-minor at \(x_0\), and fix integral lifts \(a_1,\ldots,a_u\) of their values there. For each such integral point \(x\), the map \[J\longrightarrow\mathbb A^u_{\mathcal O},\qquad g\longmapsto\text{the selected coordinates of }g x\] is smooth near the identity: \(J\) is smooth and its relative differential there is surjective. Hensel lifting supplies \(g\equiv1\pmod p\) for which the selected coordinates of \(gx\) are exactly the \(a_i\). Thus every integral orbit has a representative in the fixed affine slice \[W=V\cap\{\text{selected coordinates }=a_1,\ldots,a_u\}.\]

For any resulting point \(y\), the selected-coordinate differential is an isomorphism on \(T_{y_{K_0}}V_{K_0}\): that space has dimension \(u\), and the same minor is a unit since \(y\) reduces to \(x_0\). It follows that \(T_{y_{K_0}}W_{K_0}=0\). Such a point is isolated in the finite-type \(K_0\)-scheme \(W_{K_0}\). A noetherian scheme has only finitely many isolated points. Each generic point gives at most one integral section, again by the injection \(\mathcal O\hookrightarrow K_0\). Hence there are finitely many representatives in this slice. Taking the finite union over the special-fibre orbits and the possible ranks \(0\leq u\leq r\) proves the lemma. ◻

Proof. For a finite group \(G\), the conjugation permutation lattice and Shapiro’s lemma give \[\operatorname{HH}^1_{\mathcal O}(\mathcal OG) \cong H^1(G,\mathcal OG_{\mathrm{conj}}) \cong\bigoplus_{[g]}H^1(C_G(g),\mathcal O)=0.\] The action on the last coefficient module is trivial, and \(\operatorname{Hom}(C_G(g),\mathcal O_{\mathrm{add}})=0\) because \(C_G(g)\) is finite and \(\mathcal O\) is torsion-free. Hochschild cohomology splits over finite direct products and is Morita invariant, proving the first assertion. Thus every \(\mathcal O\)-linear derivation of \(A\) is inner.

The unit group scheme \(A^\times\) is an open subscheme of the affine space underlying \(A\), defined by invertibility of the regular-representation determinant. It is smooth. The automorphisms of \(A\) form an affine scheme of finite type: impose the multiplicative and unital equations on an invertible linear map. Let \(A^\times\) act by postcomposition with inner automorphisms. At an automorphism, compose with its inverse to identify the tangent module with the derivations of \(A\). The orbit derivative then consists of the inner derivations, so it is surjective. Lemma 6 says exactly that there are finitely many cosets modulo inner automorphisms. ◻

Proof. We apply Lemma 6 to the space of decompositions compatible with multiplication. The key point is that the tangent calculation uses cancellation in \(H\), without averaging over \(H\).

A grading \(A=\bigoplus_{h\in H}A_h\) is specified by its orthogonal projections \(e_h\in\operatorname{End}_{\mathcal O}(A)\). These satisfy \[\begin{align*} e_he_j&=\delta_{h,j}e_h,& \sum_{h\in H}e_h&=1,\\ e_h(1_A)&=\delta_{h,1}1_A,& e_l(e_g(x)e_h(y))&=0\quad(l\ne gh). \end{align*}\] The last equations need only be imposed on a fixed finite basis of \(A\). They define an affine finite-type grading scheme, on which \(A^\times\) acts by conjugation.

We compute its integral tangent module at a grading. A first-order deformation of the direct-sum decomposition is uniquely represented by an off-diagonal \(\mathcal O\)-linear map \(\delta:A\to A\), where \[e_h\delta e_h=0\quad(h\in H).\] The deformed summands are \((1+\varepsilon\delta)A_h\) over \(\mathcal O[\varepsilon]/(\varepsilon^2)\). This description follows by differentiating the equations for orthogonal idempotent projections, or by writing the summands as graphs over the original summands. It uses no division by \(|H|\).

For \(x\in A_g\) and \(y\in A_h\), the linearized multiplicative equations say that \[ \delta(xy)-\delta(x)y-x\delta(y) \tag{7}\] has zero component in every degree other than \(gh\). It also has zero \(gh\)-component. The first term is off degree \(gh\) by definition. A summand of \(\delta(x)y\) can have degree \(gh\) only if its first factor has degree \(g\), by cancellation in \(H\); that component of \(\delta(x)\) is zero. The same reasoning applies to \(x\delta(y)\). Hence Equation (7) vanishes, and \(\delta\) is a derivation.

The hypothesis gives \(\delta=[a,-]\) for some \(a\in A\). Conjugation by \(1+\varepsilon a\) produces the specified tangent to the grading. Thus the orbit derivative from the smooth group \(A^\times\) is surjective onto every integral tangent module. Lemma 6 proves the assertion. ◻

Proof. The total order. Every order in the collection has rank \(|H|\operatorname{rank}_{\mathcal O}R\). Choose a basic representative in each of its finitely many Morita classes. Any order in that class is an endomorphism order of a projective generator \(\bigoplus_i P_i^{m_i}\), with positive integer multiplicities \(m_i\) and the \(P_i\) ranging over the finitely many indecomposable projectives. Its endomorphism order contains \(M_{m_i}(\operatorname{End}(P_i))\) as a direct \(\mathcal O\)-module summand, so its rank bounds \(m_i^2\). There are consequently only finitely many ungraded isomorphism types in the collection.

The grading and the marking. Lemma 7 and Theorem 8 give finitely many labelled \(H\)-gradings on every such total order. Fix one graded representative with identity component isomorphic to \(R\). Two identifications of that component with \(R\) differ by an element of \(\operatorname{Aut}_{\mathcal O}(R)\). Inner changes extend to graded automorphisms of the total order by conjugation with a unit in degree one. Since \(\operatorname{Out}_{\mathcal O}(R)\) is finite by Lemma 7, there are finitely many remaining markings. These are precisely the based graded types in the assertion. ◻

Proof. Positive-degree cohomology of a finite group is killed by its order. Raising to \(|S|\) is an automorphism of \(\mathcal T\), since \(\mathcal T\) has no \(p\)-torsion and has unique \(p\)-power roots. This map also induces an automorphism on cohomology. It is simultaneously the zero map there, so the cohomology vanishes. ◻

Proof of Proposition 12. The imported operators retain the actual inner factors. We show first that all their presentation ambiguities lie in \(\mathcal T\), then assemble and descend the component comparisons. Only after this descent will we remove the resulting scalar cocycle.

Teichmüller-valued presentation ambiguities. Every scalar in Equation (13) belongs to \(\mathcal T\), because \(\lambda_m\) has \(p'\)-order. To represent an operation \(i\in I\), choose \(i=\operatorname{ad}(g)a\) using Equation (10) and take \(E_gD_a\). Changing this presentation is a change through \(\operatorname{Inn}(L)\), and hence introduces only a value of \(\lambda_m\), apart from the ambiguity of a central element in the choice of \(g\).

That central ambiguity is also Teichmüller-valued. If \(c\in Z(J)\) is a \(p\)-element, then \(c\in L\) and Equation (13) applied to it says \(E_c=1\): the inner operation is the identity and \(\lambda_m(c)=1\). If \(c\) has \(p'\)-order, it acts on each block side by the value of its central character. The operator \(E_c\) is the ratio of these two \(p'\)-roots of unity, and lies in \(\mathcal T\). An arbitrary central element is a product of its \(p\)- and \(p'\)-parts. Thus any two presentations give operators differing by \(\mathcal T\), rather than by an arbitrary unit of \(\mathcal O\).

Multiplication of chosen operators uses only Equation (12) and a change of presentation. Its scalar discrepancy is consequently in \(\mathcal T\). Applying this to the relation \(\alpha_s\alpha_t=\operatorname{ad}(n_{s,t})\alpha_{st}\) shows, with the actual inner factor retained, that the resulting transport has the form \[D_sD_t(v)=c_J(s,t)n'_{s,t}D_{st}(v)n_{s,t}^{-1}, \qquad c_J(s,t)\in\mathcal T.\]

Products, component permutations and the remaining factors. We now use the remaining constructions in the proof of (OpenAI 2026a, Proposition 9.1). There are boundedly many component orbits. Taking a common bounded multiple of their exponents is legitimate by tensoring successive Frobenius twists of the comparison bimodules. Frobenius preserves \(\mathcal T\), and tensor products multiply the scalar discrepancies. This operation therefore preserves their membership in \(\mathcal T\).

For the finite list of exceptional, sporadic, and relevant defining-characteristic components, a common Frobenius period allows the identity bimodule with strict group transports. The unbounded alternating-cover family also has a uniformly bounded coefficient period, as proved in that proposition, and uses the same strict transports. The bounded \(p\)-group factor \(P\) uses its identity bimodule. The central \(p'\)-factor \(Z\) has a rank-one character algebra, so its left/right discrepancy is a ratio of \(p'\)-character values and belongs to \(\mathcal T\). Permutation of factors uses the ordinary flips of bimodules and satisfies its group relations strictly. These are tensors of ordinary Morita bimodules, so no graded symmetry sign enters this step, including when \(p=2\).

Descent through the original central kernels. It remains to pass from \(P\times Z\times\prod_i J_i\) to \(\bar N\): quotient by the added factors \(C_i\) and the original kernel \(D_0\). The established construction identifies the left and right actions of every central \(p\)-element on the comparison. Quotienting by \(c-1\) on the two sides therefore gives a Morita bimodule over the corresponding quotient blocks: tensor the inverse equivalence as well and use equality of the central actions in the two inverse identities. The transport maps preserve these ideals and descend. The \(p'\)-part of the quotient kernel acts trivially on both sides in the chosen averaging sector. Thus this descent introduces no new scalar factor. Differences between lifts of the prescribed \(n_{s,t}\) vanish in the quotient. In particular, the Morita equivalence and its transport maps now live on the original identity blocks, where the defect bound of Proposition 5(ii) applies.

We have obtained an integral \((\sigma^m B_0,B_0)\)-Morita bimodule \(\mathcal M\) with \(\mathcal O\)-linear covariance maps \(D_s\) satisfying \[ D_sD_t(v)=c(s,t)a'_{s,t}D_{st}(v)a_{s,t}^{-1}, \qquad c(s,t)\in\mathcal T. \tag{14}\] The exponent remains bounded in terms of \(p,M\); the orders of the central kernels have not been bounded or used in a finiteness criterion.

Removal of the scalar error. Normalize \(D_1=\mathrm{id}\). Compare \((D_sD_t)D_v\) and \(D_s(D_tD_v)\). Covariance and Equation (5) on the two block sides cancel all the non-scalar terms and leave \[c(s,t)c(st,v)=c(t,v)c(s,tv).\] Because the maps are \(\mathcal O\)-linear, the scalar action is trivial. Thus \(c\) is a normalized \(2\)-cocycle in \(Z^2(S,\mathcal T)\). Lemma 11 makes it a coboundary. Rescaling each \(D_s\) by a normalized \(\mathcal T\)-valued \(1\)-cochain gives \(J_s\) and Equation (9), without changing the covariance equation or the exponent \(m\). ◻

Proof. The induced right module carries the required left crossed action precisely because the comparison has the exact factor identity. Put \(P=\mathcal M\otimes_B A\), a right \(A\)-module. The left \(B'\)-action is the original action on \(\mathcal M\). Define right \(A\)-linear operators \[L_s(v\otimes a)=J_s(v)\otimes u_s a.\] They are well defined on the balanced tensor product: covariance and \(u_sb=\alpha_s(b)u_s\) identify the images of \(vb\otimes a\) and \(v\otimes ba\). They are invertible, and covariance gives \(L_s b'=\alpha'_s(b')L_s\). The exact factor identity gives \[\begin{align*} L_sL_t(v\otimes a) &=a'_{s,t}J_{st}(v)a_{s,t}^{-1}\otimes a_{s,t}u_{st}a\\ &=a'_{s,t}L_{st}(v\otimes a). \end{align*}\] Thus these operators define a left \(A'\)-action.

The right \(B\)-module \(\mathcal M\) is a finitely generated projective generator. Inducing its projective summand and generator identities to \(A\) shows that \(P\) is a projective generator as a right \(A\)-module. It remains to identify its endomorphism order with \(A'\).

As a right \(B\)-module, \(P\) is the direct sum of the components \(\mathcal M\otimes_B Bu_s\). Induction–restriction adjunction gives the following isomorphisms of \(\mathcal O\)-modules: \[\operatorname{End}_A(P)\cong\operatorname{Hom}_B(\mathcal M,P) \cong\bigoplus_{s\in S} \operatorname{Hom}_B(\mathcal M,\mathcal M\otimes_B Bu_s).\] Explicitly, a \(B\)-linear map \(f\) extends uniquely to the \(A\)-endomorphism \(v\otimes a\mapsto f(v)a\). For a map in the degree-\(s\) summand, composing its extension with \(L_s^{-1}\) gives a degree-one endomorphism, hence an element of \(\operatorname{End}_B(\mathcal M)=B'\). Therefore \[\operatorname{End}_A(P)=\bigoplus_{s\in S}B'L_s.\] The relations already checked identify this order with \(A'\). The projective-generator characterization of Morita equivalence now proves the lemma. ◻

Proof. Constructing the group and its block. Use the actual automorphisms \(\alpha_s\) of \(\bar N\) and actual elements \(n_{s,t}\in N\) supplied by Proposition 5(ii). On the finite set \(\bar N\times S\) define \[ (g,s)(h,t)=\bigl(g\alpha_s(h)n_{s,t},st\bigr). \tag{15}\] Equations (4) and (5) give associativity, and their normalizations give the identity and inverses. Hence this is a finite group \(\Gamma_S\) with normal subgroup \(\bar N\) and quotient \(S\). Sending the crossed generator \(u_s\) to \((1,s)\) identifies the restriction of the unprojected crossed order with \(\mathcal O\Gamma_S\).

The block \(\bar b\) is \(S\)-stable. A stable block has a unique covering block in an extension of \(p\)-power index. Its idempotent is \(\bar b\), so our restriction is exactly \[A_S=\mathcal O\Gamma_S\bar b.\] Bounding its defect and coefficient period. If \(D\) is a defect group of this block, normal block theory gives a defect group \(D\cap\bar N\) of \(\bar b\), after conjugacy. Moreover \(D\) maps into \(S\), and in this stable \(p\)-extension it maps onto \(S\). In particular, \[|D|\leq |S|\,|D\cap\bar N|.\] Both quantities on the right are bounded by Proposition 5. Notice that the bound is applied to the group after the central descent in Section 4.

Proposition 12, applied to these same actual factors, and Lemma 13 give a Morita equivalence between \(A_S\) and \(\sigma^m A_S\), with \(m\) uniformly bounded. Thus \(\operatorname{mf}_{\mathcal O}(A_S)\leq m\). We now have an actual block with both required local bounds. Write \(M_1(p,M)\) for the defect-order bound just obtained. Proposition 3(i), applied with \(M_1(p,M)\), bounds its Cartan sum. Part (ii) of that proposition gives the claimed finite integral Morita list. There are only finitely many abstract groups \(S\) of the possible bounded orders. ◻

Proof. Pure subgroups and a fixed identity order. First suppose \(S\) is pure. Let \(e\) be a basic idempotent of \(B_0\). For every homogeneous automorphism, its translate of \(e\) is conjugate to \(e\) by a unit of \(B_0\): both idempotents represent a projective module containing one copy of every indecomposable projective type. Correcting each homogeneous unit by such a unit of \(B_0\) makes it commute with \(e\). Consequently \[C_S=eA_Se\] is a crossed order on \(R=eB_0e\), and \[ \operatorname{rank}_{\mathcal O}C_S=|S|\operatorname{rank}_{\mathcal O}R. \tag{16}\] The idempotent \(e\) is full in \(A_S\), since it is already full in \(B_0\).

Proposition 14 gives finitely many Morita classes for these total orders. For fixed \(R\) and \(S\), Proposition 10 now gives finitely many based graded types. The proof of that proposition applies here because each \(C_S\) is Morita equivalent to the actual block \(A_S\). In particular, its first Hochschild cohomology vanishes. Equation (16) also displays explicitly the rank bound used to pass from Morita classes to isomorphism classes of total orders.

Transport from arbitrary subgroups. For an arbitrary \(p\)-subgroup of \(Q\), conjugate by a character in \(\Xi\) to make it pure, as explained after Proposition 5. Conjugation is implemented by an integral homogeneous unit and transports the identity block and its full basic corner. The transported identity order is still in the same finite list. Arbitrary integral markings are already included in Proposition 10; inner differences are absorbed by conjugation in degree one, and outer differences form a finite set. Transporting back proves the assertion for the original subgroup and all its markings. ◻

Proof. The finite commutative \(\mathcal O\)-algebra \(Z_R\) is a product of complete local algebras, by henselianity of \(\mathcal O\). More than one factor would give a nontrivial central idempotent of \(R\). It is therefore local. Its residue field is a finite extension of the algebraically closed field \(k\), hence is \(k\). The residue map on units is split by the scalar Teichmüller units, and its kernel is \(U_R\), proving Equation (17). Both factors are canonical and are preserved by the stated automorphisms; these act trivially on \(\mathcal T\).

For \(u\in U_R\), the polynomial \(X^n-u\) has the root \(1\) modulo the maximal ideal, with derivative \(n\) a unit. Hensel’s lemma gives a unique root in \(U_R\). This proves the last assertion. ◻

Proof. The two factors in Equation (17) play different roles: restriction is injective on the principal-unit part, while the entire Teichmüller cohomology group is finite. On \(U_R\) the composite \(\operatorname{cor}\circ\operatorname{res}\) is multiplication by \([Q:S]\). This is an automorphism on \(U_R\), by Lemma 16, and therefore on its cohomology. Restriction is consequently injective on \(H^2(Q,U_R)\).

On \(\mathcal T\), the action is trivial and \(H^2(Q,\mathcal T)\) is finite. Indeed, let \(d=|Q|\). The \(d\)th-power map on \(\mathcal T\) is onto and has the finite kernel \(\mu_d(\mathcal O)\cap\mathcal T\). The exact sequence in cohomology and the annihilation of positive cohomology by \(d\) show that \(H^2(Q,\mathcal T)\) is a quotient of \[H^2(Q,\mu_d(\mathcal O)\cap\mathcal T),\] which is finite because both the group and the coefficient module are finite. Combining the two factors proves the lemma. ◻

Proof. Write \(A=\bigoplus_{q\in Q}Ru_q\) for one crossed order and set \[ K=O_{p'}(Q)\cap\ker(Q\longrightarrow\mathcal F). \tag{18}\] It is a normal \(p'\)-subgroup, and \([Q:K]\leq[Q:O_{p'}(Q)]|\mathcal F|\) is bounded. We isolate its twisted group algebra, pass to matrix corners, verify their Sylow restrictions, and finally count central factor systems on the bounded quotient.

Scalar factors on \(K\). Since \(K\) acts trivially modulo inner automorphisms, change the \(K\)-degree units so that they centralize \(R\). Their factors belong to \(Z_R^\times\) and form an ordinary \(2\)-cocycle for the trivial \(K\)-action. Multiplication by \(|K|\) is invertible on \(U_R\), so \(H^i(K,U_R)=0\) for \(i>0\). Removing the \(U_R\)-component by a central change of units gives units \(v_k\) with \[v_kv_l=\beta(k,l)v_{kl},\qquad \beta(k,l)\in\mathcal T.\] Their \(\mathcal O\)-span is a twisted group algebra \(E=\mathcal O_\beta K\), and the restriction to \(K\) is \(R\otimes_{\mathcal O}E\).

Every homogeneous unit of \(A\) normalizes \(E\). To verify this, write \[u_qv_ku_q^{-1}=z_kv_{qkq^{-1}},\qquad z_k\in Z_R^\times.\] The coefficient is central because both sides centralize \(R\). Comparing the products for \(k,l\in K\) shows that their \(U_R\)-components define a homomorphism \(K\to U_R\): the original factors, their conjugates, and the factors in the new \(K\)-degrees all lie in \(\mathcal T\). Such a homomorphism is trivial, since multiplication by \(|K|\) is invertible on \(U_R\). Hence every \(z_k\) belongs to \(\mathcal T\), proving normalization of \(E\).

The matrix factors of \(E\). The finitely many values of \(\beta\) generate a finite \(p'\)-subgroup of \(\mathcal T\). The usual central extension defined by \(\beta\) is therefore a finite \(p'\)-group, and \(E\) is its character summand over \(\mathcal O\). A finite \(p'\)-group algebra over \(\mathcal O\) is a product of full matrix algebras: its reduction is split semisimple, and the matrix units lift over the complete ring \(\mathcal O\). Each lifted primitive corner has rank one and is \(\mathcal O\). We obtain \[ E\cong\prod_j M_{n_j}(\mathcal O). \tag{19}\] Every \(n_j\) is prime to \(p\). Indeed these are ordinary irreducible character degrees in the indicated character sector of a finite \(p'\)-group; each such degree divides the group order.

Let \(e_j\) denote the identity of one matrix factor. The \(Q\)-orbits on these idempotents give central summands of \(A\). In each orbit summand a single \(e_j\) is full, because its homogeneous conjugates sum to that orbit idempotent. Let \(Q_j^{\mathrm{pre}}\) be its stabilizer. It contains \(K\), and the \(e_j\)-corner, with its grading coarsened by \(K\), is crossed over \[R\otimes_{\mathcal O}M_n(\mathcal O),\qquad n=n_j,\] by the group \(Q_j=Q_j^{\mathrm{pre}}/K\). The order of \(Q_j\) is bounded by \([Q:K]\).

In each \(Q_j\)-degree choose a unit \(e_ju_q\) inherited from an original \(Q\)-degree. Its conjugation preserves both \(R\) and \(M_n(\mathcal O)\), by the normalization of \(E\) proved above. Every \(\mathcal O\)-algebra automorphism of \(M_n(\mathcal O)\) is inner. For example, its images of the standard matrix idempotents split \(\mathcal O^n\) into free rank-one summands; choosing compatible basis vectors for the matrix units constructs an implementing matrix. Correct the homogeneous units by these matrices so that they centralize the matrix factor. Their multiplication factors then lie in its centralizer inside \(R\otimes_{\mathcal O}M_n(\mathcal O)\), namely \(R\). The orbit corner is consequently \[M_n(\mathcal O)\otimes_{\mathcal O}C_j,\] where \(C_j\) is a \(Q_j\)-crossed order on \(R\) with the same outer action on \(R\). Equivalently, \(C_j\) is the centralizer of the matrix factor in the orbit corner.

This passage replaces the unbounded group \(Q\) by the bounded group \(Q_j\). It has not yet supplied a finite list: the matrix corrections may change the multiplication factors of a restriction. We next recover those factors. The matrix size need not be bounded; its prime-to-\(p\) property is what the recovery uses.

Preservation of the based Sylow restriction. Let \(S\) be a Sylow \(p\)-subgroup of \(Q_j\). Schur–Zassenhaus in its inverse image gives a \(p\)-subgroup \(\widetilde S\leq Q_j^{\mathrm{pre}}\) mapping isomorphically onto \(S\). Use the original units \(u_s\) in these degrees, with original factors \(a_{s,t}\in R^\times\). Their conjugations on \(E_j=M_n(\mathcal O)\) form an honest group action, since \(a_{s,t}\) centralizes \(E_j\).

Choose implementing matrices \(c_s\in\mathrm{GL}_n(\mathcal O)\), with \(c_1=1\). We may take \(\det c_s=1\). In fact, since \(p\nmid n\), every unit of \(\mathcal O\) has an \(n\)th root: take a Teichmüller root of its residue and then use Hensel’s lemma on the principal-unit part. Multiplication by a scalar thus normalizes the determinant without changing the implemented automorphism.

Because the automorphisms form a group action, there is a scalar \(\gamma(s,t)\in\mathcal O^\times\) such that \[c_sc_tc_{st}^{-1}=\gamma(s,t)I_n.\] Taking determinants gives \(\gamma(s,t)^n=1\). Thus \(\gamma(s,t)\in\mathcal T\), since \(n\) is prime to \(p\). Associativity shows that \(\gamma\) is a normalized \(\mathcal T\)-valued \(2\)-cocycle on \(S\).

For the corrected units \(w_s=c_s^{-1}e_ju_s\), their actions on \(R\) are unchanged. The formula for their products is \[ w_sw_t=\gamma(s,t)^{-1}a_{s,t}w_{st}. \tag{20}\] For example, this follows by using \(u_sc_tu_s^{-1}=c_sc_tc_s^{-1}\) in the product on the left. Lemma 11 removes \(\gamma\) by scalar rescaling. Therefore the restriction of \(C_j\) to \(S\) is based-isomorphic to the original restriction to \(\widetilde S\). The corrected units generate the corresponding homogeneous \(R\)-components of the centralizer \(C_j\). For a fixed original unit, any two implementing matrices differ by a scalar unit; the subsequent scalar rescaling also changes only the generator of its component. Thus these choices change the presentation, not the based graded order. Hypothesis (iii) supplies a finite list for the restrictions now obtained.

Crossed systems on a bounded quotient. There are finitely many possible abstract groups \(Q_j\) and outer homomorphisms \(Q_j\to\mathcal F\). Fix one of each and choose representatives \(\alpha_q\in\operatorname{Aut}_{\mathcal O}(R)\) of its outer classes. Changing homogeneous units makes every crossed system with this outer action use these same automorphisms. If any factors satisfying the crossed identities exist, their classes under central changes of units form a torsor under \[ H^2(Q_j,Z_R^\times). \tag{21}\] Indeed, the ratio of two factor systems with these fixed actions is central by Equation (1). Equation (2) says that the ratio is a \(2\)-cocycle. A homogeneous-unit change preserving every \(\alpha_q\) must itself be central, and changes the ratio by a coboundary. The action on the centre is a genuine group action because inner automorphisms act trivially there.

On a Sylow \(p\)-subgroup, the based finiteness just proved gives finitely many restricted torsor classes. To see that the fixed representatives cause no extra ambiguity, observe that a based isomorphism between systems with the same automorphisms must change their units centrally. Lemma 17 gives finite fibres for the restriction map from Equation (21): a nonempty fibre is a coset of its finite kernel. Thus there are finitely many based crossed orders \(C_j\).

Each \(C_j\) has only finitely many block summands. The orbit corners above are matrix algebras over them, and the selected \(e_j\) was full in the corresponding orbit summand of \(A\). Hence every block summand of \(A\) is Morita equivalent to one of this finite list of block summands of the \(C_j\). Neither the number of matrix factors in Equation (19) nor their sizes had to be bounded. This proves the theorem. ◻

Proof of Corollary 2. Keep \(k=\overline{\mathbb F}_p\) and \(\mathcal O=W(k)\), and fix \(\mathcal R\) and its residue field \(\ell\) as in the corollary. Choose an embedding \(\iota:k\hookrightarrow\ell\). Since \(\ell\) is perfect, \(W(\ell)\) is a Cohen ring. The coefficient-ring theorem (The Stacks Project Authors 2026a, Theorem 10.160.8, Tag 032A) gives a local map \(W(\ell)\to\mathcal R\) inducing the identity on \(\ell\). It is injective: any nonzero ideal of the DVR \(W(\ell)\) contains a power of \(p\), whereas \(\mathcal R\) has characteristic zero. Composing with the Witt-vector map \(W(\iota)\), which is injective on Witt coordinates, gives a fixed local embedding \[\mathcal O=W(k)\xrightarrow{W(\iota)}W(\ell)\hookrightarrow\mathcal R\] whose residue map is \(\iota\). Only this coefficient embedding is used; no finiteness of \(\mathcal R\) over \(\mathcal O\) is needed.

We recall the residue-field argument of (OpenAI 2026b, sec. 2.1). For every finite group \(G\), scalar extension identifies \(Z(\ell G)=\ell\otimes_k Z(kG)\). Each local factor of the finite commutative algebra \(Z(kG)\) has nilpotent radical and residue field \(k\). After extending to \(\ell\), the extended radical is a nilpotent ideal with quotient \(\ell\), so the factor remains local. Consequently every primitive central idempotent \(\bar b\) of \(\ell G\) is uniquely of the form \(1\otimes\bar b_0\) for a primitive central idempotent \(\bar b_0\) of \(kG\). For every \(p\)-subgroup \(Q\leq G\), restriction of coefficients to \(C_G(Q)\) gives \[\operatorname{Br}^{\ell}_Q(\bar b) =1\otimes\operatorname{Br}^{k}_Q(\bar b_0).\] Faithfulness of field extension shows that these Brauer images are nonzero for the same \(Q\). The characterization of defect groups as the maximal \(p\)-subgroups with nonzero Brauer image therefore gives the same defect groups for the two residue blocks.

For either coefficient pair \((\Lambda,F)=(\mathcal O,k)\) or \((\mathcal R,\ell)\), the class sums give a \(\Lambda\)-basis of \(Z(\Lambda G)\) whose reductions give an \(F\)-basis of \(Z(FG)\). Thus \(Z(\Lambda G)\) is a finite free complete commutative \(\Lambda\)-algebra with residue algebra exactly \(Z(FG)\). Idempotents in this residue algebra lift uniquely: the derivative \(2x-1\) of \(x^2-x\) is a unit at an idempotent modulo the maximal ideal of \(\Lambda\), so the complete-ring Hensel argument applies, including when \(p=2\). Primitivity is preserved, since central decompositions lift and an idempotent reducing to zero is zero.

Now let \(b\) be a block idempotent of \(\mathcal R G\), and let \(b_0\) be the unique central idempotent of \(\mathcal OG\) lifting the corresponding \(\bar b_0\). The image of \(b_0\) in \(\mathcal R G\) is a central idempotent reducing to \(\bar b\), so uniqueness of the central lift makes it equal to \(b\). In particular \[ \mathcal R G b\cong\mathcal R\otimes_{\mathcal O}(\mathcal OG b_0) \tag{22}\] as \(\mathcal R\)-algebras, and \(b_0\) has the same defect group as \(b\).

Finally, scalar extension preserves the specified linear Morita equivalences. Indeed, for finite \(\mathcal O\)-algebras \(A,B\), a \(\mathcal O\)-linear Morita equivalence is represented by inverse \(\mathcal O\)-central Morita bimodules \({}_B P_A\) and \({}_A Q_B\) with \[P\otimes_A Q\cong B,\qquad Q\otimes_B P\cong A.\] Write \(A_{\mathcal R}=\mathcal R\otimes_{\mathcal O}A\) and similarly for \(B,P,Q\). Base change of the bimodules and their context maps gives \[\begin{aligned} P_{\mathcal R}\otimes_{A_{\mathcal R}}Q_{\mathcal R} &\cong\mathcal R\otimes_{\mathcal O}(P\otimes_A Q) \cong B_{\mathcal R},\\ Q_{\mathcal R}\otimes_{B_{\mathcal R}}P_{\mathcal R} &\cong\mathcal R\otimes_{\mathcal O}(Q\otimes_B P) \cong A_{\mathcal R}. \end{aligned}\] The Morita-context identities are preserved, as are finite projectivity and the generator property. These bimodules therefore give an \(\mathcal R\)-linear Morita equivalence on finitely generated modules.

Choose the finite list of \(\mathcal O\)-block representatives of defect order at most \(M\) from Theorem 1. The block \(\mathcal OG b_0\) in (22) has that defect bound, so it is \(\mathcal O\)-linearly Morita equivalent to one representative. The base-changed context puts \(\mathcal R G b\) in the Morita class of its scalar extension. These scalar extensions form a finite list over the fixed ring \(\mathcal R\), proving the corollary. ◻