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ring with declared local conditions. We separate the global automorphy problem into residual occupancy, deformation support, Hecke occupancy, and arithmetic realization. Local conditions are encoded by quotient rings rather than by idempotents, since irreducible components of a local deformation ring need not be clopen. For an ordinary or derived patching datum, full support descends through the augmentation quotient. Near faithfulness is shown to be equivalent to full support for finite modules over Noetherian rings. An isomorphism \\(R\\cong\\TT\\) transports support, Fitting ideals, and point fibers, but a quotient m","\\begin{abstract}\nLet \\(\\bar\\rho\\) be a residual Galois representation and let \\(R\\) be a global deformation ring with declared local conditions. We separate the global automorphy problem into residual occupancy, deformation support, Hecke occupancy, and arithmetic realization. Local conditions are encoded by quotient rings rather than by idempotents, since irreducible components of a local deformation ring need not be clopen. For an ordinary or derived patching datum, full support descends through the augmentation quotient. Near faithfulness is shown to be equivalent to full support for finite modules over Noetherian rings. An isomorphism \\(R\\cong\\TT\\) transports support, Fitting ideals, and point fibers, but a quotient map can erase an entire component. Finite Hecke alg","and arithmetic realization. Local conditions are encoded by quotient rings rather than by idempotents, since irreducible components of a local deformation ring need not be clopen. For an ordinary or derived patching datum, full support descends through the augmentation quotient. Near faithfulness is shown to be equivalent to full support for finite modules over Noetherian rings. An isomorphism \\(R\\cong\\TT\\) transports support, Fitting ideals, and point fibers, but a quotient map can erase an entire component. Finite Hecke algebras are resolved into canonical Artin packets, so simultaneous generalized eigenspaces are available without assuming semisimplicity. Nested packets in one finite carrier stabilize; for changing finite carriers, nonzero stable cores give compatible inverse limit vectors. These results give a pointwise criterion: a characteristic zero deformation point in automorphic support produces a Hecke eigensystem. With attached Galois representations, local global compatibility, and classicality, the point is automorphic. A fixed compatible system then propagates the conc…","Automorphy lifting joins several kinds of mathematics that are easy to conflate. A residual representation must first occur in an automorphic carrier. The chosen characteristic zero representation must lie on a deformation problem controlled by local conditions. A patching argument must show that the automorphic module reaches the relevant part of the deformation space. A comparison such as \\(R\\cong\\TT\\) must identify the arithmetic and Hecke coordinates. Finally, the resulting Hecke point must be classical and must carry the expected Galois representation with the required local compatibility.","The last part assembles these statements into a conditional theorem. A characteristic zero point of \\(R\\) is automorphic when it lies in the support of the automorphic carrier, the comparison with the Hecke algebra is available, and the corresponding Hecke fiber satisfies the declared realization theorem. If the point belongs to a fixed strictly compatible system, equality of Frobenius polynomials propagates the automorphic identification to every coefficient place by Chebotarev density and Brauer Nesbitt \\cite{BrauerNesbitt1941,Neukirch1999,Gee2022}.","The organization is as follows. Local quotient conditions are developed in \\cref{sec:local_conditions}. Adequacy and residual occupancy appear in \\cref{sec:adequacy}. Auxiliary group algebras are treated in \\cref{sec:auxiliary}. Ordinary and derived patching support appear in \\cref{sec:patching_support}. The effect of \\(R\\cong\\TT\\) is isolated in \\cref{sec:rt}. Finite Hecke packets and infinite occupancy are developed in \\cref{sec:hecke_packets}. Pointwise automorphy and compatible systems appear in \\cref{sec:pointwise,sec:compatible}. The coefficient prime two is discussed in \\cref{sec:two}. The final conditional theorem is stated in \\cref{sec:completion}.","Throughout, \\(S\\) is a finite set of places. Frobenius elements are geometric Frobenius elements; all Hecke polynomials, attached representations, and common algebraic spines use that same convention and the normalization specified by the declared realization theorem.","Let \\(F\\) be a number field. Fix a prime \\(p\\), a finite extension \\(E/\\QQ_p\\), its valuation ring \\(\\OO\\), uniformizer \\(\\varpi\\), and residue field \\(k\\). Let\n\\[\n\\bar\\rho:G_{F,S}\\longrightarrow\\GL_n(k)\n\\]\nbe a continuous residual representation, where \\(S\\) is a finite set of places containing every place at which the deformation problem is allowed to ramify.","\\begin{definition}[Global deformation datum]\\label{def:global_datum}\nA global deformation datum consists of the following objects.\n\\begin{enumerate}\n\\item A residual representation \\(\\bar\\rho\\).\n\\item For every \\(v\\in S\\), a framed local deformation ring \\(R_v^\\square\\).\n\\item A quotient \\(R_v^{\\tau_v}\\) representing the chosen local condition whenever such a quotient is known to exist.\n\\item A global deformation ring \\(R^\\square\\) with a continuous local map\n\\[\nR^{\\mathrm{loc}}:=\\wideotimes_{v\\in S,\\OO}R_v^\\square\n\\longrightarrow R^\\square.\n\\]\n\\item The induced global ring with local conditions\n\\[\nR^{\\tau}:=R^\\square\\wideotimes_{R^{\\mathrm{loc}}}R^{\\mathrm{loc},\\tau},\n\\qquad\nR^{\\mathrm{loc},\\tau}:=\\wideotimes_{v\\in S,\\OO}R_v^{\\tau_v}.\n\\]\n\\end{enumerate}\n\\end{definition}","l deformation datum]\\label{def:global_datum}\nA global deformation datum consists of the following objects.\n\\begin{enumerate}\n\\item A residual representation \\(\\bar\\rho\\).\n\\item For every \\(v\\in S\\), a framed local deformation ring \\(R_v^\\square\\).\n\\item A quotient \\(R_v^{\\tau_v}\\) representing the chosen local condition whenever such a quotient is known to exist.\n\\item A global deformation ring \\(R^\\square\\) with a continuous local map\n\\[\nR^{\\mathrm{loc}}:=\\wideotimes_{v\\in S,\\OO}R_v^\\square\n\\longrightarrow R^\\square.\n\\]\n\\item The induced global ring with local conditions\n\\[\nR^{\\tau}:=R^\\square\\wideotimes_{R^{\\mathrm{loc}}}R^{\\mathrm{loc},\\tau},\n\\qquad\nR^{\\mathrm{loc},\\tau}:=\\wideotimes_{v\\in S,\\OO}R_v^{\\tau_v}.\n\\]\n\\end{enumerate}\n\\end{definition}","m}\nA global deformation datum consists of the following objects.\n\\begin{enumerate}\n\\item A residual representation \\(\\bar\\rho\\).\n\\item For every \\(v\\in S\\), a framed local deformation ring \\(R_v^\\square\\).\n\\item A quotient \\(R_v^{\\tau_v}\\) representing the chosen local condition whenever such a quotient is known to exist.\n\\item A global deformation ring \\(R^\\square\\) with a continuous local map\n\\[\nR^{\\mathrm{loc}}:=\\wideotimes_{v\\in S,\\OO}R_v^\\square\n\\longrightarrow R^\\square.\n\\]\n\\item The induced global ring with local conditions\n\\[\nR^{\\tau}:=R^\\square\\wideotimes_{R^{\\mathrm{loc}}}R^{\\mathrm{loc},\\tau},\n\\qquad\nR^{\\mathrm{loc},\\tau}:=\\wideotimes_{v\\in S,\\OO}R_v^{\\tau_v}.\n\\]\n\\end{enumerate}\n\\end{definition}","every \\(v\\in S\\), a framed local deformation ring \\(R_v^\\square\\).\n\\item A quotient \\(R_v^{\\tau_v}\\) representing the chosen local condition whenever such a quotient is known to exist.\n\\item A global deformation ring \\(R^\\square\\) with a continuous local map\n\\[\nR^{\\mathrm{loc}}:=\\wideotimes_{v\\in S,\\OO}R_v^\\square\n\\longrightarrow R^\\square.\n\\]\n\\item The induced global ring with local conditions\n\\[\nR^{\\tau}:=R^\\square\\wideotimes_{R^{\\mathrm{loc}}}R^{\\mathrm{loc},\\tau},\n\\qquad\nR^{\\mathrm{loc},\\tau}:=\\wideotimes_{v\\in S,\\OO}R_v^{\\tau_v}.\n\\]\n\\end{enumerate}\n\\end{definition}","\\begin{standardfact}[Represented deformation problem]\\label{fact:represented_deformation}\nUnder the usual finiteness and Schur type hypotheses, framed lifts of \\(\\bar\\rho\\) are represented by complete Noetherian local \\(\\OO\\) algebras. A declared local deformation condition is represented by a quotient when the corresponding subfunctor is representable. Tangent and obstruction spaces are described by the appropriate first and second cohomology groups.\n\\end{standardfact}","\\begin{definition}[Residual automorphic occupancy]\\label{def:residual_occupancy}\nThe residual datum \\((\\bar\\rho,\\tau)\\) is automorphically occupied when a nonzero localized automorphic or cohomological Hecke carrier realizes the residual representation and the declared local conditions.\n\\end{definition}","\\begin{theorem}[Completed tensor quotient]\\label{thm:tensor_quotient}\nLet \\(R_1,\\ldots,R_s\\) be complete Noetherian local \\(\\OO\\) algebras with common residue field \\(k\\). Let \\(I_i\\subseteq R_i\\) be closed ideals. Put\n\\[\nR^{\\mathrm{loc}}=R_1\\wideotimes_{\\OO}\\cdots\\wideotimes_{\\OO}R_s\n\\]\nand let \\(J\\) be the closed ideal generated by the images of the \\(I_i\\). Then there is a canonical isomorphism\n\\[\n\\boxed{\n(R_1/I_1)\\wideotimes_{\\OO}\\cdots\\wideotimes_{\\OO}(R_s/I_s)\n\\cong\nR^{\\mathrm{loc}}/J.\n}\n\\]\n\\end{theorem}","gin{theorem}[Completed tensor quotient]\\label{thm:tensor_quotient}\nLet \\(R_1,\\ldots,R_s\\) be complete Noetherian local \\(\\OO\\) algebras with common residue field \\(k\\). Let \\(I_i\\subseteq R_i\\) be closed ideals. Put\n\\[\nR^{\\mathrm{loc}}=R_1\\wideotimes_{\\OO}\\cdots\\wideotimes_{\\OO}R_s\n\\]\nand let \\(J\\) be the closed ideal generated by the images of the \\(I_i\\). Then there is a canonical isomorphism\n\\[\n\\boxed{\n(R_1/I_1)\\wideotimes_{\\OO}\\cdots\\wideotimes_{\\OO}(R_s/I_s)\n\\cong\nR^{\\mathrm{loc}}/J.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nWrite \\(S=\\bigotimes_{i=1}^{s}R_i\\) for the ordinary tensor product. Its ideal of definition is\n\\[\n\\fraka=\\frakm_{\\OO}S+\\sum_i\\frakm_{R_i}S,\n\\qquad L=\\sum_i I_iS.\n\\]\nThe scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism\n\\[\nS/L\\cong\\bigotimes_{i=1}^{s}(R_i/I_i),\n\\]\ncarrying the image of \\(\\fraka\\) to the defining ideal on the right. The quotient topology on \\(S/L\\) is precisely this image ideal topology.","nary tensor product. Its ideal of definition is\n\\[\n\\fraka=\\frakm_{\\OO}S+\\sum_i\\frakm_{R_i}S,\n\\qquad L=\\sum_i I_iS.\n\\]\nThe scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism\n\\[\nS/L\\cong\\bigotimes_{i=1}^{s}(R_i/I_i),\n\\]\ncarrying the image of \\(\\fraka\\) to the defining ideal on the right. The quotient topology on \\(S/L\\) is precisely this image ideal topology.","Here completion means the actual Hausdorff completion for the additive uniformity of the indicated adic topology. For every adic ring \\(S\\) and ideal \\(L\\), there is a canonical bicontinuous ring isomorphism\n\\[\n\\widehat S\\big/\\overline{L\\widehat S}\\cong\\widehat{S/L}.\n\\]\nIndeed, the completed quotient map kills the closed extended ideal. In the other direction, the quotient map from \\(S/L\\) to \\(\\widehat S/\\overline{L\\widehat S}\\) extends to its completion. The latter quotient is complete: the adic uniformity has a countable base, and the quotient of a complete countably based additive group is complete. Its closed defining subgroup makes it Hausdorff. The two composites agree with the identity on dense quotient representatives, so continuity proves both inverse laws.","Applying this comparison to the displayed ordinary tensor isomorphism proves the claim. The closed extended ideal is exactly \\(J\\), since extension commutes with the finite sum of the factor ideals. The maps preserve coefficients and each canonical factor map. The common residue field identifies \\(S/\\fraka\\) with \\(k\\); finite generation of \\(\\fraka\\), supplied by the Noetherian factor hypotheses, identifies the completed topology with its maximal ideal topology. Proper quotients retain locality. If a condition is the unit ideal, the ring comparison remains valid with the zero ring, without treating it as a local ring with residue field \\(k\\).\n\\end{proof}","l factor map. The common residue field identifies \\(S/\\fraka\\) with \\(k\\); finite generation of \\(\\fraka\\), supplied by the Noetherian factor hypotheses, identifies the completed topology with its maximal ideal topology. Proper quotients retain locality. If a condition is the unit ideal, the ring comparison remains valid with the zero ring, without treating it as a local ring with residue field \\(k\\).\n\\end{proof}","\\begin{corollary}[Simultaneous local conditions]\\label{cor:simultaneous_local}\nA continuous point of \\(R^{\\mathrm{loc}}\\) with Hausdorff target factors continuously through \\(R^{\\mathrm{loc}}/J\\) if and only if its restriction to every \\(R_i\\) factors continuously through \\(R_i/I_i\\). Each factor is unique. If the original point is local, its quotient factor is local.\n\\end{corollary}","\\begin{proof}\nLet \\(f:R^{\\mathrm{loc}}\\to A\\) be the point. Since \\(A\\) is Hausdorff and \\(f\\) is continuous, \\(\\ker f\\) is closed. Thus it contains the closed ideal \\(J\\) if and only if it contains the algebraic ideal generated by the images of the \\(I_i\\). This is equivalent to every local restriction killing \\(I_i\\). The universal property of the actual ideal quotient gives the factors, and the quotient topology gives their continuity. Surjectivity of the quotient map gives uniqueness. If \\(f\\) reflects units, lift an element of the quotient to the original ring to see that its factor also reflects units.\n\\end{proof}","ontinuous, \\(\\ker f\\) is closed. Thus it contains the closed ideal \\(J\\) if and only if it contains the algebraic ideal generated by the images of the \\(I_i\\). This is equivalent to every local restriction killing \\(I_i\\). The universal property of the actual ideal quotient gives the factors, and the quotient topology gives their continuity. Surjectivity of the quotient map gives uniqueness. If \\(f\\) reflects units, lift an element of the quotient to the original ring to see that its factor also reflects units.\n\\end{proof}","\\begin{proposition}[Global base change for local conditions]\\label{prop:global_local_basechange}\nIn the global deformation datum of \\cref{def:global_datum}, let \\(R^\\square\\) be a complete Noetherian local \\(\\OO\\) algebra receiving a continuous local \\(\\OO\\) algebra map from \\(R^{\\mathrm{loc}}\\). Then\n\\[\nR^{\\tau}\n=\nR^\\square\\wideotimes_{R^{\\mathrm{loc}}}R^{\\mathrm{loc},\\tau}\n\\]\nrepresents the base change of the global deformation problem to the simultaneous local conditions. A continuous local \\(\\OO\\) algebra point of \\(R^\\square\\) with complete Hausdorff coefficient target factors uniquely and continuously through \\(R^{\\tau}\\) exactly when all of its local restrictions satisfy the declared quotient conditions.\n\\end{proposition}","he global deformation datum of \\cref{def:global_datum}, let \\(R^\\square\\) be a complete Noetherian local \\(\\OO\\) algebra receiving a continuous local \\(\\OO\\) algebra map from \\(R^{\\mathrm{loc}}\\). Then\n\\[\nR^{\\tau}\n=\nR^\\square\\wideotimes_{R^{\\mathrm{loc}}}R^{\\mathrm{loc},\\tau}\n\\]\nrepresents the base change of the global deformation problem to the simultaneous local conditions. A continuous local \\(\\OO\\) algebra point of \\(R^\\square\\) with complete Hausdorff coefficient target factors uniquely and continuously through \\(R^{\\tau}\\) exactly when all of its local restrictions satisfy the declared quotient conditions.\n\\end{proposition}","um of \\cref{def:global_datum}, let \\(R^\\square\\) be a complete Noetherian local \\(\\OO\\) algebra receiving a continuous local \\(\\OO\\) algebra map from \\(R^{\\mathrm{loc}}\\). Then\n\\[\nR^{\\tau}\n=\nR^\\square\\wideotimes_{R^{\\mathrm{loc}}}R^{\\mathrm{loc},\\tau}\n\\]\nrepresents the base change of the global deformation problem to the simultaneous local conditions. A continuous local \\(\\OO\\) algebra point of \\(R^\\square\\) with complete Hausdorff coefficient target factors uniquely and continuously through \\(R^{\\tau}\\) exactly when all of its local restrictions satisfy the declared quotient conditions.\n\\end{proposition}","eiving a continuous local \\(\\OO\\) algebra map from \\(R^{\\mathrm{loc}}\\). Then\n\\[\nR^{\\tau}\n=\nR^\\square\\wideotimes_{R^{\\mathrm{loc}}}R^{\\mathrm{loc},\\tau}\n\\]\nrepresents the base change of the global deformation problem to the simultaneous local conditions. A continuous local \\(\\OO\\) algebra point of \\(R^\\square\\) with complete Hausdorff coefficient target factors uniquely and continuously through \\(R^{\\tau}\\) exactly when all of its local restrictions satisfy the declared quotient conditions.\n\\end{proposition}","\\begin{proof}\nPut \\(A=R^{\\mathrm{loc}}\\), \\(B=R^\\square\\), and let \\(J\\subseteq A\\) be the closed local condition ideal. By \\cref{thm:tensor_quotient}, the independently constructed ring \\(R^{\\mathrm{loc},\\tau}\\) is canonically \\(A/J\\). The ordinary tensor identity\n\\[\nB\\otimes_A(A/J)\\cong B/JB\n\\]\ncarries the defining ideal of the tensor to the image of \\(\\frakm_B\\). Here the structural map is local by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective.","\\begin{proof}\nPut \\(A=R^{\\mathrm{loc}}\\), \\(B=R^\\square\\), and let \\(J\\subseteq A\\) be the closed local condition ideal. By \\cref{thm:tensor_quotient}, the independently constructed ring \\(R^{\\mathrm{loc},\\tau}\\) is canonically \\(A/J\\). The ordinary tensor identity\n\\[\nB\\otimes_A(A/J)\\cong B/JB\n\\]\ncarries the defining ideal of the tensor to the image of \\(\\frakm_B\\). Here the structural map is local by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective. It is also unchan","\\begin{proof}\nPut \\(A=R^{\\mathrm{loc}}\\), \\(B=R^\\square\\), and let \\(J\\subseteq A\\) be the closed local condition ideal. By \\cref{thm:tensor_quotient}, the independently constructed ring \\(R^{\\mathrm{loc},\\tau}\\) is canonically \\(A/J\\). The ordinary tensor identity\n\\[\nB\\otimes_A(A/J)\\cong B/JB\n\\]\ncarries the defining ideal of the tensor to the image of \\(\\frakm_B\\). Here the structural map is local by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective. It is also unchanged by transporting the te","\\begin{proof}\nPut \\(A=R^{\\mathrm{loc}}\\), \\(B=R^\\square\\), and let \\(J\\subseteq A\\) be the closed local condition ideal. By \\cref{thm:tensor_quotient}, the independently constructed ring \\(R^{\\mathrm{loc},\\tau}\\) is canonically \\(A/J\\). The ordinary tensor identity\n\\[\nB\\otimes_A(A/J)\\cong B/JB\n\\]\ncarries the defining ideal of the tensor to the image of \\(\\frakm_B\\). Here the structural map is local by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective. It is also unchanged by transporting the tensor factor from \\(A/J\\) to the actual completed tensor of the local quotients. The transported ideal is exactly its natural defi","\\begin{proof}\nPut \\(A=R^{\\mathrm{loc}}\\), \\(B=R^\\square\\), and let \\(J\\subseteq A\\) be the closed local condition ideal. By \\cref{thm:tensor_quotient}, the independently constructed ring \\(R^{\\mathrm{loc},\\tau}\\) is canonically \\(A/J\\). The ordinary tensor identity\n\\[\nB\\otimes_A(A/J)\\cong B/JB\n\\]\ncarries the defining ideal of the tensor to the image of \\(\\frakm_B\\). Here the structural map is local by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective. It is also unchanged by transporting the tensor factor from \\(A/J\\) to the actual completed tensor of the local quotients. The transported ideal is exactly its natural defining ideal.","and let \\(J\\subseteq A\\) be the closed local condition ideal. By \\cref{thm:tensor_quotient}, the independently constructed ring \\(R^{\\mathrm{loc},\\tau}\\) is canonically \\(A/J\\). The ordinary tensor identity\n\\[\nB\\otimes_A(A/J)\\cong B/JB\n\\]\ncarries the defining ideal of the tensor to the image of \\(\\frakm_B\\). Here the structural map is local by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective. It is also unchanged by transporting the tensor factor from \\(A/J\\) to the actual completed tensor of the local quotients. The transported ideal is exactly its natural defining ideal.","teq A\\) be the closed local condition ideal. By \\cref{thm:tensor_quotient}, the independently constructed ring \\(R^{\\mathrm{loc},\\tau}\\) is canonically \\(A/J\\). The ordinary tensor identity\n\\[\nB\\otimes_A(A/J)\\cong B/JB\n\\]\ncarries the defining ideal of the tensor to the image of \\(\\frakm_B\\). Here the structural map is local by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective. It is also unchanged by transporting the tensor factor from \\(A/J\\) to the actual completed tensor of the local quotients. The transported ideal is exactly its natural defining ideal.","is canonically \\(A/J\\). The ordinary tensor identity\n\\[\nB\\otimes_A(A/J)\\cong B/JB\n\\]\ncarries the defining ideal of the tensor to the image of \\(\\frakm_B\\). Here the structural map is local by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective. It is also unchanged by transporting the tensor factor from \\(A/J\\) to the actual completed tensor of the local quotients. The transported ideal is exactly its natural defining ideal.","ies the defining ideal of the tensor to the image of \\(\\frakm_B\\). Here the structural map is local by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective. It is also unchanged by transporting the tensor factor from \\(A/J\\) to the actual completed tensor of the local quotients. The transported ideal is exactly its natural defining ideal.","l by the stated global deformation datum, so it carries \\(\\frakm_A\\) into \\(\\frakm_B\\). Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism\n\\[\nR^\\tau\\cong B/\\overline{JB}.\n\\]\nThe construction commutes with the canonical map from \\(B\\), which is therefore surjective. It is also unchanged by transporting the tensor factor from \\(A/J\\) to the actual completed tensor of the local quotients. The transported ideal is exactly its natural defining ideal.","For a continuous algebra point \\(x:B\\to C\\) with complete Hausdorff target, the kernel of \\(x\\) is closed. Consequently \\(x\\) has a unique continuous factor through \\(R^\\tau\\) if and only if it kills \\(JB\\), equivalently if its restriction to \\(A\\) kills \\(J\\). By \\cref{cor:simultaneous_local}, this is exactly the requirement that all local restrictions satisfy their quotient conditions. The factor is an algebra map because its composite with the canonical global map is \\(x\\). If \\(x\\) is local, surjectivity of that map shows that its factor reflects units and is local. Uniqueness also proves naturality under coefficient preserving maps of targets. This identifies the represented base change; it does not supply representability of","For a continuous algebra point \\(x:B\\to C\\) with complete Hausdorff target, the kernel of \\(x\\) is closed. Consequently \\(x\\) has a unique continuous factor through \\(R^\\tau\\) if and only if it kills \\(JB\\), equivalently if its restriction to \\(A\\) kills \\(J\\). By \\cref{cor:simultaneous_local}, this is exactly the requirement that all local restrictions satisfy their quotient conditions. The factor is an algebra map because its composite with the canonical global map is \\(x\\). If \\(x\\) is local, surjectivity of that map shows that its factor reflects units and is local. Uniqueness also proves naturality under coefficient preserving maps of targets. This identifies the represented base change; it does not supply representability of a new arithmetic deformation problem.\n\\end{proof}","he kernel of \\(x\\) is closed. Consequently \\(x\\) has a unique continuous factor through \\(R^\\tau\\) if and only if it kills \\(JB\\), equivalently if its restriction to \\(A\\) kills \\(J\\). By \\cref{cor:simultaneous_local}, this is exactly the requirement that all local restrictions satisfy their quotient conditions. The factor is an algebra map because its composite with the canonical global map is \\(x\\). If \\(x\\) is local, surjectivity of that map shows that its factor reflects units and is local. Uniqueness also proves naturality under coefficient preserving maps of targets. This identifies the represented base change; it does not supply representability of a new arithmetic deformation problem.\n\\end{proof}","of \\(x\\) is closed. Consequently \\(x\\) has a unique continuous factor through \\(R^\\tau\\) if and only if it kills \\(JB\\), equivalently if its restriction to \\(A\\) kills \\(J\\). By \\cref{cor:simultaneous_local}, this is exactly the requirement that all local restrictions satisfy their quotient conditions. The factor is an algebra map because its composite with the canonical global map is \\(x\\). If \\(x\\) is local, surjectivity of that map shows that its factor reflects units and is local. Uniqueness also proves naturality under coefficient preserving maps of targets. This identifies the represented base change; it does not supply representability of a new arithmetic deformation problem.\n\\end{proof}","The local coefficient points in \\cref{prop:global_local_basechange} are test rings in the represented deformation category. Later field valued points \\(x:R\\to E'\\) are algebraic or generic specializations, and continuity of \\(\\rho_x\\) is imposed separately. Such a field specialization need not be a local ring map: under \\(\\OO\\to E\\), a uniformizer becomes a unit.","\\begin{proposition}[Idempotents in a local ring]\\label{prop:local_idempotent}\nIf \\(R\\) is a commutative local ring, then its only idempotents are zero and one.\n\\end{proposition}","\\begin{proof}\nLet \\(e^2=e\\). Since \\(e+(1-e)=1\\), at least one of \\(e\\) and \\(1-e\\) is a unit. If \\(e\\) is a unit, then \\(e=1\\). If \\(1-e\\) is a unit, then \\(e=0\\).\n\\end{proof}","\\begin{remark}\\label{rem:components_not_idempotents}\nAn irreducible component of \\(\\Spec R\\) for a local deformation ring need not be open and closed. It therefore need not be selected by an idempotent. Components and local conditions should be described by quotient ideals and module support. Idempotent packet formulas are reserved for finite Artin algebras or other settings with an actual product decomposition.\n\\end{remark}","\\begin{standardfact}[Adequate image]\\label{fact:adequacy}\nLet \\(G\\subseteq\\GL(V)\\) act absolutely irreducibly over a finite field \\(k\\) of characteristic \\(p\\). Under the standard definition, adequacy imposes vanishing conditions on \\(H^1(G,k)\\) and \\(H^1(G,\\ad^0V)\\), a scalar condition, and a generalized eigenspace detection condition for every irreducible submodule of \\(\\ad^0V\\).\n\\end{standardfact}","Let \\(G=S_3\\), let \\(k=\\FF_7\\), and let \\(V\\) be the standard two dimensional representation. In the basis\n\\[\nv_1=(1,-1,0),\n\\qquad\nv_2=(0,1,-1),\n\\]\na transposition and a three cycle act by\n\\[\ns=\n\\begin{pmatrix}\n6&1","Let \\(G=S_3\\), let \\(k=\\FF_7\\), and let \\(V\\) be the standard two dimensional representation. In the basis\n\\[\nv_1=(1,-1,0),\n\\qquad\nv_2=(0,1,-1),\n\\]\na transposition and a three cycle act by\n\\[\ns=\n\\begin{pmatrix}\n6&1\\\\\n0&1\n\\end{pmatrix},\n\\qquad\nt=\n\\begin{pmatrix}\n0&6\\\\\n1&6\n\\end{pmatrix}.\n\\]","\\begin{theorem}[Finite adequacy certificate]\\label{thm:s3_adequacy}\nThe standard representation of \\(S_3\\) over \\(\\FF_7\\) is adequate. More precisely,\n\\[\nH^0(S_3,\\ad^0V)=0,\n\\qquad\nH^1(S_3,\\FF_7)=0,\n\\qquad\nH^1(S_3,\\ad^0V)=0,\n\\]\nand\n\\[\n\\ad^0V\\cong\\operatorname{sgn}\\oplus V.\n\\]\nThe sign summand is detected by an eigenprojector of a three cycle, and the standard summand is detected by an eigenprojector of a transposition.\n\\end{theorem}","\\begin{proof}\nThe coordinate map\n\\[\n(a,b)\\longmapsto(a,-a+b,-b)\n\\]\nidentifies the displayed matrices with the permutation action on the sum zero subspace of \\(\\FF_7^3\\). This action is faithful. The four matrices \\(1,s,t,st\\) span \\(M_2\\) over every field of characteristic seven: the determinant of their coordinate matrix is \\(4\\ne0\\). A subspace stable under the group is therefore stable under every endomorphism, so the representation is absolutely irreducible. The characteristic polynomials of the identity, transpositions, and three cycles are respectively\n\\[\n(X-1)^2,\\qquad (X-1)(X+1),\\qquad (X-2)(X-4).\n\\]\nThus all eigenvalues already lie in \\(\\FF_7\\).","). This action is faithful. The four matrices \\(1,s,t,st\\) span \\(M_2\\) over every field of characteristic seven: the determinant of their coordinate matrix is \\(4\\ne0\\). A subspace stable under the group is therefore stable under every endomorphism, so the representation is absolutely irreducible. The characteristic polynomials of the identity, transpositions, and three cycles are respectively\n\\[\n(X-1)^2,\\qquad (X-1)(X+1),\\qquad (X-2)(X-4).\n\\]\nThus all eigenvalues already lie in \\(\\FF_7\\).","d of characteristic seven: the determinant of their coordinate matrix is \\(4\\ne0\\). A subspace stable under the group is therefore stable under every endomorphism, so the representation is absolutely irreducible. The characteristic polynomials of the identity, transpositions, and three cycles are respectively\n\\[\n(X-1)^2,\\qquad (X-1)(X+1),\\qquad (X-2)(X-4).\n\\]\nThus all eigenvalues already lie in \\(\\FF_7\\).","A trace zero matrix fixed by conjugation by both \\(s\\) and \\(t\\) is zero, as follows by comparing its four entries. This proves \\(H^0(S_3,\\ad^0V)=0\\). For any \\(\\FF_7[S_3]\\) module \\(M\\) and one cocycle \\(f\\), put\n\\[\nu=-6^{-1}\\sum_{h\\in S_3}f(h).\n\\]\nSumming \\(f(gh)=g f(h)+f(g)\\) over \\(h\\) gives \\(gu-u=f(g)\\). Hence every one cocycle is a coboundary and \\(H^1(S_3,M)=0\\), giving both asserted first cohomology vanishings.","Write\n\\[\nW(a,b,c)=\\begin{pmatrix}a&b\\\\c&-a\\end{pmatrix},\n\\qquad\nZ=W(2,3,4),\\quad A=W(1,1,2),\\quad B=W(1,5,6).\n\\]\nThe three coordinate columns have determinant \\(3\\ne0\\). Under conjugation,\n\\[\nsZ=-Z,\\quad tZ=Z,\\qquad\nsA=-A,\\quad sB=A+B,\\quad tA=B,\\quad tB=-A-B.\n\\]\nConsequently \\((z,(a,b))\\mapsto zZ+aA+bB\\) is an equivariant isomorphism from \\(\\operatorname{sgn}\\oplus V\\) to \\(\\ad^0V\\).","c&-a\\end{pmatrix},\n\\qquad\nZ=W(2,3,4),\\quad A=W(1,1,2),\\quad B=W(1,5,6).\n\\]\nThe three coordinate columns have determinant \\(3\\ne0\\). Under conjugation,\n\\[\nsZ=-Z,\\quad tZ=Z,\\qquad\nsA=-A,\\quad sB=A+B,\\quad tA=B,\\quad tB=-A-B.\n\\]\nConsequently \\((z,(a,b))\\mapsto zZ+aA+bB\\) is an equivariant isomorphism from \\(\\operatorname{sgn}\\oplus V\\) to \\(\\ad^0V\\).","The sign witness is \\(W_{\\mathrm{sgn}}=Z\\), while\n\\(W_{\\mathrm{std}}=W(6,4,3)=3A+3B\\) belongs to the standard summand. The corresponding projectors and trace pairings are\n\\[\nP_{t,2}=\\begin{pmatrix}2&4","The sign witness is \\(W_{\\mathrm{sgn}}=Z\\), while\n\\(W_{\\mathrm{std}}=W(6,4,3)=3A+3B\\) belongs to the standard summand. The corresponding projectors and trace pairings are\n\\[\nP_{t,2}=\\begin{pmatrix}2&4\\\\3&6\\end{pmatrix},\n\\qquad \\Tr(P_{t,2}W_{\\mathrm{sgn}})=3\\ne0,\n\\]\n\\[\nP_{s,1}=\\begin{pmatrix}0&4","3&6\\end{pmatrix},\n\\qquad \\Tr(P_{t,2}W_{\\mathrm{sgn}})=3\\ne0,\n\\]\n\\[\nP_{s,1}=\\begin{pmatrix}0&4\\\\0&1\\end{pmatrix},\n\\qquad \\Tr(P_{s,1}W_{\\mathrm{std}})=6\\ne0.\n\\]\nFor an operator \\(D\\) with \\((D-\\alpha I)(D-\\beta I)=0\\) and \\(\\alpha\\ne\\beta\\), the canonical projector is\n\\[\nP_{D,\\alpha}=(\\alpha-\\beta)^{-1}(D-\\beta I).\n\\]\nIts image equals \\(\\ker((D-\\alpha I)^n)\\) for every \\(n\\ge1\\), and it kills \\(\\ker((D-\\beta I)^n)\\). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.","0&1\\end{pmatrix},\n\\qquad \\Tr(P_{s,1}W_{\\mathrm{std}})=6\\ne0.\n\\]\nFor an operator \\(D\\) with \\((D-\\alpha I)(D-\\beta I)=0\\) and \\(\\alpha\\ne\\beta\\), the canonical projector is\n\\[\nP_{D,\\alpha}=(\\alpha-\\beta)^{-1}(D-\\beta I).\n\\]\nIts image equals \\(\\ker((D-\\alpha I)^n)\\) for every \\(n\\ge1\\), and it kills \\(\\ker((D-\\beta I)^n)\\). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.","For detection on every irreducible submodule, one can check the stronger assertion that canonical projectors detect every nonzero matrix. In addition to the two projectors above, use\n\\[\nP_{s,-1}=\\begin{pmatrix}1&3\\\\0&0\\end{pmatrix},\n\\qquad\nP_{tst^{-1},1}=\\begin{pmatrix}1&0\\\\4&0\\end{pmatrix}.\n\\]\nThe four trace pairings with these matrices have trivial common kernel in \\(M_2(\\FF_7)\\), by row reduction. Every irreducible adjoint submodule contains a nonzero matrix, so some canonical projector has nonzero trace pairing on that submodule. This verifies the last adequacy clause and completes the certificate.\n\\end{proof}","4&0\\end{pmatrix}.\n\\]\nThe four trace pairings with these matrices have trivial common kernel in \\(M_2(\\FF_7)\\), by row reduction. Every irreducible adjoint submodule contains a nonzero matrix, so some canonical projector has nonzero trace pairing on that submodule. This verifies the last adequacy clause and completes the certificate.\n\\end{proof}","Let \\(\\Delta\\) be a finite abelian \\(p\\) group and let\n\\[\n\\Lambda=\\OO[\\Delta].\n\\]\nThe augmentation map is\n\\[\n\\epsilon:\\Lambda\\longrightarrow\\OO,\n\\qquad\n\\sum_{g\\in\\Delta}a_gg\\longmapsto\\sum_{g\\in\\Delta}a_g.\n\\]\nIts kernel is the augmentation ideal \\(\\fraka\\).","\\begin{proposition}[Functorial group algebra transfer]\\label{prop:group_algebra_transfer}\nA group homomorphism \\(q:\\Delta'\\to\\Delta\\) induces an \\(\\OO\\) algebra homomorphism\n\\[\nq_*:\\OO[\\Delta']\\longrightarrow\\OO[\\Delta].\n\\]\nIf \\(q\\) is surjective, then \\(q_*\\) is surjective. Augmentation commutes with the transfer:\n\\[\n\\epsilon_\\Delta q_*=\\epsilon_{\\Delta'}.\n\\]\n\\end{proposition}","\\begin{proof}\nExtend \\(q\\) linearly from the group basis. Multiplicativity follows from \\(q(gh)=q(g)q(h)\\). Surjectivity follows because every basis element of \\(\\OO[\\Delta]\\) has a preimage. Both augmentation maps send every group element to one, which gives the final identity.\n\\end{proof}","\\begin{theorem}[Residual augmentation algebra]\\label{thm:residual_augmentation_algebra}\nSuppose\n\\[\n\\Delta\\cong\n\\prod_{i=1}^{r}C_{p^{n_i}}.\n\\]\nChoose generators \\(\\gamma_i\\), and let \\(X_i=\\gamma_i-1\\). Then\n\\[\n\\boxed{\nk[\\Delta]\n\\cong\nk[X_1,\\ldots,X_r]\n\\big/\n(X_1^{p^{n_1}},\\ldots,X_r^{p^{n_r}}).\n}\n\\]\nThe augmentation ideal is generated by the \\(X_i\\), and \\(k[\\Delta]\\) is local.\n\\end{theorem}","\\begin{proof}\nIn characteristic \\(p\\),\n\\[\n(1+X_i)^{p^{n_i}}-1=X_i^{p^{n_i}}.\n\\]\nThus the assignment \\(X_i\\mapsto\\gamma_i-1\\) gives a surjective homomorphism from the displayed truncated polynomial ring to \\(k[\\Delta]\\). Both sides have dimension \\(\\prod_ip^{n_i}=\\abs{\\Delta}\\) over \\(k\\), so the map is an isomorphism. The quotient by the ideal generated by the \\(X_i\\) is \\(k\\). Since each \\(X_i\\) is nilpotent, that ideal is the unique maximal ideal.\n\\end{proof}","\\begin{corollary}[No residual character splitting]\\label{cor:no_residual_character_splitting}\nThe residual group algebra \\(k[\\Delta]\\) has no nontrivial idempotents. In particular, character projectors do not split a finite \\(p\\) group algebra in residue characteristic \\(p\\).\n\\end{corollary}","\\begin{proof}\nThe first assertion follows from locality in \\cref{thm:residual_augmentation_algebra}. For the general finite \\(p\\) group assertion, let \\(e\\in k[G]\\) be idempotent and write \\(\\epsilon:k[G]\\to k\\) for augmentation. Suppose \\(\\epsilon(e)=0\\) and \\(e\\ne0\\). The \\(\\FF_p\\) span \\(W\\) of the left translates \\(ge\\) is a nonzero finite dimensional \\(\\FF_p\\) vector space, stable under \\(G\\). Its cardinality is divisible by \\(p\\). The orbit counting formula implies that its fixed point set also has cardinality divisible by \\(p\\). Since zero is fixed, there is a nonzero fixed vector \\(v\\in W\\).","llows from locality in \\cref{thm:residual_augmentation_algebra}. For the general finite \\(p\\) group assertion, let \\(e\\in k[G]\\) be idempotent and write \\(\\epsilon:k[G]\\to k\\) for augmentation. Suppose \\(\\epsilon(e)=0\\) and \\(e\\ne0\\). The \\(\\FF_p\\) span \\(W\\) of the left translates \\(ge\\) is a nonzero finite dimensional \\(\\FF_p\\) vector space, stable under \\(G\\). Its cardinality is divisible by \\(p\\). The orbit counting formula implies that its fixed point set also has cardinality divisible by \\(p\\). Since zero is fixed, there is a nonzero fixed vector \\(v\\in W\\).","ssertion, let \\(e\\in k[G]\\) be idempotent and write \\(\\epsilon:k[G]\\to k\\) for augmentation. Suppose \\(\\epsilon(e)=0\\) and \\(e\\ne0\\). The \\(\\FF_p\\) span \\(W\\) of the left translates \\(ge\\) is a nonzero finite dimensional \\(\\FF_p\\) vector space, stable under \\(G\\). Its cardinality is divisible by \\(p\\). The orbit counting formula implies that its fixed point set also has cardinality divisible by \\(p\\). Since zero is fixed, there is a nonzero fixed vector \\(v\\in W\\).","on:k[G]\\to k\\) for augmentation. Suppose \\(\\epsilon(e)=0\\) and \\(e\\ne0\\). The \\(\\FF_p\\) span \\(W\\) of the left translates \\(ge\\) is a nonzero finite dimensional \\(\\FF_p\\) vector space, stable under \\(G\\). Its cardinality is divisible by \\(p\\). The orbit counting formula implies that its fixed point set also has cardinality divisible by \\(p\\). Since zero is fixed, there is a nonzero fixed vector \\(v\\in W\\).","A vector in the regular representation fixed by all left translations has constant coefficients. Consequently \\(ve=\\epsilon(e)v=0\\). On the other hand, \\(e^2=e\\) implies \\(we=w\\) for every \\(w\\in W\\), hence \\(ve=v\\), a contradiction. Thus every idempotent of augmentation zero is zero. Since \\(\\epsilon(e)^2=\\epsilon(e)\\) in the field \\(k\\), either \\(\\epsilon(e)=0\\) or \\(\\epsilon(e)=1\\); in the latter case apply the preceding argument to \\(1-e\\).\n\\end{proof}","ed by all left translations has constant coefficients. Consequently \\(ve=\\epsilon(e)v=0\\). On the other hand, \\(e^2=e\\) implies \\(we=w\\) for every \\(w\\in W\\), hence \\(ve=v\\), a contradiction. Thus every idempotent of augmentation zero is zero. Since \\(\\epsilon(e)^2=\\epsilon(e)\\) in the field \\(k\\), either \\(\\epsilon(e)=0\\) or \\(\\epsilon(e)=1\\); in the latter case apply the preceding argument to \\(1-e\\).\n\\end{proof}","For \\(M\\otimes_{\\Lambda,\\epsilon}\\OO\\), we use a right \\(\\Lambda\\) module \\(M\\). The left module convention is \\(\\OO\\otimes_{\\Lambda,\\epsilon}M\\). Throughout, \\(\\fraka M\\) denotes the augmentation relation submodule: it is generated by \\(ma\\) for right modules and by \\(am\\) for left modules, with \\(a\\in\\fraka\\). This convention also applies to nonabelian groups in \\cref{cor:dyadic_coinvariant_recovery}. For abelian groups, both tensor orders agree over the commutative group algebra.","\\begin{theorem}[Coinvariant recovery]\\label{thm:coinvariant_recovery}\nLet \\(M\\) be a free \\(\\Lambda\\) module of rank \\(d\\). Then\n\\[\n\\boxed{\nM/\\fraka M\n\\cong\nM\\otimes_{\\Lambda,\\epsilon}\\OO\n\\cong\n\\OO^d.\n}\n\\]\nThe statement is valid for every coefficient prime, including two.\n\\end{theorem}","\\begin{proof}\nChoose a \\(\\Lambda\\) basis of \\(M\\). Tensoring the resulting isomorphism \\(M\\cong\\Lambda^d\\) along augmentation gives \\(\\OO^d\\). Tensoring by \\(\\Lambda/\\fraka\\) is the same as quotienting by \\(\\fraka M\\).\n\\end{proof}","\\begin{standardfact}[Iwasawa limit]\\label{fact:iwasawa_limit}\nFor \\(\\Delta_m=(\\ZZ/p^m\\ZZ)^r\\), the inverse limit of \\(\\OO[\\Delta_m]\\) is the Iwasawa algebra\n\\[\n\\OO[[\\ZZ_p^r]]\\cong\\OO[[X_1,\\ldots,X_r]],\n\\]\nwhere \\(X_i\\) corresponds to a chosen topological generator minus one \\cite{RibesZalesskii2010}.\n\\end{standardfact}","The augmentation formalism is uniform in \\(p\\). The difficult coefficient two questions lie elsewhere. Residual image hypotheses, local deformation geometry, and classicality may change at two even though coinvariant recovery remains exact.","\\begin{definition}[Support and near faithfulness]\\label{def:support_near_faithful}\nLet \\(R\\) be a commutative ring and let \\(M\\) be an \\(R\\) module. Its support is\n\\[\n\\Supp_R(M)\n=\n\\{\\frakp\\in\\Spec R:M_{\\frakp}\\neq0\\}.\n\\]\nIf \\(R\\) is Noetherian and \\(M\\) is finite, then \\(M\\) is \\emph{nearly faithful} when \\(\\Ann_R(M)\\) is a nilpotent ideal.\n\\end{definition}","\\begin{theorem}[Finite module support]\\label{thm:finite_module_support}\nLet \\(R\\) be a commutative ring and let \\(M\\) be a finite \\(R\\) module. Then\n\\[\n\\boxed{\n\\Supp_R(M)\n=\nV\\bigl(\\Ann_R(M)\\bigr).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nLet \\(\\frakp\\in\\Spec R\\). If \\(\\Ann_R(M)\\not\\subseteq\\frakp\\), choose\n\\(a\\in\\Ann_R(M)\\setminus\\frakp\\). The element \\(a\\) becomes a unit in\n\\(R_{\\frakp}\\), so \\(M_{\\frakp}=0\\).","Conversely, suppose \\(M_{\\frakp}=0\\). Choose generators\n\\(m_1,\\ldots,m_d\\) of \\(M\\). For each \\(i\\), there is an element\n\\(s_i\\notin\\frakp\\) such that \\(s_im_i=0\\). The product\n\\(s=s_1\\cdots s_d\\) lies outside \\(\\frakp\\) and annihilates every generator.\nThus \\(s\\in\\Ann_R(M)\\setminus\\frakp\\). This proves the equality.\n\\end{proof}","\\begin{theorem}[Fitting support criterion]\\label{thm:fitting_support}\nLet \\(R\\) be Noetherian and let \\(M\\) be a finite \\(R\\) module. Then\n\\[\n\\boxed{\n\\Supp_R(M)\n=\nV\\bigl(\\Fitt_0^R(M)\\bigr).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nIf \\(M=0\\), the statement follows from the usual convention \\(\\Fitt_0(0)=R\\). Assume \\(M\\neq0\\), and choose a finite presentation\n\\[\nR^m\\xrightarrow{A}R^n\\longrightarrow M\\longrightarrow0\n\\]\nwith \\(n\\geq1\\).\nThe zeroth Fitting ideal is generated by the \\(n\\) by \\(n\\) minors of \\(A\\). After localization at a prime \\(\\frakp\\), the module \\(M_{\\frakp}\\) vanishes exactly when \\(A_{\\frakp}\\) is surjective. A map onto the free module \\(R_{\\frakp}^n\\) is surjective exactly when one maximal minor is a unit. Thus \\(M_{\\frakp}=0\\) exactly when \\(\\Fitt_0^R(M)\\not\\subseteq\\frakp\\).\n\\end{proof}","from the usual convention \\(\\Fitt_0(0)=R\\). Assume \\(M\\neq0\\), and choose a finite presentation\n\\[\nR^m\\xrightarrow{A}R^n\\longrightarrow M\\longrightarrow0\n\\]\nwith \\(n\\geq1\\).\nThe zeroth Fitting ideal is generated by the \\(n\\) by \\(n\\) minors of \\(A\\). After localization at a prime \\(\\frakp\\), the module \\(M_{\\frakp}\\) vanishes exactly when \\(A_{\\frakp}\\) is surjective. A map onto the free module \\(R_{\\frakp}^n\\) is surjective exactly when one maximal minor is a unit. Thus \\(M_{\\frakp}=0\\) exactly when \\(\\Fitt_0^R(M)\\not\\subseteq\\frakp\\).\n\\end{proof}","nite presentation\n\\[\nR^m\\xrightarrow{A}R^n\\longrightarrow M\\longrightarrow0\n\\]\nwith \\(n\\geq1\\).\nThe zeroth Fitting ideal is generated by the \\(n\\) by \\(n\\) minors of \\(A\\). After localization at a prime \\(\\frakp\\), the module \\(M_{\\frakp}\\) vanishes exactly when \\(A_{\\frakp}\\) is surjective. A map onto the free module \\(R_{\\frakp}^n\\) is surjective exactly when one maximal minor is a unit. Thus \\(M_{\\frakp}=0\\) exactly when \\(\\Fitt_0^R(M)\\not\\subseteq\\frakp\\).\n\\end{proof}","tarrow{A}R^n\\longrightarrow M\\longrightarrow0\n\\]\nwith \\(n\\geq1\\).\nThe zeroth Fitting ideal is generated by the \\(n\\) by \\(n\\) minors of \\(A\\). After localization at a prime \\(\\frakp\\), the module \\(M_{\\frakp}\\) vanishes exactly when \\(A_{\\frakp}\\) is surjective. A map onto the free module \\(R_{\\frakp}^n\\) is surjective exactly when one maximal minor is a unit. Thus \\(M_{\\frakp}=0\\) exactly when \\(\\Fitt_0^R(M)\\not\\subseteq\\frakp\\).\n\\end{proof}","\\begin{theorem}[Higher Fitting strata]\\label{thm:higher_fitting_strata}\nLet \\(R\\) be Noetherian and let \\(M\\) be finite. For \\(r\\geq1\\), put\n\\[\n\\mu_{\\frakp}(M)\n=\n\\dim_{\\kappa(\\frakp)}\n\\left(M\\otimes_R\\kappa(\\frakp)\\right).\n\\]\nThen\n\\[\n\\boxed{\n\\left\\{\n\\frakp\\in\\Spec R:\n\\mu_{\\frakp}(M)\\geq r\n\\right\\}\n=\nV\\bigl(\\Fitt_{r-1}^R(M)\\bigr).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nUse a finite presentation with target rank \\(n\\). If \\(r>n\\), every fiber has dimension at most \\(n\\), so the left hand locus is empty. The convention \\(\\Fitt_{r-1}(M)=R\\) for \\(r-1\\geq n\\) makes the right hand locus empty as well. It remains to consider \\(1\\leq r\\leq n\\). At \\(\\frakp\\), the fiber dimension is\n\\[\nn-\\rank A(\\frakp).\n\\]\nIt is at least \\(r\\) exactly when the rank is at most \\(n-r\\), which is equivalent to the vanishing of every minor of size \\(n-r+1\\). Those minors generate \\(\\Fitt_{r-1}^R(M)\\).\n\\end{proof}","proof}\nUse a finite presentation with target rank \\(n\\). If \\(r>n\\), every fiber has dimension at most \\(n\\), so the left hand locus is empty. The convention \\(\\Fitt_{r-1}(M)=R\\) for \\(r-1\\geq n\\) makes the right hand locus empty as well. It remains to consider \\(1\\leq r\\leq n\\). At \\(\\frakp\\), the fiber dimension is\n\\[\nn-\\rank A(\\frakp).\n\\]\nIt is at least \\(r\\) exactly when the rank is at most \\(n-r\\), which is equivalent to the vanishing of every minor of size \\(n-r+1\\). Those minors generate \\(\\Fitt_{r-1}^R(M)\\).\n\\end{proof}","every fiber has dimension at most \\(n\\), so the left hand locus is empty. The convention \\(\\Fitt_{r-1}(M)=R\\) for \\(r-1\\geq n\\) makes the right hand locus empty as well. It remains to consider \\(1\\leq r\\leq n\\). At \\(\\frakp\\), the fiber dimension is\n\\[\nn-\\rank A(\\frakp).\n\\]\nIt is at least \\(r\\) exactly when the rank is at most \\(n-r\\), which is equivalent to the vanishing of every minor of size \\(n-r+1\\). Those minors generate \\(\\Fitt_{r-1}^R(M)\\).\n\\end{proof}","\\(n\\), so the left hand locus is empty. The convention \\(\\Fitt_{r-1}(M)=R\\) for \\(r-1\\geq n\\) makes the right hand locus empty as well. It remains to consider \\(1\\leq r\\leq n\\). At \\(\\frakp\\), the fiber dimension is\n\\[\nn-\\rank A(\\frakp).\n\\]\nIt is at least \\(r\\) exactly when the rank is at most \\(n-r\\), which is equivalent to the vanishing of every minor of size \\(n-r+1\\). Those minors generate \\(\\Fitt_{r-1}^R(M)\\).\n\\end{proof}","\\begin{proposition}[Fitting ideals and base change]\\label{prop:fitting_base_change}\nLet \\(R\\to S\\) be a ring map and let \\(M\\) be a finitely presented \\(R\\) module. Then for every \\(j\\geq0\\),\n\\[\n\\boxed{\n\\Fitt_j^S(M\\otimes_RS)\n=\n\\Fitt_j^R(M)S.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nBase change carries a finite presentation matrix for \\(M\\) to a presentation matrix for \\(M\\otimes_RS\\). Every defining minor is carried to the corresponding minor over \\(S\\).\n\\end{proof}","\\begin{theorem}[Near faithfulness is full support]\\label{thm:near_faithful_full_support}\nLet \\(R\\) be Noetherian and let \\(M\\) be a finite \\(R\\) module. The following conditions are equivalent.\n\\begin{enumerate}\n\\item \\(M\\) is nearly faithful.\n\\item \\(\\Supp_R(M)=\\Spec R\\).\n\\item \\(\\Ann_R(M)\\subseteq\\nil(R)\\).\n\\end{enumerate}\nIf \\(R\\) is reduced, these conditions are equivalent to faithfulness.\n\\end{theorem}","\\begin{proof}\nBy \\cref{thm:finite_module_support}, full support is equivalent to\n\\[\nV\\bigl(\\Ann_R(M)\\bigr)=\\Spec R.\n\\]\nThis holds exactly when \\(\\Ann_R(M)\\) is contained in every prime ideal, hence in the nilradical. Because \\(R\\) is Noetherian, the ideal \\(\\Ann_R(M)\\) is finitely generated. An ideal generated by finitely many nilpotent elements is nilpotent. This proves the equivalence of the first three conditions. If \\(R\\) is reduced, its nilradical is zero.\n\\end{proof}",":finite_module_support}, full support is equivalent to\n\\[\nV\\bigl(\\Ann_R(M)\\bigr)=\\Spec R.\n\\]\nThis holds exactly when \\(\\Ann_R(M)\\) is contained in every prime ideal, hence in the nilradical. Because \\(R\\) is Noetherian, the ideal \\(\\Ann_R(M)\\) is finitely generated. An ideal generated by finitely many nilpotent elements is nilpotent. This proves the equivalence of the first three conditions. If \\(R\\) is reduced, its nilradical is zero.\n\\end{proof}","\\begin{corollary}[Support on every irreducible component]\\label{cor:every_component_support}\nUnder the hypotheses of \\cref{thm:near_faithful_full_support}, a nearly faithful module is nonzero at the generic point of every irreducible component of \\(\\Spec R\\).\n\\end{corollary}","\\begin{proof}\nEvery generic point is a prime of \\(R\\), and full support contains every prime.\n\\end{proof}","\\begin{theorem}[Support after a quotient]\\label{thm:quotient_support}\nLet \\(R\\) be Noetherian, let \\(I\\subseteq R\\) be an ideal, and let \\(M\\) be a finite \\(R\\) module. Under the canonical identification\n\\[\n\\Spec(R/I)\\cong V(I)\\subseteq\\Spec R,\n\\]\none has\n\\[\n\\boxed{\n\\Supp_{R/I}(M/IM)\n=\n\\Supp_R(M)\\cap V(I).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nLet \\(\\frakp\\supseteq I\\). Localizing gives\n\\[\n(M/IM)_{\\frakp}\n\\cong\nM_{\\frakp}/I_{\\frakp}M_{\\frakp}.\n\\]\nIf \\(M_{\\frakp}=0\\), the quotient is zero. If \\(M_{\\frakp}\\neq0\\), then\n\\(I_{\\frakp}\\) lies in the maximal ideal of the local ring \\(R_{\\frakp}\\). Nakayama's lemma shows that\n\\(M_{\\frakp}/I_{\\frakp}M_{\\frakp}\\neq0\\). The stated equality follows.\n\\end{proof}","\\begin{definition}[Ordinary patching datum]\\label{def:ordinary_patching_datum}\nAn ordinary patching datum consists of a Noetherian ring \\(R_\\infty\\), an ideal\n\\(\\fraka\\subseteq R_\\infty\\), a finite \\(R_\\infty\\) module \\(M_\\infty\\), and identifications\n\\[\nR\\cong R_\\infty/\\fraka,\n\\qquad\nM\\cong M_\\infty/\\fraka M_\\infty.\n\\]\nIn automorphy lifting applications, \\(R_\\infty\\) is a patched deformation ring and \\(M_\\infty\\) is a patched automorphic module. The definition itself contains no automorphy assertion.\n\\end{definition}","definition}[Ordinary patching datum]\\label{def:ordinary_patching_datum}\nAn ordinary patching datum consists of a Noetherian ring \\(R_\\infty\\), an ideal\n\\(\\fraka\\subseteq R_\\infty\\), a finite \\(R_\\infty\\) module \\(M_\\infty\\), and identifications\n\\[\nR\\cong R_\\infty/\\fraka,\n\\qquad\nM\\cong M_\\infty/\\fraka M_\\infty.\n\\]\nIn automorphy lifting applications, \\(R_\\infty\\) is a patched deformation ring and \\(M_\\infty\\) is a patched automorphic module. The definition itself contains no automorphy assertion.\n\\end{definition}","\\begin{theorem}[Ordinary patching support descent]\\label{thm:ordinary_support_descent}\nLet \\((R_\\infty,\\fraka,M_\\infty,R,M)\\) be an ordinary patching datum. If\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty,\n\\]\nthen\n\\[\n\\boxed{\n\\Supp_R(M)=\\Spec R.\n}\n\\]\nEquivalently, if \\(M_\\infty\\) is nearly faithful, then \\(M\\) is nearly faithful. If \\(R\\) is reduced, then \\(M\\) is faithful.\n\\end{theorem}","\\begin{proof}\nApply \\cref{thm:quotient_support} with \\(I=\\fraka\\). The support of the quotient is\n\\[\n\\Spec R_\\infty\\cap V(\\fraka)=V(\\fraka),\n\\]\nwhich identifies with \\(\\Spec R\\). The remaining statements follow from\n\\cref{thm:near_faithful_full_support}.\n\\end{proof}","\\begin{corollary}[Componentwise finite level occupancy]\\label{cor:componentwise_occupancy}\nIn the setting of \\cref{thm:ordinary_support_descent}, every irreducible component of\n\\(\\Spec R\\) is contained in the support of \\(M\\). In particular, no component of the finite level deformation ring is erased by the quotient from the patched level.\n\\end{corollary}","\\begin{proof}\nThe module \\(M\\) has full support by \\cref{thm:ordinary_support_descent}.\n\\end{proof}","\\begin{example}[Freeness over auxiliary variables is not enough]\\label{ex:freeness_not_support}\nLet \\(k\\) be a field, let \\(S=k[[t]]\\), and put\n\\[\nR=S[[x]]/(x(x-t)).\n\\]\nThe ring \\(R\\) is reduced and has two irreducible components. Let\n\\[\nM=R/(x)\\cong S.\n\\]\nThen \\(M\\) is free of rank one over \\(S\\), but\n\\[\n\\Supp_R(M)=V(x)\n\\]\nis only one component of \\(\\Spec R\\). Thus freeness over the auxiliary power series ring does not imply full support over the deformation ring.\n\\end{example}","\\begin{proof}\nThe two nonassociate prime factors \\(x\\) and \\(x-t\\) occur with multiplicity one, so the principal quotient is reduced. The displayed isomorphism makes \\(M\\) free over \\(S\\). Its annihilator as an \\(R\\) module is \\((x)\\), hence\n\\[\n\\Supp_R(M)=V(x)\n\\]\nby \\cref{thm:finite_module_support}. The generic point \\((x-t)\\) of the other component does not contain \\(x\\), so it is not in the support.\n\\end{proof}","\\begin{definition}[Derived support]\\label{def:derived_support}\nLet \\(R\\) be a commutative ring and let \\(C\\) be a perfect complex of \\(R\\) modules. Its derived support is\n\\[\n\\Supp_R(C)\n=\n\\left\\{\n\\frakp\\in\\Spec R:\nC\\mathbin{\\mathop{\\otimes}\\limits_R^{\\mathbf L}}\\kappa(\\frakp)\n\\not\\simeq0\n\\right\\}.\n\\]\n\\end{definition}","\\begin{standardfact}[Cohomological description of derived support]\\label{fact:derived_support_cohomology}\nIf \\(R\\) is Noetherian and \\(C\\) is perfect, then\n\\[\n\\Supp_R(C)\n=\n\\bigcup_i\\Supp_R H^i(C).\n\\]\nIn particular, derived support is closed and is empty exactly when \\(C\\simeq0\\) \\cite{Weibel1994,Eisenbud1995}.\n\\end{standardfact}","\\begin{theorem}[Derived quotient support]\\label{thm:derived_quotient_support}\nLet \\(R\\) be a commutative ring, let \\(I\\subseteq R\\) be an ideal, and let \\(C\\) be a perfect complex over \\(R\\). Under\n\\(\\Spec(R/I)\\cong V(I)\\), one has\n\\[\n\\boxed{\n\\Supp_{R/I}\n\\left(\nC\\mathbin{\\mathop{\\otimes}\\limits_R^{\\mathbf L}}R/I\n\\right)\n=\n\\Supp_R(C)\\cap V(I).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nChoose a bounded complex \\(P\\) of finite projective \\(R\\) modules representing \\(C\\). Scalar extension to a commutative \\(R\\) algebra \\(S\\) is computed degree by degree as \\(S\\otimes_RP\\); the canonical symmetry of tensor products gives the displayed order \\(C\\otimes_R^{\\mathbf L}S\\). Scalar extension preserves the bounds, finite generation, and projectivity of the terms. A derived isomorphism between two such projective models lifts to a homotopy equivalence, which scalar extension preserves. The resulting derived object and its residue field fibers are therefore independent of the chosen model.","Let \\(q\\in\\Spec(R/I)\\), and let \\(\\frakp\\) be its inverse image in \\(R\\). The composite point \\(R\\to R/I\\to\\kappa(q)\\) has kernel \\(\\frakp\\), so it factors through a field embedding \\(\\kappa(\\frakp)\\to\\kappa(q)\\). The ordinary tensor associator in every degree gives an isomorphism of complexes\n\\[\n\\kappa(q)\\otimes_{R/I}\\bigl((R/I)\\otimes_RP\\bigr)\n\\cong\\kappa(q)\\otimes_RP\n\\cong\\kappa(q)\\otimes_{\\kappa(\\frakp)}\n\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the su","Let \\(q\\in\\Spec(R/I)\\), and let \\(\\frakp\\) be its inverse image in \\(R\\). The composite point \\(R\\to R/I\\to\\kappa(q)\\) has kernel \\(\\frakp\\), so it factors through a field embedding \\(\\kappa(\\frakp)\\to\\kappa(q)\\). The ordinary tensor associator in every degree gives an isomorphism of complexes\n\\[\n\\kappa(q)\\otimes_{R/I}\\bigl((R/I)\\otimes_RP\\bigr)\n\\cong\\kappa(q)\\otimes_RP\n\\cong\\kappa(q)\\otimes_{\\kappa(\\frakp)}\n\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on co","Let \\(q\\in\\Spec(R/I)\\), and let \\(\\frakp\\) be its inverse image in \\(R\\). The composite point \\(R\\to R/I\\to\\kappa(q)\\) has kernel \\(\\frakp\\), so it factors through a field embedding \\(\\kappa(\\frakp)\\to\\kappa(q)\\). The ordinary tensor associator in every degree gives an isomorphism of complexes\n\\[\n\\kappa(q)\\otimes_{R/I}\\bigl((R/I)\\otimes_RP\\bigr)\n\\cong\\kappa(q)\\otimes_RP\n\\cong\\kappa(q)\\otimes_{\\kappa(\\frakp)}\n\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on corresponding primes. The image","Let \\(q\\in\\Spec(R/I)\\), and let \\(\\frakp\\) be its inverse image in \\(R\\). The composite point \\(R\\to R/I\\to\\kappa(q)\\) has kernel \\(\\frakp\\), so it factors through a field embedding \\(\\kappa(\\frakp)\\to\\kappa(q)\\). The ordinary tensor associator in every degree gives an isomorphism of complexes\n\\[\n\\kappa(q)\\otimes_{R/I}\\bigl((R/I)\\otimes_RP\\bigr)\n\\cong\\kappa(q)\\otimes_RP\n\\cong\\kappa(q)\\otimes_{\\kappa(\\frakp)}\n\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on corresponding primes. The image of \\(\\Spec(R/I)\\to\\Spec R\\) is exactly \\(V(I)\\","Let \\(q\\in\\Spec(R/I)\\), and let \\(\\frakp\\) be its inverse image in \\(R\\). The composite point \\(R\\to R/I\\to\\kappa(q)\\) has kernel \\(\\frakp\\), so it factors through a field embedding \\(\\kappa(\\frakp)\\to\\kappa(q)\\). The ordinary tensor associator in every degree gives an isomorphism of complexes\n\\[\n\\kappa(q)\\otimes_{R/I}\\bigl((R/I)\\otimes_RP\\bigr)\n\\cong\\kappa(q)\\otimes_RP\n\\cong\\kappa(q)\\otimes_{\\kappa(\\frakp)}\n\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on corresponding primes. The image of \\(\\Spec(R/I)\\to\\Spec R\\) is exactly \\(V(I)\\), giving the stated a","Let \\(q\\in\\Spec(R/I)\\), and let \\(\\frakp\\) be its inverse image in \\(R\\). The composite point \\(R\\to R/I\\to\\kappa(q)\\) has kernel \\(\\frakp\\), so it factors through a field embedding \\(\\kappa(\\frakp)\\to\\kappa(q)\\). The ordinary tensor associator in every degree gives an isomorphism of complexes\n\\[\n\\kappa(q)\\otimes_{R/I}\\bigl((R/I)\\otimes_RP\\bigr)\n\\cong\\kappa(q)\\otimes_RP\n\\cong\\kappa(q)\\otimes_{\\kappa(\\frakp)}\n\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on corresponding primes. The image of \\(\\Spec(R/I)\\to\\Spec R\\) is exactly \\(V(I)\\), giving the stated ambient spectrum equality, including the case \\(I=R\\).\n\\end{proof}","omorphism of complexes\n\\[\n\\kappa(q)\\otimes_{R/I}\\bigl((R/I)\\otimes_RP\\bigr)\n\\cong\\kappa(q)\\otimes_RP\n\\cong\\kappa(q)\\otimes_{\\kappa(\\frakp)}\n\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on corresponding primes. The image of \\(\\Spec(R/I)\\to\\Spec R\\) is exactly \\(V(I)\\), giving the stated ambient spectrum equality, including the case \\(I=R\\).\n\\end{proof}","of complexes\n\\[\n\\kappa(q)\\otimes_{R/I}\\bigl((R/I)\\otimes_RP\\bigr)\n\\cong\\kappa(q)\\otimes_RP\n\\cong\\kappa(q)\\otimes_{\\kappa(\\frakp)}\n\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on corresponding primes. The image of \\(\\Spec(R/I)\\to\\Spec R\\) is exactly \\(V(I)\\), giving the stated ambient spectrum equality, including the case \\(I=R\\).\n\\end{proof}","ong\\kappa(q)\\otimes_{\\kappa(\\frakp)}\n\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on corresponding primes. The image of \\(\\Spec(R/I)\\to\\Spec R\\) is exactly \\(V(I)\\), giving the stated ambient spectrum equality, including the case \\(I=R\\).\n\\end{proof}","\\bigl(\\kappa(\\frakp)\\otimes_RP\\bigr).\n\\]\nExtension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on corresponding primes. The image of \\(\\Spec(R/I)\\to\\Spec R\\) is exactly \\(V(I)\\), giving the stated ambient spectrum equality, including the case \\(I=R\\).\n\\end{proof}","y flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left hand complex is thus nonzero in the derived category exactly when the residue field fiber of \\(C\\) at \\(\\frakp\\) is nonzero. This proves the support identity on corresponding primes. The image of \\(\\Spec(R/I)\\to\\Spec R\\) is exactly \\(V(I)\\), giving the stated ambient spectrum equality, including the case \\(I=R\\).\n\\end{proof}","\\begin{definition}[Derived patching datum]\\label{def:derived_patching_datum}\nA derived patching datum consists of a Noetherian ring \\(R_\\infty\\), an ideal\n\\(\\fraka\\), a perfect complex \\(C_\\infty\\) over \\(R_\\infty\\), and identifications\n\\[\nR\\cong R_\\infty/\\fraka,\n\\qquad\nC\\simeq\nC_\\infty\n\\mathbin{\\mathop{\\otimes}\\limits_{R_\\infty}^{\\mathbf L}}R.\n\\]\n\\end{definition}","\\begin{theorem}[Derived patching support descent]\\label{thm:derived_support_descent}\nLet \\((R_\\infty,\\fraka,C_\\infty,R,C)\\) be a derived patching datum. If\n\\[\n\\Supp_{R_\\infty}(C_\\infty)=\\Spec R_\\infty,\n\\]\nthen\n\\[\n\\boxed{\n\\Supp_R(C)=\\Spec R.\n}\n\\]\nFor every point \\(x:R\\to E'\\) to a field, the derived fiber\n\\[\nC\\mathbin{\\mathop{\\otimes}\\limits_{R,x}^{\\mathbf L}}E'\n\\]\nis nonzero.\n\\end{theorem}","\\begin{proof}\nApply \\cref{thm:derived_quotient_support} to the patched complex and use the actual ring identification \\(R\\cong R_\\infty/\\fraka\\). Full patched support restricts to \\(V(\\fraka)\\), which corresponds to all of \\(\\Spec R\\). The supplied derived isomorphism with \\(C\\) preserves this support by the projective model argument in that theorem.","For a field valued point \\(x:R\\to E'\\), put \\(\\frakp=\\ker x\\). The point factors as \\(R\\to\\kappa(\\frakp)\\to E'\\), where the second map is a field embedding. Tensor association on a projective model identifies the fiber at \\(x\\) with the extension to \\(E'\\) of the fiber at \\(\\frakp\\). The latter is nonzero by full support. Faithful flatness of the field extension reflects exactness in every degree, so the fiber at \\(x\\) is nonzero as well. This applies to every field valued point, with no closed point restriction.\n\\end{proof}","point \\(x:R\\to E'\\), put \\(\\frakp=\\ker x\\). The point factors as \\(R\\to\\kappa(\\frakp)\\to E'\\), where the second map is a field embedding. Tensor association on a projective model identifies the fiber at \\(x\\) with the extension to \\(E'\\) of the fiber at \\(\\frakp\\). The latter is nonzero by full support. Faithful flatness of the field extension reflects exactness in every degree, so the fiber at \\(x\\) is nonzero as well. This applies to every field valued point, with no closed point restriction.\n\\end{proof}","\\begin{corollary}[A nonzero cohomological degree survives]\\label{cor:derived_degree_survives}\nUnder the hypotheses of \\cref{thm:derived_support_descent}, for every field valued point \\(x:R\\to E'\\), there is an integer \\(i\\) such that\n\\[\nH^i\n\\left(\nC\\mathbin{\\mathop{\\otimes}\\limits_{R,x}^{\\mathbf L}}E'\n\\right)\n\\neq0.\n\\]\n\\end{corollary}","\\section{Transport through an isomorphism \\texorpdfstring{\\(R\\cong\\TT\\)}{R equals T}}\\label{sec:rt}","\\begin{theorem}[Support and Fitting transport]\\label{thm:rt_support_transport}\nLet \\(\\varphi:R\\to\\TT\\) be an isomorphism of Noetherian rings. Let \\(M\\) be a finite \\(\\TT\\) module, viewed as an \\(R\\) module through \\(\\varphi\\). Then the homeomorphism\n\\[\n\\varphi^*:\\Spec\\TT\\longrightarrow\\Spec R\n\\]\nsatisfies\n\\[\n\\boxed{\n\\varphi^*\\bigl(\\Supp_{\\TT}(M)\\bigr)\n=\n\\Supp_R(M).\n}\n\\]\nFor every \\(j\\geq0\\),\n\\[\n\\boxed{\n\\varphi\n\\left(\n\\Fitt_j^R(M)\n\\right)\n=\n\\Fitt_j^{\\TT}(M).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nLocalization commutes with transport along an isomorphism, so a localization of \\(M\\) vanishes on one side exactly when the corresponding localization vanishes on the other. This proves the support identity. A finite presentation matrix over \\(R\\) is carried entry by entry to a presentation matrix over \\(\\TT\\). The minors generating each Fitting ideal are therefore carried to the corresponding minors.\n\\end{proof}","\\begin{corollary}[Full support is invariant under \\texorpdfstring{\\(R\\cong\\TT\\)}{R equals T}]\\label{cor:rt_full_support}\nIn the setting of \\cref{thm:rt_support_transport}, \\(M\\) has full \\(R\\) support if and only if it has full \\(\\TT\\) support.\n\\end{corollary}","\\begin{proposition}[A quotient can erase a component]\\label{prop:quotient_erases_component}\nLet\n\\[\nR=k[[x,y]]/(xy)\n\\]\nand let\n\\[\n\\TT=R/(x)\\cong k[[y]].\n\\]\nViewed as an \\(R\\) module, \\(\\TT\\) has support \\(V(x)\\), which misses the generic point \\((y)\\) of the other irreducible component. Thus a surjection \\(R\\twoheadrightarrow\\TT\\) does not by itself imply full component occupancy.\n\\end{proposition}","\\begin{proof}\nThe annihilator of the \\(R\\) module \\(\\TT\\) is \\((x)\\). Hence its support is \\(V(x)\\) by \\cref{thm:finite_module_support}. The prime \\((y)\\) does not contain \\(x\\), so it is not in this support.\n\\end{proof}","\\begin{theorem}[Field fiber criterion]\\label{thm:field_fiber_criterion}\nLet \\(R\\) be a commutative ring, let \\(M\\) be a finite \\(R\\) module, and let\n\\(x:R\\to E'\\) be a homomorphism to a field. Put \\(\\frakp=\\ker x\\). Then\n\\[\n\\boxed{\nM\\otimes_{R,x}E'\\neq0\n\\quad\\Longleftrightarrow\\quad\n\\frakp\\in\\Supp_R(M).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nThe map \\(x\\) factors through the local ring \\(R_{\\frakp}\\) and then through an embedding of the residue field \\(\\kappa(\\frakp)\\) into \\(E'\\). Since \\(M\\) is finite, Nakayama's lemma gives\n\\[\nM_{\\frakp}\\neq0\n\\quad\\Longleftrightarrow\\quad\nM\\otimes_R\\kappa(\\frakp)\\neq0.\n\\]\nExtension of scalars from \\(\\kappa(\\frakp)\\) to \\(E'\\) preserves nonzero vector spaces.\n\\end{proof}","\\begin{corollary}[Supported deformation points give nonzero Hecke fibers]\\label{cor:supported_hecke_fiber}\nLet \\(R\\cong\\TT\\), and let \\(M\\) be a finite Hecke module. If a point\n\\(x:R\\to E'\\) lies in \\(\\Supp_R(M)\\), then the corresponding Hecke fiber\n\\[\nM\\otimes_{\\TT,x}E'\n\\]\nis nonzero.\n\\end{corollary}","\\begin{standardfact}[Artin decomposition]\\label{fact:artin_decomposition}\nLet \\(E'\\) be a field and let \\(\\TT\\) be a finite dimensional commutative\n\\(E'\\) algebra. Then\n\\[\n\\boxed{\n\\TT\n\\cong\n\\prod_{\\frakm\\in\\Max\\TT}\\TT_{\\frakm}.\n}\n\\]\nEach factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents \\((e_{\\frakm})\\) such that\n\\[\ne_{\\frakm}e_{\\mathfrak n}=0\n\\quad(\\frakm\\neq\\mathfrak n),\n\\qquad\n\\sum_{\\frakm}e_{\\frakm}=1,\n\\]\nand \\(e_{\\frakm}\\TT\\cong\\TT_{\\frakm}\\) \\cite{AtiyahMacdonald1969,Matsumura1989}.\n\\end{standardfact}","Let \\(E'\\) be a field and let \\(\\TT\\) be a finite dimensional commutative\n\\(E'\\) algebra. Then\n\\[\n\\boxed{\n\\TT\n\\cong\n\\prod_{\\frakm\\in\\Max\\TT}\\TT_{\\frakm}.\n}\n\\]\nEach factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents \\((e_{\\frakm})\\) such that\n\\[\ne_{\\frakm}e_{\\mathfrak n}=0\n\\quad(\\frakm\\neq\\mathfrak n),\n\\qquad\n\\sum_{\\frakm}e_{\\frakm}=1,\n\\]\nand \\(e_{\\frakm}\\TT\\cong\\TT_{\\frakm}\\) \\cite{AtiyahMacdonald1969,Matsumura1989}.\n\\end{standardfact}","\\begin{theorem}[Canonical Hecke packet decomposition]\\label{thm:canonical_hecke_packets}\nLet \\(\\TT\\) be a finite dimensional commutative algebra over a field \\(E'\\), and let \\(V\\) be a finite \\(\\TT\\) module. Then\n\\[\n\\boxed{\nV\n=\n\\bigoplus_{\\frakm\\in\\Max\\TT}e_{\\frakm}V.\n}\n\\]\nFor every maximal ideal \\(\\frakm\\), the canonical map gives\n\\[\n\\boxed{\ne_{\\frakm}V\\cong V_{\\frakm}.}\n\\]\nThe packet \\(e_{\\frakm}V\\) is nonzero exactly when \\(\\frakm\\in\\Supp_{\\TT}(V)\\).\n\\end{theorem}","packet decomposition]\\label{thm:canonical_hecke_packets}\nLet \\(\\TT\\) be a finite dimensional commutative algebra over a field \\(E'\\), and let \\(V\\) be a finite \\(\\TT\\) module. Then\n\\[\n\\boxed{\nV\n=\n\\bigoplus_{\\frakm\\in\\Max\\TT}e_{\\frakm}V.\n}\n\\]\nFor every maximal ideal \\(\\frakm\\), the canonical map gives\n\\[\n\\boxed{\ne_{\\frakm}V\\cong V_{\\frakm}.}\n\\]\nThe packet \\(e_{\\frakm}V\\) is nonzero exactly when \\(\\frakm\\in\\Supp_{\\TT}(V)\\).\n\\end{theorem}","\\begin{proof}\nThe idempotent relations in \\cref{fact:artin_decomposition} give the direct sum decomposition. Localization at \\(\\frakm\\) kills every factor except the \\(\\frakm\\) factor, where \\(e_{\\frakm}\\) becomes one. Thus the localization map identifies \\(e_{\\frakm}V\\) with \\(V_{\\frakm}\\). The final statement is the definition of support.\n\\end{proof}","\\begin{corollary}[Simultaneous generalized eigenspaces]\\label{cor:simultaneous_generalized_eigenspaces}\nAssume that \\(E'\\) is algebraically closed and that \\(\\TT\\) is generated by commuting operators \\(T_1,\\ldots,T_d\\) on \\(V\\). For a maximal ideal\n\\[\n\\frakm\n=\n(T_1-\\lambda_1,\\ldots,T_d-\\lambda_d),\n\\]\nthere is an integer \\(N\\geq1\\) such that\n\\[\n\\boxed{\ne_{\\frakm}V\n=\n\\left\\{\nv\\in V:\n(T_i-\\lambda_i)^Nv=0\n\\text{ for every }i\n\\right\\}.\n}\n\\]\nThus the canonical Artin packet is the simultaneous generalized eigenspace.\n\\end{corollary}","\\begin{proof}\nThe maximal ideal of the Artin local algebra \\(\\TT_{\\frakm}\\) is nilpotent. Hence a common power of every \\(T_i-\\lambda_i\\) kills the local packet. On every other local factor, at least one \\(T_i-\\lambda_i\\) is a unit, so no nonzero vector can satisfy all the displayed nilpotence conditions.","Conversely, suppose every \\((T_i-\\lambda_i)^N\\) kills \\(v\\). Every monomial of total degree greater than \\(d(N-1)\\) contains one factor to exponent at least \\(N\\). Therefore a power of \\(\\frakm\\) kills \\(v\\), which places \\(v\\) in the \\(\\frakm\\) primary packet.\n\\end{proof}","\\begin{corollary}[Rank and trace detection in characteristic zero]\\label{cor:rank_trace_detection}\nAssume that \\(E'\\) has characteristic zero. For every maximal Hecke packet,\n\\[\n\\boxed{\n\\Tr_{E'}(e_{\\frakm}|V)\n=\n\\rank_{E'}(e_{\\frakm}|V)\n=\n\\dim_{E'}e_{\\frakm}V.\n}\n\\]\nIn particular, the packet is occupied exactly when its projector has positive rank, equivalently nonzero trace.\n\\end{corollary}","\\begin{proof}\nAn idempotent linear map has minimal polynomial dividing \\(X(X-1)\\), which has distinct roots. It is diagonalizable with eigenvalues zero and one. Its trace and rank both equal the multiplicity of the eigenvalue one. A positive integer is nonzero in characteristic zero.\n\\end{proof}","\\begin{example}[A nonsemisimple Hecke packet]\\label{ex:nonsemisimple_hecke_packet}\nLet\n\\[\n\\TT\n=\nE'[u]/(u^2)\n\\times\nE'[v]/(v^3)\n\\]\nand let \\(V=\\TT\\) be the regular module. The two canonical central idempotents split\n\\[\nV=V_u\\oplus V_v\n\\]\nwith dimensions two and three. Multiplication by \\(u\\) is nilpotent on \\(V_u\\), and multiplication by \\(v\\) is nilpotent on \\(V_v\\). Neither local factor is semisimple, but the two maximal Hecke packets are isolated exactly by the Artin idempotents.\n\\end{example}","\\begin{proof}\nThe product decomposition gives the two central idempotents \\((1,0)\\) and \\((0,1)\\). Their ranges are the two local factors. The classes of \\(u\\) and \\(v\\) are nilpotent of orders two and three, respectively.\n\\end{proof}","\\begin{theorem}[Nested packets in one finite carrier]\\label{thm:nested_finite_carrier}\nLet \\(V\\) be a finite dimensional vector space. Let \\(e_1,e_2,\\ldots\\) be idempotents such that\n\\[\ne_{m+1}e_m=e_me_{m+1}=e_{m+1}\n\\]\nfor every \\(m\\). If \\(e_mV\\neq0\\) for every \\(m\\), then the descending chain stabilizes and\n\\[\n\\boxed{\n\\bigcap_{m\\geq1}e_mV\n\\neq0.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nThe displayed relations give\n\\[\ne_{m+1}V\\subseteq e_mV.\n\\]\nThe nonnegative dimensions form a nonincreasing sequence of positive integers, so they eventually become constant. Once two consecutive dimensions agree, the corresponding inclusion is equality. Thus the ranges stabilize at a nonzero subspace, which is their intersection.\n\\end{proof}","\\begin{corollary}[Finite witness of incompatibility]\\label{cor:finite_witness_incompatibility}\nUnder the hypotheses of \\cref{thm:nested_finite_carrier}, if\n\\[\n\\bigcap_{m\\geq1}e_mV=0,\n\\]\nthen \\(e_mV=0\\) for some finite \\(m\\).\n\\end{corollary}","\\begin{definition}[Stable core]\\label{def:stable_core}\nLet\n\\[\nV_1\\longleftarrow V_2\\longleftarrow V_3\\longleftarrow\\cdots\n\\]\nbe an inverse system of finite dimensional vector spaces. For \\(n\\geq m\\), write\n\\(f_{n,m}:V_n\\to V_m\\). The stable core at level \\(m\\) is\n\\[\nY_m\n=\n\\bigcap_{n\\geq m}\\im(f_{n,m}).\n\\]\n\\end{definition}","\\begin{lemma}[Surjectivity on stable cores]\\label{lem:stable_core_surjectivity}\nFor every \\(m\\), the transition map \\(V_{m+1}\\to V_m\\) induces a surjection\n\\[\n\\boxed{\nY_{m+1}\\twoheadrightarrow Y_m.\n}\n\\]\n\\end{lemma}","\\begin{proof}\nFix \\(y\\in Y_m\\). For \\(n\\geq m+1\\), let\n\\[\nA_n\n=\n\\left\\{\nz\\in\\im(f_{n,m+1}):\nf_{m+1,m}(z)=y\n\\right\\}.\n\\]\nEach \\(A_n\\) is a nonempty affine subspace of \\(V_{m+1}\\), and the sequence\n\\((A_n)\\) is descending. A descending sequence of affine subspaces in a finite dimensional vector space stabilizes whenever every term is nonempty. Choose an element in its stable value. It lies in \\(Y_{m+1}\\) and maps to \\(y\\).\n\\end{proof}","y\\in Y_m\\). For \\(n\\geq m+1\\), let\n\\[\nA_n\n=\n\\left\\{\nz\\in\\im(f_{n,m+1}):\nf_{m+1,m}(z)=y\n\\right\\}.\n\\]\nEach \\(A_n\\) is a nonempty affine subspace of \\(V_{m+1}\\), and the sequence\n\\((A_n)\\) is descending. A descending sequence of affine subspaces in a finite dimensional vector space stabilizes whenever every term is nonempty. Choose an element in its stable value. It lies in \\(Y_{m+1}\\) and maps to \\(y\\).\n\\end{proof}","\\begin{theorem}[Nonzero stable core criterion]\\label{thm:nonzero_stable_core}\nLet \\((V_m)\\) be an inverse system of finite dimensional vector spaces. If\n\\[\nY_m\\neq0\n\\]\nfor every \\(m\\), then\n\\[\n\\boxed{\n\\varprojlim_mV_m\n\\neq0.\n}\n\\]\nMore precisely, the inverse limit contains a compatible sequence whose first coordinate is any prescribed nonzero element of \\(Y_1\\).\n\\end{theorem}","\\begin{proof}\nChoose \\(0\\neq y_1\\in Y_1\\). By \\cref{lem:stable_core_surjectivity}, choose recursively\n\\(y_{m+1}\\in Y_{m+1}\\) mapping to \\(y_m\\). Every \\(y_m\\) is nonzero because it maps to the nonzero vector \\(y_1\\). The sequence \\((y_m)\\) is a nonzero element of the inverse limit.\n\\end{proof}","\\begin{corollary}[Surjective systems]\\label{cor:surjective_inverse_system}\nIf every transition map \\(V_{m+1}\\to V_m\\) is surjective and every \\(V_m\\) is nonzero, then the inverse limit is nonzero.\n\\end{corollary}","\\begin{proof}\nSurjectivity gives \\(Y_m=V_m\\). Apply \\cref{thm:nonzero_stable_core}.\n\\end{proof}","\\begin{example}[Tail escape in an infinite carrier]\\label{ex:tail_escape}\nLet \\(V=\\ell^2(\\NN)\\), and let \\(P_m\\) be the orthogonal projection onto the closed span of\n\\[\ne_m,e_{m+1},e_{m+2},\\ldots.\n\\]\nEvery range \\(P_mV\\) is nonzero and the ranges are nested, but\n\\[\n\\boxed{\n\\bigcap_{m\\geq1}P_mV=0.\n}\n\\]\nThus the finite carrier hypothesis in \\cref{thm:nested_finite_carrier} cannot be removed.\n\\end{example}","\\begin{proof}\nA vector in every tail range has its first \\(m-1\\) coordinates equal to zero for every \\(m\\). All of its coordinates vanish.\n\\end{proof}","\\begin{definition}[Automorphic Hecke datum]\\label{def:automorphic_hecke_datum}\nAn automorphic Hecke datum consists of the following objects.\n\\begin{enumerate}\n\\item A commutative \\(\\OO\\) algebra \\(\\TT\\).\n\\item A finite \\(\\TT\\) module \\(H\\), or a perfect \\(\\TT\\) complex \\(C\\), obtained from an automorphic or cohomological carrier.\n\\item For every unramified place \\(v\\) outside a finite set, a polynomial\n\\[\nP_v(X)\\in\\TT[X].\n\\]\n\\item A declared characteristic zero locus on which a nonzero field fiber is known to be classical and is known to carry an attached Galois representation whose Frobenius polynomial at \\(v\\) is the specialization of \\(P_v(X)\\).\n\\end{enumerate}\n\\end{definition}","efinition}[Automorphic Hecke datum]\\label{def:automorphic_hecke_datum}\nAn automorphic Hecke datum consists of the following objects.\n\\begin{enumerate}\n\\item A commutative \\(\\OO\\) algebra \\(\\TT\\).\n\\item A finite \\(\\TT\\) module \\(H\\), or a perfect \\(\\TT\\) complex \\(C\\), obtained from an automorphic or cohomological carrier.\n\\item For every unramified place \\(v\\) outside a finite set, a polynomial\n\\[\nP_v(X)\\in\\TT[X].\n\\]\n\\item A declared characteristic zero locus on which a nonzero field fiber is known to be classical and is known to carry an attached Galois representation whose Frobenius polynomial at \\(v\\) is the specialization of \\(P_v(X)\\).\n\\end{enumerate}\n\\end{definition}","te \\(\\TT\\) module \\(H\\), or a perfect \\(\\TT\\) complex \\(C\\), obtained from an automorphic or cohomological carrier.\n\\item For every unramified place \\(v\\) outside a finite set, a polynomial\n\\[\nP_v(X)\\in\\TT[X].\n\\]\n\\item A declared characteristic zero locus on which a nonzero field fiber is known to be classical and is known to carry an attached Galois representation whose Frobenius polynomial at \\(v\\) is the specialization of \\(P_v(X)\\).\n\\end{enumerate}\n\\end{definition}","erfect \\(\\TT\\) complex \\(C\\), obtained from an automorphic or cohomological carrier.\n\\item For every unramified place \\(v\\) outside a finite set, a polynomial\n\\[\nP_v(X)\\in\\TT[X].\n\\]\n\\item A declared characteristic zero locus on which a nonzero field fiber is known to be classical and is known to carry an attached Galois representation whose Frobenius polynomial at \\(v\\) is the specialization of \\(P_v(X)\\).\n\\end{enumerate}\n\\end{definition}","\\begin{standardfact}[Chebotarev and Brauer Nesbitt identification]\\label{fact:chebotarev_brauer_nesbitt}\nLet \\(E'\\) be a Hausdorff topological field of characteristic zero, with continuous ring operations. Let\n\\[\n\\rho_1,\\rho_2:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nbe continuous semisimple representations. If their characteristic polynomials agree at every Frobenius element outside a density zero exceptional set, then\n\\[\n\\boxed{\\rho_1\\cong\\rho_2.}\n\\]\nThis follows from Chebotarev density and the Brauer Nesbitt theorem\n\\cite{BrauerNesbitt1941,Neukirch1999,Gee2022}.\n\\end{standardfact}","[Chebotarev and Brauer Nesbitt identification]\\label{fact:chebotarev_brauer_nesbitt}\nLet \\(E'\\) be a Hausdorff topological field of characteristic zero, with continuous ring operations. Let\n\\[\n\\rho_1,\\rho_2:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nbe continuous semisimple representations. If their characteristic polynomials agree at every Frobenius element outside a density zero exceptional set, then\n\\[\n\\boxed{\\rho_1\\cong\\rho_2.}\n\\]\nThis follows from Chebotarev density and the Brauer Nesbitt theorem\n\\cite{BrauerNesbitt1941,Neukirch1999,Gee2022}.\n\\end{standardfact}","\\begin{theorem}[Support gives a Hecke eigensystem]\\label{thm:support_gives_hecke_system}\nLet \\(R\\cong\\TT\\), let \\(H\\) be a finite \\(\\TT\\) module, and let\n\\(x:R\\to E'\\) be a field valued point. If\n\\[\n\\ker x\\in\\Supp_R(H),\n\\]\nthen\n\\[\n\\boxed{\nH_x:=H\\otimes_{\\TT,x}E'\\neq0.\n}\n\\]\nEvery element of \\(\\TT\\) acts on \\(H_x\\) through the scalar prescribed by \\(x\\).\n\\end{theorem}","\\begin{proof}\nNonvanishing follows from \\cref{cor:supported_hecke_fiber}. The tensor product relation\n\\[\nth\\otimes1=h\\otimes x(t)\n\\]\nshows that \\(t\\in\\TT\\) acts through \\(x(t)\\).\n\\end{proof}","\\begin{theorem}[Pointwise Galois realization]\\label{thm:pointwise_galois_realization}\nLet \\(E'\\) be a Hausdorff topological field of characteristic zero, with continuous ring operations. Require both \\(\\rho_x=x\\circ\\rho_R\\) and the attached representation below to be continuous. Assume the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nw","\\begin{theorem}[Pointwise Galois realization]\\label{thm:pointwise_galois_realization}\nLet \\(E'\\) be a Hausdorff topological field of characteristic zero, with continuous ring operations. Require both \\(\\rho_x=x\\circ\\rho_R\\) and the attached representation below to be continuous. Assume the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nf","\\begin{theorem}[Pointwise Galois realization]\\label{thm:pointwise_galois_realization}\nLet \\(E'\\) be a Hausdorff topological field of characteristic zero, with continuous ring operations. Require both \\(\\rho_x=x\\circ\\rho_R\\) and the attached representation below to be continuous. Assume the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_","\\begin{theorem}[Pointwise Galois realization]\\label{thm:pointwise_galois_realization}\nLet \\(E'\\) be a Hausdorff topological field of characteristic zero, with continuous ring operations. Require both \\(\\rho_x=x\\circ\\rho_R\\) and the attached representation below to be continuous. Assume the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irre…","galois_realization}\nLet \\(E'\\) be a Hausdorff topological field of characteristic zero, with continuous ring operations. Require both \\(\\rho_x=x\\circ\\rho_R\\) and the attached representation below to be continuous. Assume the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","pological field of characteristic zero, with continuous ring operations. Require both \\(\\rho_x=x\\circ\\rho_R\\) and the attached representation below to be continuous. Assume the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","us ring operations. Require both \\(\\rho_x=x\\circ\\rho_R\\) and the attached representation below to be continuous. Assume the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","uire both \\(\\rho_x=x\\circ\\rho_R\\) and the attached representation below to be continuous. Assume the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","tion below to be continuous. Assume the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","me the following data.\n\\begin{enumerate}\n\\item A represented deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","deformation ring \\(R\\) with universal representation\n\\[\n\\rho_R:G_{F,S}\\longrightarrow\\GL_n(R).\n\\]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","]\n\\item An isomorphism \\(R\\cong\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","g\\TT\\).\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}",")\\).\n\\item A characteristic zero point \\(x:R\\to E'\\) lying in \\(\\Supp_R(H)\\).\n\\item A classical automorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","tomorphic representation \\(\\pi_x\\) occurring in the fiber \\(H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","H_x\\), together with a semisimple attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","e attached representation\n\\[\n\\rho_{\\pi_x}:G_{F,S}\\longrightarrow\\GL_n(E')\n\\]\nwhose unramified characteristic polynomial is \\(x(P_v(X))\\).\n\\item Equality\n\\[\n\\det\\left(X-\\rho_x(\\Frob_v)\\right)=x(P_v(X))\n\\]\nfor \\(\\rho_x=x\\circ\\rho_R\\) at every unramified place in a Chebotarev dense set.\n\\end{enumerate}\nThen\n\\[\n\\boxed{\n\\rho_x^{\\mathrm{ss}}\n\\cong\n\\rho_{\\pi_x}.\n}\n\\]\nIf \\(\\rho_x\\) is irreducible, then \\(\\rho_x\\cong\\rho_{\\pi_x}\\).\n\\end{theorem}","\\begin{proof}\nThe coefficients of the characteristic polynomials of \\(\\rho_x\\) and \\(\\rho_{\\pi_x}\\) are continuous and constant on conjugacy classes. Their equality on the declared dense union of Frobenius conjugacy classes therefore extends to every element of \\(G_{F,S}\\), by Hausdorffness. A semisimplification has the same characteristic polynomials as the original representation. The algebraic Brauer Nesbitt input in \\cref{fact:chebotarev_brauer_nesbitt} thus gives \\(\\rho_x^{\\mathrm{ss}}\\cong\\rho_{\\pi_x}\\). If \\(\\rho_x\\) is irreducible, it is already semisimple.\n\\end{proof}","x\\) and \\(\\rho_{\\pi_x}\\) are continuous and constant on conjugacy classes. Their equality on the declared dense union of Frobenius conjugacy classes therefore extends to every element of \\(G_{F,S}\\), by Hausdorffness. A semisimplification has the same characteristic polynomials as the original representation. The algebraic Brauer Nesbitt input in \\cref{fact:chebotarev_brauer_nesbitt} thus gives \\(\\rho_x^{\\mathrm{ss}}\\cong\\rho_{\\pi_x}\\). If \\(\\rho_x\\) is irreducible, it is already semisimple.\n\\end{proof}","nstant on conjugacy classes. Their equality on the declared dense union of Frobenius conjugacy classes therefore extends to every element of \\(G_{F,S}\\), by Hausdorffness. A semisimplification has the same characteristic polynomials as the original representation. The algebraic Brauer Nesbitt input in \\cref{fact:chebotarev_brauer_nesbitt} thus gives \\(\\rho_x^{\\mathrm{ss}}\\cong\\rho_{\\pi_x}\\). If \\(\\rho_x\\) is irreducible, it is already semisimple.\n\\end{proof}","\\begin{corollary}[Full support gives pointwise automorphy on the realized locus]\\label{cor:full_support_pointwise_automorphy}\nIn the setting of \\cref{thm:pointwise_galois_realization}, suppose\n\\[\n\\Supp_R(H)=\\Spec R.\n\\]\nFor every characteristic zero point \\(x:R\\to E'\\) with \\(\\ker x\\in U\\subseteq\\Spec R\\), assume the coefficient topology and continuity hypotheses of \\cref{thm:pointwise_galois_realization} and the declared classicality and Galois realization input. Then every such point is automorphic in the precise sense of \\cref{thm:pointwise_galois_realization}.\n\\end{corollary}","\\begin{proof}\nEvery point of \\(U\\) lies in the support, so \\cref{thm:support_gives_hecke_system} gives a nonzero Hecke fiber. Apply the declared realization input and then \\cref{thm:pointwise_galois_realization}.\n\\end{proof}","\\begin{theorem}[Derived pointwise realization]\\label{thm:derived_pointwise_realization}\nLet \\(e:R\\cong\\TT\\), and let \\(C\\) be a perfect automorphic Hecke complex with full derived support. Choose a bounded finite projective model for \\(C\\), supported in degrees \\([a,b]\\). For every characteristic zero point \\(x:R\\to E'\\), write \\(y=x\\circ e^{-1}\\). Then some integer \\(i\\in[a,b]\\) has\n\\[\nH^i\\left(C\\mathbin{\\mathop{\\otimes}\\limits_{\\TT,y}^{\\mathbf L}}E'\\right)\\neq0.\n\\]\nEvery \\(t\\in\\TT\\) acts on this cohomology through \\(y(t)\\).","zation}\nLet \\(e:R\\cong\\TT\\), and let \\(C\\) be a perfect automorphic Hecke complex with full derived support. Choose a bounded finite projective model for \\(C\\), supported in degrees \\([a,b]\\). For every characteristic zero point \\(x:R\\to E'\\), write \\(y=x\\circ e^{-1}\\). Then some integer \\(i\\in[a,b]\\) has\n\\[\nH^i\\left(C\\mathbin{\\mathop{\\otimes}\\limits_{\\TT,y}^{\\mathbf L}}E'\\right)\\neq0.\n\\]\nEvery \\(t\\in\\TT\\) acts on this cohomology through \\(y(t)\\).","e a perfect automorphic Hecke complex with full derived support. Choose a bounded finite projective model for \\(C\\), supported in degrees \\([a,b]\\). For every characteristic zero point \\(x:R\\to E'\\), write \\(y=x\\circ e^{-1}\\). Then some integer \\(i\\in[a,b]\\) has\n\\[\nH^i\\left(C\\mathbin{\\mathop{\\otimes}\\limits_{\\TT,y}^{\\mathbf L}}E'\\right)\\neq0.\n\\]\nEvery \\(t\\in\\TT\\) acts on this cohomology through \\(y(t)\\).","For the realization conclusion, give \\(E'\\) a Hausdorff topological field structure. Supply a universal representation \\(\\rho_R\\), a continuous specialization \\(\\rho_x=x\\circ\\rho_R\\), and a semisimplification of \\(\\rho_x\\). Suppose its characteristic polynomials on a Chebotarev dense set are \\(y(P_v(X))\\). Let the declared classicality range contain \\([a,b]\\). Assume that every nonzero Hecke eigensystem in that range is realized by a classical automorphic representation occurring in that degree, with a continuous semisimple attached representation whose characteristic polynomials on the same dense set are \\(y(P_v(X))\\). With the standard Chebotarev density and Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}, the spec","For the realization conclusion, give \\(E'\\) a Hausdorff topological field structure. Supply a universal representation \\(\\rho_R\\), a continuous specialization \\(\\rho_x=x\\circ\\rho_R\\), and a semisimplification of \\(\\rho_x\\). Suppose its characteristic polynomials on a Chebotarev dense set are \\(y(P_v(X))\\). Let the declared classicality range contain \\([a,b]\\). Assume that every nonzero Hecke eigensystem in that range is realized by a classical automorphic representation occurring in that degree, with a continuous semisimple attached representation whose characteristic polynomials on the same dense set are \\(y(P_v(X))\\). With the standard Chebotarev density and Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}, the specialization \\(\\rho_x\\) is automorphic after semisimplification. If \\(\\rho_x\\) is irredu","For the realization conclusion, give \\(E'\\) a Hausdorff topological field structure. Supply a universal representation \\(\\rho_R\\), a continuous specialization \\(\\rho_x=x\\circ\\rho_R\\), and a semisimplification of \\(\\rho_x\\). Suppose its characteristic polynomials on a Chebotarev dense set are \\(y(P_v(X))\\). Let the declared classicality range contain \\([a,b]\\). Assume that every nonzero Hecke eigensystem in that range is realized by a classical automorphic representation occurring in that degree, with a continuous semisimple attached representation whose characteristic polynomials on the same dense set are \\(y(P_v(X))\\). With the standard Chebotarev density and Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}, the specialization \\(\\rho_x\\) is automorphic after semisimplification. If \\(\\rho_x\\) is irreducible, it is itself isomorphic to the attached repres","For the realization conclusion, give \\(E'\\) a Hausdorff topological field structure. Supply a universal representation \\(\\rho_R\\), a continuous specialization \\(\\rho_x=x\\circ\\rho_R\\), and a semisimplification of \\(\\rho_x\\). Suppose its characteristic polynomials on a Chebotarev dense set are \\(y(P_v(X))\\). Let the declared classicality range contain \\([a,b]\\). Assume that every nonzero Hecke eigensystem in that range is realized by a classical automorphic representation occurring in that degree, with a continuous semisimple attached representation whose characteristic polynomials on the same dense set are \\(y(P_v(X))\\). With the standard Chebotarev density and Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}, the specialization \\(\\rho_x\\) is automorphic after semisimplification. If \\(\\rho_x\\) is irreducible, it is itself isomorphic to the attached representation.\n\\end{theorem}","rho_x\\). Suppose its characteristic polynomials on a Chebotarev dense set are \\(y(P_v(X))\\). Let the declared classicality range contain \\([a,b]\\). Assume that every nonzero Hecke eigensystem in that range is realized by a classical automorphic representation occurring in that degree, with a continuous semisimple attached representation whose characteristic polynomials on the same dense set are \\(y(P_v(X))\\). With the standard Chebotarev density and Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}, the specialization \\(\\rho_x\\) is automorphic after semisimplification. If \\(\\rho_x\\) is irreducible, it is itself isomorphic to the attached representation.\n\\end{theorem}","[a,b]\\). Assume that every nonzero Hecke eigensystem in that range is realized by a classical automorphic representation occurring in that degree, with a continuous semisimple attached representation whose characteristic polynomials on the same dense set are \\(y(P_v(X))\\). With the standard Chebotarev density and Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}, the specialization \\(\\rho_x\\) is automorphic after semisimplification. If \\(\\rho_x\\) is irreducible, it is itself isomorphic to the attached representation.\n\\end{theorem}","that range is realized by a classical automorphic representation occurring in that degree, with a continuous semisimple attached representation whose characteristic polynomials on the same dense set are \\(y(P_v(X))\\). With the standard Chebotarev density and Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}, the specialization \\(\\rho_x\\) is automorphic after semisimplification. If \\(\\rho_x\\) is irreducible, it is itself isomorphic to the attached representation.\n\\end{theorem}","\\begin{proof}\nFull derived support gives a nonzero fiber at \\(y\\). Scalar extension of the chosen projective model preserves its degree bounds. Its cohomology therefore vanishes below \\(a\\) and above \\(b\\). Since vanishing in every degree is equivalent to being zero in the derived category, some degree \\(i\\in[a,b]\\) has nonzero cohomology. Choose a nonzero vector in that cohomology module.","Multiplication by \\(t\\in\\TT\\) is a cochain endomorphism of the original model. In the scalar extension the tensor relation identifies this endomorphism with \\(y(t)\\) times the identity. Passing to the derived category and then to cohomology preserves this equality. Thus the chosen vector is a common Hecke eigenvector with character \\(y\\).","The interval inclusion puts this eigensystem in the declared classicality range. Apply the realization input to obtain an occurring automorphic representation and its continuous semisimple attached representation. Its Frobenius characteristic polynomials agree with those of \\(\\rho_x\\) on the prescribed dense set. Continuity extends the equality to every group element. The supplied semisimplification preserves these polynomials, so the general Brauer Nesbitt input gives an invertible linear intertwiner. If \\(\\rho_x\\) is irreducible, it is already semisimple and the same comparison applies directly.\n\\end{proof}","btain an occurring automorphic representation and its continuous semisimple attached representation. Its Frobenius characteristic polynomials agree with those of \\(\\rho_x\\) on the prescribed dense set. Continuity extends the equality to every group element. The supplied semisimplification preserves these polynomials, so the general Brauer Nesbitt input gives an invertible linear intertwiner. If \\(\\rho_x\\) is irreducible, it is already semisimple and the same comparison applies directly.\n\\end{proof}","\\begin{definition}[Algebraic Frobenius spine]\\label{def:frobenius_spine}\nAn algebraic Frobenius spine consists of a number field \\(L\\), a finite set of places \\(S\\) of \\(F\\), and polynomials\n\\[\nP_v(X)\\in L[X]\n\\]\nfor every finite place \\(v\\notin S\\), together with any additional local labels required by the declared compatibility problem.\n\\end{definition}","\\begin{definition}[Finite companion forest]\\label{def:companion_forest}\nFix a coefficient place \\(\\lambda\\) of \\(L\\). A finite companion forest for the spine is an inverse system of finite sets\n\\[\nX_{\\lambda,1}\n\\longleftarrow\nX_{\\lambda,2}\n\\longleftarrow\nX_{\\lambda,3}\n\\longleftarrow\\cdots\n\\]\nwhose points are admissible realizations modulo \\(\\lambda^m\\), with edges given by reduction. A complete realization is admissible for this forest when all its finite reductions belong to the displayed levels; the empty level obstruction below concerns this declared admissibility problem.\n\\end{definition}","\\begin{theorem}[Companion branch criterion]\\label{thm:companion_branch_criterion}\nSuppose every level \\(X_{\\lambda,m}\\) is finite and nonempty, and every point reduces along the displayed forest to the fixed residual root. Then\n\\[\n\\boxed{\n\\varprojlim_mX_{\\lambda,m}\n\\neq\\varnothing.\n}\n\\]\n\\end{theorem}","\\begin{corollary}[Empty finite companion obstruction]\\label{cor:empty_companion_obstruction}\nIf \\(X_{\\lambda,m}=\\varnothing\\) for some \\(m\\), then no complete \\(\\lambda\\) adic realization admissible for this forest exists.\n\\end{corollary}","Fix a topological group \\(\\Gamma\\), a finite matrix index set \\(D\\), and a set \\(\\Lambda\\) of coefficient indices. For each \\(\\lambda\\), let \\(\\OO_\\lambda\\) be a commutative ring, complete and separated for a finitely generated ideal \\(I_\\lambda\\), with finite residue ring \\(\\OO_\\lambda/I_\\lambda\\). Give \\(\\OO_\\lambda\\) its ideal adic topology and every quotient its discrete topology. Let\n\\[\ni_\\lambda:\\OO_\\lambda\\longrightarrow K_\\lambda,\n\\qquad e_\\lambda:L\\longrightarrow K_\\lambda\n\\]\nbe supplied ring homomorphisms, where \\(L\\) and \\(K_\\lambda\\) are fields, \\(K_\\lambda\\) has a declared topology, and \\(i_\\lambda\\) is continuous.","bda\\), let \\(\\OO_\\lambda\\) be a commutative ring, complete and separated for a finitely generated ideal \\(I_\\lambda\\), with finite residue ring \\(\\OO_\\lambda/I_\\lambda\\). Give \\(\\OO_\\lambda\\) its ideal adic topology and every quotient its discrete topology. Let\n\\[\ni_\\lambda:\\OO_\\lambda\\longrightarrow K_\\lambda,\n\\qquad e_\\lambda:L\\longrightarrow K_\\lambda\n\\]\nbe supplied ring homomorphisms, where \\(L\\) and \\(K_\\lambda\\) are fields, \\(K_\\lambda\\) has a declared topology, and \\(i_\\lambda\\) is continuous.","\\(\\OO_\\lambda\\) be a commutative ring, complete and separated for a finitely generated ideal \\(I_\\lambda\\), with finite residue ring \\(\\OO_\\lambda/I_\\lambda\\). Give \\(\\OO_\\lambda\\) its ideal adic topology and every quotient its discrete topology. Let\n\\[\ni_\\lambda:\\OO_\\lambda\\longrightarrow K_\\lambda,\n\\qquad e_\\lambda:L\\longrightarrow K_\\lambda\n\\]\nbe supplied ring homomorphisms, where \\(L\\) and \\(K_\\lambda\\) are fields, \\(K_\\lambda\\) has a declared topology, and \\(i_\\lambda\\) is continuous.","utative ring, complete and separated for a finitely generated ideal \\(I_\\lambda\\), with finite residue ring \\(\\OO_\\lambda/I_\\lambda\\). Give \\(\\OO_\\lambda\\) its ideal adic topology and every quotient its discrete topology. Let\n\\[\ni_\\lambda:\\OO_\\lambda\\longrightarrow K_\\lambda,\n\\qquad e_\\lambda:L\\longrightarrow K_\\lambda\n\\]\nbe supplied ring homomorphisms, where \\(L\\) and \\(K_\\lambda\\) are fields, \\(K_\\lambda\\) has a declared topology, and \\(i_\\lambda\\) is continuous.","finitely generated ideal \\(I_\\lambda\\), with finite residue ring \\(\\OO_\\lambda/I_\\lambda\\). Give \\(\\OO_\\lambda\\) its ideal adic topology and every quotient its discrete topology. Let\n\\[\ni_\\lambda:\\OO_\\lambda\\longrightarrow K_\\lambda,\n\\qquad e_\\lambda:L\\longrightarrow K_\\lambda\n\\]\nbe supplied ring homomorphisms, where \\(L\\) and \\(K_\\lambda\\) are fields, \\(K_\\lambda\\) has a declared topology, and \\(i_\\lambda\\) is continuous.","For every positive integer \\(n\\), let \\(X_{\\lambda,n}\\) be a finite nonempty set of conjugacy classes of actual continuous representations\n\\[\n\\Gamma\\longrightarrow\\GL_D(\\OO_\\lambda/I_\\lambda^n).\n\\]\nConjugacy means change of basis by an invertible matrix over that quotient. Assume reduction preserves these sets and that \\(X_{\\lambda,1}\\) consists of the declared residual class. Each admissible class must have a continuous representative. The auxiliary level zero is the unique class over the trivial quotient and imposes no extra occupancy hypothesis.","Choose elements \\(\\gamma_v\\in\\Gamma\\) on a sample set \\(\\Omega\\), with declared good subsets \\(\\Omega_\\lambda\\subseteq\\Omega\\). In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each \\(v\\in\\Omega_\\lambda\\), supply an integral polynomial \\(P_{\\lambda,v}\\in\\OO_\\lambda[X]\\) such that\n\\[\ni_\\lambda(P_{\\lambda,v})=e_\\lambda(P_v)\\quad\\text{in }K_\\lambda[X].\n\\]\nHere and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in \\(X_{\\lambda,n}\\) has characteristic polynomial \\(P_{\\lambda,v}\\bmod I_\\lambda^n\\) at \\(\\gamma_v\\). Also supply a subgroup \\(H_\\lambda\\leq\\Gamma\\) on which all these finite representations are trivia","Choose elements \\(\\gamma_v\\in\\Gamma\\) on a sample set \\(\\Omega\\), with declared good subsets \\(\\Omega_\\lambda\\subseteq\\Omega\\). In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each \\(v\\in\\Omega_\\lambda\\), supply an integral polynomial \\(P_{\\lambda,v}\\in\\OO_\\lambda[X]\\) such that\n\\[\ni_\\lambda(P_{\\lambda,v})=e_\\lambda(P_v)\\quad\\text{in }K_\\lambda[X].\n\\]\nHere and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in \\(X_{\\lambda,n}\\) has characteristic polynomial \\(P_{\\lambda,v}\\bmod I_\\lambda^n\\) at \\(\\gamma_v\\). Also supply a subgroup \\(H_\\lambda\\leq\\Gamma\\) on which all these finite representations are trivial. This is an inertia condi","Choose elements \\(\\gamma_v\\in\\Gamma\\) on a sample set \\(\\Omega\\), with declared good subsets \\(\\Omega_\\lambda\\subseteq\\Omega\\). In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each \\(v\\in\\Omega_\\lambda\\), supply an integral polynomial \\(P_{\\lambda,v}\\in\\OO_\\lambda[X]\\) such that\n\\[\ni_\\lambda(P_{\\lambda,v})=e_\\lambda(P_v)\\quad\\text{in }K_\\lambda[X].\n\\]\nHere and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in \\(X_{\\lambda,n}\\) has characteristic polynomial \\(P_{\\lambda,v}\\bmod I_\\lambda^n\\) at \\(\\gamma_v\\). Also supply a subgroup \\(H_\\lambda\\leq\\Gamma\\) on which all these finite representations are trivial. This is an inertia condition when \\(H_\\lambda\\) is the specified inertia subgroup.","thmetic application these are the chosen Frobenius elements at the permitted places. For each \\(v\\in\\Omega_\\lambda\\), supply an integral polynomial \\(P_{\\lambda,v}\\in\\OO_\\lambda[X]\\) such that\n\\[\ni_\\lambda(P_{\\lambda,v})=e_\\lambda(P_v)\\quad\\text{in }K_\\lambda[X].\n\\]\nHere and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in \\(X_{\\lambda,n}\\) has characteristic polynomial \\(P_{\\lambda,v}\\bmod I_\\lambda^n\\) at \\(\\gamma_v\\). Also supply a subgroup \\(H_\\lambda\\leq\\Gamma\\) on which all these finite representations are trivial. This is an inertia condition when \\(H_\\lambda\\) is the specified inertia subgroup.","us elements at the permitted places. For each \\(v\\in\\Omega_\\lambda\\), supply an integral polynomial \\(P_{\\lambda,v}\\in\\OO_\\lambda[X]\\) such that\n\\[\ni_\\lambda(P_{\\lambda,v})=e_\\lambda(P_v)\\quad\\text{in }K_\\lambda[X].\n\\]\nHere and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in \\(X_{\\lambda,n}\\) has characteristic polynomial \\(P_{\\lambda,v}\\bmod I_\\lambda^n\\) at \\(\\gamma_v\\). Also supply a subgroup \\(H_\\lambda\\leq\\Gamma\\) on which all these finite representations are trivial. This is an inertia condition when \\(H_\\lambda\\) is the specified inertia subgroup.","or each \\(v\\in\\Omega_\\lambda\\), supply an integral polynomial \\(P_{\\lambda,v}\\in\\OO_\\lambda[X]\\) such that\n\\[\ni_\\lambda(P_{\\lambda,v})=e_\\lambda(P_v)\\quad\\text{in }K_\\lambda[X].\n\\]\nHere and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in \\(X_{\\lambda,n}\\) has characteristic polynomial \\(P_{\\lambda,v}\\bmod I_\\lambda^n\\) at \\(\\gamma_v\\). Also supply a subgroup \\(H_\\lambda\\leq\\Gamma\\) on which all these finite representations are trivial. This is an inertia condition when \\(H_\\lambda\\) is the specified inertia subgroup.","y an integral polynomial \\(P_{\\lambda,v}\\in\\OO_\\lambda[X]\\) such that\n\\[\ni_\\lambda(P_{\\lambda,v})=e_\\lambda(P_v)\\quad\\text{in }K_\\lambda[X].\n\\]\nHere and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in \\(X_{\\lambda,n}\\) has characteristic polynomial \\(P_{\\lambda,v}\\bmod I_\\lambda^n\\) at \\(\\gamma_v\\). Also supply a subgroup \\(H_\\lambda\\leq\\Gamma\\) on which all these finite representations are trivial. This is an inertia condition when \\(H_\\lambda\\) is the specified inertia subgroup.","quad\\text{in }K_\\lambda[X].\n\\]\nHere and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in \\(X_{\\lambda,n}\\) has characteristic polynomial \\(P_{\\lambda,v}\\bmod I_\\lambda^n\\) at \\(\\gamma_v\\). Also supply a subgroup \\(H_\\lambda\\leq\\Gamma\\) on which all these finite representations are trivial. This is an inertia condition when \\(H_\\lambda\\) is the specified inertia subgroup.","Finally, put\n\\[\nB_\\lambda=\\OO_\\lambda[x_{g,a,b}:g\\in\\Gamma,\\ a,b\\in D].\n\\]\nFor each declared local condition index \\(u\\), supply an ideal \\(J_{\\lambda,u}\\subseteq B_\\lambda\\). Evaluation at a representation sends \\(x_{g,a,b}\\) to its \\((a,b)\\) matrix entry at \\(g\\). Assume that evaluation at every admissible finite representative kills \\(J_{\\lambda,u}\\), with polynomial coefficients reduced modulo \\(I_\\lambda^n\\). For this class based theorem, the equations must hold in every admissible frame. A condition imposed only in one chosen frame needs a separate compatible frame argument. No arbitrary arithmetic condition is identified with such an ideal without a separate argument.","ally, put\n\\[\nB_\\lambda=\\OO_\\lambda[x_{g,a,b}:g\\in\\Gamma,\\ a,b\\in D].\n\\]\nFor each declared local condition index \\(u\\), supply an ideal \\(J_{\\lambda,u}\\subseteq B_\\lambda\\). Evaluation at a representation sends \\(x_{g,a,b}\\) to its \\((a,b)\\) matrix entry at \\(g\\). Assume that evaluation at every admissible finite representative kills \\(J_{\\lambda,u}\\), with polynomial coefficients reduced modulo \\(I_\\lambda^n\\). For this class based theorem, the equations must hold in every admissible frame. A condition imposed only in one chosen frame needs a separate compatible frame argument. No arbitrary arithmetic condition is identified with such an ideal without a separate argument.","\\begin{theorem}[Conditional simultaneous companion completion]\\label{thm:simultaneous_companion_completion}\nUnder the coefficient and finite admissibility hypotheses above, one can choose continuous representations\n\\[\n\\rho_\\lambda:\\Gamma\\longrightarrow\\GL_D(\\OO_\\lambda)\n\\qquad(\\lambda\\in\\Lambda)\n\\]\nwhose reduction class belongs to \\(X_{\\lambda,n}\\) for every positive \\(n\\). They retain the declared residual class. After applying \\(i_\\lambda\\), they are continuous, trivial on \\(H_\\lambda\\), and satisfy\n\\[\n\\det\\!\\left(X- i_\\lambda(\\rho_\\lambda(\\gamma_v))\\right)\n=e_\\lambda(P_v),\\qquad v\\in\\Omega_\\lambda.\n\\]\nTheir integral evaluation points factor through every actual quotient \\(B_\\lambda/J_{\\lambda,u}\\).\n\\end{theorem}","neous companion completion]\\label{thm:simultaneous_companion_completion}\nUnder the coefficient and finite admissibility hypotheses above, one can choose continuous representations\n\\[\n\\rho_\\lambda:\\Gamma\\longrightarrow\\GL_D(\\OO_\\lambda)\n\\qquad(\\lambda\\in\\Lambda)\n\\]\nwhose reduction class belongs to \\(X_{\\lambda,n}\\) for every positive \\(n\\). They retain the declared residual class. After applying \\(i_\\lambda\\), they are continuous, trivial on \\(H_\\lambda\\), and satisfy\n\\[\n\\det\\!\\left(X- i_\\lambda(\\rho_\\lambda(\\gamma_v))\\right)\n=e_\\lambda(P_v),\\qquad v\\in\\Omega_\\lambda.\n\\]\nTheir integral evaluation points factor through every actual quotient \\(B_\\lambda/J_{\\lambda,u}\\).\n\\end{theorem}","ous_companion_completion}\nUnder the coefficient and finite admissibility hypotheses above, one can choose continuous representations\n\\[\n\\rho_\\lambda:\\Gamma\\longrightarrow\\GL_D(\\OO_\\lambda)\n\\qquad(\\lambda\\in\\Lambda)\n\\]\nwhose reduction class belongs to \\(X_{\\lambda,n}\\) for every positive \\(n\\). They retain the declared residual class. After applying \\(i_\\lambda\\), they are continuous, trivial on \\(H_\\lambda\\), and satisfy\n\\[\n\\det\\!\\left(X- i_\\lambda(\\rho_\\lambda(\\gamma_v))\\right)\n=e_\\lambda(P_v),\\qquad v\\in\\Omega_\\lambda.\n\\]\nTheir integral evaluation points factor through every actual quotient \\(B_\\lambda/J_{\\lambda,u}\\).\n\\end{theorem}","nder the coefficient and finite admissibility hypotheses above, one can choose continuous representations\n\\[\n\\rho_\\lambda:\\Gamma\\longrightarrow\\GL_D(\\OO_\\lambda)\n\\qquad(\\lambda\\in\\Lambda)\n\\]\nwhose reduction class belongs to \\(X_{\\lambda,n}\\) for every positive \\(n\\). They retain the declared residual class. After applying \\(i_\\lambda\\), they are continuous, trivial on \\(H_\\lambda\\), and satisfy\n\\[\n\\det\\!\\left(X- i_\\lambda(\\rho_\\lambda(\\gamma_v))\\right)\n=e_\\lambda(P_v),\\qquad v\\in\\Omega_\\lambda.\n\\]\nTheir integral evaluation points factor through every actual quotient \\(B_\\lambda/J_{\\lambda,u}\\).\n\\end{theorem}","a\\in\\Lambda)\n\\]\nwhose reduction class belongs to \\(X_{\\lambda,n}\\) for every positive \\(n\\). They retain the declared residual class. After applying \\(i_\\lambda\\), they are continuous, trivial on \\(H_\\lambda\\), and satisfy\n\\[\n\\det\\!\\left(X- i_\\lambda(\\rho_\\lambda(\\gamma_v))\\right)\n=e_\\lambda(P_v),\\qquad v\\in\\Omega_\\lambda.\n\\]\nTheir integral evaluation points factor through every actual quotient \\(B_\\lambda/J_{\\lambda,u}\\).\n\\end{theorem}","Completeness and separatedness identify \\(\\GL_D(\\OO_\\lambda)\\) with the compatible inverse limit of the quotient matrix groups. Lift entries and inverse entries to construct this identification. The two reduction and reconstruction identities hold entrywise. Finite quotient continuity implies continuity for the ideal adic topology, including the matrix inverse, so the reconstructed representation is continuous.","Each coefficient of its characteristic polynomial agrees with the prescribed integral coefficient modulo every positive power of \\(I_\\lambda\\). Separatedness gives equality of the full polynomials. The same argument applied to the matrix entries of \\(\\rho_\\lambda(g)-1\\) gives triviality on \\(H_\\lambda\\). Continuous coefficient change and the matching equations for \\(P_{\\lambda,v}\\) give the asserted common spine over the local fields.","For \\(p\\in J_{\\lambda,u}\\), evaluation commutes with reduction. Its value is zero modulo every positive power of \\(I_\\lambda\\), hence is zero integrally by separatedness. Therefore the evaluation homomorphism kills the whole ideal and factors uniquely through \\(B_\\lambda/J_{\\lambda,u}\\). Choose a realization for each \\(\\lambda\\).\n\\end{proof}","\\begin{theorem}[One member propagation in a fixed system]\\label{thm:one_member_propagation}\nLet\n\\[\n\\{\\rho_\\lambda\\}_{\\lambda\\in\\Lambda}\n\\]\nbe a semisimple strictly compatible system with Frobenius spine \\((P_v)\\). Suppose that for one coefficient place \\(\\lambda_0\\), the representation \\(\\rho_{\\lambda_0}\\) is identified with the Galois representation attached to an automorphic representation \\(\\pi\\). Assume that the compatible system attached to \\(\\pi\\) has the same polynomials \\(P_v(X)\\). Then for every \\(\\lambda\\in\\Lambda\\),\n\\[\n\\boxed{\n\\rho_\\lambda\n\\cong\n\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}.\n}\n\\]\n\\end{theorem}","stem]\\label{thm:one_member_propagation}\nLet\n\\[\n\\{\\rho_\\lambda\\}_{\\lambda\\in\\Lambda}\n\\]\nbe a semisimple strictly compatible system with Frobenius spine \\((P_v)\\). Suppose that for one coefficient place \\(\\lambda_0\\), the representation \\(\\rho_{\\lambda_0}\\) is identified with the Galois representation attached to an automorphic representation \\(\\pi\\). Assume that the compatible system attached to \\(\\pi\\) has the same polynomials \\(P_v(X)\\). Then for every \\(\\lambda\\in\\Lambda\\),\n\\[\n\\boxed{\n\\rho_\\lambda\n\\cong\n\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}.\n}\n\\]\n\\end{theorem}","ation}\nLet\n\\[\n\\{\\rho_\\lambda\\}_{\\lambda\\in\\Lambda}\n\\]\nbe a semisimple strictly compatible system with Frobenius spine \\((P_v)\\). Suppose that for one coefficient place \\(\\lambda_0\\), the representation \\(\\rho_{\\lambda_0}\\) is identified with the Galois representation attached to an automorphic representation \\(\\pi\\). Assume that the compatible system attached to \\(\\pi\\) has the same polynomials \\(P_v(X)\\). Then for every \\(\\lambda\\in\\Lambda\\),\n\\[\n\\boxed{\n\\rho_\\lambda\n\\cong\n\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}.\n}\n\\]\n\\end{theorem}","a\\}_{\\lambda\\in\\Lambda}\n\\]\nbe a semisimple strictly compatible system with Frobenius spine \\((P_v)\\). Suppose that for one coefficient place \\(\\lambda_0\\), the representation \\(\\rho_{\\lambda_0}\\) is identified with the Galois representation attached to an automorphic representation \\(\\pi\\). Assume that the compatible system attached to \\(\\pi\\) has the same polynomials \\(P_v(X)\\). Then for every \\(\\lambda\\in\\Lambda\\),\n\\[\n\\boxed{\n\\rho_\\lambda\n\\cong\n\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}.\n}\n\\]\n\\end{theorem}","\\]\nbe a semisimple strictly compatible system with Frobenius spine \\((P_v)\\). Suppose that for one coefficient place \\(\\lambda_0\\), the representation \\(\\rho_{\\lambda_0}\\) is identified with the Galois representation attached to an automorphic representation \\(\\pi\\). Assume that the compatible system attached to \\(\\pi\\) has the same polynomials \\(P_v(X)\\). Then for every \\(\\lambda\\in\\Lambda\\),\n\\[\n\\boxed{\n\\rho_\\lambda\n\\cong\n\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nFor every \\(\\lambda\\), the two semisimple representations have the same unramified characteristic polynomials at all places outside one finite set. Apply \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{proof}","\\begin{proposition}[Finite Frobenius separator]\\label{prop:finite_frobenius_separator}\nLet \\(t_1,\\ldots,t_r\\) be linearly independent functions on a set \\(\\Omega\\) with values in a field \\(E'\\). Then there are points\n\\(\\omega_1,\\ldots,\\omega_r\\in\\Omega\\) such that the matrix\n\\[\n\\left(t_i(\\omega_j)\\right)_{j,i}\n\\]\nis invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely many Frobenius traces recover their multiplicities in any known semisimple carrier.\n\\end{proposition}","\\begin{proof}\nLet\n\\[\nW\n=\n\\operatorname{span}\n\\left\\{\n(t_1(\\omega),\\ldots,t_r(\\omega)):\\omega\\in\\Omega\n\\right\\}\n\\subseteq E'^r.\n\\]\nIf \\(W\\neq E'^r\\), a nonzero linear functional on \\(E'^r\\) would vanish on every evaluation vector, giving a nontrivial linear relation among the functions. Thus \\(W=E'^r\\). Choose \\(r\\) evaluation vectors forming a basis.\n\\end{proof}","\\begin{proposition}[Failure of integral sign projectors]\\label{prop:dyadic_sign_projectors}\nLet \\(C_2=\\langle g\\rangle\\). Over \\(\\QQ_2\\), the two sign projectors are\n\\[\n\\boxed{\ne_+=\\frac{1+g}{2},\n\\qquad\ne_-=\\frac{1-g}{2}.\n}\n\\]\nNeither belongs to \\(\\ZZ_2[C_2]\\). After reduction modulo two,\n\\[\n\\boxed{\n\\FF_2[C_2]\n\\cong\n\\FF_2[X]/(X^2),\n\\qquad\nX=g-1.\n}\n\\]\nThis ring is local and has no nontrivial idempotents.\n\\end{proposition}","\\begin{proof}\nThe displayed rational elements are the unique primitive idempotents because they project to the two characters of \\(C_2\\). Their coefficients contain \\(1/2\\), so they are not integral. In characteristic two,\n\\[\n(g-1)^2=g^2-2g+1=0.\n\\]\nThe map \\(X\\mapsto g-1\\) gives the displayed isomorphism. The quotient by \\((X)\\) is \\(\\FF_2\\), and \\((X)\\) is nilpotent. Thus the ring is local.\n\\end{proof}","\\begin{corollary}[Dyadic coinvariant recovery]\\label{cor:dyadic_coinvariant_recovery}\nLet \\(\\Delta\\) be a finite two group, let \\(\\Lambda=\\OO[\\Delta]\\) with \\(\\OO\\) a finite extension of \\(\\ZZ_2\\), and let \\(M\\) be finite free over \\(\\Lambda\\). Then\n\\[\n\\boxed{\nM\\otimes_{\\Lambda,\\epsilon}\\OO\n\\cong\nM/\\fraka M\n}\n\\]\nis free over \\(\\OO\\) of the same \\(\\Lambda\\) rank. No semisimple character decomposition is required.\n\\end{corollary}","\\begin{proof}\nUse the right module convention fixed in \\cref{sec:auxiliary}. Quotienting by all balancing relations identifies \\(M\\otimes_{\\Lambda,\\epsilon}\\OO\\) with \\(M/\\fraka M\\), without any commutativity assumption on \\(\\Delta\\). Applying \\(\\epsilon\\) to the coordinates of a finite \\(\\Lambda\\) basis gives an \\(\\OO\\) basis with the same index set.\n\\end{proof}","\\begin{definition}[Ordinary completion datum at a point]\\label{def:ordinary_completion_datum}\nAn ordinary completion datum at a characteristic zero point \\(x:R\\to E'\\) consists of the following objects.\n\\begin{enumerate}\n\\item A represented global deformation ring \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality","\\begin{definition}[Ordinary completion datum at a point]\\label{def:ordinary_completion_datum}\nAn ordinary completion datum at a characteristic zero point \\(x:R\\to E'\\) consists of the following objects.\n\\begin{enumerate}\n\\item A represented global deformation ring \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for","\\begin{definition}[Ordinary completion datum at a point]\\label{def:ordinary_completion_datum}\nAn ordinary completion datum at a characteristic zero point \\(x:R\\to E'\\) consists of the following objects.\n\\begin{enumerate}\n\\item A represented global deformation ring \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation w","\\begin{definition}[Ordinary completion datum at a point]\\label{def:ordinary_completion_datum}\nAn ordinary completion datum at a characteristic zero point \\(x:R\\to E'\\) consists of the following objects.\n\\begin{enumerate}\n\\item A represented global deformation ring \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation;…","point]\\label{def:ordinary_completion_datum}\nAn ordinary completion datum at a characteristic zero point \\(x:R\\to E'\\) consists of the following objects.\n\\begin{enumerate}\n\\item A represented global deformation ring \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of bot…","datum at a characteristic zero point \\(x:R\\to E'\\) consists of the following objects.\n\\begin{enumerate}\n\\item A represented global deformation ring \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\…","llowing objects.\n\\begin{enumerate}\n\\item A represented global deformation ring \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\) on the same declared Chebotarev dense union of Frobenius conjugacy …","ented global deformation ring \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplification, den…","l deformation ring \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplification, density, and g…","ng \\(R\\) with declared local quotient conditions and universal representation \\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplification, density, and general Brauer Ne…","\\(\\rho_R\\).\n\\item An ordinary patching datum\n\\[\n(R_\\infty,\\fraka,M_\\infty,R,H)\n\\]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplification, density, and general Brauer Nesbitt inputs. The coefficient topology and continuity requirements are retaine…","]\nwhose finite level module \\(H\\) is an automorphic Hecke carrier.\n\\item Full patched support\n\\[\n\\Supp_{R_\\infty}(M_\\infty)=\\Spec R_\\infty.\n\\]\n\\item An isomorphism\n\\[\nR\\cong\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplification, density, and general Brauer Nesbitt inputs. The coefficient topology and continuity requirements are retained in \\cref{thm:ordinary_conditional_completion}.\n\\end{enumerate}\n\\end{definition…","g\\TT\n\\]\nwith the Hecke algebra acting on \\(H\\). The \\(R\\) action on \\(H\\) supplied by the patched control isomorphism is the restriction of this \\(\\TT\\) action along the displayed isomorphism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplification, density, and general Brauer Nesbitt inputs. The coefficient topology and continuity requirements are retained in \\cref{thm:ordinary_conditional_completion}.\n\\end{enumerate}\n\\end{definition}","phism.\n\\item An automorphic Hecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplification, density, and general Brauer Nesbitt inputs. The coefficient topology and continuity requirements are retained in \\cref{thm:ordinary_conditional_completion}.\n\\end{enumerate}\n\\end{definition}","ecke datum \\((\\TT,H,(P_v))\\) and a pointwise classicality and Galois realization theorem satisfying all the inputs of \\cref{thm:pointwise_galois_realization} for the nonzero fiber at \\(x\\): an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and \\(\\rho_x=x\\circ\\rho_R\\) with \\(x(P_v(X))\\) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplification, density, and general Brauer Nesbitt inputs. The coefficient topology and continuity requirements are retained in \\cref{thm:ordinary_conditional_completion}.\n\\end{enumerate}\n\\end{definition}","\\begin{theorem}[Ordinary conditional automorphic completion]\\label{thm:ordinary_conditional_completion}\nLet \\(x:R\\to E'\\) admit an ordinary completion datum. Assume \\(E'\\) is a Hausdorff topological field of characteristic zero with continuous ring operations, and that both \\(\\rho_x=x\\circ\\rho_R\\) and its attached comparison representation are continuous. Then\n\\[\n\\boxed{\nH\\otimes_{\\TT,x}E'\\neq0.\n}\n\\]\nThe specialization\n\\[\n\\rho_x=x\\circ\\rho_R\n\\]\nis identified after semisimplification with the Galois representation attached to a classical automorphic representation in that fiber. If \\(\\rho_x\\) is irreducible, it is itself automorphic.\n\\end{theorem}","conditional automorphic completion]\\label{thm:ordinary_conditional_completion}\nLet \\(x:R\\to E'\\) admit an ordinary completion datum. Assume \\(E'\\) is a Hausdorff topological field of characteristic zero with continuous ring operations, and that both \\(\\rho_x=x\\circ\\rho_R\\) and its attached comparison representation are continuous. Then\n\\[\n\\boxed{\nH\\otimes_{\\TT,x}E'\\neq0.\n}\n\\]\nThe specialization\n\\[\n\\rho_x=x\\circ\\rho_R\n\\]\nis identified after semisimplification with the Galois representation attached to a classical automorphic representation in that fiber. If \\(\\rho_x\\) is irreducible, it is itself automorphic.\n\\end{theorem}","ological field of characteristic zero with continuous ring operations, and that both \\(\\rho_x=x\\circ\\rho_R\\) and its attached comparison representation are continuous. Then\n\\[\n\\boxed{\nH\\otimes_{\\TT,x}E'\\neq0.\n}\n\\]\nThe specialization\n\\[\n\\rho_x=x\\circ\\rho_R\n\\]\nis identified after semisimplification with the Galois representation attached to a classical automorphic representation in that fiber. If \\(\\rho_x\\) is irreducible, it is itself automorphic.\n\\end{theorem}","\\begin{proof}\nFull patched support descends to full support of \\(H\\) over \\(R\\) by\n\\cref{thm:ordinary_support_descent}. The isomorphism \\(R\\cong\\TT\\) transports that support by \\cref{thm:rt_support_transport}. The field fiber is nonzero by\n\\cref{thm:support_gives_hecke_system}. The declared classicality and realization theorem supplies an automorphic representation in the fiber and an attached Galois representation. The comparison with \\(\\rho_x\\) follows from \\cref{thm:pointwise_galois_realization}.\n\\end{proof}","s to full support of \\(H\\) over \\(R\\) by\n\\cref{thm:ordinary_support_descent}. The isomorphism \\(R\\cong\\TT\\) transports that support by \\cref{thm:rt_support_transport}. The field fiber is nonzero by\n\\cref{thm:support_gives_hecke_system}. The declared classicality and realization theorem supplies an automorphic representation in the fiber and an attached Galois representation. The comparison with \\(\\rho_x\\) follows from \\cref{thm:pointwise_galois_realization}.\n\\end{proof}","\\begin{definition}[Derived completion datum at a point]\\label{def:derived_completion_datum}\nA derived completion datum is defined in the same way, with a perfect patched complex\n\\(C_\\infty\\) of full derived support and a finite level automorphic complex\n\\[\nC\n\\simeq\nC_\\infty\n\\mathbin{\\mathop{\\otimes}\\limits_{R_\\infty}^{\\mathbf L}}R\n\\]\nin place of the ordinary modules. Under the Hecke ring isomorphism \\(e:R\\cong\\TT\\), require an actual derived carrier identification between the scalar transport of this finite level complex and the automorphic Hecke complex. This identification retains the \\(R\\) action induced by patched control as the restriction of the \\(\\TT\\) action along \\(e\\).\n\\end{definition}","n{definition}[Derived completion datum at a point]\\label{def:derived_completion_datum}\nA derived completion datum is defined in the same way, with a perfect patched complex\n\\(C_\\infty\\) of full derived support and a finite level automorphic complex\n\\[\nC\n\\simeq\nC_\\infty\n\\mathbin{\\mathop{\\otimes}\\limits_{R_\\infty}^{\\mathbf L}}R\n\\]\nin place of the ordinary modules. Under the Hecke ring isomorphism \\(e:R\\cong\\TT\\), require an actual derived carrier identification between the scalar transport of this finite level complex and the automorphic Hecke complex. This identification retains the \\(R\\) action induced by patched control as the restriction of the \\(\\TT\\) action along \\(e\\).\n\\end{definition}","ll derived support and a finite level automorphic complex\n\\[\nC\n\\simeq\nC_\\infty\n\\mathbin{\\mathop{\\otimes}\\limits_{R_\\infty}^{\\mathbf L}}R\n\\]\nin place of the ordinary modules. Under the Hecke ring isomorphism \\(e:R\\cong\\TT\\), require an actual derived carrier identification between the scalar transport of this finite level complex and the automorphic Hecke complex. This identification retains the \\(R\\) action induced by patched control as the restriction of the \\(\\TT\\) action along \\(e\\).\n\\end{definition}","meq\nC_\\infty\n\\mathbin{\\mathop{\\otimes}\\limits_{R_\\infty}^{\\mathbf L}}R\n\\]\nin place of the ordinary modules. Under the Hecke ring isomorphism \\(e:R\\cong\\TT\\), require an actual derived carrier identification between the scalar transport of this finite level complex and the automorphic Hecke complex. This identification retains the \\(R\\) action induced by patched control as the restriction of the \\(\\TT\\) action along \\(e\\).\n\\end{definition}","{\\mathop{\\otimes}\\limits_{R_\\infty}^{\\mathbf L}}R\n\\]\nin place of the ordinary modules. Under the Hecke ring isomorphism \\(e:R\\cong\\TT\\), require an actual derived carrier identification between the scalar transport of this finite level complex and the automorphic Hecke complex. This identification retains the \\(R\\) action induced by patched control as the restriction of the \\(\\TT\\) action along \\(e\\).\n\\end{definition}","\\begin{theorem}[Derived conditional automorphic completion]\\label{thm:derived_conditional_completion}\nLet \\(x:R\\to E'\\) admit a derived completion datum, with Hecke identification \\(e:R\\cong\\TT\\), and put \\(y=x\\circ e^{-1}\\). Then\n\\[\n\\boxed{\nC\\mathbin{\\mathop{\\otimes}\\limits_{\\TT,y}^{\\mathbf L}}E'\\not\\simeq0.\n}\n\\]\nSome integer cohomological degree contains a nonzero Hecke eigensystem with character \\(y\\). Its degree lies between the bounds of any chosen finite projective model for \\(C\\).","in{theorem}[Derived conditional automorphic completion]\\label{thm:derived_conditional_completion}\nLet \\(x:R\\to E'\\) admit a derived completion datum, with Hecke identification \\(e:R\\cong\\TT\\), and put \\(y=x\\circ e^{-1}\\). Then\n\\[\n\\boxed{\nC\\mathbin{\\mathop{\\otimes}\\limits_{\\TT,y}^{\\mathbf L}}E'\\not\\simeq0.\n}\n\\]\nSome integer cohomological degree contains a nonzero Hecke eigensystem with character \\(y\\). Its degree lies between the bounds of any chosen finite projective model for \\(C\\).","l_completion}\nLet \\(x:R\\to E'\\) admit a derived completion datum, with Hecke identification \\(e:R\\cong\\TT\\), and put \\(y=x\\circ e^{-1}\\). Then\n\\[\n\\boxed{\nC\\mathbin{\\mathop{\\otimes}\\limits_{\\TT,y}^{\\mathbf L}}E'\\not\\simeq0.\n}\n\\]\nSome integer cohomological degree contains a nonzero Hecke eigensystem with character \\(y\\). Its degree lies between the bounds of any chosen finite projective model for \\(C\\).","t topology, continuous specialization, supplied semisimplification, and common Frobenius polynomial hypotheses in \\cref{thm:derived_pointwise_realization}. If the declared classicality and Galois realization theorem applies to this eigensystem and supplies an occurring classical automorphic representation with a continuous semisimple attached representation and those Frobenius polynomials, then \\(\\rho_x\\) is automorphic after semisimplification. If \\(\\rho_x\\) is irreducible, it is itself isomorphic to the attached representation. This pointwise applicability condition does not require a uniform classicality assertion in other degrees.\n\\end{theorem}","plification, and common Frobenius polynomial hypotheses in \\cref{thm:derived_pointwise_realization}. If the declared classicality and Galois realization theorem applies to this eigensystem and supplies an occurring classical automorphic representation with a continuous semisimple attached representation and those Frobenius polynomials, then \\(\\rho_x\\) is automorphic after semisimplification. If \\(\\rho_x\\) is irreducible, it is itself isomorphic to the attached representation. This pointwise applicability condition does not require a uniform classicality assertion in other degrees.\n\\end{theorem}","\\begin{proof}\nFull support descends from the patched perfect complex by \\cref{thm:derived_support_descent}. Scalar transport through \\(e\\), followed by the actual derived isomorphism with the automorphic Hecke complex, gives full support for \\(C\\). Its fiber at \\(y\\) is therefore nonzero. The bounded model and induced Hecke action argument in \\cref{thm:derived_pointwise_realization} supplies a nonzero cohomology eigenvector in an integer degree within the model's bounds.","If realization applies to this eigensystem, it supplies the occurring automorphic object and a continuous semisimple attached representation with the declared Frobenius polynomials. The representation comparison in \\cref{thm:derived_pointwise_realization} then identifies the supplied semisimplification of \\(\\rho_x\\) with that attached representation. Only applicability to this eigensystem is used; a classicality theorem for every degree in a prescribed interval is unnecessary.\n\\end{proof}","\\begin{theorem}[Compatible system completion from one automorphic member]\\label{thm:system_completion}\nLet\n\\[\n\\{\\rho_\\lambda\\}_{\\lambda\\in\\Lambda}\n\\]\nbe a semisimple strictly compatible system with common Frobenius spine. Suppose the member at \\(\\lambda_0\\) admits an ordinary or derived completion datum and is identified with the Galois representation attached to an automorphic representation \\(\\pi\\). Assume that the compatible system attached to \\(\\pi\\) has the same Frobenius spine. Then\n\\[\n\\boxed{\n\\rho_\\lambda\n\\cong\n\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}\n\\quad\n\\text{for every }\\lambda\\in\\Lambda.\n}\n\\]\n\\end{theorem}","automorphic member]\\label{thm:system_completion}\nLet\n\\[\n\\{\\rho_\\lambda\\}_{\\lambda\\in\\Lambda}\n\\]\nbe a semisimple strictly compatible system with common Frobenius spine. Suppose the member at \\(\\lambda_0\\) admits an ordinary or derived completion datum and is identified with the Galois representation attached to an automorphic representation \\(\\pi\\). Assume that the compatible system attached to \\(\\pi\\) has the same Frobenius spine. Then\n\\[\n\\boxed{\n\\rho_\\lambda\n\\cong\n\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}\n\\quad\n\\text{for every }\\lambda\\in\\Lambda.\n}\n\\]\n\\end{theorem}","etion}\nLet\n\\[\n\\{\\rho_\\lambda\\}_{\\lambda\\in\\Lambda}\n\\]\nbe a semisimple strictly compatible system with common Frobenius spine. Suppose the member at \\(\\lambda_0\\) admits an ordinary or derived completion datum and is identified with the Galois representation attached to an automorphic representation \\(\\pi\\). Assume that the compatible system attached to \\(\\pi\\) has the same Frobenius spine. Then\n\\[\n\\boxed{\n\\rho_\\lambda\n\\cong\n\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}\n\\quad\n\\text{for every }\\lambda\\in\\Lambda.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nThe completion theorem at \\(\\lambda_0\\) gives the automorphic representation \\(\\pi\\). Apply \\cref{thm:one_member_propagation} to the two compatible systems.\n\\end{proof}","\\begin{theorem}[Five gate formulation]\\label{thm:five_gate_completion}\nFix a characteristic zero deformation point \\(x:R\\to E'\\). Give \\(E'\\) a Hausdorff topological field structure. The conditional argument uses the following five gates.\n\\begin{enumerate}\n\\item \\textbf{Residual gate.} Residual occupancy in the declared automorphic carrier and representability of the chosen deformation problem are supplied arithmetic inputs. In particular, the deformation ring \\(R\\) and its universal representation \\(\\rho_R\\) are available.\n\\item \\textbf{Patching gate.} An applicable patching theorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\it","\\begin{theorem}[Five gate formulation]\\label{thm:five_gate_completion}\nFix a characteristic zero deformation point \\(x:R\\to E'\\). Give \\(E'\\) a Hausdorff topological field structure. The conditional argument uses the following five gates.\n\\begin{enumerate}\n\\item \\textbf{Residual gate.} Residual occupancy in the declared automorphic carrier and representability of the chosen deformation problem are supplied arithmetic inputs. In particular, the deformation ring \\(R\\) and its universal representation \\(\\rho_R\\) are available.\n\\item \\textbf{Patching gate.} An applicable patching theorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\item \\textbf{Co","on}\nFix a characteristic zero deformation point \\(x:R\\to E'\\). Give \\(E'\\) a Hausdorff topological field structure. The conditional argument uses the following five gates.\n\\begin{enumerate}\n\\item \\textbf{Residual gate.} Residual occupancy in the declared automorphic carrier and representability of the chosen deformation problem are supplied arithmetic inputs. In particular, the deformation ring \\(R\\) and its universal representation \\(\\rho_R\\) are available.\n\\item \\textbf{Patching gate.} An applicable patching theorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\item \\textbf{Comparison gate.} Let \\(H\\) be a finite ordinary Hecke module or a perfect Hecke complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does trans…","on point \\(x:R\\to E'\\). Give \\(E'\\) a Hausdorff topological field structure. The conditional argument uses the following five gates.\n\\begin{enumerate}\n\\item \\textbf{Residual gate.} Residual occupancy in the declared automorphic carrier and representability of the chosen deformation problem are supplied arithmetic inputs. In particular, the deformation ring \\(R\\) and its universal representation \\(\\rho_R\\) are available.\n\\item \\textbf{Patching gate.} An applicable patching theorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\item \\textbf{Comparison gate.} Let \\(H\\) be a finite ordinary Hecke module or a perfect Hecke complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with th…","e chosen deformation problem are supplied arithmetic inputs. In particular, the deformation ring \\(R\\) and its universal representation \\(\\rho_R\\) are available.\n\\item \\textbf{Patching gate.} An applicable patching theorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\item \\textbf{Comparison gate.} Let \\(H\\) be a finite ordinary Hecke module or a perfect Hecke complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived","ic inputs. In particular, the deformation ring \\(R\\) and its universal representation \\(\\rho_R\\) are available.\n\\item \\textbf{Patching gate.} An applicable patching theorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\item \\textbf{Comparison gate.} Let \\(H\\) be a finite ordinary Hecke module or a perfect Hecke complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realiza…","ing \\(R\\) and its universal representation \\(\\rho_R\\) are available.\n\\item \\textbf{Patching gate.} An applicable patching theorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\item \\textbf{Comparison gate.} Let \\(H\\) be a finite ordinary Hecke module or a perfect Hecke complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theor…","em \\textbf{Patching gate.} An applicable patching theorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\item \\textbf{Comparison gate.} Let \\(H\\) be a finite ordinary Hecke module or a perfect Hecke complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classi…","An applicable patching theorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\item \\textbf{Comparison gate.} Let \\(H\\) be a finite ordinary Hecke module or a perfect Hecke complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representat…","eorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write \\(M\\) for the resulting finite level carrier over \\(R\\).\n\\item \\textbf{Comparison gate.} Let \\(H\\) be a finite ordinary Hecke module or a perfect Hecke complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a conti…","le or a perfect Hecke complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a continuous semisimple attached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard…","complex over \\(\\TT\\). Supply a ring map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a continuous semisimple attached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general B…","ng map \\(f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a continuous semisimple attached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fac…","f:R\\to\\TT\\) satisfying\n\\[\n(f^*)^{-1}\\bigl(\\Supp_R(M)\\bigr)\\subseteq\\Supp_{\\TT}(H).\n\\]\nUse ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a continuous semisimple attached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebota…","he carrier. Actual scalar extension satisfies this inclusion, as does transport through a ring isomorphism with the corresponding carrier identification.\n\\item \\textbf{Fiber gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a continuous semisimple attached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{enumerate}\nThen \\(\\rho_x^{\\mathrm{ss}}\\) is isomorphic to the representation attached to \\(\\pi\\). If \\(\\rh…","gate.} Supply a Hecke point \\(y:\\TT\\to E'\\) with \\(y\\circ f=x\\). Then the fiber of \\(H\\) at \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a continuous semisimple attached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{enumerate}\nThen \\(\\rho_x^{\\mathrm{ss}}\\) is isomorphic to the representation attached to \\(\\pi\\). If \\(\\rho_x\\) is irreducible, the same conclusion holds for \\(\\rho_x\\) itself.","t \\(y\\) is nonzero. It contains a nonzero Hecke eigenvector, in some integer cohomological degree in the derived case, with character \\(y\\).\n\\item \\textbf{Realization gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a continuous semisimple attached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{enumerate}\nThen \\(\\rho_x^{\\mathrm{ss}}\\) is isomorphic to the representation attached to \\(\\pi\\). If \\(\\rho_x\\) is irreducible, the same conclusion holds for \\(\\rho_x\\) itself.","ation gate.} The declared classicality theorem applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a continuous semisimple attached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{enumerate}\nThen \\(\\rho_x^{\\mathrm{ss}}\\) is isomorphic to the representation attached to \\(\\pi\\). If \\(\\rho_x\\) is irreducible, the same conclusion holds for \\(\\rho_x\\) itself.","applies to the resulting eigensystem and supplies an occurring classical automorphic representation \\(\\pi\\), with a continuous semisimple attached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{enumerate}\nThen \\(\\rho_x^{\\mathrm{ss}}\\) is isomorphic to the representation attached to \\(\\pi\\). If \\(\\rho_x\\) is irreducible, the same conclusion holds for \\(\\rho_x\\) itself.","tached representation. Require the specialization \\(\\rho_x=x\\circ\\rho_R\\) to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{enumerate}\nThen \\(\\rho_x^{\\mathrm{ss}}\\) is isomorphic to the representation attached to \\(\\pi\\). If \\(\\rho_x\\) is irreducible, the same conclusion holds for \\(\\rho_x\\) itself.","to be continuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{enumerate}\nThen \\(\\rho_x^{\\mathrm{ss}}\\) is isomorphic to the representation attached to \\(\\pi\\). If \\(\\rho_x\\) is irreducible, the same conclusion holds for \\(\\rho_x\\) itself.","nuous and supply its semisimplification. On the same Chebotarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{enumerate}\nThen \\(\\rho_x^{\\mathrm{ss}}\\) is isomorphic to the representation attached to \\(\\pi\\). If \\(\\rho_x\\) is irreducible, the same conclusion holds for \\(\\rho_x\\) itself.","otarev dense set, assume that the characteristic polynomials of both \\(\\rho_x\\) and the attached representation are \\(y(P_v(X))\\). Retain the standard density and general Brauer Nesbitt inputs of \\cref{fact:chebotarev_brauer_nesbitt}.\n\\end{enumerate}\nThen \\(\\rho_x^{\\mathrm{ss}}\\) is isomorphic to the representation attached to \\(\\pi\\). If \\(\\rho_x\\) is irreducible, the same conclusion holds for \\(\\rho_x\\) itself.","For propagation, suppose both a semisimple compatible family \\(\\{\\rho_\\lambda\\}\\) containing \\(\\rho_x\\) and a family \\(\\{\\rho_{\\pi,\\lambda}\\}\\) attached to this same \\(\\pi\\) already exist. Require the distinguished members to be \\(\\rho_x\\) and its attached representation, respectively. Assume the coefficient, continuity, density, semisimplification and common algebraic Frobenius spine hypotheses of \\cref{thm:one_member_propagation}, with that spine agreeing with the specialized Hecke polynomials at the distinguished place. Then\n\\[\n\\rho_\\lambda\\cong\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}\n\\qquad\\text{for every }\\lambda.\n\\]\n\\end{theorem}","da}\\}\\) attached to this same \\(\\pi\\) already exist. Require the distinguished members to be \\(\\rho_x\\) and its attached representation, respectively. Assume the coefficient, continuity, density, semisimplification and common algebraic Frobenius spine hypotheses of \\cref{thm:one_member_propagation}, with that spine agreeing with the specialized Hecke polynomials at the distinguished place. Then\n\\[\n\\rho_\\lambda\\cong\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}\n\\qquad\\text{for every }\\lambda.\n\\]\n\\end{theorem}","The comparison inclusion transports this full support to \\(H\\). More specifically, \\(y\\circ f=x\\) identifies the contraction of \\(\\ker y\\) with \\(\\ker x\\), so support at the source point gives support at the Hecke point. In the ordinary case, \\cref{thm:field_fiber_criterion} gives a nonzero actual tensor fiber. In the derived case, the support criterion gives a nonzero actual derived fiber and hence a nonzero cohomology vector. Its integer degree lies within the bounds of the chosen finite projective model. In both cases, multiplication on the original carrier induces the scalar \\(y(t)\\) on the specialized vector for every \\(t\\in\\TT\\).","ives support at the Hecke point. In the ordinary case, \\cref{thm:field_fiber_criterion} gives a nonzero actual tensor fiber. In the derived case, the support criterion gives a nonzero actual derived fiber and hence a nonzero cohomology vector. Its integer degree lies within the bounds of the chosen finite projective model. In both cases, multiplication on the original carrier induces the scalar \\(y(t)\\) on the specialized vector for every \\(t\\in\\TT\\).","ary case, \\cref{thm:field_fiber_criterion} gives a nonzero actual tensor fiber. In the derived case, the support criterion gives a nonzero actual derived fiber and hence a nonzero cohomology vector. Its integer degree lies within the bounds of the chosen finite projective model. In both cases, multiplication on the original carrier induces the scalar \\(y(t)\\) on the specialized vector for every \\(t\\in\\TT\\).","Finally, retain the same \\(\\pi\\) in its attached family. The distinguished member identities connect both families to the pointwise construction, while the common algebraic spine gives the required Frobenius polynomials at every coefficient place. Apply \\cref{thm:one_member_propagation} to these two existing families. The construction does not produce a missing companion or an attached compatible family.\n\\end{proof}","An isomorphism \\(R\\cong\\TT\\) transports support and Fitting data & An \\(R\\cong\\TT\\) theorem outside published hypotheses","The preserved release records complete mapped Lean statements for all 49 internal numbered claims of its revised manuscript. The present traditional edition retains those proof bodies, exposes the local-map and coefficient-target hypotheses in \\cref{prop:global_local_basechange}, makes the \\(r>n\\) convention explicit in the proof of \\cref{thm:higher_fitting_strata}, states all the pointwise comparison inputs in \\cref{def:ordinary_completion_datum}, and declares a finite ramification set and coherent geometric Frobenius normalization. These editorial clarifications are recorded separately from the inherited eight amendments; they are not counted as newly compiled Lean certificates. No new statement amendment was needed for the completion and tensor results. The eight inherited amendments remain explicitly separate from the original wording, and the cited arithmetic inputs remain outside the internal claim census. Formal proof coverage is recorded claim by claim in the roadmap. These","The new checks for \\cref{thm:residual_augmentation_algebra,cor:no_residual_character_splitting,cor:rank_trace_detection,prop:dyadic_sign_projectors} construct the full multivariable truncated polynomial algebra isomorphism and identify its augmentation generators. The residual group algebra is local for abelian \\(p\\) groups in characteristic \\(p\\). For each supplied Hecke idempotent, the trace is the image dimension cast into the coefficient field; occupancy is equivalent to positive rank and, in characteristic zero, nonzero trace. The dyadic calculation gives the two rational sign projectors, proves that neither lies in the integral group algebra, and identifies the residual algebra with \\(\\FF_2[X]/(X^2)\\).","for abelian \\(p\\) groups in characteristic \\(p\\). For each supplied Hecke idempotent, the trace is the image dimension cast into the coefficient field; occupancy is equivalent to positive rank and, in characteristic zero, nonzero trace. The dyadic calculation gives the two rational sign projectors, proves that neither lies in the integral group algebra, and identifies the residual algebra with \\(\\FF_2[X]/(X^2)\\).","Corollary~\\ref{cor:no_residual_character_splitting} is now checked for every finite \\(p\\) group, including nonabelian groups, over every field of characteristic \\(p\\). The proof uses a finite prime field orbit span and a nonzero fixed vector. The original unrestricted statement is restored, and the earlier explicit abelian restriction is no longer needed.","The checks for \\cref{prop:quotient_erases_component,thm:canonical_hecke_packets,cor:simultaneous_generalized_eigenspaces} now include the actual formal power series quotient and the canonical Hecke packet maps. In the node \\(k[[x,y]]/(xy)\\), quotienting by \\((x)\\) gives \\(k[[y]]\\), with support \\(V(x)\\); Lean verifies both irreducible components and the exclusion of the generic point \\((y)\\). The components can still meet, so this is not a claim that their supports are disjoint.","For the Hecke results, the canonical ring decomposition in \\cref{fact:artin_decomposition} remains a declared external input. Its coordinates are the actual localization maps. The module direct sum, the packet identification with the localized module, and the formula \\(v\\mapsto v/1\\) are constructed in Lean. Nilpotence in the selected Artin local factor, together with invertibility of a generator on each other factor, gives a single positive exponent for the simultaneous generalized eigenspace. The result includes the explicit generators \\(T_i-\\lambda_i\\), agrees with the packet used in the trace theorem, and does not assume semisimplicity.","the actual localization maps. The module direct sum, the packet identification with the localized module, and the formula \\(v\\mapsto v/1\\) are constructed in Lean. Nilpotence in the selected Artin local factor, together with invertibility of a generator on each other factor, gives a single positive exponent for the simultaneous generalized eigenspace. The result includes the explicit generators \\(T_i-\\lambda_i\\), agrees with the packet used in the trace theorem, and does not assume semisimplicity.","The original comparison wording did not explicitly supply a compatible Hecke point over the fixed coefficient field. Support preservation alone is insufficient: extension from \\(\\mathbb Q\\) to \\(\\mathbb Q(i)\\) preserves full support, but there is no ring map \\(\\mathbb Q(i)\\to\\mathbb Q\\) extending the identity. The revision therefore states \\(y\\circ f=x\\) and exposes the arithmetic comparison and existing-family inputs. Residual occupancy, representability and the applicable patching theorem remain supplied arithmetic provenance. The original full statement remains partial. The completed tensor quotients and intrinsic Fitting ideals are now checked as described above, including the full Fitting transport clause.","ted. Perfect complexes are actual derived category objects equipped with bounded finite projective models and derived isomorphisms to those models. Scalar extension is constructed on cochain complexes. Its associativity, preservation of homotopy equivalences, and independence under actual derived isomorphisms are checked. The quotient support formula uses the actual map of spectra and its image \\(V(I)\\). Every field valued point factors through the residue field at its kernel, and faithful flatness of field extensions preserves and reflects acyclicity. The surviving integer degree is expressed using the actual derived cohomology functor, with a corresponding witness in the specialized cochain model. These proofs do not require the external cohomological support description in \\cref{fact:derived_support_cohomology}. The external classicality theorems remain supplied arithmetic inputs; the conditional derived realization argument is checked as described above.","The adequacy checks prove \\cref{thm:s3_adequacy} as originally stated. They construct the faithful standard representation from the actual permutation module, prove irreducibility after extension to the algebraic closure, and split every group element's characteristic polynomial over \\(\\FF_7\\). The trace zero adjoint is the kernel of matrix trace with its conjugation action, and its sign plus standard decomposition is an actual equivariant linear isomorphism. The averaging proof is connected to Mathlib's actual first group cohomology, while the invariant calculation gives actual degree zero vanishing. Canonical projectors are identified with generalized eigenspace projectors for every positive exponent. Their trace pairings detect every nonzero endomorphism and hence every irreducible adjoint submodule. The two printed constituent witnesses and their traces are checked exactly. The adequacy definition remains a cited convention; no external adequacy theorem is assumed in this f","\\node[anchor=west,font=\\sffamily\\bfseries] at (0,2.8) {sign constituent};\n\\node[anchor=west] at (0,1.8) {$W=\\begin{pmatrix}2&3\\\\4&5\\end{pmatrix}$};\n\\node[anchor=west] at (4,1.8) {$P_{t,2}=\\begin{pmatrix}2&4","4&5\\end{pmatrix}$};\n\\node[anchor=west] at (4,1.8) {$P_{t,2}=\\begin{pmatrix}2&4\\\\3&6\\end{pmatrix}$};\n\\node[anchor=west] at (8.2,1.8) {$\\operatorname{Tr}(PW)=3$};\n\\draw[black!30] (0,.9)--(11.3,.9);\n\\node[anchor=west,font=\\sffamily\\bfseries] at (0,.3) {standard constituent};\n\\node[anchor=west] at (0,-.7) {$W=\\begin{pmatrix}6&4","3&6\\end{pmatrix}$};\n\\node[anchor=west] at (8.2,1.8) {$\\operatorname{Tr}(PW)=3$};\n\\draw[black!30] (0,.9)--(11.3,.9);\n\\node[anchor=west,font=\\sffamily\\bfseries] at (0,.3) {standard constituent};\n\\node[anchor=west] at (0,-.7) {$W=\\begin{pmatrix}6&4","3&6\\end{pmatrix}$};\n\\node[anchor=west] at (8.2,1.8) {$\\operatorname{Tr}(PW)=3$};\n\\draw[black!30] (0,.9)--(11.3,.9);\n\\node[anchor=west,font=\\sffamily\\bfseries] at (0,.3) {standard constituent};\n\\node[anchor=west] at (0,-.7) {$W=\\begin{pmatrix}6&4\\\\3&1\\end{pmatrix}$};\n\\node[anchor=west] at (4,-.7) {$P_{s,1}=\\begin{pmatrix}0&4","3&1\\end{pmatrix}$};\n\\node[anchor=west] at (4,-.7) {$P_{s,1}=\\begin{pmatrix}0&4\\\\0&1\\end{pmatrix}$};\n\\node[anchor=west] at (8.2,-.7) {$\\operatorname{Tr}(PW)=6$};","0&1\\end{pmatrix}$};\n\\node[anchor=west] at (8.2,-.7) {$\\operatorname{Tr}(PW)=6$};","\\begin{tikzpicture}[x=1cm,y=1cm,font=\\small,>=Stealth]\n\\node[anchor=east] at (-.18,1.1) {$\\alpha_j$};\n\\draw[black!35,fill=black!8] (0.0,0.76) rectangle ++(.80,.67);\n\\node at (0.4,1.095) {1};\n\\draw[black!35,fill=black!12] (0.88,0.76) rectangle ++(.80,.67);\n\\node at (1.28,1.095) {2};\n\\draw[black!35,fill=black!4] (1.76,0.76) rectangle ++(.80,.67);\n\\node at (2.16,1.095) {0};\n\\draw[black!35,fill=black!4] (2.64,0.76) rectangle 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{$c$};\n\\end{tikzpicture}","\\node (B) at (0,2.2) {$B$};\\node (O) at (6.5,2.2) {$\\mathbb Z_5$};\n\\node (BJ) at (0,0) {$B/J$};\\node (On) at (6.5,0) {$\\mathbb Z/5^n\\mathbb Z$};\n\\draw[->] (B)--(O) node[midway,above] {evaluation at $A$};\n\\draw[->] (B)--(BJ) node[midway,left] {quotient};\n\\draw[->] (O)--(On) node[midway,right] {reduction};\n\\draw[->] (BJ)--(On) node[midway,below] {evaluation at $A_n$};\n\\draw[->,dashed] (BJ)--(O) node[pos=.4,above,sloped,font=\\footnotesize] {unique factor};","\\begin{tikzpicture}[x=1cm,y=1cm,font=\\small,>=Stealth]\n\\draw[->,black!30] (-2.65,0)--(2.7,0);\\draw[->,black!30] (0,-2.45)--(0,2.55);\n\\draw[line width=1.4pt] (-2.4,0)--(2.4,0);\\draw[line width=1.4pt] (0,-2.2)--(0,2.2);\n\\fill (0,0) circle (.085);\\node[anchor=west] at (.18,.35) {$\\mu=2$};\n\\fill (1.65,0) circle (.065);\\node[anchor=north] at (1.65,-.15) {$\\mu=1$};\n\\draw[fill=white] (1.25,1.3) circle (.065);\\node[anchor=west] at (1.4,1.3) {$\\mu=0$};\n\\node[anchor=west,font=\\footnotesize] at (.15,-1.7) {$V(x)$};\\node[anchor=south,font=\\footnotesize] at (-1.6,.1) {$V(y)$};\n\\node[anchor=west] at (2.72,0) {$x$};\\node[anchor=south] at (0,2.57) {$y$};\n\\end{tikzpicture}","mall,>=Stealth]\n\\draw[->,black!30] (-2.65,0)--(2.7,0);\\draw[->,black!30] (0,-2.45)--(0,2.55);\n\\draw[line width=1.4pt] (-2.4,0)--(2.4,0);\\draw[line width=1.4pt] (0,-2.2)--(0,2.2);\n\\fill (0,0) circle (.085);\\node[anchor=west] at (.18,.35) {$\\mu=2$};\n\\fill (1.65,0) circle (.065);\\node[anchor=north] at (1.65,-.15) {$\\mu=1$};\n\\draw[fill=white] (1.25,1.3) circle (.065);\\node[anchor=west] at (1.4,1.3) {$\\mu=0$};\n\\node[anchor=west,font=\\footnotesize] at (.15,-1.7) {$V(x)$};\\node[anchor=south,font=\\footnotesize] at (-1.6,.1) {$V(y)$};\n\\node[anchor=west] at (2.72,0) {$x$};\\node[anchor=south] at (0,2.57) {$y$};\n\\end{tikzpicture}","black!30] (0,-2.45)--(0,2.55);\n\\draw[line width=1.4pt] (-2.4,0)--(2.4,0);\\draw[line width=1.4pt] (0,-2.2)--(0,2.2);\n\\fill (0,0) circle (.085);\\node[anchor=west] at (.18,.35) {$\\mu=2$};\n\\fill (1.65,0) circle (.065);\\node[anchor=north] at (1.65,-.15) {$\\mu=1$};\n\\draw[fill=white] (1.25,1.3) circle (.065);\\node[anchor=west] at (1.4,1.3) {$\\mu=0$};\n\\node[anchor=west,font=\\footnotesize] at (.15,-1.7) {$V(x)$};\\node[anchor=south,font=\\footnotesize] at (-1.6,.1) {$V(y)$};\n\\node[anchor=west] at (2.72,0) {$x$};\\node[anchor=south] at (0,2.57) {$y$};\n\\end{tikzpicture}","0)--(2.4,0);\\draw[line width=1.4pt] (0,-2.2)--(0,2.2);\n\\fill (0,0) circle (.085);\\node[anchor=west] at (.18,.35) {$\\mu=2$};\n\\fill (1.65,0) circle (.065);\\node[anchor=north] at (1.65,-.15) {$\\mu=1$};\n\\draw[fill=white] (1.25,1.3) circle (.065);\\node[anchor=west] at (1.4,1.3) {$\\mu=0$};\n\\node[anchor=west,font=\\footnotesize] at (.15,-1.7) {$V(x)$};\\node[anchor=south,font=\\footnotesize] at (-1.6,.1) {$V(y)$};\n\\node[anchor=west] at (2.72,0) {$x$};\\node[anchor=south] at (0,2.57) {$y$};\n\\end{tikzpicture}",".2)--(0,2.2);\n\\fill (0,0) circle (.085);\\node[anchor=west] at (.18,.35) {$\\mu=2$};\n\\fill (1.65,0) circle (.065);\\node[anchor=north] at (1.65,-.15) {$\\mu=1$};\n\\draw[fill=white] (1.25,1.3) circle (.065);\\node[anchor=west] at (1.4,1.3) {$\\mu=0$};\n\\node[anchor=west,font=\\footnotesize] at (.15,-1.7) {$V(x)$};\\node[anchor=south,font=\\footnotesize] at (-1.6,.1) {$V(y)$};\n\\node[anchor=west] at (2.72,0) {$x$};\\node[anchor=south] at (0,2.57) {$y$};\n\\end{tikzpicture}",");\\node[anchor=west] at (.18,.35) {$\\mu=2$};\n\\fill (1.65,0) circle (.065);\\node[anchor=north] at (1.65,-.15) {$\\mu=1$};\n\\draw[fill=white] (1.25,1.3) circle (.065);\\node[anchor=west] at (1.4,1.3) {$\\mu=0$};\n\\node[anchor=west,font=\\footnotesize] at (.15,-1.7) {$V(x)$};\\node[anchor=south,font=\\footnotesize] at (-1.6,.1) {$V(y)$};\n\\node[anchor=west] at (2.72,0) {$x$};\\node[anchor=south] at (0,2.57) {$y$};\n\\end{tikzpicture}","\\draw[->,black!35] (-2.5,0)--(2.5,0);\n\\draw[->,black!35] (0,-2.3)--(0,2.3) node[above] {second coordinate};\n\\draw[->,line width=1.2pt] (0,0)--(1.5,1.5);\n\\draw[->,line width=1.2pt] (0,0)--(1.5,-1.5);\n\\node[anchor=south west] at (1.5,1.5) {$(1,1)$};\n\\node[anchor=north west] at (1.5,-1.5) {$(1,-1)$};\n\\node[anchor=west] at (3.3,.7) {$m_1+m_2=5$};\n\\node[anchor=west] at (3.3,-.2) {$m_1-m_2=1$};\n\\node[anchor=west,font=\\sffamily\\bfseries] at (3.3,-1.3) {solution: $m_1=3$, $m_2=2$};","2.3)--(0,2.3) node[above] {second coordinate};\n\\draw[->,line width=1.2pt] (0,0)--(1.5,1.5);\n\\draw[->,line width=1.2pt] (0,0)--(1.5,-1.5);\n\\node[anchor=south west] at (1.5,1.5) {$(1,1)$};\n\\node[anchor=north west] at (1.5,-1.5) {$(1,-1)$};\n\\node[anchor=west] at (3.3,.7) {$m_1+m_2=5$};\n\\node[anchor=west] at (3.3,-.2) {$m_1-m_2=1$};\n\\node[anchor=west,font=\\sffamily\\bfseries] at (3.3,-1.3) {solution: $m_1=3$, $m_2=2$};","2.3) node[above] {second coordinate};\n\\draw[->,line width=1.2pt] (0,0)--(1.5,1.5);\n\\draw[->,line width=1.2pt] (0,0)--(1.5,-1.5);\n\\node[anchor=south west] at (1.5,1.5) {$(1,1)$};\n\\node[anchor=north west] at (1.5,-1.5) {$(1,-1)$};\n\\node[anchor=west] at (3.3,.7) {$m_1+m_2=5$};\n\\node[anchor=west] at (3.3,-.2) {$m_1-m_2=1$};\n\\node[anchor=west,font=\\sffamily\\bfseries] at (3.3,-1.3) {solution: $m_1=3$, $m_2=2$};","[3pt]\n\\textbf{Support} & At which prime ideals does the supplied module or complex survive?\\\\[3pt]\n\\textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation?\n\\end{tabularx}\n\\end{insight}\n\n\\head{Three models, rather than one overloaded analogy}\n\\begin{tabularx}{\\linewidth}{@{}p{.20\\linewidth}Y>{\\raggedright\\arraybackslash}p{.15\\linewidth}@{}}\\toprule\n\\textbf{Model} & \\textbf{What it lets you see} & \\textbf{Lessons}\\\\\\midrule\nA formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \\pagerefV{3}--\\pagerefV{5}\\\\\nA presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \\pagerefV{6}--\\pagerefV{8}\\\\\nA finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \\pagerefV{12}--\\pagerefV{14}\\\\\\bottomrule\n\\end{tabularx}\n\nThe models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a c…","\\head{Why arrows reverse when points appear}\nA ring map $f:R\\to T$ sends a prime of $T$ to its inverse image in $R$:\n\\[\nf^*:\\Spec T\\longrightarrow\\Spec R,\\qquad \\mathfrak q\\longmapsto f^{-1}(\\mathfrak q).\n\\]\nSimilarly, a point $y:T\\to E'$ gives the point $y\\circ f:R\\to E'$. The equality $y\\circ f=x$ is an actual requirement in the final theorem, not a consequence of drawing a connecting arrow.","\\head{Two kinds of point}\nLocal quotient factorization is stated for continuous local coefficient-algebra maps to deformation test rings. The later field-valued point $x:R\\to E'$ uses its kernel and scalar fiber. A characteristic-zero field point is generally not a local ring map: the coefficient uniformizer becomes a unit. These two uses of ``point'' must retain their own categories.","% ---------------------------------------------------------------------------\n\\vpage{3}{Local conditions remove directions}{MODEL A / A FORMAL THICKENING}{Main text: Theorem 3.1 and Corollary 3.2\\quad \\href{p12-route:visual-3}{Main statement 3.1}}\nFix $\\OO=\\ZZ_5$. Take two independent local variables and impose two different kinds of condition:\n\\[\nR_1=\\OO[[x]],\\quad I_1=(x^2),\\qquad\nR_2=\\OO[[y]],\\quad I_2=(y).\n\\]\nTheir completed local tensor is $A=\\OO[[x,y]]$. The closed generated ideal is $J=(x^2,y)$, and both routes through Theorem 3.1 give\n\\[\n(R_1/I_1)\\whot_{\\OO}(R_2/I_2)\n\\cong A/J\n\\cong \\OO[\\varepsilon]/(\\varepsilon^2).\n\\]","---------------------------\n\\vpage{3}{Local conditions remove directions}{MODEL A / A FORMAL THICKENING}{Main text: Theorem 3.1 and Corollary 3.2\\quad \\href{p12-route:visual-3}{Main statement 3.1}}\nFix $\\OO=\\ZZ_5$. Take two independent local variables and impose two different kinds of condition:\n\\[\nR_1=\\OO[[x]],\\quad I_1=(x^2),\\qquad\nR_2=\\OO[[y]],\\quad I_2=(y).\n\\]\nTheir completed local tensor is $A=\\OO[[x,y]]$. The closed generated ideal is $J=(x^2,y)$, and both routes through Theorem 3.1 give\n\\[\n(R_1/I_1)\\whot_{\\OO}(R_2/I_2)\n\\cong A/J\n\\cong \\OO[\\varepsilon]/(\\varepsilon^2).\n\\]","nditions remove directions}{MODEL A / A FORMAL THICKENING}{Main text: Theorem 3.1 and Corollary 3.2\\quad \\href{p12-route:visual-3}{Main statement 3.1}}\nFix $\\OO=\\ZZ_5$. Take two independent local variables and impose two different kinds of condition:\n\\[\nR_1=\\OO[[x]],\\quad I_1=(x^2),\\qquad\nR_2=\\OO[[y]],\\quad I_2=(y).\n\\]\nTheir completed local tensor is $A=\\OO[[x,y]]$. The closed generated ideal is $J=(x^2,y)$, and both routes through Theorem 3.1 give\n\\[\n(R_1/I_1)\\whot_{\\OO}(R_2/I_2)\n\\cong A/J\n\\cong \\OO[\\varepsilon]/(\\varepsilon^2).\n\\]","NG}{Main text: Theorem 3.1 and Corollary 3.2\\quad \\href{p12-route:visual-3}{Main statement 3.1}}\nFix $\\OO=\\ZZ_5$. Take two independent local variables and impose two different kinds of condition:\n\\[\nR_1=\\OO[[x]],\\quad I_1=(x^2),\\qquad\nR_2=\\OO[[y]],\\quad I_2=(y).\n\\]\nTheir completed local tensor is $A=\\OO[[x,y]]$. The closed generated ideal is $J=(x^2,y)$, and both routes through Theorem 3.1 give\n\\[\n(R_1/I_1)\\whot_{\\OO}(R_2/I_2)\n\\cong A/J\n\\cong \\OO[\\varepsilon]/(\\varepsilon^2).\n\\]","\\vfig[.96]{monomial_quotient}\n\\captiontext{An exact finite window of monomials over $\\OO$. Crosses mean that the monomial is zero in the quotient; they do not represent a small numerical coefficient. All terms outside the window satisfy the same divisibility rule.}","\\head{The calculation behind the image}\nFor $f=\\sum_{i,j\\ge0}a_{ij}x^iy^j$, reduction gives\n\\[\nf\\longmapsto a_{00}+a_{10}\\varepsilon.\n\\]\nEvery term involving $y$ vanishes. Every term involving $x^2$ vanishes. The class of $x$ does \\emph{not} vanish: it becomes a nonzero element whose square is zero. The quotient is a free $\\OO$-module with basis $1,\\varepsilon$.","\\head{Why a point picture alone would miss this}\nAfter passing to the special fiber, $k[\\varepsilon]/(\\varepsilon^2)$ has the same underlying one-point spectrum as $k$. The extra direction is visible in the algebra and in maps from test rings, not as a second point. It is also not a second component or a second Artin packet.","\\takeaway{The condition $y=0$ deletes a variable. The condition $x^2=0$ retains a first-order nilpotent direction. Both are quotient conditions, but their effects are different.}","% ---------------------------------------------------------------------------\n\\vpage{4}{A point is a test of the equations}{MODEL A / FACTORIZATION AND OBSTRUCTION}{Main text: Proposition 3.3; this explicit example also exhibits a failed infinitesimal lift\\quad \\href{p12-route:visual-4}{Main statement 3.3}}\nLet $R^\\square=\\OO[[u,v]]$ and send $x\\mapsto u$, $y\\mapsto v-u$. The local conditions become\n\\[\nu^2=0,\\qquad v-u=0,\n\\quad\\text{so}\\quad\nR^\\tau=\\OO[[u,v]]/(u^2,v-u)\\cong\\OO[\\varepsilon]/(\\varepsilon^2).\n\\]\nProbe this ring using $D_3=\\FF_5[\\delta]/(\\delta^3)$, with the fixed coefficient map $\\ZZ_5\\to\\FF_5$. A continuous local point is specified by\n\\[\nu\\longmapsto a\\delta+b\\delta^2,\\qquad\nv\\longmapsto c\\delta+d\\delta^2,\n\\qquad a,b,c,d\\in\\FF_5.\n\\]\nIt factors through $R^\\tau$ precisely when\n\\[\na^2=0,\\quad c=a,\\quad d=b\n\\quad\\Longleftrightarrow\\quad a=c=0,\\ d=b.\n\\]","------------------------------------------------------------------------\n\\vpage{4}{A point is a test of the equations}{MODEL A / FACTORIZATION AND OBSTRUCTION}{Main text: Proposition 3.3; this explicit example also exhibits a failed infinitesimal lift\\quad \\href{p12-route:visual-4}{Main statement 3.3}}\nLet $R^\\square=\\OO[[u,v]]$ and send $x\\mapsto u$, $y\\mapsto v-u$. The local conditions become\n\\[\nu^2=0,\\qquad v-u=0,\n\\quad\\text{so}\\quad\nR^\\tau=\\OO[[u,v]]/(u^2,v-u)\\cong\\OO[\\varepsilon]/(\\varepsilon^2).\n\\]\nProbe this ring using $D_3=\\FF_5[\\delta]/(\\delta^3)$, with the fixed coefficient map $\\ZZ_5\\to\\FF_5$. A continuous local point is specified by\n\\[\nu\\longmapsto a\\delta+b\\delta^2,\\qquad\nv\\longmapsto c\\delta+d\\delta^2,\n\\qquad a,b,c,d\\in\\FF_5.\n\\]\nIt factors through $R^\\tau$ precisely when\n\\[\na^2=0,\\quad c=a,\\quad d=b\n\\quad\\Longleftrightarrow\\quad a=c=0,\\ d=b.\n\\]","ORIZATION AND OBSTRUCTION}{Main text: Proposition 3.3; this explicit example also exhibits a failed infinitesimal lift\\quad \\href{p12-route:visual-4}{Main statement 3.3}}\nLet $R^\\square=\\OO[[u,v]]$ and send $x\\mapsto u$, $y\\mapsto v-u$. The local conditions become\n\\[\nu^2=0,\\qquad v-u=0,\n\\quad\\text{so}\\quad\nR^\\tau=\\OO[[u,v]]/(u^2,v-u)\\cong\\OO[\\varepsilon]/(\\varepsilon^2).\n\\]\nProbe this ring using $D_3=\\FF_5[\\delta]/(\\delta^3)$, with the fixed coefficient map $\\ZZ_5\\to\\FF_5$. A continuous local point is specified by\n\\[\nu\\longmapsto a\\delta+b\\delta^2,\\qquad\nv\\longmapsto c\\delta+d\\delta^2,\n\\qquad a,b,c,d\\in\\FF_5.\n\\]\nIt factors through $R^\\tau$ precisely when\n\\[\na^2=0,\\quad c=a,\\quad d=b\n\\quad\\Longleftrightarrow\\quad a=c=0,\\ d=b.\n\\]","licit example also exhibits a failed infinitesimal lift\\quad \\href{p12-route:visual-4}{Main statement 3.3}}\nLet $R^\\square=\\OO[[u,v]]$ and send $x\\mapsto u$, $y\\mapsto v-u$. The local conditions become\n\\[\nu^2=0,\\qquad v-u=0,\n\\quad\\text{so}\\quad\nR^\\tau=\\OO[[u,v]]/(u^2,v-u)\\cong\\OO[\\varepsilon]/(\\varepsilon^2).\n\\]\nProbe this ring using $D_3=\\FF_5[\\delta]/(\\delta^3)$, with the fixed coefficient map $\\ZZ_5\\to\\FF_5$. A continuous local point is specified by\n\\[\nu\\longmapsto a\\delta+b\\delta^2,\\qquad\nv\\longmapsto c\\delta+d\\delta^2,\n\\qquad a,b,c,d\\in\\FF_5.\n\\]\nIt factors through $R^\\tau$ precisely when\n\\[\na^2=0,\\quad c=a,\\quad d=b\n\\quad\\Longleftrightarrow\\quad a=c=0,\\ d=b.\n\\]","d \\href{p12-route:visual-4}{Main statement 3.3}}\nLet $R^\\square=\\OO[[u,v]]$ and send $x\\mapsto u$, $y\\mapsto v-u$. The local conditions become\n\\[\nu^2=0,\\qquad v-u=0,\n\\quad\\text{so}\\quad\nR^\\tau=\\OO[[u,v]]/(u^2,v-u)\\cong\\OO[\\varepsilon]/(\\varepsilon^2).\n\\]\nProbe this ring using $D_3=\\FF_5[\\delta]/(\\delta^3)$, with the fixed coefficient map $\\ZZ_5\\to\\FF_5$. A continuous local point is specified by\n\\[\nu\\longmapsto a\\delta+b\\delta^2,\\qquad\nv\\longmapsto c\\delta+d\\delta^2,\n\\qquad a,b,c,d\\in\\FF_5.\n\\]\nIt factors through $R^\\tau$ precisely when\n\\[\na^2=0,\\quad c=a,\\quad d=b\n\\quad\\Longleftrightarrow\\quad a=c=0,\\ d=b.\n\\]","local conditions become\n\\[\nu^2=0,\\qquad v-u=0,\n\\quad\\text{so}\\quad\nR^\\tau=\\OO[[u,v]]/(u^2,v-u)\\cong\\OO[\\varepsilon]/(\\varepsilon^2).\n\\]\nProbe this ring using $D_3=\\FF_5[\\delta]/(\\delta^3)$, with the fixed coefficient map $\\ZZ_5\\to\\FF_5$. A continuous local point is specified by\n\\[\nu\\longmapsto a\\delta+b\\delta^2,\\qquad\nv\\longmapsto c\\delta+d\\delta^2,\n\\qquad a,b,c,d\\in\\FF_5.\n\\]\nIt factors through $R^\\tau$ precisely when\n\\[\na^2=0,\\quad c=a,\\quad d=b\n\\quad\\Longleftrightarrow\\quad a=c=0,\\ d=b.\n\\]","\\[\nu^2=0,\\qquad v-u=0,\n\\quad\\text{so}\\quad\nR^\\tau=\\OO[[u,v]]/(u^2,v-u)\\cong\\OO[\\varepsilon]/(\\varepsilon^2).\n\\]\nProbe this ring using $D_3=\\FF_5[\\delta]/(\\delta^3)$, with the fixed coefficient map $\\ZZ_5\\to\\FF_5$. A continuous local point is specified by\n\\[\nu\\longmapsto a\\delta+b\\delta^2,\\qquad\nv\\longmapsto c\\delta+d\\delta^2,\n\\qquad a,b,c,d\\in\\FF_5.\n\\]\nIt factors through $R^\\tau$ precisely when\n\\[\na^2=0,\\quad c=a,\\quad d=b\n\\quad\\Longleftrightarrow\\quad a=c=0,\\ d=b.\n\\]","\\begin{minipage}[t]{.48\\linewidth}\\centering\n{\\sffamily\\small\\bfseries First-order coefficients}\\par\n\\vfig[.68]{points_slopes}\n\\captiontext{Only the slope pair $(a,c)=(0,0)$ survives. Each square initially has 25 choices of $(b,d)$.}\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{.48\\linewidth}\\centering\n{\\sffamily\\small\\bfseries Remaining second-order coefficients}\\par\n\\vfig[.68]{points_curvatures}\n\\captiontext{Once $a=c=0$, the five diagonal choices $d=b$ give all accepted points.}\n\\end{minipage}","[t]{.48\\linewidth}\\centering\n{\\sffamily\\small\\bfseries First-order coefficients}\\par\n\\vfig[.68]{points_slopes}\n\\captiontext{Only the slope pair $(a,c)=(0,0)$ survives. Each square initially has 25 choices of $(b,d)$.}\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{.48\\linewidth}\\centering\n{\\sffamily\\small\\bfseries Remaining second-order coefficients}\\par\n\\vfig[.68]{points_curvatures}\n\\captiontext{Once $a=c=0$, the five diagonal choices $d=b$ give all accepted points.}\n\\end{minipage}","mily\\small\\bfseries First-order coefficients}\\par\n\\vfig[.68]{points_slopes}\n\\captiontext{Only the slope pair $(a,c)=(0,0)$ survives. Each square initially has 25 choices of $(b,d)$.}\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{.48\\linewidth}\\centering\n{\\sffamily\\small\\bfseries Remaining second-order coefficients}\\par\n\\vfig[.68]{points_curvatures}\n\\captiontext{Once $a=c=0$, the five diagonal choices $d=b$ give all accepted points.}\n\\end{minipage}","\\head{An obstruction you can calculate}\nThere are $5^4=625$ candidate $D_3$-points and exactly five accepted points. Over $D_2=\\FF_5[\\delta]/(\\delta^2)$, however, the five maps $u=v=a\\delta$ all satisfy the equations. Reduction of \\emph{every} accepted $D_3$-point gives the zero $D_2$-point. The four nonzero tangent directions therefore do not lift to order three.","% ---------------------------------------------------------------------------\n\\vpage{5}{Completion retains every precision}{MODEL A / COMPATIBLE COEFFICIENTS}{Main text: Theorem 3.1; reconstruction and finite occupancy in Section 10\\quad \\href{p12-route:visual-5}{Main statement 10.5}}\nFor $B=\\ZZ_5[\\varepsilon]/(\\varepsilon^2)$, consider the finite quotient rings\n\\[\nB_n=(\\ZZ/5^n\\ZZ)[\\varepsilon]/(\\varepsilon^2),\n\\qquad |B_n|=5^{2n},\\qquad B\\cong\\varprojlim_n B_n.\n\\]\nA point of the limit is a \\emph{single compatible choice} at every level, not unrelated choices from the nonempty finite rings.","\\head{Two explicit 5-adic numbers}\nChoose the canonical digits\n\\[\n\\alpha_j=(j^2+1)\\bmod5,\\qquad \\beta_j=(2j+1)\\bmod5,\n\\]\nwith values in $\\{0,1,2,3,4\\}$. Set $a=\\sum_{j\\ge0}\\alpha_j5^j$ and $b=\\sum_{j\\ge0}\\beta_j5^j$. The element $a+\\varepsilon b$ is determined by the following coefficient array.","\\vfig[.87]{adic_digits}\n\\captiontext{Cutting before column $5^n$ produces reduction modulo $5^n$. The digit formulas define every column, including those not drawn.}","\\begin{center}\\small\n\\begin{tabular}{@{}lrrrrr@{}}\\toprule\n$n$ & 1 & 2 & 3 & 4 & 5","midrule\n$5^n$ & 5 & 25 & 125 & 625 & 3125","$a\\bmod5^n$ & 1 & 11 & 11 & 11 & 1261","$b\\bmod5^n$ & 1 & 16 & 16 & 266 & 2766","\\head{Where the topology enters}\nThe $5$-adic and $(5,\\varepsilon)$-adic topologies on $B$ agree: for every $n$,\n\\[\n(5,\\varepsilon)^{n+1}\\subseteq 5^nB\\subseteq(5,\\varepsilon)^n.\n\\]\nFor the general quotient theorem, the condition ideal must be closed. A continuous map to a Hausdorff target has closed kernel: vanishing on the original generators therefore kills their closure.","% ---------------------------------------------------------------------------\n\\vpage{6}{A matrix becomes a support locus}{MODEL B / FITTING STRATA}{Main text: Theorems 6.2, 6.3 and 6.4\\quad \\href{p12-route:visual-6}{Main statement 6.3}}\nNow change models. Let $R=k[x,y]$ for a field $k$, and present a module by\n\\[\nR^2\\xrightarrow{A}R^2\\longrightarrow M\\longrightarrow0,\n\\qquad\nA=\\begin{pmatrix}x&0","% ---------------------------------------------------------------------------\n\\vpage{6}{A matrix becomes a support locus}{MODEL B / FITTING STRATA}{Main text: Theorems 6.2, 6.3 and 6.4\\quad \\href{p12-route:visual-6}{Main statement 6.3}}\nNow change models. Let $R=k[x,y]$ for a field $k$, and present a module by\n\\[\nR^2\\xrightarrow{A}R^2\\longrightarrow M\\longrightarrow0,\n\\qquad\nA=\\begin{pmatrix}x&0\\\\0&y\\end{pmatrix}.\n\\]\nThe relations are $xe_1=0$ and $ye_2=0$, so $M\\cong R/(x)\\oplus R/(y)$.","0&y\\end{pmatrix}.\n\\]\nThe relations are $xe_1=0$ and $ye_2=0$, so $M\\cong R/(x)\\oplus R/(y)$.","\\begin{minipage}[c]{.48\\linewidth}\n\\vfig[.98]{support_axes}\n\\end{minipage}\\hfill\n\\begin{minipage}[c]{.48\\linewidth}\n{\\sffamily\\bfseries The minor calculation}\\par\n\\[\n\\begin{aligned}\n\\Fitt_0(M)&=(xy),\\\\\n\\Fitt_1(M)&=(x,y),\\\\\n\\Fitt_j(M)&=R\\quad(j\\ge2).\n\\end{aligned}\n\\]\nAt a prime $\\p$, reduce the matrix to its residue field:\n\\[\n\\mu_{\\p}(M)=2-\\rank A(\\p).\n\\]\nTwo independent relations disappear at the origin. Only one disappears along either punctured axis.\n\\end{minipage}\n\\captiontext{For the drawing, take $k=\\RR$. It shows real closed points, not every point of $\\Spec R$. The displayed ideals and rank criterion describe the full algebraic loci over any field.}","\\Fitt_j(M)&=R\\quad(j\\ge2).\n\\end{aligned}\n\\]\nAt a prime $\\p$, reduce the matrix to its residue field:\n\\[\n\\mu_{\\p}(M)=2-\\rank A(\\p).\n\\]\nTwo independent relations disappear at the origin. Only one disappears along either punctured axis.\n\\end{minipage}\n\\captiontext{For the drawing, take $k=\\RR$. It shows real closed points, not every point of $\\Spec R$. The displayed ideals and rank criterion describe the full algebraic loci over any field.}","midrule\n$x\\ne0$, $y\\ne0$ & 2 & 0 & neither generator","$x=0$, $y\\ne0$ & 1 & 1 & the $e_1$ direction","$x\\ne0$, $y=0$ & 1 & 1 & the $e_2$ direction","$x=y=0$ & 0 & 2 & both directions","bottomrule\n\\end{tabular}\n\\end{center}\nThus $\\Supp(M)=V(xy)=V(x)\\cup V(y)$, whereas the locus where the fiber needs at least two generators is $V(x,y)$. The latter is a \\emph{higher} Fitting stratum. For $r>2$, the locus $\\mu_{\\p}(M)\\ge r$ is empty, corresponding to the unit Fitting ideal.","% ---------------------------------------------------------------------------\n\\vpage{7}{The same support can hide more algebra}{MODEL B / NILPOTENT STRUCTURE}{Main text: Theorems 6.3--6.4; distinguish an ideal from its vanishing set\\quad \\href{p12-route:visual-7}{Main statement 6.4}}\nCompare the previous module with a slightly different presentation:\n\\[\nM=\\coker\\begin{pmatrix}x&0\\\\0&y\\end{pmatrix},\n\\qquad\nM'=\\coker\\begin{pmatrix}x^2&0\\\\0&y\\end{pmatrix}.\n\\]\nTheir support sets coincide. Their residue-field fiber dimensions also coincide at every prime. But their Fitting ideals do not.","\\begin{center}\n\\begin{tabular}{@{}lll@{}}\\toprule\n & $M$ & $M'$","midrule\nZeroth Fitting ideal & $(xy)$ & $(x^2y)$","First Fitting ideal & $(x,y)$ & $(x^2,y)$","Support as a set & $V(x)\\cup V(y)$ & $V(x)\\cup V(y)$","Fiber dimension at $(x,y)$ & 2 & 2","\\head{Magnify transversely, rather than draw a thicker line}\nTake the slice $y=1$. The second summand vanishes because $y$ becomes a unit. What remains is\n\\[\nM|_{y=1}\\cong k[x]/(x),\\qquad\nM'|_{y=1}\\cong k[x]/(x^2).\n\\]\nThe first has basis $1$; the second has basis $1,x$. Multiplication by $x$ distinguishes them exactly:","\\begin{center}\n\\begin{tikzpicture}[x=1cm,y=1cm,font=\\small,>=Stealth]\n\\node[font=\\sffamily\\bfseries] at (1.3,1.8) {reduced slice};\n\\node at (.2,.85) {$1$};\\node at (2.4,.85) {$0$};\n\\draw[->] (.5,.85)--(2.1,.85) node[midway,above] {$\\times x$};\n\\node[font=\\sffamily\\bfseries] at (7.2,1.8) {nilpotent slice};\n\\node at (5,.85) {$1$};\\node at (7.2,.85) {$x$};\\node at (9.4,.85) {$0$};\n\\draw[->] (5.3,.85)--(6.9,.85) node[midway,above] {$\\times x$};\n\\draw[->] (7.5,.85)--(9.1,.85) node[midway,above] {$\\times x$};\n\\node at (1.3,-.05) {$[x]=[0]$};\n\\node at (7.2,-.05) {$[x]=\\begin{pmatrix}0&0","}[x=1cm,y=1cm,font=\\small,>=Stealth]\n\\node[font=\\sffamily\\bfseries] at (1.3,1.8) {reduced slice};\n\\node at (.2,.85) {$1$};\\node at (2.4,.85) {$0$};\n\\draw[->] (.5,.85)--(2.1,.85) node[midway,above] {$\\times x$};\n\\node[font=\\sffamily\\bfseries] at (7.2,1.8) {nilpotent slice};\n\\node at (5,.85) {$1$};\\node at (7.2,.85) {$x$};\\node at (9.4,.85) {$0$};\n\\draw[->] (5.3,.85)--(6.9,.85) node[midway,above] {$\\times x$};\n\\draw[->] (7.5,.85)--(9.1,.85) node[midway,above] {$\\times x$};\n\\node at (1.3,-.05) {$[x]=[0]$};\n\\node at (7.2,-.05) {$[x]=\\begin{pmatrix}0&0","es] at (1.3,1.8) {reduced slice};\n\\node at (.2,.85) {$1$};\\node at (2.4,.85) {$0$};\n\\draw[->] (.5,.85)--(2.1,.85) node[midway,above] {$\\times x$};\n\\node[font=\\sffamily\\bfseries] at (7.2,1.8) {nilpotent slice};\n\\node at (5,.85) {$1$};\\node at (7.2,.85) {$x$};\\node at (9.4,.85) {$0$};\n\\draw[->] (5.3,.85)--(6.9,.85) node[midway,above] {$\\times x$};\n\\draw[->] (7.5,.85)--(9.1,.85) node[midway,above] {$\\times x$};\n\\node at (1.3,-.05) {$[x]=[0]$};\n\\node at (7.2,-.05) {$[x]=\\begin{pmatrix}0&0","\\node at (.2,.85) {$1$};\\node at (2.4,.85) {$0$};\n\\draw[->] (.5,.85)--(2.1,.85) node[midway,above] {$\\times x$};\n\\node[font=\\sffamily\\bfseries] at (7.2,1.8) {nilpotent slice};\n\\node at (5,.85) {$1$};\\node at (7.2,.85) {$x$};\\node at (9.4,.85) {$0$};\n\\draw[->] (5.3,.85)--(6.9,.85) node[midway,above] {$\\times x$};\n\\draw[->] (7.5,.85)--(9.1,.85) node[midway,above] {$\\times x$};\n\\node at (1.3,-.05) {$[x]=[0]$};\n\\node at (7.2,-.05) {$[x]=\\begin{pmatrix}0&0","(2.4,.85) {$0$};\n\\draw[->] (.5,.85)--(2.1,.85) node[midway,above] {$\\times x$};\n\\node[font=\\sffamily\\bfseries] at (7.2,1.8) {nilpotent slice};\n\\node at (5,.85) {$1$};\\node at (7.2,.85) {$x$};\\node at (9.4,.85) {$0$};\n\\draw[->] (5.3,.85)--(6.9,.85) node[midway,above] {$\\times x$};\n\\draw[->] (7.5,.85)--(9.1,.85) node[midway,above] {$\\times x$};\n\\node at (1.3,-.05) {$[x]=[0]$};\n\\node at (7.2,-.05) {$[x]=\\begin{pmatrix}0&0\\\\1&0\\end{pmatrix}$};\n\\end{tikzpicture}\n\\end{center}","\\head{A test ring detects the higher-stratum thickening}\nLet $D=k[\\delta]/(\\delta^2)$ and send $x\\mapsto\\delta$, $y\\mapsto0$. The ideal $(x^2,y)$ maps to zero, but $(x,y)$ does not. Therefore this $D$-point lies in the scheme defined by the first Fitting ideal of $M'$, but not in the corresponding scheme for $M$.","\\takeaway{Taking $V(I)$ forgets nilpotent thickness. Fitting \\emph{ideals} can retain information that both the support set and the fiber-dimension function fail to distinguish.}","% ---------------------------------------------------------------------------\n\\vpage{8}{Base change is an algebraic pullback}{MODEL B / SPECIALIZATION AND TRANSPORT}{Main text: Proposition 6.5 and Theorem 7.1\\quad \\href{p12-route:visual-8}{Main statement 6.5}}\nReturn to $M=\\coker\\operatorname{diag}(x,y)$ over $R=k[x,y]$. Pull back along the diagonal $R\\to k[t]$, $x\\mapsto t$, $y\\mapsto t$. The base-changed presentation is literal:\n\\[\nM\\otimes_R k[t]\\cong\\coker\\begin{pmatrix}t&0","----------------------------------------\n\\vpage{8}{Base change is an algebraic pullback}{MODEL B / SPECIALIZATION AND TRANSPORT}{Main text: Proposition 6.5 and Theorem 7.1\\quad \\href{p12-route:visual-8}{Main statement 6.5}}\nReturn to $M=\\coker\\operatorname{diag}(x,y)$ over $R=k[x,y]$. Pull back along the diagonal $R\\to k[t]$, $x\\mapsto t$, $y\\mapsto t$. The base-changed presentation is literal:\n\\[\nM\\otimes_R k[t]\\cong\\coker\\begin{pmatrix}t&0\\\\0&t\\end{pmatrix}\n\\cong\\bigl(k[t]/(t)\\bigr)^2.\n\\]","\\begin{minipage}[c]{.44\\linewidth}\n\\vfig[.94]{diagonal_slice}\n\\end{minipage}\\hfill\n\\begin{minipage}[c]{.51\\linewidth}\n\\head{Both relations vanish together}\n\\[\n(xy)\\longmapsto(t^2),\\qquad\n(x,y)\\longmapsto(t).\n\\]\nThe support is the origin, but the origin has fiber dimension two. Replacing $(t^2)$ by its radical $(t)$ would lose the zeroth Fitting ideal.","This map is not flat: it is the quotient by $x-y$. The Fitting base-change theorem still applies; it does not require flatness.\n\\end{minipage}\n\\captiontext{A real-point shadow of the diagonal slice. The double algebraic contribution is certified by the minor $t^2$, not by the visual thickness of the intersection point.}","\\head{An isomorphism changes coordinates, not occupancy}\nOver a field of characteristic other than two, the isomorphism\n\\[\nk[x,y]\\xrightarrow{\\sim}k[a,b],\\qquad x\\mapsto a+b,\\quad y\\mapsto a-b\n\\]\nsends the Fitting ideals to $(a^2-b^2)$ and $(a,b)$. The two axes become the two lines $a=b$ and $a=-b$. No previously missing module vector is created by this relabeling.","\\head{An extra presentation generator changes nothing}\nReplacing $\\operatorname{diag}(x,y)$ by $\\operatorname{diag}(x,y,1)$ adds a generator killed by the relation $1$. The new maximal minor is $xy$, and the new next-size minors generate $(xy,x,y)=(x,y)$. The module and its intrinsic Fitting ideals are unchanged. This is a small visible instance of presentation independence of Fitting ideals.","% ---------------------------------------------------------------------------\n\\vpage{9}{A branch is not an Artin packet}{LOCAL GEOMETRY / AN EXACT FINITE SPECTRUM}{Main text: Proposition 3.4 and Proposition 7.3\\quad \\href{p12-route:visual-9}{Main statement 7.3}}\nConsider the complete local node\n\\[\nR=k[[x,y]]/(xy),\\qquad T=R/(x)\\cong k[[y]].\n\\]\nEvery prime of $R$ contains $x$ or $y$. Since $k[[x]]$ and $k[[y]]$ are discrete valuation rings, the following diagram lists \\emph{all three} prime ideals of $R$ and their specialization relations.","-----------------------------------------------------------------------\n\\vpage{9}{A branch is not an Artin packet}{LOCAL GEOMETRY / AN EXACT FINITE SPECTRUM}{Main text: Proposition 3.4 and Proposition 7.3\\quad \\href{p12-route:visual-9}{Main statement 7.3}}\nConsider the complete local node\n\\[\nR=k[[x,y]]/(xy),\\qquad T=R/(x)\\cong k[[y]].\n\\]\nEvery prime of $R$ contains $x$ or $y$. Since $k[[x]]$ and $k[[y]]$ are discrete valuation rings, the following diagram lists \\emph{all three} prime ideals of $R$ and their specialization relations.","Artin packet}{LOCAL GEOMETRY / AN EXACT FINITE SPECTRUM}{Main text: Proposition 3.4 and Proposition 7.3\\quad \\href{p12-route:visual-9}{Main statement 7.3}}\nConsider the complete local node\n\\[\nR=k[[x,y]]/(xy),\\qquad T=R/(x)\\cong k[[y]].\n\\]\nEvery prime of $R$ contains $x$ or $y$. Since $k[[x]]$ and $k[[y]]$ are discrete valuation rings, the following diagram lists \\emph{all three} prime ideals of $R$ and their specialization relations.","\\head{What the quotient removes}\nAs an $R$-module, $T$ has annihilator $(x)$, hence\n\\[\n\\Supp_R(T)=V(x)=\\{(x),(x,y)\\}.\n\\]\nThe other generic point $(y)$ is missing because it does not contain $x$. The quotient has not created a disconnected projector summand inside the original local ring. It has changed the ring by imposing an equation.","\\head{Why no idempotent selects just this branch}\nIn a commutative local ring, $e^2=e$ implies $e=0$ or $e=1$. Indeed, $e+(1-e)=1$ forces at least one of $e,1-e$ to be a unit; canceling in $e(1-e)=0$ gives the conclusion. Thus this node has no nontrivial idempotent even though it has two irreducible components.","By contrast, a product of two Artin local rings has two disjoint maximal points, with canonical idempotents $(1,0)$ and $(0,1)$. Those idempotents act on \\emph{any supplied module over the product}. Their two ranges are the packets studied on \\pagerefV{12}.","% ---------------------------------------------------------------------------\n\\vpage{10}{Free over what? Supported where?}{PATCHING / A COUNTEREXAMPLE WITH A VISIBLE SPECIAL FIBER}{Main text: Theorems 6.7, 6.9 and 6.11; Example 6.13\\quad \\href{p12-route:visual-10}{Main statement 6.11}}\nLet\n\\[\nS=k[[t]],\\qquad R_\\infty=S[[x]]/(x(x-t)),\\qquad\nM_\\infty=R_\\infty/(x)\\cong S.\n\\]\nThe module is free of rank one over $S$. Yet over $R_\\infty$ its support is only $V(x)$; the generic point $(x-t)$ of the other branch is not occupied.","-----------------------------------------------------------------\n\\vpage{10}{Free over what? Supported where?}{PATCHING / A COUNTEREXAMPLE WITH A VISIBLE SPECIAL FIBER}{Main text: Theorems 6.7, 6.9 and 6.11; Example 6.13\\quad \\href{p12-route:visual-10}{Main statement 6.11}}\nLet\n\\[\nS=k[[t]],\\qquad R_\\infty=S[[x]]/(x(x-t)),\\qquad\nM_\\infty=R_\\infty/(x)\\cong S.\n\\]\nThe module is free of rank one over $S$. Yet over $R_\\infty$ its support is only $V(x)$; the generic point $(x-t)$ of the other branch is not occupied.","---------------------------------------------------\n\\vpage{10}{Free over what? Supported where?}{PATCHING / A COUNTEREXAMPLE WITH A VISIBLE SPECIAL FIBER}{Main text: Theorems 6.7, 6.9 and 6.11; Example 6.13\\quad \\href{p12-route:visual-10}{Main statement 6.11}}\nLet\n\\[\nS=k[[t]],\\qquad R_\\infty=S[[x]]/(x(x-t)),\\qquad\nM_\\infty=R_\\infty/(x)\\cong S.\n\\]\nThe module is free of rank one over $S$. Yet over $R_\\infty$ its support is only $V(x)$; the generic point $(x-t)$ of the other branch is not occupied.","--------------------\n\\vpage{10}{Free over what? Supported where?}{PATCHING / A COUNTEREXAMPLE WITH A VISIBLE SPECIAL FIBER}{Main text: Theorems 6.7, 6.9 and 6.11; Example 6.13\\quad \\href{p12-route:visual-10}{Main statement 6.11}}\nLet\n\\[\nS=k[[t]],\\qquad R_\\infty=S[[x]]/(x(x-t)),\\qquad\nM_\\infty=R_\\infty/(x)\\cong S.\n\\]\nThe module is free of rank one over $S$. Yet over $R_\\infty$ its support is only $V(x)$; the generic point $(x-t)$ of the other branch is not occupied.","e{10}{Free over what? Supported where?}{PATCHING / A COUNTEREXAMPLE WITH A VISIBLE SPECIAL FIBER}{Main text: Theorems 6.7, 6.9 and 6.11; Example 6.13\\quad \\href{p12-route:visual-10}{Main statement 6.11}}\nLet\n\\[\nS=k[[t]],\\qquad R_\\infty=S[[x]]/(x(x-t)),\\qquad\nM_\\infty=R_\\infty/(x)\\cong S.\n\\]\nThe module is free of rank one over $S$. Yet over $R_\\infty$ its support is only $V(x)$; the generic point $(x-t)$ of the other branch is not occupied.","\\begin{minipage}[c]{.47\\linewidth}\n\\vfig[.94]{patching_branches}\n\\end{minipage}\\hfill\n\\begin{minipage}[c]{.49\\linewidth}\n\\head{Remove the auxiliary variable}\nSet $t=0$. Then\n\\[\nR=R_\\infty/(t)\\cong k[[x]]/(x^2),\n\\]\n\\[\nM=M_\\infty/tM_\\infty\\cong k.\n\\]\nThe two branches meet in a nonreduced special fiber. The relation becomes $x^2=0$, not $x=0$.\n\\end{minipage}\n\\captiontext{The drawing is a real algebraic shadow of $x(x-t)=0$. It is not the complete local spectrum. The special fiber's nilpotent structure is recorded by its quotient algebra, not by the point drawn at the intersection.}","e}[c]{.47\\linewidth}\n\\vfig[.94]{patching_branches}\n\\end{minipage}\\hfill\n\\begin{minipage}[c]{.49\\linewidth}\n\\head{Remove the auxiliary variable}\nSet $t=0$. Then\n\\[\nR=R_\\infty/(t)\\cong k[[x]]/(x^2),\n\\]\n\\[\nM=M_\\infty/tM_\\infty\\cong k.\n\\]\nThe two branches meet in a nonreduced special fiber. The relation becomes $x^2=0$, not $x=0$.\n\\end{minipage}\n\\captiontext{The drawing is a real algebraic shadow of $x(x-t)=0$. It is not the complete local spectrum. The special fiber's nilpotent structure is recorded by its quotient algebra, not by the point drawn at the intersection.}","\\head{Full support need not mean faithful action}\nThe ring $R$ has one prime, $(x)$. The nonzero module $M=k$ therefore has full support. But $x$ annihilates $M$, so the action is not faithful. It is \\emph{nearly faithful}: the annihilator $(x)$ is nilpotent. This distinction disappears over a reduced ring.","\\head{The actual descent theorem is one-way}\nFor a finite module over a Noetherian ring,\n\\[\n\\Supp_{R_\\infty/(t)}(M_\\infty/tM_\\infty)\n=\\Supp_{R_\\infty}(M_\\infty)\\cap V(t).\n\\]\nIf the initial support is full, the resulting support is full on the quotient. Our example shows that full support at the quotient can occur even when initial support was not full. It also shows why freeness over $S$ alone is not the required patching hypothesis.","% ---------------------------------------------------------------------------\n\\vpage{11}{A vanishing differential changes the fiber}{DERIVED SUPPORT / AN ACTUAL TWO-TERM COMPLEX}{Main text: Definition 6.14 and Theorem 6.16\\quad \\href{p12-route:visual-11}{Main statement 6.16}}\nTake $R=k[t]$ and the perfect complex\n\\[\nC=\\bigl[R\\xrightarrow{\\;t\\;}R\\bigr]\n\\qquad\\text{in cohomological degrees }0,1.\n\\]\nSpecialization at $t=a$ gives $C_a=[k\\xrightarrow{a}k]$. When $a\\ne0$, the differential is invertible and the complex is acyclic. When $a=0$, the differential is zero and both cohomology groups survive.","----------------------------------------------------------\n\\vpage{11}{A vanishing differential changes the fiber}{DERIVED SUPPORT / AN ACTUAL TWO-TERM COMPLEX}{Main text: Definition 6.14 and Theorem 6.16\\quad \\href{p12-route:visual-11}{Main statement 6.16}}\nTake $R=k[t]$ and the perfect complex\n\\[\nC=\\bigl[R\\xrightarrow{\\;t\\;}R\\bigr]\n\\qquad\\text{in cohomological degrees }0,1.\n\\]\nSpecialization at $t=a$ gives $C_a=[k\\xrightarrow{a}k]$. When $a\\ne0$, the differential is invertible and the complex is acyclic. When $a=0$, the differential is zero and both cohomology groups survive.","----------------------------------------------\n\\vpage{11}{A vanishing differential changes the fiber}{DERIVED SUPPORT / AN ACTUAL TWO-TERM COMPLEX}{Main text: Definition 6.14 and Theorem 6.16\\quad \\href{p12-route:visual-11}{Main statement 6.16}}\nTake $R=k[t]$ and the perfect complex\n\\[\nC=\\bigl[R\\xrightarrow{\\;t\\;}R\\bigr]\n\\qquad\\text{in cohomological degrees }0,1.\n\\]\nSpecialization at $t=a$ gives $C_a=[k\\xrightarrow{a}k]$. When $a\\ne0$, the differential is invertible and the complex is acyclic. When $a=0$, the differential is zero and both cohomology groups survive.","--------------\n\\vpage{11}{A vanishing differential changes the fiber}{DERIVED SUPPORT / AN ACTUAL TWO-TERM COMPLEX}{Main text: Definition 6.14 and Theorem 6.16\\quad \\href{p12-route:visual-11}{Main statement 6.16}}\nTake $R=k[t]$ and the perfect complex\n\\[\nC=\\bigl[R\\xrightarrow{\\;t\\;}R\\bigr]\n\\qquad\\text{in cohomological degrees }0,1.\n\\]\nSpecialization at $t=a$ gives $C_a=[k\\xrightarrow{a}k]$. When $a\\ne0$, the differential is invertible and the complex is acyclic. When $a=0$, the differential is zero and both cohomology groups survive.","IVED SUPPORT / AN ACTUAL TWO-TERM COMPLEX}{Main text: Definition 6.14 and Theorem 6.16\\quad \\href{p12-route:visual-11}{Main statement 6.16}}\nTake $R=k[t]$ and the perfect complex\n\\[\nC=\\bigl[R\\xrightarrow{\\;t\\;}R\\bigr]\n\\qquad\\text{in cohomological degrees }0,1.\n\\]\nSpecialization at $t=a$ gives $C_a=[k\\xrightarrow{a}k]$. When $a\\ne0$, the differential is invertible and the complex is acyclic. When $a=0$, the differential is zero and both cohomology groups survive.","\\vfig[.81]{derived_fibers}\n\\captiontext{A few fibers over $k=\\QQ$ are displayed. The statement for every $a\\ne0$ follows from invertibility of the scalar $a$; it is not inferred from the sample points.}","\\head{Compute before and after specialization}\nBefore specialization,\n\\[\nH^0(C)=0,\\qquad H^1(C)=k[t]/(t).\n\\]\nAfter \\emph{derived} specialization at zero,\n\\[\nC\\otimes_R^{\\mathbf L}k=[k\\xrightarrow{0}k],\n\\quad H^0\\cong k,\\quad H^1\\cong k.\n\\]\nIf one retained only $H^1(C)$ and took its ordinary tensor product with $k$, one would see a single $k$ and miss the extra cohomology in degree zero. The projective complex keeps the information needed for specialization.","\\head{The support is still a precise locus}\nAt any prime not containing $t$, the residue-field differential is invertible. At the prime $(t)$ it is zero. Therefore\n\\[\n\\Supp_R(C)=V(t).\n\\]\nTheorem 6.16 identifies support after derived quotient with the intersection of original support and the quotient locus.","% ---------------------------------------------------------------------------\n\\vpage{12}{A packet contains an entire Jordan chain}{MODEL C / A NONSEMISIMPLE HECKE ALGEBRA}{Main text: Theorem 8.2, Corollary 8.3 and Example 8.6\\quad \\href{p12-route:visual-12}{Main statement 8.2}}\nWork over $\\QQ$ and let\n\\[\nT=\\QQ[z]\\big/\\bigl((z-1)^2(z+1)^3\\bigr),\\qquad V=T.\n\\]\nThe Chinese remainder theorem gives\n\\[\nT\\cong \\underbrace{\\QQ[\\varepsilon]/(\\varepsilon^2)}_{T_+,\\ z=1+\\varepsilon}\n\\times\n\\underbrace{\\QQ[\\eta]/(\\eta^3)}_{T_-,\\ z=-1+\\eta}.\n\\]\nMultiplication by $z$ is an operator $H$. In the ordered local bases $1,\\varepsilon$ and $1,\\eta,\\eta^2$, its matrix is\n\\[\nH=\\left(\\begin{array}{cc|ccc}\n1&0&0&0&0","HECKE ALGEBRA}{Main text: Theorem 8.2, Corollary 8.3 and Example 8.6\\quad \\href{p12-route:visual-12}{Main statement 8.2}}\nWork over $\\QQ$ and let\n\\[\nT=\\QQ[z]\\big/\\bigl((z-1)^2(z+1)^3\\bigr),\\qquad V=T.\n\\]\nThe Chinese remainder theorem gives\n\\[\nT\\cong \\underbrace{\\QQ[\\varepsilon]/(\\varepsilon^2)}_{T_+,\\ z=1+\\varepsilon}\n\\times\n\\underbrace{\\QQ[\\eta]/(\\eta^3)}_{T_-,\\ z=-1+\\eta}.\n\\]\nMultiplication by $z$ is an operator $H$. In the ordered local bases $1,\\varepsilon$ and $1,\\eta,\\eta^2$, its matrix is\n\\[\nH=\\left(\\begin{array}{cc|ccc}\n1&0&0&0&0","text: Theorem 8.2, Corollary 8.3 and Example 8.6\\quad \\href{p12-route:visual-12}{Main statement 8.2}}\nWork over $\\QQ$ and let\n\\[\nT=\\QQ[z]\\big/\\bigl((z-1)^2(z+1)^3\\bigr),\\qquad V=T.\n\\]\nThe Chinese remainder theorem gives\n\\[\nT\\cong \\underbrace{\\QQ[\\varepsilon]/(\\varepsilon^2)}_{T_+,\\ z=1+\\varepsilon}\n\\times\n\\underbrace{\\QQ[\\eta]/(\\eta^3)}_{T_-,\\ z=-1+\\eta}.\n\\]\nMultiplication by $z$ is an operator $H$. In the ordered local bases $1,\\varepsilon$ and $1,\\eta,\\eta^2$, its matrix is\n\\[\nH=\\left(\\begin{array}{cc|ccc}\n1&0&0&0&0","3 and Example 8.6\\quad \\href{p12-route:visual-12}{Main statement 8.2}}\nWork over $\\QQ$ and let\n\\[\nT=\\QQ[z]\\big/\\bigl((z-1)^2(z+1)^3\\bigr),\\qquad V=T.\n\\]\nThe Chinese remainder theorem gives\n\\[\nT\\cong \\underbrace{\\QQ[\\varepsilon]/(\\varepsilon^2)}_{T_+,\\ z=1+\\varepsilon}\n\\times\n\\underbrace{\\QQ[\\eta]/(\\eta^3)}_{T_-,\\ z=-1+\\eta}.\n\\]\nMultiplication by $z$ is an operator $H$. In the ordered local bases $1,\\varepsilon$ and $1,\\eta,\\eta^2$, its matrix is\n\\[\nH=\\left(\\begin{array}{cc|ccc}\n1&0&0&0&0","ad \\href{p12-route:visual-12}{Main statement 8.2}}\nWork over $\\QQ$ and let\n\\[\nT=\\QQ[z]\\big/\\bigl((z-1)^2(z+1)^3\\bigr),\\qquad V=T.\n\\]\nThe Chinese remainder theorem gives\n\\[\nT\\cong \\underbrace{\\QQ[\\varepsilon]/(\\varepsilon^2)}_{T_+,\\ z=1+\\varepsilon}\n\\times\n\\underbrace{\\QQ[\\eta]/(\\eta^3)}_{T_-,\\ z=-1+\\eta}.\n\\]\nMultiplication by $z$ is an operator $H$. In the ordered local bases $1,\\varepsilon$ and $1,\\eta,\\eta^2$, its matrix is\n\\[\nH=\\left(\\begin{array}{cc|ccc}\n1&0&0&0&0","in statement 8.2}}\nWork over $\\QQ$ and let\n\\[\nT=\\QQ[z]\\big/\\bigl((z-1)^2(z+1)^3\\bigr),\\qquad V=T.\n\\]\nThe Chinese remainder theorem gives\n\\[\nT\\cong \\underbrace{\\QQ[\\varepsilon]/(\\varepsilon^2)}_{T_+,\\ z=1+\\varepsilon}\n\\times\n\\underbrace{\\QQ[\\eta]/(\\eta^3)}_{T_-,\\ z=-1+\\eta}.\n\\]\nMultiplication by $z$ is an operator $H$. In the ordered local bases $1,\\varepsilon$ and $1,\\eta,\\eta^2$, its matrix is\n\\[\nH=\\left(\\begin{array}{cc|ccc}\n1&0&0&0&0\\\\1&1&0&0&0\\\\\\hline\n0&0&-1&0&0\\\\0&0&1&-1&0\\\\0&0&0&1&-1\n\\end{array}\\right).\n\\]","\\vfig[.98]{jordan_packets}\n\\captiontext{These arrows are the actual nilpotent actions $H-1$ and $H+1$. Each nonzero displayed basis vector belongs to the indicated packet. Only the last nonzero vector in each chain is an ordinary eigenvector.}","\\head{The projectors can be written explicitly}\nModulo $(z-1)^2(z+1)^3$, put\n\\[\ne_+(z)=\\frac{(5-3z)(z+1)^3}{16},\\qquad e_-(z)=1-e_+(z).\n\\]\nThen $e_+\\equiv1\\pmod{(z-1)^2}$ and $e_+\\equiv0\\pmod{(z+1)^3}$. Consequently\n\\[\ne_+(H)=\\operatorname{diag}(1,1,0,0,0),\\qquad\n e_-(H)=\\operatorname{diag}(0,0,1,1,1).\n\\]\nThe packet dimensions are two and three. The ordinary eigenspaces each have dimension one; the generalized eigenspaces have dimensions two and three.","% ---------------------------------------------------------------------------\n\\vpage{13}{A possible packet can still be empty}{MODEL C / OCCUPANCY IS MODULE DATA}{Main text: Theorem 7.4, Theorem 8.2 and Corollary 8.4\\quad \\href{p12-route:visual-13}{Main statement 7.4}}\nKeep the algebra $T=T_+\\times T_-$ from \\pagerefV{12}, but compare two different $T$-modules:\n\\[\nV=T,\\qquad W=T_+,\n\\]\nwhere $T$ acts on $W$ through projection to its first factor. The algebra still has two maximal ideals, but only one of them is in $\\Supp_T(W)$. Here $T$ is a supplied algebra; its action on $W$ is \\emph{not} assumed faithful.","----------------------------------------------------------------------\n\\vpage{13}{A possible packet can still be empty}{MODEL C / OCCUPANCY IS MODULE DATA}{Main text: Theorem 7.4, Theorem 8.2 and Corollary 8.4\\quad \\href{p12-route:visual-13}{Main statement 7.4}}\nKeep the algebra $T=T_+\\times T_-$ from \\pagerefV{12}, but compare two different $T$-modules:\n\\[\nV=T,\\qquad W=T_+,\n\\]\nwhere $T$ acts on $W$ through projection to its first factor. The algebra still has two maximal ideals, but only one of them is in $\\Supp_T(W)$. Here $T$ is a supplied algebra; its action on $W$ is \\emph{not} assumed faithful.","empty}{MODEL C / OCCUPANCY IS MODULE DATA}{Main text: Theorem 7.4, Theorem 8.2 and Corollary 8.4\\quad \\href{p12-route:visual-13}{Main statement 7.4}}\nKeep the algebra $T=T_+\\times T_-$ from \\pagerefV{12}, but compare two different $T$-modules:\n\\[\nV=T,\\qquad W=T_+,\n\\]\nwhere $T$ acts on $W$ through projection to its first factor. The algebra still has two maximal ideals, but only one of them is in $\\Supp_T(W)$. Here $T$ is a supplied algebra; its action on $W$ is \\emph{not} assumed faithful.","CUPANCY IS MODULE DATA}{Main text: Theorem 7.4, Theorem 8.2 and Corollary 8.4\\quad \\href{p12-route:visual-13}{Main statement 7.4}}\nKeep the algebra $T=T_+\\times T_-$ from \\pagerefV{12}, but compare two different $T$-modules:\n\\[\nV=T,\\qquad W=T_+,\n\\]\nwhere $T$ acts on $W$ through projection to its first factor. The algebra still has two maximal ideals, but only one of them is in $\\Supp_T(W)$. Here $T$ is a supplied algebra; its action on $W$ is \\emph{not} assumed faithful.","m 7.4, Theorem 8.2 and Corollary 8.4\\quad \\href{p12-route:visual-13}{Main statement 7.4}}\nKeep the algebra $T=T_+\\times T_-$ from \\pagerefV{12}, but compare two different $T$-modules:\n\\[\nV=T,\\qquad W=T_+,\n\\]\nwhere $T$ acts on $W$ through projection to its first factor. The algebra still has two maximal ideals, but only one of them is in $\\Supp_T(W)$. Here $T$ is a supplied algebra; its action on $W$ is \\emph{not} assumed faithful.","\\begin{center}\n\\begin{tikzpicture}[x=1cm,y=1cm,font=\\small]\n\\node[anchor=east] at (-.25,1.65) {$V=T$};\\node[anchor=east] at (-.25,.55) {$W=T_+$};\n\\foreach \\j in {0,...,4}{\\draw[fill=black!15,draw=black!35] (\\j*.95,1.3) rectangle ++(.75,.7);}\n\\foreach \\j in {0,1}{\\draw[fill=black!15,draw=black!35] (\\j*.95,.2) rectangle ++(.75,.7);}\n\\foreach \\j in {2,3,4}{\\draw[black!30] (\\j*.95,.2) rectangle ++(.75,.7);\\node[text=black!40] at (\\j*.95+.375,.55) {$\\times$};}\n\\draw[dashed] (1.80,-.15)--(1.80,2.35);\n\\node[font=\\footnotesize] at (.85,2.48) {$T_+$: 2 slots};\n\\node[font=\\footnotesize] at (3.25,2.48) {$T_-$: 3 slots};\n\\node[font=\\footnotesize,anchor=west] at (5.15,1.65) {dimensions $2+3$};\n\\node[font=\\footnotesize,anchor=west] at (5.15,.55) {dimensions $2+0$};\n\\end{tikzpicture}\n\\end{center}\n\\capti","\\begin{center}\n\\begin{tikzpicture}[x=1cm,y=1cm,font=\\small]\n\\node[anchor=east] at (-.25,1.65) {$V=T$};\\node[anchor=east] at (-.25,.55) {$W=T_+$};\n\\foreach \\j in {0,...,4}{\\draw[fill=black!15,draw=black!35] (\\j*.95,1.3) rectangle ++(.75,.7);}\n\\foreach \\j in {0,1}{\\draw[fill=black!15,draw=black!35] (\\j*.95,.2) rectangle ++(.75,.7);}\n\\foreach \\j in {2,3,4}{\\draw[black!30] (\\j*.95,.2) rectangle ++(.75,.7);\\node[text=black!40] at (\\j*.95+.375,.55) {$\\times$};}\n\\draw[dashed] (1.80,-.15)--(1.80,2.35);\n\\node[font=\\footnotesize] at (.85,2.48) {$T_+$: 2 slots};\n\\node[font=\\footnotesize] at (3.25,2.48) {$T_-$: 3 slots};\n\\node[font=\\footnotesize,anchor=west] at (5.15,1.65) {dimensions $2+3$};\n\\node[font=\\footnotesize,anchor=west] at (5.15,.55) {dimensions $2+0$};\n\\end{tikzpicture}\n\\end{center}\n\\captiontext{Squares refer to the local bases of","nt=\\small]\n\\node[anchor=east] at (-.25,1.65) {$V=T$};\\node[anchor=east] at (-.25,.55) {$W=T_+$};\n\\foreach \\j in {0,...,4}{\\draw[fill=black!15,draw=black!35] (\\j*.95,1.3) rectangle ++(.75,.7);}\n\\foreach \\j in {0,1}{\\draw[fill=black!15,draw=black!35] (\\j*.95,.2) rectangle ++(.75,.7);}\n\\foreach \\j in {2,3,4}{\\draw[black!30] (\\j*.95,.2) rectangle ++(.75,.7);\\node[text=black!40] at (\\j*.95+.375,.55) {$\\times$};}\n\\draw[dashed] (1.80,-.15)--(1.80,2.35);\n\\node[font=\\footnotesize] at (.85,2.48) {$T_+$: 2 slots};\n\\node[font=\\footnotesize] at (3.25,2.48) {$T_-$: 3 slots};\n\\node[font=\\footnotesize,anchor=west] at (5.15,1.65) {dimensions $2+3$};\n\\node[font=\\footnotesize,anchor=west] at (5.15,.55) {dimensions $2+0$};\n\\end{tikzpicture}\n\\end{center}\n\\captiontext{Squares refer to the local bases of the regular module. Crossed reference slots are absent from $W$; they are not additional basis vectors of $W$.}","$};\n\\foreach \\j in {0,...,4}{\\draw[fill=black!15,draw=black!35] (\\j*.95,1.3) rectangle ++(.75,.7);}\n\\foreach \\j in {0,1}{\\draw[fill=black!15,draw=black!35] (\\j*.95,.2) rectangle ++(.75,.7);}\n\\foreach \\j in {2,3,4}{\\draw[black!30] (\\j*.95,.2) rectangle ++(.75,.7);\\node[text=black!40] at (\\j*.95+.375,.55) {$\\times$};}\n\\draw[dashed] (1.80,-.15)--(1.80,2.35);\n\\node[font=\\footnotesize] at (.85,2.48) {$T_+$: 2 slots};\n\\node[font=\\footnotesize] at (3.25,2.48) {$T_-$: 3 slots};\n\\node[font=\\footnotesize,anchor=west] at (5.15,1.65) {dimensions $2+3$};\n\\node[font=\\footnotesize,anchor=west] at (5.15,.55) {dimensions $2+0$};\n\\end{tikzpicture}\n\\end{center}\n\\captiontext{Squares refer to the local bases of the regular module. Crossed reference slots are absent from $W$; they are not additional basis vectors of $W$.}","!35] (\\j*.95,1.3) rectangle ++(.75,.7);}\n\\foreach \\j in {0,1}{\\draw[fill=black!15,draw=black!35] (\\j*.95,.2) rectangle ++(.75,.7);}\n\\foreach \\j in {2,3,4}{\\draw[black!30] (\\j*.95,.2) rectangle ++(.75,.7);\\node[text=black!40] at (\\j*.95+.375,.55) {$\\times$};}\n\\draw[dashed] (1.80,-.15)--(1.80,2.35);\n\\node[font=\\footnotesize] at (.85,2.48) {$T_+$: 2 slots};\n\\node[font=\\footnotesize] at (3.25,2.48) {$T_-$: 3 slots};\n\\node[font=\\footnotesize,anchor=west] at (5.15,1.65) {dimensions $2+3$};\n\\node[font=\\footnotesize,anchor=west] at (5.15,.55) {dimensions $2+0$};\n\\end{tikzpicture}\n\\end{center}\n\\captiontext{Squares refer to the local bases of the regular module. Crossed reference slots are absent from $W$; they are not additional basis vectors of $W$.}","draw=black!35] (\\j*.95,.2) rectangle ++(.75,.7);}\n\\foreach \\j in {2,3,4}{\\draw[black!30] (\\j*.95,.2) rectangle ++(.75,.7);\\node[text=black!40] at (\\j*.95+.375,.55) {$\\times$};}\n\\draw[dashed] (1.80,-.15)--(1.80,2.35);\n\\node[font=\\footnotesize] at (.85,2.48) {$T_+$: 2 slots};\n\\node[font=\\footnotesize] at (3.25,2.48) {$T_-$: 3 slots};\n\\node[font=\\footnotesize,anchor=west] at (5.15,1.65) {dimensions $2+3$};\n\\node[font=\\footnotesize,anchor=west] at (5.15,.55) {dimensions $2+0$};\n\\end{tikzpicture}\n\\end{center}\n\\captiontext{Squares refer to the local bases of the regular module. Crossed reference slots are absent from $W$; they are not additional basis vectors of $W$.}","{\\draw[black!30] (\\j*.95,.2) rectangle ++(.75,.7);\\node[text=black!40] at (\\j*.95+.375,.55) {$\\times$};}\n\\draw[dashed] (1.80,-.15)--(1.80,2.35);\n\\node[font=\\footnotesize] at (.85,2.48) {$T_+$: 2 slots};\n\\node[font=\\footnotesize] at (3.25,2.48) {$T_-$: 3 slots};\n\\node[font=\\footnotesize,anchor=west] at (5.15,1.65) {dimensions $2+3$};\n\\node[font=\\footnotesize,anchor=west] at (5.15,.55) {dimensions $2+0$};\n\\end{tikzpicture}\n\\end{center}\n\\captiontext{Squares refer to the local bases of the regular module. Crossed reference slots are absent from $W$; they are not additional basis vectors of $W$.}","de[font=\\footnotesize] at (.85,2.48) {$T_+$: 2 slots};\n\\node[font=\\footnotesize] at (3.25,2.48) {$T_-$: 3 slots};\n\\node[font=\\footnotesize,anchor=west] at (5.15,1.65) {dimensions $2+3$};\n\\node[font=\\footnotesize,anchor=west] at (5.15,.55) {dimensions $2+0$};\n\\end{tikzpicture}\n\\end{center}\n\\captiontext{Squares refer to the local bases of the regular module. Crossed reference slots are absent from $W$; they are not additional basis vectors of $W$.}","slots};\n\\node[font=\\footnotesize] at (3.25,2.48) {$T_-$: 3 slots};\n\\node[font=\\footnotesize,anchor=west] at (5.15,1.65) {dimensions $2+3$};\n\\node[font=\\footnotesize,anchor=west] at (5.15,.55) {dimensions $2+0$};\n\\end{tikzpicture}\n\\end{center}\n\\captiontext{Squares refer to the local bases of the regular module. Crossed reference slots are absent from $W$; they are not additional basis vectors of $W$.}","\\begin{center}\n\\begin{tabular}{@{}lcc@{}}\\toprule\n & regular module $V$ & first-factor module $W$","midrule\n$\\dim e_+(-)$ & 2 & 2","$\\dim e_-(-)$ & 3 & 0","Support & both maximal points & only the $+$ point","Trace of $e_-$ & 3 & 0","\\head{A generalized packet is not its residue-field fiber}\nAt the point $z=1$, the full local packet $e_+V$ has dimension two, but its scalar fiber is\n\\[\nV\\otimes_{T,\\,z\\mapsto1}\\QQ\n\\cong T_+/(\\varepsilon)\\cong\\QQ.\n\\]\nThe packet has the basis $1,\\varepsilon$. Its scalar fiber kills $\\varepsilon$. The ordinary eigenspace is instead the kernel of multiplication by $\\varepsilon$, spanned by $\\varepsilon$. In this example the last two spaces have the same dimension, but they are obtained by different operations.","For $W$, the fiber at $z=-1$ is zero. At $z=1$ it is nonzero. The field-fiber criterion detects exactly the support of the supplied module.","% ---------------------------------------------------------------------------\n\\vpage{14}{At two, division is the obstruction}{A COEFFICIENT-PRIME BOUNDARY}{Main text: Section 5, Remark 8.5 and Section 11\\quad \\href{p12-route:visual-14}{Main statement 11.1}}\nLet $C_2=\\langle g:g^2=1\\rangle$. Over $\\QQ_2$, the two familiar sign projectors are\n\\[\ne_+=\\frac{1+g}{2},\\qquad e_-=\\frac{1-g}{2},\n\\qquad \\QQ_2[C_2]\\cong\\QQ_2\\times\\QQ_2.\n\\]\nThey do not belong to $\\ZZ_2[C_2]$. Therefore one cannot first use them integrally and then reduce modulo two.","---------------------\n\\vpage{14}{At two, division is the obstruction}{A COEFFICIENT-PRIME BOUNDARY}{Main text: Section 5, Remark 8.5 and Section 11\\quad \\href{p12-route:visual-14}{Main statement 11.1}}\nLet $C_2=\\langle g:g^2=1\\rangle$. Over $\\QQ_2$, the two familiar sign projectors are\n\\[\ne_+=\\frac{1+g}{2},\\qquad e_-=\\frac{1-g}{2},\n\\qquad \\QQ_2[C_2]\\cong\\QQ_2\\times\\QQ_2.\n\\]\nThey do not belong to $\\ZZ_2[C_2]$. Therefore one cannot first use them integrally and then reduce modulo two.","\\head{Compute the actual residual algebra}\nOver $\\FF_2$, put $\\delta=g-1=g+1$. Then $\\delta^2=g^2-2g+1=0$ and\n\\[\n\\FF_2[C_2]\\cong\\FF_2[\\delta]/(\\delta^2).\n\\]\nThe two sign eigenvalues have merged. The ring is local and retains a nilpotent direction instead of two residual projector pieces.","midrule\n$0$ & $0$ & yes","$1$ & $1$ & yes","$\\delta$ & $0$ & no","$1+\\delta$ & $1$ & no","bottomrule\n\\end{tabular}\n\\end{minipage}\\hfill\n\\begin{minipage}[c]{.47\\linewidth}\nIn the basis $1,\\delta$, multiplication by $g=1+\\delta$ is\n\\[\nG=\\begin{pmatrix}1&0","bottomrule\n\\end{tabular}\n\\end{minipage}\\hfill\n\\begin{minipage}[c]{.47\\linewidth}\nIn the basis $1,\\delta$, multiplication by $g=1+\\delta$ is\n\\[\nG=\\begin{pmatrix}1&0\\\\1&1\\end{pmatrix},\\qquad\nG^2=I\\quad\\text{over }\\FF_2.\n\\]\nIt is not the identity: $G-I$ is nonzero and square-zero.\n\\end{minipage}","1&1\\end{pmatrix},\\qquad\nG^2=I\\quad\\text{over }\\FF_2.\n\\]\nIt is not the identity: $G-I$ is nonzero and square-zero.\n\\end{minipage}","\\head{A trace can vanish while a packet is occupied}\nThe identity projector on $\\FF_2^2$ has rank two and trace $2=0$. Hence nonzero rank and nonzero trace are not equivalent in this characteristic. The localization and rank criteria still detect the nonzero space. The characteristic-zero qualification in Corollary 8.4 is essential.","\\head{Recovery without averaging}\nThe augmentation map sends $g$ to $1$ and has kernel $(g-1)$. Thus\n\\[\n\\ZZ_2[C_2]/(g-1)\\cong\\ZZ_2,\n\\qquad\n\\bigl(\\ZZ_2[C_2]^r\\bigr)/(g-1)\\ZZ_2[C_2]^r\\cong\\ZZ_2^r.\n\\]\nThese quotient identities require no division by two. They explain why augmentation and coinvariants remain usable even when sign-projector formulas do not.","% ---------------------------------------------------------------------------\n\\vpage{15}{Nonzero at every level is not enough}{INVERSE SYSTEMS / THE STABLE CORE}{Main text: Definition 8.9, Lemma 8.10 and Theorem 8.11\\quad \\href{p12-route:visual-15}{Main statement 8.11}}\nLet every level be $V_m=k^3$. For a transition matrix $F$, compatibility means $v_m=Fv_{m+1}$. The directions visible at level $m$ after looking $h$ levels ahead form $\\im(F^h)$.","------------------------------------------------------------\n\\vpage{15}{Nonzero at every level is not enough}{INVERSE SYSTEMS / THE STABLE CORE}{Main text: Definition 8.9, Lemma 8.10 and Theorem 8.11\\quad \\href{p12-route:visual-15}{Main statement 8.11}}\nLet every level be $V_m=k^3$. For a transition matrix $F$, compatibility means $v_m=Fv_{m+1}$. The directions visible at level $m$ after looking $h$ levels ahead form $\\im(F^h)$.","--------------------------------------\n\\vpage{15}{Nonzero at every level is not enough}{INVERSE SYSTEMS / THE STABLE CORE}{Main text: Definition 8.9, Lemma 8.10 and Theorem 8.11\\quad \\href{p12-route:visual-15}{Main statement 8.11}}\nLet every level be $V_m=k^3$. For a transition matrix $F$, compatibility means $v_m=Fv_{m+1}$. The directions visible at level $m$ after looking $h$ levels ahead form $\\im(F^h)$.","\\begin{minipage}[t]{.48\\linewidth}\\centering\n{\\sffamily\\bfseries A surviving direction}\\par\n\\[\nP=\\begin{pmatrix}1&0&0\\\\0&0&0\\\\0&0&0\\end{pmatrix},\\qquad P^2=P.\n\\]\n\\vfig[.45]{core_projection}\n\\captiontext{At every positive horizon, precisely the $e_1$ direction remains visible.}\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{.48\\linewidth}\\centering\n{\\sffamily\\bfseries Every direction eventually dies}\\par\n\\[\nN=\\begin{pmatrix}0&1&0","0&0&0\\end{pmatrix},\\qquad P^2=P.\n\\]\n\\vfig[.45]{core_projection}\n\\captiontext{At every positive horizon, precisely the $e_1$ direction remains visible.}\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{.48\\linewidth}\\centering\n{\\sffamily\\bfseries Every direction eventually dies}\\par\n\\[\nN=\\begin{pmatrix}0&1&0","0&0&0\\end{pmatrix},\\qquad P^2=P.\n\\]\n\\vfig[.45]{core_projection}\n\\captiontext{At every positive horizon, precisely the $e_1$ direction remains visible.}\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{.48\\linewidth}\\centering\n{\\sffamily\\bfseries Every direction eventually dies}\\par\n\\[\nN=\\begin{pmatrix}0&1&0\\\\0&0&1\\\\0&0&0\\end{pmatrix},\\qquad N^3=0.\n\\]\n\\vfig[.45]{core_nilpotent}\n\\captiontext{Image dimensions are $3,2,1,0,0,\\ldots$, although every level still has dimension three.}\n\\end{minipage}","0&0&0\\end{pmatrix},\\qquad N^3=0.\n\\]\n\\vfig[.45]{core_nilpotent}\n\\captiontext{Image dimensions are $3,2,1,0,0,\\ldots$, although every level still has dimension three.}\n\\end{minipage}","\\head{Now compute the limits, not just their level dimensions}\nFor the $P$-system, the limit consists of the constant compatible sequences $(a e_1,a e_1,\\ldots)$, so it is isomorphic to $k$. For the $N$-system, every compatible sequence satisfies\n\\[\nv_m=N^3v_{m+3}=0\\quad\\text{for every }m,\n\\]\nand the limit is zero. This is an all-level argument from the identity $N^3=0$, not a finite experimental guess.","\\head{What the stable core measures}\nIn a general inverse system of finite-dimensional spaces,\n\\[\nY_m=\\bigcap_{n\\ge m}\\im(f_{n,m})\n\\]\ncontains exactly the vectors visible from arbitrarily remote levels. The paper proves that the transition maps restrict to surjections $Y_{m+1}\\twoheadrightarrow Y_m$. A prescribed nonzero vector in $Y_1$ can then be lifted recursively to a coherent nonzero sequence.","\\begin{minipage}[t]{.44\\linewidth}\\centering\n{\\sffamily\\bfseries Fixed carrier $k^5$}\\par\\vspace{4pt}\n\\vfig[.75]{nested_finite}\n\\captiontext{The projectors retain initial basis directions with dimensions $5,4,4,2,1,1,\\ldots$. In this explicitly defined example the projector is constant from level five onward.}\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{.52\\linewidth}\\centering\n{\\sffamily\\bfseries Infinite carrier $\\ell^2(\\mathbb N)$}\\par\\vspace{4pt}\n\\vfig[.91]{infinite_tail}\n\\captiontext{Row $m$ retains all basis vectors $e_j$ with $j\\ge m$. Every row continues infinitely to the right. The display is a finite window, not a truncation of the actual carrier.}\n\\end{minipage}","[t]{.44\\linewidth}\\centering\n{\\sffamily\\bfseries Fixed carrier $k^5$}\\par\\vspace{4pt}\n\\vfig[.75]{nested_finite}\n\\captiontext{The projectors retain initial basis directions with dimensions $5,4,4,2,1,1,\\ldots$. In this explicitly defined example the projector is constant from level five onward.}\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{.52\\linewidth}\\centering\n{\\sffamily\\bfseries Infinite carrier $\\ell^2(\\mathbb N)$}\\par\\vspace{4pt}\n\\vfig[.91]{infinite_tail}\n\\captiontext{Row $m$ retains all basis vectors $e_j$ with $j\\ge m$. Every row continues infinitely to the right. The display is a finite window, not a truncation of the actual carrier.}\n\\end{minipage}","{4pt}\n\\vfig[.75]{nested_finite}\n\\captiontext{The projectors retain initial basis directions with dimensions $5,4,4,2,1,1,\\ldots$. In this explicitly defined example the projector is constant from level five onward.}\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{.52\\linewidth}\\centering\n{\\sffamily\\bfseries Infinite carrier $\\ell^2(\\mathbb N)$}\\par\\vspace{4pt}\n\\vfig[.91]{infinite_tail}\n\\captiontext{Row $m$ retains all basis vectors $e_j$ with $j\\ge m$. Every row continues infinitely to the right. The display is a finite window, not a truncation of the actual carrier.}\n\\end{minipage}","}\n\\captiontext{The projectors retain initial basis directions with dimensions $5,4,4,2,1,1,\\ldots$. In this explicitly defined example the projector is constant from level five onward.}\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{.52\\linewidth}\\centering\n{\\sffamily\\bfseries Infinite carrier $\\ell^2(\\mathbb N)$}\\par\\vspace{4pt}\n\\vfig[.91]{infinite_tail}\n\\captiontext{Row $m$ retains all basis vectors $e_j$ with $j\\ge m$. Every row continues infinitely to the right. The display is a finite window, not a truncation of the actual carrier.}\n\\end{minipage}","ext{The projectors retain initial basis directions with dimensions $5,4,4,2,1,1,\\ldots$. In this explicitly defined example the projector is constant from level five onward.}\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{.52\\linewidth}\\centering\n{\\sffamily\\bfseries Infinite carrier $\\ell^2(\\mathbb N)$}\\par\\vspace{4pt}\n\\vfig[.91]{infinite_tail}\n\\captiontext{Row $m$ retains all basis vectors $e_j$ with $j\\ge m$. Every row continues infinitely to the right. The display is a finite window, not a truncation of the actual carrier.}\n\\end{minipage}","\\head{Why the finite chain eventually stops changing}\nSuppose idempotents satisfy\n\\[\ne_{m+1}e_m=e_me_{m+1}=e_{m+1}.\n\\]\nTheir ranges are nested. In dimension $d$, nonzero ranges allow at most $d-1$ strict dimension drops. There may be long gaps between drops, but only finitely many drops. The chain therefore becomes constant and has nonzero intersection.","\\begin{insight}\nA single repeated dimension does \\emph{not} certify final stabilization. The finite example deliberately has a plateau $4,4$ before another drop. Finite dimension bounds the number of strict drops, not the time of the last one.\n\\end{insight}","\\head{Why every infinite tail can be nonzero while their intersection vanishes}\nLet $P_m$ project onto the closed span of $e_m,e_{m+1},\\ldots$. Each range is nonzero. But a vector belonging to every range has its $j$-th coordinate zero once $m>j$. Every coordinate is therefore zero:\n\\[\n\\bigcap_{m\\ge1}P_m\\ell^2(\\mathbb N)=\\{0\\}.\n\\]","Comparison & A ring map $f:R\\to T$ with the stated support comparison, or the suitable isomorphism and carrier identification. & Support is transported at compatible points.","Fiber & A Hecke point $y:T\\to E'$ over the same field, with $y\\circ f=x$. & A nonzero scalar fiber and an actual eigensystem; in the derived case, one survives in a cohomological degree.","\\head{Support alone does not supply a point over the same field}\nThe inclusion $\\QQ\\to\\QQ(i)$ preserves full support of the nonzero scalar-extended carrier. But no $\\QQ$-algebra map $\\QQ(i)\\to\\QQ$ exists: the image of $i$ would be a rational number $a$ with $a^2=-1$. Thus one cannot infer the required Hecke point $y$ merely from a support statement.","\\head{Propagation needs two existing families and the same automorphic object}\nAfter producing an automorphic object $\\pi$, retain \\emph{that same} $\\pi$. The propagation statement assumes both the original compatible family and the family attached to $\\pi$ already exist, with the common Frobenius polynomial data and the stated comparison hypotheses. It then identifies corresponding members. It does not construct a missing family.","\\head{Three short proof-reading tests}\n\\textbf{Can a module have full support but fail to be faithful?} Yes. On \\pagerefV{10}, the module $k$ over $k[[x]]/(x^2)$ has a nonzero nilpotent annihilator and full support.","\\textbf{Do equal support sets force equal Fitting ideals?} No. On \\pagerefV{7}, $(xy)$ and $(x^2y)$ have the same vanishing set but different infinitesimal behavior.","\\textbf{Do nonzero finite levels force a nonzero inverse limit?} No. On \\pagerefV{15}, $N^3=0$ forces every compatible sequence to vanish, although every level is $k^3$.","% ---------------------------------------------------------------------------\n\\setlength{\\parskip}{5pt}\n\\vpage{19}{Complete the actual topology}{TOPOLOGICAL ALGEBRA / THE COMPARISON MAP}{Main text: Theorem 3.1, Corollary 3.2 and Proposition 3.3\\quad \\href{p12-route:visual-19}{Main statement 3.1}}\nThe hat in $\\whot$ names an actual completion. For the finite family of local rings, start with the ordinary tensor and its ideal of definition:\n\\[\n S=\\bigotimes_{i=1}^s R_i,\\qquad\n \\mathfrak a=\\m_{\\OO}S+\\sum_i\\m_{R_i}S,\\qquad L=\\sum_i I_iS.\n\\]\nThe comparison uses the $\\mathfrak a$-adic additive uniformity on $S$, the image ideal topology on $S/L$, and the closed extended ideal $J=\\overline{L\\widehat S}$.","--------------------------------\n\\setlength{\\parskip}{5pt}\n\\vpage{19}{Complete the actual topology}{TOPOLOGICAL ALGEBRA / THE COMPARISON MAP}{Main text: Theorem 3.1, Corollary 3.2 and Proposition 3.3\\quad \\href{p12-route:visual-19}{Main statement 3.1}}\nThe hat in $\\whot$ names an actual completion. For the finite family of local rings, start with the ordinary tensor and its ideal of definition:\n\\[\n S=\\bigotimes_{i=1}^s R_i,\\qquad\n \\mathfrak a=\\m_{\\OO}S+\\sum_i\\m_{R_i}S,\\qquad L=\\sum_i I_iS.\n\\]\nThe comparison uses the $\\mathfrak a$-adic additive uniformity on $S$, the image ideal topology on $S/L$, and the closed extended ideal $J=\\overline{L\\widehat S}$.","HE COMPARISON MAP}{Main text: Theorem 3.1, Corollary 3.2 and Proposition 3.3\\quad \\href{p12-route:visual-19}{Main statement 3.1}}\nThe hat in $\\whot$ names an actual completion. For the finite family of local rings, start with the ordinary tensor and its ideal of definition:\n\\[\n S=\\bigotimes_{i=1}^s R_i,\\qquad\n \\mathfrak a=\\m_{\\OO}S+\\sum_i\\m_{R_i}S,\\qquad L=\\sum_i I_iS.\n\\]\nThe comparison uses the $\\mathfrak a$-adic additive uniformity on $S$, the image ideal topology on $S/L$, and the closed extended ideal $J=\\overline{L\\widehat S}$.","Corollary 3.2 and Proposition 3.3\\quad \\href{p12-route:visual-19}{Main statement 3.1}}\nThe hat in $\\whot$ names an actual completion. For the finite family of local rings, start with the ordinary tensor and its ideal of definition:\n\\[\n S=\\bigotimes_{i=1}^s R_i,\\qquad\n \\mathfrak a=\\m_{\\OO}S+\\sum_i\\m_{R_i}S,\\qquad L=\\sum_i I_iS.\n\\]\nThe comparison uses the $\\mathfrak a$-adic additive uniformity on $S$, the image ideal topology on $S/L$, and the closed extended ideal $J=\\overline{L\\widehat S}$.","3\\quad \\href{p12-route:visual-19}{Main statement 3.1}}\nThe hat in $\\whot$ names an actual completion. For the finite family of local rings, start with the ordinary tensor and its ideal of definition:\n\\[\n S=\\bigotimes_{i=1}^s R_i,\\qquad\n \\mathfrak a=\\m_{\\OO}S+\\sum_i\\m_{R_i}S,\\qquad L=\\sum_i I_iS.\n\\]\nThe comparison uses the $\\mathfrak a$-adic additive uniformity on $S$, the image ideal topology on $S/L$, and the closed extended ideal $J=\\overline{L\\widehat S}$.","n statement 3.1}}\nThe hat in $\\whot$ names an actual completion. For the finite family of local rings, start with the ordinary tensor and its ideal of definition:\n\\[\n S=\\bigotimes_{i=1}^s R_i,\\qquad\n \\mathfrak a=\\m_{\\OO}S+\\sum_i\\m_{R_i}S,\\qquad L=\\sum_i I_iS.\n\\]\nThe comparison uses the $\\mathfrak a$-adic additive uniformity on $S$, the image ideal topology on $S/L$, and the closed extended ideal $J=\\overline{L\\widehat S}$.","\\vfig[.82]{completion_square}\n\\captiontext{The square names the actual quotient and completion maps. The lower identification is bicontinuous and preserves the coefficient map and every canonical factor map. The square does not assert that completion leaves $S$ unchanged.}","\\head{The explicit model revisited}\nFor $R_1=\\ZZ_5[[x]]$ and $R_2=\\ZZ_5[[y]]$, the completion of their ordinary tensor in this topology is $\\ZZ_5[[x,y]]$. With $L$ generated by $x^2,y$, the lower quotient is $\\ZZ_5[\\varepsilon]/(\\varepsilon^2)$ with its $(5,\\varepsilon)$-adic topology. The containment on \\pagerefV{5} identifies this with its $5$-adic topology.","midrule\nEmpty family & $S=\\OO$ and $\\mathfrak a=\\m_\\OO$. The scalar adic topology remains; an empty sum of factor ideals does not erase it.","All $I_i=0$ & $J=0$; the comparison returns the completed tensor itself.","Some $I_i=R_i$ & $J=\\widehat S$ and both sides are the zero ring. This comparison does not give a local ring with residue field $k$.","For global base change, the structural map $A\\to B$ belongs to the continuous local coefficient-algebra setting. Then $B\\whot_A(A/J)\\cong B/\\overline{JB}$. For a continuous algebra point to a complete Hausdorff target, closedness of its kernel gives the factorization test. Locality of a point, when supplied, passes to its unique quotient factor.","% ---------------------------------------------------------------------------\n\\vpage{20}{A finite adequacy calculation}{RESIDUAL IMAGE / ONE EXPLICIT CERTIFICATE}{Main text: Theorem 4.2; label \\texttt{thm:s3\\_adequacy}. Adequacy convention: external Standard Fact 4.1.\\quad \\href{p12-route:visual-20}{Main statement 4.2}}\nOver $k=\\FF_7$, the standard representation of $S_3$ has matrices\n\\[\n s=\\begin{pmatrix}6&1","% ---------------------------------------------------------------------------\n\\vpage{20}{A finite adequacy calculation}{RESIDUAL IMAGE / ONE EXPLICIT CERTIFICATE}{Main text: Theorem 4.2; label \\texttt{thm:s3\\_adequacy}. Adequacy convention: external Standard Fact 4.1.\\quad \\href{p12-route:visual-20}{Main statement 4.2}}\nOver $k=\\FF_7$, the standard representation of $S_3$ has matrices\n\\[\n s=\\begin{pmatrix}6&1\\\\0&1\\end{pmatrix},\\qquad\n t=\\begin{pmatrix}0&6\\\\1&6\\end{pmatrix}.\n\\]\nAll entries and calculations on this page are modulo seven. Direct multiplication gives $s^2=t^3=I$ and $sts=t^{-1}$. Their eigenvalues are $\\{-1,1\\}$ and $\\{2,4\\}$. Distinct eigenvalues allow an exact projector formula\n\\[\nP_{D,\\alpha}=(\\alpha-\\beta)^{-1}(D-\\beta I).\n\\]","1&6\\end{pmatrix}.\n\\]\nAll entries and calculations on this page are modulo seven. Direct multiplication gives $s^2=t^3=I$ and $sts=t^{-1}$. Their eigenvalues are $\\{-1,1\\}$ and $\\{2,4\\}$. Distinct eigenvalues allow an exact projector formula\n\\[\nP_{D,\\alpha}=(\\alpha-\\beta)^{-1}(D-\\beta I).\n\\]","Write $W(a,b,c)=\\left(\\begin{smallmatrix}a&b\\\\c&-a\\end{smallmatrix}\\right)$. The manuscript exhibits $Z=W(2,3,4)$, $A=W(1,1,2)$ and $B=W(1,5,6)$; their coordinate determinant is $3$. Thus they are a basis of the trace-zero adjoint. Conjugation splits this space as $\\operatorname{sgn}\\oplus V$. In the drawing the standard witness is $W(6,4,3)=3A+3B$.","c&-a\\end{smallmatrix}\\right)$. The manuscript exhibits $Z=W(2,3,4)$, $A=W(1,1,2)$ and $B=W(1,5,6)$; their coordinate determinant is $3$. Thus they are a basis of the trace-zero adjoint. Conjugation splits this space as $\\operatorname{sgn}\\oplus V$. In the drawing the standard witness is $W(6,4,3)=3A+3B$.","\\head{Why the cohomology argument is uniform in the module}\nSince $|S_3|=6$ is a unit in $\\FF_7$, a cocycle $f:S_3\\to M$ is a coboundary: set $u=-6^{-1}\\sum_h f(h)$ and sum $f(gh)=g f(h)+f(g)$ to obtain $gu-u=f(g)$. This proves $H^1(S_3,M)=0$ for every $\\FF_7[S_3]$-module $M$. The printed invariant calculation separately proves $H^0(S_3,\\operatorname{ad}^0V)=0$.","\\head{What makes this a finite certificate}\nThe manuscript also checks absolute irreducibility and projector detection on every irreducible adjoint submodule. In this example $7\\nmid2$, and every group element's eigenvalues lie in $\\FF_7$, as required by the stated convention. The two displayed pairings alone are a worked part of that complete certificate.","\\takeaway{The calculation is exact for $S_3/\\FF_7$. Applying an automorphy lifting theorem still requires its actual residual image hypotheses and residual occupancy. No coefficient-prime-two adequacy theorem follows from this picture.}","% ---------------------------------------------------------------------------\n\\vpage{21}{Augmentation in two variables}{AUXILIARY ALGEBRA / AN EXACT MONOMIAL BASIS}{Main text: Proposition 5.1, Theorem 5.2 and Theorem 5.4\\quad \\href{p12-route:visual-21}{Main statement 5.2}}\nTake $\\Delta=C_4\\times C_2$ in residue characteristic two. If $g$ and $h$ generate its two factors, put $X=g-1$, $Y=h-1$. Then\n\\[\n\\FF_2[\\Delta]\\cong\\FF_2[X,Y]/(X^4,Y^2).\n\\]\nThe identity $(1+X)^4-1=X^4$ and its $Y$ analogue provide the relations. The eight surviving monomials provide the exact dimension $|\\Delta|=8$ and establish the isomorphism after surjectivity.","--------------------------------------------------------------------------\n\\vpage{21}{Augmentation in two variables}{AUXILIARY ALGEBRA / AN EXACT MONOMIAL BASIS}{Main text: Proposition 5.1, Theorem 5.2 and Theorem 5.4\\quad \\href{p12-route:visual-21}{Main statement 5.2}}\nTake $\\Delta=C_4\\times C_2$ in residue characteristic two. If $g$ and $h$ generate its two factors, put $X=g-1$, $Y=h-1$. Then\n\\[\n\\FF_2[\\Delta]\\cong\\FF_2[X,Y]/(X^4,Y^2).\n\\]\nThe identity $(1+X)^4-1=X^4$ and its $Y$ analogue provide the relations. The eight surviving monomials provide the exact dimension $|\\Delta|=8$ and establish the isomorphism after surjectivity.","----------------\n\\vpage{21}{Augmentation in two variables}{AUXILIARY ALGEBRA / AN EXACT MONOMIAL BASIS}{Main text: Proposition 5.1, Theorem 5.2 and Theorem 5.4\\quad \\href{p12-route:visual-21}{Main statement 5.2}}\nTake $\\Delta=C_4\\times C_2$ in residue characteristic two. If $g$ and $h$ generate its two factors, put $X=g-1$, $Y=h-1$. Then\n\\[\n\\FF_2[\\Delta]\\cong\\FF_2[X,Y]/(X^4,Y^2).\n\\]\nThe identity $(1+X)^4-1=X^4$ and its $Y$ analogue provide the relations. The eight surviving monomials provide the exact dimension $|\\Delta|=8$ and establish the isomorphism after surjectivity.","21}{Augmentation in two variables}{AUXILIARY ALGEBRA / AN EXACT MONOMIAL BASIS}{Main text: Proposition 5.1, Theorem 5.2 and Theorem 5.4\\quad \\href{p12-route:visual-21}{Main statement 5.2}}\nTake $\\Delta=C_4\\times C_2$ in residue characteristic two. If $g$ and $h$ generate its two factors, put $X=g-1$, $Y=h-1$. Then\n\\[\n\\FF_2[\\Delta]\\cong\\FF_2[X,Y]/(X^4,Y^2).\n\\]\nThe identity $(1+X)^4-1=X^4$ and its $Y$ analogue provide the relations. The eight surviving monomials provide the exact dimension $|\\Delta|=8$ and establish the isomorphism after surjectivity.","position 5.1, Theorem 5.2 and Theorem 5.4\\quad \\href{p12-route:visual-21}{Main statement 5.2}}\nTake $\\Delta=C_4\\times C_2$ in residue characteristic two. If $g$ and $h$ generate its two factors, put $X=g-1$, $Y=h-1$. Then\n\\[\n\\FF_2[\\Delta]\\cong\\FF_2[X,Y]/(X^4,Y^2).\n\\]\nThe identity $(1+X)^4-1=X^4$ and its $Y$ analogue provide the relations. The eight surviving monomials provide the exact dimension $|\\Delta|=8$ and establish the isomorphism after surjectivity.","\\head{The residual ring is local}\nThe augmentation ideal is $(X,Y)$, its quotient is $\\FF_2$, and $(X,Y)^5=0$: a degree-five monomial has $X$-exponent at least four or $Y$-exponent at least two. Every element outside $(X,Y)$ has the form $1+n$ for nilpotent $n$, with inverse the finite geometric series. Thus $(X,Y)$ is the unique maximal ideal, and the only idempotents are $0,1$.","\\head{A quotient of groups gives a quotient of algebras}\nMap $C_4\\times C_2\\to C_2$ by $g\\mapsto\\bar g$, $h\\mapsto1$. The induced algebra map sends $X$ to $\\bar X$ with $\\bar X^2=0$ and sends $Y$ to zero. It is surjective. Augmentation commutes with this map because every group basis element is sent to another group element of augmentation one.","\\head{Recover a free carrier, with its base ring named}\nFor $\\Lambda=\\ZZ_2[\\Delta]$ and the right module $M=\\Lambda^r$,\n\\[\nM\\otimes_{\\Lambda,\\epsilon}\\ZZ_2\n\\cong M/M\\ker\\epsilon\\cong\\ZZ_2^r.\n\\]\nThe quotient applies augmentation to each basis coordinate. This remains valid for nonabelian finite two groups with the stated right-module convention. It establishes the rank over the coefficient ring after coinvariants, without establishing full support over a separate deformation ring.","-------------------------------------\n\\vpage{22}{A finite forest selects a branch}{FINITE OCCUPANCY / MAPS NEED NOT BE SURJECTIVE}{Main text: Theorem 10.3 and Corollary 10.4\\quad \\href{p12-route:visual-22}{Main statement 10.3}}\nA finite nonempty set at every precision, with total reduction maps, can select a compatible branch even when some nodes have no deeper lift. Here is an explicit system:\n\\[\nX_1=\\{r\\},\\quad X_2=\\{a,b\\},\\quad X_3=\\{c,d\\},\\quad\nX_n=\\{e_n\\}\\quad(n\\ge4).\n\\]\nBoth $a,b$ reduce to $r$; both $c,d$ reduce to $a$; $e_4$ reduces to $c$; and $e_{n+1}$ reduces to $e_n$ for every $n\\ge4$.","TE OCCUPANCY / MAPS NEED NOT BE SURJECTIVE}{Main text: Theorem 10.3 and Corollary 10.4\\quad \\href{p12-route:visual-22}{Main statement 10.3}}\nA finite nonempty set at every precision, with total reduction maps, can select a compatible branch even when some nodes have no deeper lift. Here is an explicit system:\n\\[\nX_1=\\{r\\},\\quad X_2=\\{a,b\\},\\quad X_3=\\{c,d\\},\\quad\nX_n=\\{e_n\\}\\quad(n\\ge4).\n\\]\nBoth $a,b$ reduce to $r$; both $c,d$ reduce to $a$; $e_4$ reduces to $c$; and $e_{n+1}$ reduces to $e_n$ for every $n\\ge4$.","APS NEED NOT BE SURJECTIVE}{Main text: Theorem 10.3 and Corollary 10.4\\quad \\href{p12-route:visual-22}{Main statement 10.3}}\nA finite nonempty set at every precision, with total reduction maps, can select a compatible branch even when some nodes have no deeper lift. Here is an explicit system:\n\\[\nX_1=\\{r\\},\\quad X_2=\\{a,b\\},\\quad X_3=\\{c,d\\},\\quad\nX_n=\\{e_n\\}\\quad(n\\ge4).\n\\]\nBoth $a,b$ reduce to $r$; both $c,d$ reduce to $a$; $e_4$ reduces to $c$; and $e_{n+1}$ reduces to $e_n$ for every $n\\ge4$.","OT BE SURJECTIVE}{Main text: Theorem 10.3 and Corollary 10.4\\quad \\href{p12-route:visual-22}{Main statement 10.3}}\nA finite nonempty set at every precision, with total reduction maps, can select a compatible branch even when some nodes have no deeper lift. Here is an explicit system:\n\\[\nX_1=\\{r\\},\\quad X_2=\\{a,b\\},\\quad X_3=\\{c,d\\},\\quad\nX_n=\\{e_n\\}\\quad(n\\ge4).\n\\]\nBoth $a,b$ reduce to $r$; both $c,d$ reduce to $a$; $e_4$ reduces to $c$; and $e_{n+1}$ reduces to $e_n$ for every $n\\ge4$.","}{Main text: Theorem 10.3 and Corollary 10.4\\quad \\href{p12-route:visual-22}{Main statement 10.3}}\nA finite nonempty set at every precision, with total reduction maps, can select a compatible branch even when some nodes have no deeper lift. Here is an explicit system:\n\\[\nX_1=\\{r\\},\\quad X_2=\\{a,b\\},\\quad X_3=\\{c,d\\},\\quad\nX_n=\\{e_n\\}\\quad(n\\ge4).\n\\]\nBoth $a,b$ reduce to $r$; both $c,d$ reduce to $a$; $e_4$ reduces to $c$; and $e_{n+1}$ reduces to $e_n$ for every $n\\ge4$.","n text: Theorem 10.3 and Corollary 10.4\\quad \\href{p12-route:visual-22}{Main statement 10.3}}\nA finite nonempty set at every precision, with total reduction maps, can select a compatible branch even when some nodes have no deeper lift. Here is an explicit system:\n\\[\nX_1=\\{r\\},\\quad X_2=\\{a,b\\},\\quad X_3=\\{c,d\\},\\quad\nX_n=\\{e_n\\}\\quad(n\\ge4).\n\\]\nBoth $a,b$ reduce to $r$; both $c,d$ reduce to $a$; $e_4$ reduces to $c$; and $e_{n+1}$ reduces to $e_n$ for every $n\\ge4$.","0.3 and Corollary 10.4\\quad \\href{p12-route:visual-22}{Main statement 10.3}}\nA finite nonempty set at every precision, with total reduction maps, can select a compatible branch even when some nodes have no deeper lift. Here is an explicit system:\n\\[\nX_1=\\{r\\},\\quad X_2=\\{a,b\\},\\quad X_3=\\{c,d\\},\\quad\nX_n=\\{e_n\\}\\quad(n\\ge4).\n\\]\nBoth $a,b$ reduce to $r$; both $c,d$ reduce to $a$; $e_4$ reduces to $c$; and $e_{n+1}$ reduces to $e_n$ for every $n\\ge4$.","orollary 10.4\\quad \\href{p12-route:visual-22}{Main statement 10.3}}\nA finite nonempty set at every precision, with total reduction maps, can select a compatible branch even when some nodes have no deeper lift. Here is an explicit system:\n\\[\nX_1=\\{r\\},\\quad X_2=\\{a,b\\},\\quad X_3=\\{c,d\\},\\quad\nX_n=\\{e_n\\}\\quad(n\\ge4).\n\\]\nBoth $a,b$ reduce to $r$; both $c,d$ reduce to $a$; $e_4$ reduces to $c$; and $e_{n+1}$ reduces to $e_n$ for every $n\\ge4$.","ref{p12-route:visual-22}{Main statement 10.3}}\nA finite nonempty set at every precision, with total reduction maps, can select a compatible branch even when some nodes have no deeper lift. Here is an explicit system:\n\\[\nX_1=\\{r\\},\\quad X_2=\\{a,b\\},\\quad X_3=\\{c,d\\},\\quad\nX_n=\\{e_n\\}\\quad(n\\ge4).\n\\]\nBoth $a,b$ reduce to $r$; both $c,d$ reduce to $a$; $e_4$ reduces to $c$; and $e_{n+1}$ reduces to $e_n$ for every $n\\ge4$.","sual-22}{Main statement 10.3}}\nA finite nonempty set at every precision, with total reduction maps, can select a compatible branch even when some nodes have no deeper lift. Here is an explicit system:\n\\[\nX_1=\\{r\\},\\quad X_2=\\{a,b\\},\\quad X_3=\\{c,d\\},\\quad\nX_n=\\{e_n\\}\\quad(n\\ge4).\n\\]\nBoth $a,b$ reduce to $r$; both $c,d$ reduce to $a$; $e_4$ reduces to $c$; and $e_{n+1}$ reduces to $e_n$ for every $n\\ge4$.","\\vfig[.66]{finite_forest}\n\\captiontext{Every displayed edge is an actual reduction map. The formula specifies the infinite continuation. The chosen branch is $r,a,c,e_4,e_5,\\ldots$. The nodes $b,d$ are admissible at their own precision and have no successor.}","\\head{What the compactness argument proves}\nFor general finite levels, the reduction forest is a locally finite rooted tree with vertices at every depth. An infinite branch exists. Surjectivity is sufficient but is not necessary: the map $X_3\\to X_2$ above misses $b$, and $X_4\\to X_3$ misses $d$.","% ---------------------------------------------------------------------------\n\\vpage{23}{Classes are not matrix frames}{COMPANION COMPLETION / ACTUAL REPRESENTATIVES}{Main text: revised Theorem 10.5 and its preceding admissibility hypotheses\\quad \\href{p12-route:visual-23}{Main statement 10.5}}\nLet $\\Gamma=C_2$ with the discrete topology, and represent its generator by $D=\\operatorname{diag}(1,-1)$ over $\\ZZ_5$. At precision $n$, its conjugacy class is an admissible class in $\\GL_2(\\ZZ/5^n\\ZZ)$. These classes are compatible. Arbitrary representatives need not be.","---------------------------------------------------------------------\n\\vpage{23}{Classes are not matrix frames}{COMPANION COMPLETION / ACTUAL REPRESENTATIVES}{Main text: revised Theorem 10.5 and its preceding admissibility hypotheses\\quad \\href{p12-route:visual-23}{Main statement 10.5}}\nLet $\\Gamma=C_2$ with the discrete topology, and represent its generator by $D=\\operatorname{diag}(1,-1)$ over $\\ZZ_5$. At precision $n$, its conjugacy class is an admissible class in $\\GL_2(\\ZZ/5^n\\ZZ)$. These classes are compatible. Arbitrary representatives need not be.","-----------------------------------------------\n\\vpage{23}{Classes are not matrix frames}{COMPANION COMPLETION / ACTUAL REPRESENTATIVES}{Main text: revised Theorem 10.5 and its preceding admissibility hypotheses\\quad \\href{p12-route:visual-23}{Main statement 10.5}}\nLet $\\Gamma=C_2$ with the discrete topology, and represent its generator by $D=\\operatorname{diag}(1,-1)$ over $\\ZZ_5$. At precision $n$, its conjugacy class is an admissible class in $\\GL_2(\\ZZ/5^n\\ZZ)$. These classes are compatible. Arbitrary representatives need not be.","age{23}{Classes are not matrix frames}{COMPANION COMPLETION / ACTUAL REPRESENTATIVES}{Main text: revised Theorem 10.5 and its preceding admissibility hypotheses\\quad \\href{p12-route:visual-23}{Main statement 10.5}}\nLet $\\Gamma=C_2$ with the discrete topology, and represent its generator by $D=\\operatorname{diag}(1,-1)$ over $\\ZZ_5$. At precision $n$, its conjugacy class is an admissible class in $\\GL_2(\\ZZ/5^n\\ZZ)$. These classes are compatible. Arbitrary representatives need not be.","\\vfig[.70]{frames_example}\n\\captiontext{The upper representatives are conjugate to the same diagonal involution, but their reductions are unequal matrices. The lower choices use $D$ consistently at both precisions. All entries displayed are canonical residues.}","\\head{Calculate the mismatch}\nFor $Q=\\left(\\begin{smallmatrix}1&1\\\\0&1\\end{smallmatrix}\\right)$,\n\\[\nQDQ^{-1}=\\begin{pmatrix}1&-2","0&1\\end{smallmatrix}\\right)$,\n\\[\nQDQ^{-1}=\\begin{pmatrix}1&-2\\\\0&-1\\end{pmatrix}\n\\equiv\\begin{pmatrix}1&23\\\\0&24\\end{pmatrix}\\pmod{25}.\n\\]\nIt reduces to $\\left(\\begin{smallmatrix}1&3","0&24\\end{pmatrix}\\pmod{25}.\n\\]\nIt reduces to $\\left(\\begin{smallmatrix}1&3\\\\0&4\\end{smallmatrix}\\right)$ modulo five, not the chosen diagonal matrix $\\left(\\begin{smallmatrix}1&0","0&4\\end{smallmatrix}\\right)$ modulo five, not the chosen diagonal matrix $\\left(\\begin{smallmatrix}1&0\\\\0&4\\end{smallmatrix}\\right)$. Nevertheless, their residual conjugacy classes agree.","the finitely many admissible classes therefore has only finitely many representatives. Take \\emph{all} actual continuous representatives of the admissible classes at each level. Reduction is a total map between these finite nonempty sets because admissibility is preserved under reduction. A branch then gives strictly compatible matrices. Completeness reconstructs entries and inverse entries over $\\OO_\\lambda$; finite-level continuity gives continuity in the ideal-adic topology.","% ---------------------------------------------------------------------------\n\\vpage{24}{Equations survive every precision}{ADIC EFFECTIVITY / DECLARED POLYNOMIAL CONDITIONS}{Main text: revised Theorem 10.5 and Corollary 3.2\\quad \\href{p12-route:visual-24}{Main statement 10.5}}\nThe preceding involution gives an exact polynomial condition. For a matrix $A=(a_{ij})$, require all entries of $A^2-I$ to vanish. In\n\\[\nB=\\ZZ_5[a_{11},a_{12},a_{21},a_{22}],\n\\]\nthe ideal $J$ is generated by\n\\[\n\\begin{aligned}\na_{11}^2+a_{12}a_{21}-1,&\\qquad a_{12}(a_{11}+a_{22}),","--------------------------------------------------------------\n\\vpage{24}{Equations survive every precision}{ADIC EFFECTIVITY / DECLARED POLYNOMIAL CONDITIONS}{Main text: revised Theorem 10.5 and Corollary 3.2\\quad \\href{p12-route:visual-24}{Main statement 10.5}}\nThe preceding involution gives an exact polynomial condition. For a matrix $A=(a_{ij})$, require all entries of $A^2-I$ to vanish. In\n\\[\nB=\\ZZ_5[a_{11},a_{12},a_{21},a_{22}],\n\\]\nthe ideal $J$ is generated by\n\\[\n\\begin{aligned}\na_{11}^2+a_{12}a_{21}-1,&\\qquad a_{12}(a_{11}+a_{22}),","---------\n\\vpage{24}{Equations survive every precision}{ADIC EFFECTIVITY / DECLARED POLYNOMIAL CONDITIONS}{Main text: revised Theorem 10.5 and Corollary 3.2\\quad \\href{p12-route:visual-24}{Main statement 10.5}}\nThe preceding involution gives an exact polynomial condition. For a matrix $A=(a_{ij})$, require all entries of $A^2-I$ to vanish. In\n\\[\nB=\\ZZ_5[a_{11},a_{12},a_{21},a_{22}],\n\\]\nthe ideal $J$ is generated by\n\\[\n\\begin{aligned}\na_{11}^2+a_{12}a_{21}-1,&\\qquad a_{12}(a_{11}+a_{22}),","Equations survive every precision}{ADIC EFFECTIVITY / DECLARED POLYNOMIAL CONDITIONS}{Main text: revised Theorem 10.5 and Corollary 3.2\\quad \\href{p12-route:visual-24}{Main statement 10.5}}\nThe preceding involution gives an exact polynomial condition. For a matrix $A=(a_{ij})$, require all entries of $A^2-I$ to vanish. In\n\\[\nB=\\ZZ_5[a_{11},a_{12},a_{21},a_{22}],\n\\]\nthe ideal $J$ is generated by\n\\[\n\\begin{aligned}\na_{11}^2+a_{12}a_{21}-1,&\\qquad a_{12}(a_{11}+a_{22}),\\\\\na_{21}(a_{11}+a_{22}),&\\qquad a_{21}a_{12}+a_{22}^2-1.\n\\end{aligned}\n\\]\nFor $D$ and every conjugate $QDQ^{-1}$ these equations vanish. The condition is valid in every admissible frame. By contrast, the equation $a_{12}=0$ holds for $D$ but fails for the conjugate displayed on \\pagerefV{23}.","a_{21}(a_{11}+a_{22}),&\\qquad a_{21}a_{12}+a_{22}^2-1.\n\\end{aligned}\n\\]\nFor $D$ and every conjugate $QDQ^{-1}$ these equations vanish. The condition is valid in every admissible frame. By contrast, the equation $a_{12}=0$ holds for $D$ but fails for the conjugate displayed on \\pagerefV{23}.","\\head{The reconstruction argument is coefficientwise}\nLet $A_n$ be compatible matrices modulo $5^n$, each satisfying these four equations. Completeness reconstructs $A\\in M_2(\\ZZ_5)$. For each polynomial $f\\in J$, evaluation commutes with reduction, so\n\\[\nf(A)\\in\\bigcap_{n\\ge1}5^n\\ZZ_5=\\{0\\}.\n\\]\nSeparatedness supplies the last equality. Thus the actual evaluation map $B\\to\\ZZ_5$ kills $J$ and factors uniquely through $B/J$. If inverse matrices were reconstructed as well, this is an actual representation in $\\GL_2(\\ZZ_5)$.","on argument is coefficientwise}\nLet $A_n$ be compatible matrices modulo $5^n$, each satisfying these four equations. Completeness reconstructs $A\\in M_2(\\ZZ_5)$. For each polynomial $f\\in J$, evaluation commutes with reduction, so\n\\[\nf(A)\\in\\bigcap_{n\\ge1}5^n\\ZZ_5=\\{0\\}.\n\\]\nSeparatedness supplies the last equality. Thus the actual evaluation map $B\\to\\ZZ_5$ kills $J$ and factors uniquely through $B/J$. If inverse matrices were reconstructed as well, this is an actual representation in $\\GL_2(\\ZZ_5)$.","ing these four equations. Completeness reconstructs $A\\in M_2(\\ZZ_5)$. For each polynomial $f\\in J$, evaluation commutes with reduction, so\n\\[\nf(A)\\in\\bigcap_{n\\ge1}5^n\\ZZ_5=\\{0\\}.\n\\]\nSeparatedness supplies the last equality. Thus the actual evaluation map $B\\to\\ZZ_5$ kills $J$ and factors uniquely through $B/J$. If inverse matrices were reconstructed as well, this is an actual representation in $\\GL_2(\\ZZ_5)$.","% ---------------------------------------------------------------------------\n\\vpage{25}{Recover multiplicities from traces}{A FINITE SEPARATOR / A KNOWN SEMISIMPLE CARRIER}{Main text: Proposition 10.9\\quad \\href{p12-route:visual-25}{Main statement 10.9}}\nSuppose a known semisimple carrier is a sum of two known constituents with multiplicities $m_1,m_2$. Consider the independent trace functions\n\\[\nt_1(\\omega)=1,\\qquad t_2(\\omega)=\\omega,\\qquad \\omega\\in\\QQ.\n\\]\nAt the two sample points $\\omega=1,-1$, their evaluation matrix is\n\\[\nA=\\begin{pmatrix}1&1","cover multiplicities from traces}{A FINITE SEPARATOR / A KNOWN SEMISIMPLE CARRIER}{Main text: Proposition 10.9\\quad \\href{p12-route:visual-25}{Main statement 10.9}}\nSuppose a known semisimple carrier is a sum of two known constituents with multiplicities $m_1,m_2$. Consider the independent trace functions\n\\[\nt_1(\\omega)=1,\\qquad t_2(\\omega)=\\omega,\\qquad \\omega\\in\\QQ.\n\\]\nAt the two sample points $\\omega=1,-1$, their evaluation matrix is\n\\[\nA=\\begin{pmatrix}1&1","SEPARATOR / A KNOWN SEMISIMPLE CARRIER}{Main text: Proposition 10.9\\quad \\href{p12-route:visual-25}{Main statement 10.9}}\nSuppose a known semisimple carrier is a sum of two known constituents with multiplicities $m_1,m_2$. Consider the independent trace functions\n\\[\nt_1(\\omega)=1,\\qquad t_2(\\omega)=\\omega,\\qquad \\omega\\in\\QQ.\n\\]\nAt the two sample points $\\omega=1,-1$, their evaluation matrix is\n\\[\nA=\\begin{pmatrix}1&1\\\\1&-1\\end{pmatrix},\\qquad\\det A=-2.\n\\]\nIt is invertible over $\\QQ$. If the measured trace vector is $(5,1)$, solve\n\\[\nA\\begin{pmatrix}m_1","1&-1\\end{pmatrix},\\qquad\\det A=-2.\n\\]\nIt is invertible over $\\QQ$. If the measured trace vector is $(5,1)$, solve\n\\[\nA\\begin{pmatrix}m_1","1&-1\\end{pmatrix},\\qquad\\det A=-2.\n\\]\nIt is invertible over $\\QQ$. If the measured trace vector is $(5,1)$, solve\n\\[\nA\\begin{pmatrix}m_1\\\\m_2\\end{pmatrix}=\\begin{pmatrix}5\\\\1\\end{pmatrix}\n\\quad\\Longrightarrow\\quad\n\\begin{pmatrix}m_1\\\\m_2\\end{pmatrix}=\\begin{pmatrix}3\\\\2\\end{pmatrix}.\n\\]","\\vfig[.46]{trace_separator}\n\\captiontext{The vectors $(1,1)$ and $(1,-1)$ are the two independent rows of the exact evaluation matrix. The coefficients $3,2$ are recovered by the displayed linear equations. This illustration uses functions on $\\QQ$, not a newly constructed arithmetic Frobenius set.}","\\head{Why finitely many evaluations suffice in general}\nIndependent functions $t_1,\\ldots,t_r$ on $\\Omega$ have evaluation vectors spanning $E'^r$. Otherwise an annihilating nonzero linear functional would give a forbidden relation among the functions. Choosing $r$ independent evaluations gives an invertible matrix that recovers coefficients in their known sum.","12.6 & A compatible Hecke point $y\\circ f=x$, actual ordinary or derived carriers, and both existing families for the same automorphic object.","\\head{Final editorial clarifications}\nThe final companions expose finite $S$, a coherent Frobenius normalization, the local coefficient-test category in 3.3, the $r>n$ Fitting boundary, and both compared Frobenius polynomial identities in ordinary completion. These clarifications are distinct from the eight inherited amendments; no new Lean replay certifies them.","midrule\n6.12 & unchecked & The reversed containment is false: in the node of V--9, $M=R$ has full support, which is not contained in either single component."]}