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omega","display":"\\omega^j","kind":"use","occurrences":[1342,1343]},{"id":"symbol-03c0ea7172492518","tex":"\\omega_\\pi","aliases":"ω_π omega pi","display":"\\omega_\\pi","kind":"use","occurrences":[124,435]},{"id":"symbol-b1c5a4dcc1c6e1a4","tex":"\\omega_\\pi^{dM}","aliases":"ω_π^dM omega pi","display":"\\omega_\\pi^{dM}","kind":"use","occurrences":[434]},{"id":"symbol-0146061874d89d11","tex":"\\omega_{\\pi_0}","aliases":"ω_π_0 omega pi","display":"\\omega_{\\pi_0}","kind":"use","occurrences":[135]},{"id":"symbol-d486dbdd61bfee4d","tex":"\\omega_{\\pi_0}^{-1}","aliases":"ω_π_0^-1 omega pi","display":"\\omega_{\\pi_0}^{-1}","kind":"use","occurrences":[142]},{"id":"symbol-e2f59fa8d4077552","tex":"\\omega_{\\pi_0\\otimes\\chi}","aliases":"ω_π_0⊗χ omega pi otimes chi","display":"\\omega_{\\pi_0\\otimes\\chi}","kind":"use","occurrences":[134]},{"id":"symbol-5ecbb9401ee325c3","tex":"\\omega_{\\pi\\otimes\\chi}","aliases":"ω_π⊗χ omega pi otimes chi","display":"\\omega_{\\pi\\otimes\\chi}","kind":"use","occurrences":[124]},{"id":"symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","occurrences":[568,707,732,764,887,938,941,968,1031,1035,1270]},{"id":"symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","occurrences":[128,240,241]},{"id":"symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"use","occurrences":[1273]},{"id":"symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","occurrences":[1273]},{"id":"symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","occurrences":[1273]},{"id":"symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","occurrences":[536,812,815]},{"id":"symbol-b7027dbff3dd2bf8","tex":"\\prod_i","aliases":"∏_i prod","display":"\\prod_i","kind":"use","occurrences":[674,857,1020]},{"id":"symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","occurrences":[493,855]},{"id":"symbol-5c21a30e63e2813a","tex":"\\sum_{i=0}^{d-1}","aliases":"∑_i=0^d-1 sum","display":"\\sum_{i=0}^{d-1}","kind":"use","occurrences":[1112]},{"id":"symbol-b26eca2dd8854316","tex":"\\sum_{j=0}^{e-1}","aliases":"∑_j=0^e-1 sum","display":"\\sum_{j=0}^{e-1}","kind":"use","occurrences":[905]},{"id":"symbol-d0b76425559f0527","tex":"\\sum_{j=0}^{r-1}","aliases":"∑_j=0^r-1 sum","display":"\\sum_{j=0}^{r-1}","kind":"use","occurrences":[892]},{"id":"symbol-41e0c211ba80d17c","tex":"\\sum_{k=0}^{n}","aliases":"∑_k=0^n sum","display":"\\sum_{k=0}^{n}","kind":"use","occurrences":[95]},{"id":"symbol-90c396c1f06e154a","tex":"\\sum_t","aliases":"∑_t sum","display":"\\sum_t","kind":"use","occurrences":[1148]},{"id":"symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","occurrences":[803]},{"id":"symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","occurrences":[162]},{"id":"symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"use","occurrences":[902,1006]},{"id":"symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","occurrences":[1012,1239,1266,1273,1345]},{"id":"symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","occurrences":[618,665,679,994]},{"id":"symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","occurrences":[487,502,508,559,565,710,892,1109,1110,1125,1126,1127,1128,1139,1141,1142,1143,1148,1152,1154,1162,1164,1165,1167,1169,1179,1183,1199,1209,1213,1225,1271,1273,1393,1394,1399]},{"id":"symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","occurrences":[521]},{"id":"symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","occurrences":[494,550,1087,1153]},{"id":"symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","occurrences":[483,534,1272]},{"id":"symbol-3c62641491514988","tex":"\\bigoplus_i","aliases":"⊕_i bigoplus","display":"\\bigoplus_i","kind":"use","occurrences":[493]},{"id":"symbol-050a45cb4caa57c9","tex":"\\bigoplus_{j=0}^{p-1}","aliases":"⊕_j=0^p-1 bigoplus","display":"\\bigoplus_{j=0}^{p-1}","kind":"use","occurrences":[869]},{"id":"symbol-43e6f080801d0972","tex":"\\bigoplus_{\\chi\\in X}","aliases":"⊕_χ∈ X bigoplus chi in","display":"\\bigoplus_{\\chi\\in X}","kind":"use","occurrences":[203,209]},{"id":"symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","occurrences":[282,365]},{"id":"symbol-63c03a011ef19538","tex":"\\bigcap_v","aliases":"⋂_v bigcap","display":"\\bigcap_v","kind":"use","occurrences":[365]},{"id":"symbol-eaf0c3abbbfc3011","tex":"\\bigcap_\\tau","aliases":"⋂_τ bigcap tau","display":"\\bigcap_\\tau","kind":"use","occurrences":[282]},{"id":"symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","occurrences":[673,789]}],"contexts":["\\bibitem{KimYamauchi2016}\nH. H. Kim and T. Yamauchi, A uniform structure on subgroups of \\(GL_n(\\mathbb F_q)\\) and its application to a conditional construction of Artin representations of \\(GL_n\\), \\emph{Journal of the Ramanujan Mathematical Society} 32(1) (2017), 75--99. Checked preprint \\href{https://arxiv.org/abs/1403.5535}{arXiv:1403.5535v3}, 7 April 2016. Theorem 1.1 is conditional on its five inputs. The inherited citation key records the preprint year.","\\bibitem{Ramakrishnan2002}\nD. Ramakrishnan, Modularity of solvable Artin representations of \\(GO(4)\\) type, \\emph{International Mathematics Research Notices} 2002(1), 1--54. DOI: \\href{https://doi.org/10.1155/S1073792802000016}{10.1155/S1073792802000016}.","\\bibitem{Langlands1980}\nR. P. Langlands, \\emph{Base Change for \\(GL(2)\\)}, Annals of Mathematics Studies 96, Princeton University Press, 1980.","\\bibitem{Henniart2000}\nG. Henniart, Une preuve simple des conjectures de Langlands pour \\(GL(n)\\) sur un corps p-adique, \\emph{Inventiones Mathematicae} 139(2) (2000), 439--455.","\\bibitem{Brauer1947}\nR. Brauer, On Artin's \\(L\\)-series with general group characters, \\emph{Annals of Mathematics} 48 (1947), 502--514.","\\bibitem{CimpoeasNicolae2020}\nM. Cimpoea\\c{s} and F. Nicolae, Artin \\(L\\)-functions of almost monomial Galois groups, \\emph{Forum Mathematicum} 32(4) (2020).","\\bibitem{DimitrovWiese2020}\nM. Dimitrov and G. Wiese, Unramifiedness of Galois representations attached to Hilbert modular forms mod \\(p\\) of weight one, \\emph{Journal of the Institute of Mathematics of Jussieu} 19(2) (2020), 281--306. DOI: \\href{https://doi.org/10.1017/S1474748018000026}{10.1017/S1474748018000026}. Checked preprint version 4, 28 November 2017. The full Frobenius polynomial comparison also requires the diamond character determinant normalization.","\\bibitem{Deligne1973}\nP. Deligne, Les constantes des \\'equations fonctionnelles des fonctions \\(L\\), in \\emph{Modular Functions of One Variable II}, Lecture Notes in Mathematics 349, Springer, 1973, 501--597.","\\bibitem{DeligneHenniart1981}\nP. Deligne and G. Henniart, Sur la variation, par torsion, des constantes locales d'\\'equations fonctionnelles de fonctions \\(L\\), \\emph{Inventiones Mathematicae} 64(1) (1981), 89--118. Theorem 4.6, p. 110, supplies the stable epsilon ratio used here.","\\bibitem{Wang2026}\nJ. Wang, Strong Artin conjecture for generalized octahedral representations in \\(GL_3\\), \\href{https://arxiv.org/abs/2609.38231}{arXiv:2609.38231v1}, 28 September 2026. Theorem 1.1 and Corollary 5.4 state the special three and six dimensional cases mentioned here. Statements checked. The full proof has not been independently audited for this reader edition.\n\\end{thebibliography}","Let \\(F\\) be a number field, let \\(G_F=\\Gal(\\overline F/F)\\), and let\n\\[\n \\rho:G_F\\longrightarrow\\GL_n(\\CC)\n\\]\nbe a continuous representation with finite image. Continuity is taken for the usual complex topology. Such a representation factors through the Galois group of a finite extension. Throughout the conjecture discussion, \\(\\rho\\) is irreducible. The trivial one dimensional representation is excluded when discussing entireness of an Artin \\(L\\) function, since its \\(L\\) function is the Dedekind zeta function.","ine F/F)\\), and let\n\\[\n \\rho:G_F\\longrightarrow\\GL_n(\\CC)\n\\]\nbe a continuous representation with finite image. Continuity is taken for the usual complex topology. Such a representation factors through the Galois group of a finite extension. Throughout the conjecture discussion, \\(\\rho\\) is irreducible. The trivial one dimensional representation is excluded when discussing entireness of an Artin \\(L\\) function, since its \\(L\\) function is the Dedekind zeta function.","ho:G_F\\longrightarrow\\GL_n(\\CC)\n\\]\nbe a continuous representation with finite image. Continuity is taken for the usual complex topology. Such a representation factors through the Galois group of a finite extension. Throughout the conjecture discussion, \\(\\rho\\) is irreducible. The trivial one dimensional representation is excluded when discussing entireness of an Artin \\(L\\) function, since its \\(L\\) function is the Dedekind zeta function.","At a finite place where \\(\\rho\\) is unramified, fix the Frobenius convention used by the normalized local Langlands correspondence. Write\n\\[\n P_{\\rho,v}(T)=\\det\\bigl(TI_n-\\rho(\\Frob_v)\\bigr).\n\\]\nFor an unramified automorphic local component \\(\\pi_v\\), write \\(P_{\\pi,v}(T)\\) for the monic polynomial of its normalized Satake conjugacy class. The convention is chosen so that an attached pair has the same polynomial. This keeps a possible geometric versus arithmetic Frobenius inversion from entering a comparison silently.","\\begin{conjecture}[Strong Artin, unramified form]\\label{conj:strong_artin_weak}\nEvery irreducible finite image complex representation \\(\\rho\\) of \\(G_F\\) has a cuspidal automorphic representation \\(\\pi\\) of \\(\\GL_n(\\Adele_F)\\) such that\n\\[\n P_{\\pi,v}(T)=P_{\\rho,v}(T)\n\\]\nfor all finite places outside a finite set.\n\\end{conjecture}","\\begin{conjecture}[Strong Artin, all local form]\\label{conj:strong_artin_full}\nThe representation \\(\\pi\\) can be chosen so that\n\\[\n \\rec_{F_v}(\\pi_v)\\cong\\rho|_{W_{F_v}}\n\\]\nfor every place \\(v\\), with zero monodromy at finite places.\n\\end{conjecture}","The solvable two dimensional sectors treated by Langlands and Tunnell, further examples such as the solvable \\(\\mathrm{GO}(4)\\) range, and other established cases show that the conjecture is far from an empty expectation \\cite{Langlands1980,Tunnell1981,Ramakrishnan2002}. A recent preprint of Wang states the generalized octahedral \\(\\GL_3\\) case with projective image \\(C_3^2\\rtimes\\mathrm{SL}(2,3)\\), together with a primitive solvable six dimensional consequence \\cite{Wang2026}. Its stated results have been checked against the submitted source. Its full proof has not been independently audited for this edition. These particular sectors must therefore not be described as untouched simply because the general problem remains open.","The finite representation arguments have a parallel shape. If \\(H\\triangleleft G\\) and \\(G/H\\cong C_p\\), an irreducible representation either restricts to a full orbit of \\(p\\) subgroup types or restricts irreducibly to one invariant type. In the first case induction reconstructs the representation. In the second case an extension must be selected. The second case is where two torsors meet. Each torsor exists, and each has \\(p\\) elements. There is no preferred origin in the statement that they are torsors under the same character group.","uments have a parallel shape. If \\(H\\triangleleft G\\) and \\(G/H\\cong C_p\\), an irreducible representation either restricts to a full orbit of \\(p\\) subgroup types or restricts irreducibly to one invariant type. In the first case induction reconstructs the representation. In the second case an extension must be selected. The second case is where two torsors meet. Each torsor exists, and each has \\(p\\) elements. There is no preferred origin in the statement that they are torsors under the same character group.","The core argument retains the detailed residual realization analysis of the earlier project source. Stable subgroup types carry multiplicity spaces on which the cyclic quotient acts. Their ordered character multiplicities are recorded in polynomials\n\\[\n P_{\\tau,V}(z)=\\sum_{a\\in\\ZZ/p\\ZZ}m_{\\tau,a}(V)z^a.\n\\]\nA quotient twist rotates every such polynomial by the same amount. Moving subgroup types carry induced blocks and do not retain a quotient label. This gives a complete extension comparison, not just a determinant or total trace test.","At each place, a finite alignment set records the quotient labels that attach the local automorphic parameter to the target. The all local cyclic step is equivalent to a common label in all these sets. A nonempty exact alignment set is a coset of a subgroup of \\(C_p\\), so it is a singleton or the whole label group. It may also be empty. The reduction makes the missing statement precise. It does not show that the local requirements are compatible.","The same trace pairing gives an especially direct proof for an ordinary semisimple \\(\\ell\\) adic realization. Chebotarev and continuity extend a finite Frobenius trace alphabet to every image element. The semisimple image algebra then embeds the image in a finite Cartesian power of that alphabet. Thus the image is finite. The existence of the \\(\\ell\\) adic representation remains the input. The finite image conclusion is the output.","A third phenomenon occurs at the quotient prime \\(p\\). The order \\(p\\) characters all reduce to the trivial character, and the unramified cyclic base change identities force the automorphic and target polynomial systems to agree modulo a coefficient prime above \\(p\\). This congruence cannot choose a characteristic zero label. It is the beginning of a lifting problem. It supplies neither a residual Galois representation for an arbitrary automorphic descent nor an unrestricted automorphy lifting theorem for repeated weight zero parameters.","Let \\(G\\) be a finite group, let \\(H\\triangleleft G\\), and assume\n\\[\n G/H\\cong C_p\n\\]\nfor a prime \\(p\\). Let \\(X=\\Hom(G/H,\\CC^\\times)\\), which is cyclic of order \\(p\\).","\\begin{proposition}[Galois extension torsor]\\label{prop:galois_torsor}\nLet \\(\\tau\\) be an irreducible complex representation of \\(H\\) that is invariant under \\(G\\), and suppose \\(\\tau\\) extends to an irreducible representation of \\(G\\). Then the set of isomorphism classes of extensions of \\(\\tau\\) to \\(G\\) is an \\(X\\) torsor under tensor product.\n\\end{proposition}","\\begin{proof}\nFix one extension \\(\\rho\\). Every \\(\\chi\\in X\\) is trivial on \\(H\\), so \\(\\rho\\otimes\\chi\\) restricts to \\(\\tau\\). Conversely, let \\(\\rho'\\) be another extension. Since both restrictions are irreducible and isomorphic, Schur's lemma shows that after one choice of an \\(H\\) intertwiner, the ratio between the two actions is scalar on each coset. Those scalars define a character of \\(G/H\\), so \\(\\rho'\\cong\\rho\\otimes\\chi\\).","oof}\nFix one extension \\(\\rho\\). Every \\(\\chi\\in X\\) is trivial on \\(H\\), so \\(\\rho\\otimes\\chi\\) restricts to \\(\\tau\\). Conversely, let \\(\\rho'\\) be another extension. Since both restrictions are irreducible and isomorphic, Schur's lemma shows that after one choice of an \\(H\\) intertwiner, the ratio between the two actions is scalar on each coset. Those scalars define a character of \\(G/H\\), so \\(\\rho'\\cong\\rho\\otimes\\chi\\).","The action is free. If \\(\\rho\\cong\\rho\\otimes\\chi\\), an intertwiner restricts to a scalar on \\(H\\). Commuting it through one element outside \\(H\\) gives \\(\\chi=1\\).\n\\end{proof}","\\begin{standardfact}[Prime cyclic cuspidal descent and fibre]\\label{fact:base_change_fibre}\nLet \\(E/F\\) be cyclic of prime degree, and let \\(\\Pi\\) be an invariant unitary cuspidal representation of \\(\\GL_n(\\Adele_E)\\). Then \\(\\Pi\\) has a cuspidal descent to \\(F\\). Its cuspidal descents form a torsor under the Hecke characters corresponding by class field theory to \\(\\Gal(E/F)\\). In particular, distinct quotient twists of one descent remain distinct. We use the prime cyclic construction of Arthur and Clozel and the fibre statement and cyclic descent clarification of Clozel and Rajan \\cite{ArthurClozel1989,ClozelRajan2020}. The assertion concerns a cuspidal base change. It is not a claim that quotient twisting acts freely on every cuspidal representation.\n\\end{standardfact}","\\begin{definition}[Attached cyclic descent]\\label{def:attached_descent}\nLet \\(E/F\\) be cyclic of prime degree \\(p\\), and let \\(\\rho\\) be an irreducible finite image representation of \\(G_F\\) whose restriction \\(\\tau=\\rho|_{G_E}\\) is irreducible. Suppose \\(\\tau\\) is automorphic with cuspidal representation \\(\\Pi\\) of \\(\\GL_n(\\Adele_E)\\). An attached cyclic descent is a cuspidal descent \\(\\pi\\) of \\(\\Pi\\) such that\n\\[\n \\rec_{F_v}(\\pi_v)\n \\cong\n \\rho|_{W_{F_v}}\n\\]\nfor every place \\(v\\) of \\(F\\).\n\\end{definition}","begin{definition}[Attached cyclic descent]\\label{def:attached_descent}\nLet \\(E/F\\) be cyclic of prime degree \\(p\\), and let \\(\\rho\\) be an irreducible finite image representation of \\(G_F\\) whose restriction \\(\\tau=\\rho|_{G_E}\\) is irreducible. Suppose \\(\\tau\\) is automorphic with cuspidal representation \\(\\Pi\\) of \\(\\GL_n(\\Adele_E)\\). An attached cyclic descent is a cuspidal descent \\(\\pi\\) of \\(\\Pi\\) such that\n\\[\n \\rec_{F_v}(\\pi_v)\n \\cong\n \\rho|_{W_{F_v}}\n\\]\nfor every place \\(v\\) of \\(F\\).\n\\end{definition}","Attached cyclic descent]\\label{def:attached_descent}\nLet \\(E/F\\) be cyclic of prime degree \\(p\\), and let \\(\\rho\\) be an irreducible finite image representation of \\(G_F\\) whose restriction \\(\\tau=\\rho|_{G_E}\\) is irreducible. Suppose \\(\\tau\\) is automorphic with cuspidal representation \\(\\Pi\\) of \\(\\GL_n(\\Adele_E)\\). An attached cyclic descent is a cuspidal descent \\(\\pi\\) of \\(\\Pi\\) such that\n\\[\n \\rec_{F_v}(\\pi_v)\n \\cong\n \\rho|_{W_{F_v}}\n\\]\nfor every place \\(v\\) of \\(F\\).\n\\end{definition}","be cyclic of prime degree \\(p\\), and let \\(\\rho\\) be an irreducible finite image representation of \\(G_F\\) whose restriction \\(\\tau=\\rho|_{G_E}\\) is irreducible. Suppose \\(\\tau\\) is automorphic with cuspidal representation \\(\\Pi\\) of \\(\\GL_n(\\Adele_E)\\). An attached cyclic descent is a cuspidal descent \\(\\pi\\) of \\(\\Pi\\) such that\n\\[\n \\rec_{F_v}(\\pi_v)\n \\cong\n \\rho|_{W_{F_v}}\n\\]\nfor every place \\(v\\) of \\(F\\).\n\\end{definition}","of prime degree \\(p\\), and let \\(\\rho\\) be an irreducible finite image representation of \\(G_F\\) whose restriction \\(\\tau=\\rho|_{G_E}\\) is irreducible. Suppose \\(\\tau\\) is automorphic with cuspidal representation \\(\\Pi\\) of \\(\\GL_n(\\Adele_E)\\). An attached cyclic descent is a cuspidal descent \\(\\pi\\) of \\(\\Pi\\) such that\n\\[\n \\rec_{F_v}(\\pi_v)\n \\cong\n \\rho|_{W_{F_v}}\n\\]\nfor every place \\(v\\) of \\(F\\).\n\\end{definition}","\\begin{proof}\nLocal class field theory identifies a quotient character \\(\\chi\\in X\\) with the corresponding Hecke character. Local Langlands is compatible with character twisting, \\cite{HarrisTaylor2001}. Hence\n\\[\n \\rec(\\pi_v\\otimes\\chi_v)\n \\cong\n \\rec(\\pi_v)\\otimes\\chi_v\n \\cong\n \\rho_v\\otimes\\chi_v.\n\\]\nThe result holds at every place.\n\\end{proof}","Let \\(\\xi\\) be a generator of \\(X\\), and identify its values with \\(p\\)th roots of unity. For a matrix \\(A\\in\\GL_n(\\CC)\\), write\n\\[\n \\det(TI-A)\n =\n \\sum_{k=0}^{n}(-1)^k e_k(A)T^{n-k}.\n\\]","\\begin{proposition}[Coefficient rotation]\\label{prop:coefficient_rotation}\nFor \\(\\zeta\\in\\CC^\\times\\),\n\\[\n \\boxed{\n e_k(\\zeta A)=\\zeta^k e_k(A).\n }\n\\]\n\\end{proposition}","\\begin{proof}\nThe coefficient \\(e_k(A)\\) is the trace of \\(\\bigwedge^k A\\). Scalar multiplication by \\(\\zeta\\) acts on \\(\\bigwedge^k\\CC^n\\) by \\(\\zeta^k\\).\n\\end{proof}","\\begin{definition}[Active coefficient degrees]\\label{def:active_degrees}\nDefine\n\\[\n S(A)=\\{k\\in\\{1,\\ldots,n\\}:e_k(A)\\neq0\\}.\n\\]\n\\end{definition}","\\begin{theorem}[One polynomial separator]\\label{thm:one_polynomial_separator}\nLet \\(\\zeta\\) range over \\(\\mu_p\\). The twists \\(\\zeta A\\) have pairwise distinct characteristic polynomials exactly when \\(S(A)\\) contains a degree not divisible by \\(p\\).\n\\end{theorem}","\\begin{proof}\nBy \\cref{prop:coefficient_rotation}, the stabilizer of the characteristic polynomial is\n\\[\n \\{\\zeta\\in\\mu_p:\\zeta^k=1\\text{ for every }k\\in S(A)\\}.\n\\]\nSince \\(p\\) is prime, a nontrivial \\(p\\)th root satisfies \\(\\zeta^k=1\\) exactly when \\(p\\mid k\\).\n\\end{proof}","\\begin{corollary}[Trace and determinant selectors]\\label{cor:trace_det_selector}\nA nonzero trace separates all cyclic twists. The determinant separates all cyclic twists when \\(p\\nmid n\\).\n\\end{corollary}","\\begin{proof}\nUse degree one for the trace and degree \\(n\\) for the determinant in \\cref{thm:one_polynomial_separator}.\n\\end{proof}","\\begin{theorem}[Determinant selector for extension torsors]\\label{thm:determinant_selector}\nAssume \\(p\\nmid n\\). In the Galois extension torsor of an \\(n\\) dimensional irreducible representation, the determinant distinguishes all \\(p\\) extensions. In the automorphic descent torsor, the central character distinguishes all \\(p\\) descents. Both transformations are equivariant under the \\(n\\)th power map on \\(X\\).\n\\end{theorem}","eorem}[Determinant selector for extension torsors]\\label{thm:determinant_selector}\nAssume \\(p\\nmid n\\). In the Galois extension torsor of an \\(n\\) dimensional irreducible representation, the determinant distinguishes all \\(p\\) extensions. In the automorphic descent torsor, the central character distinguishes all \\(p\\) descents. Both transformations are equivariant under the \\(n\\)th power map on \\(X\\).\n\\end{theorem}","\\begin{proof}\nFor a quotient character \\(\\chi\\),\n\\[\n \\det(\\rho\\otimes\\chi)=\\det(\\rho)\\chi^n.\n\\]\nThe central character satisfies the identical formula\n\\[\n \\omega_{\\pi\\otimes\\chi}=\\omega_\\pi\\chi^n.\n\\]\nSince \\(X\\) has prime order \\(p\\) and \\(p\\nmid n\\), the map \\(\\chi\\mapsto\\chi^n\\) is an automorphism of \\(X\\).\n\\end{proof}","\\begin{theorem}[Unique determinant matched descent]\\label{thm:determinant_matched_descent}\nUse the cyclic Artin setup of \\cref{def:attached_descent}, and assume \\(p\\nmid n\\). Let \\(\\pi_0\\) be any cuspidal descent of \\(\\Pi\\). Then there is a unique quotient character \\(\\chi\\in X\\) such that\n\\[\n \\boxed{\n \\omega_{\\pi_0\\otimes\\chi}\n =\n \\det\\rho\n }\n\\]\nunder global class field theory.\n\\end{theorem}","\\begin{proof}\nThe base change of \\(\\omega_{\\pi_0}\\) is the central character of \\(\\Pi\\). Since \\(\\Pi\\) is attached to \\(\\tau=\\rho|_{G_E}\\), that character equals \\(\\det\\tau\\). The base change of \\(\\det\\rho\\) is also \\(\\det\\tau\\). Hence\n\\[\n \\delta=(\\det\\rho)\\omega_{\\pi_0}^{-1}\n\\]\nis a character killed by base change to \\(E\\), so \\(\\delta\\in X\\). We need \\(\\chi^n=\\delta\\). Since \\(X\\) has order \\(p\\) and \\(p\\nmid n\\), the \\(n\\)th power map on \\(X\\) is bijective. Thus there is one and only one such \\(\\chi\\).\n\\end{proof}","roof}\nThe base change of \\(\\omega_{\\pi_0}\\) is the central character of \\(\\Pi\\). Since \\(\\Pi\\) is attached to \\(\\tau=\\rho|_{G_E}\\), that character equals \\(\\det\\tau\\). The base change of \\(\\det\\rho\\) is also \\(\\det\\tau\\). Hence\n\\[\n \\delta=(\\det\\rho)\\omega_{\\pi_0}^{-1}\n\\]\nis a character killed by base change to \\(E\\), so \\(\\delta\\in X\\). We need \\(\\chi^n=\\delta\\). Since \\(X\\) has order \\(p\\) and \\(p\\nmid n\\), the \\(n\\)th power map on \\(X\\) is bijective. Thus there is one and only one such \\(\\chi\\).\n\\end{proof}","nge of \\(\\omega_{\\pi_0}\\) is the central character of \\(\\Pi\\). Since \\(\\Pi\\) is attached to \\(\\tau=\\rho|_{G_E}\\), that character equals \\(\\det\\tau\\). The base change of \\(\\det\\rho\\) is also \\(\\det\\tau\\). Hence\n\\[\n \\delta=(\\det\\rho)\\omega_{\\pi_0}^{-1}\n\\]\nis a character killed by base change to \\(E\\), so \\(\\delta\\in X\\). We need \\(\\chi^n=\\delta\\). Since \\(X\\) has order \\(p\\) and \\(p\\nmid n\\), the \\(n\\)th power map on \\(X\\) is bijective. Thus there is one and only one such \\(\\chi\\).\n\\end{proof}","0}\\) is the central character of \\(\\Pi\\). Since \\(\\Pi\\) is attached to \\(\\tau=\\rho|_{G_E}\\), that character equals \\(\\det\\tau\\). The base change of \\(\\det\\rho\\) is also \\(\\det\\tau\\). Hence\n\\[\n \\delta=(\\det\\rho)\\omega_{\\pi_0}^{-1}\n\\]\nis a character killed by base change to \\(E\\), so \\(\\delta\\in X\\). We need \\(\\chi^n=\\delta\\). Since \\(X\\) has order \\(p\\) and \\(p\\nmid n\\), the \\(n\\)th power map on \\(X\\) is bijective. Thus there is one and only one such \\(\\chi\\).\n\\end{proof}","is attached to \\(\\tau=\\rho|_{G_E}\\), that character equals \\(\\det\\tau\\). The base change of \\(\\det\\rho\\) is also \\(\\det\\tau\\). Hence\n\\[\n \\delta=(\\det\\rho)\\omega_{\\pi_0}^{-1}\n\\]\nis a character killed by base change to \\(E\\), so \\(\\delta\\in X\\). We need \\(\\chi^n=\\delta\\). Since \\(X\\) has order \\(p\\) and \\(p\\nmid n\\), the \\(n\\)th power map on \\(X\\) is bijective. Thus there is one and only one such \\(\\chi\\).\n\\end{proof}","\\begin{corollary}[Rank prime sieve]\\label{cor:rank_prime_sieve}\nAt a cyclic prime step of degree \\(p\\), determinant data leaves no torsor ambiguity unless \\(p\\mid n\\). The only quotient primes invisible to determinant selection are prime divisors of the rank.\n\\end{corollary}","\\begin{remark}\\label{rem:determinant_not_attachment}\nMatching determinant and central character selects a unique candidate descent when \\(p\\nmid n\\). It does not prove that the remaining local characteristic polynomials match. The selection theorem removes the finite torsor ambiguity, not the global realization problem.\n\\end{remark}","\\begin{proposition}[Determinant character of an algebraic representation]\\label{prop:algebraic_det_character}\nLet\n\\[\n r:\\GL_n(\\CC)\\longrightarrow\\GL_N(\\CC)\n\\]\nbe an irreducible algebraic representation. Suppose the center acts with weight \\(w\\), so\n\\[\n r(zI_n)=z^wI_N.\n\\]\nThere is an integer \\(a\\) such that\n\\[\n\\boxed{\n \\det r(g)=\\det(g)^a\n}\n\\]\nand\n\\[\n\\boxed{\n wN=an.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nEvery algebraic character of \\(\\GL_n\\) is an integral power of the determinant. Thus \\(\\det\\circ r=\\det^a\\) for one integer \\(a\\). Evaluate both sides at \\(zI_n\\). The left side is \\(z^{wN}\\), while the right side is \\(z^{an}\\).\n\\end{proof}","\\begin{theorem}[Algebraic central characters are blind when \\(p\\mid n\\)]\\label{thm:central_character_no_go}\nLet \\(\\chi\\) have order \\(p\\), and assume \\(p\\mid n\\). For every irreducible algebraic representation \\(r\\) of \\(\\GL_n\\),\n\\[\n\\boxed{\n \\det r(\\rho\\otimes\\chi)=\\det r(\\rho).\n}\n\\]\nThe same statement holds on every irreducible summand of an arbitrary finite dimensional algebraic representation.\n\\end{theorem}","\\begin{proof}\nThe scalar twist acts on \\(r(\\rho)\\) by \\(\\chi^w\\). Hence its determinant changes by \\(\\chi^{wN}\\). By \\cref{prop:algebraic_det_character}, \\(wN=an\\), which is divisible by \\(p\\). Thus \\(\\chi^{wN}=1\\). Apply the argument to each irreducible summand.\n\\end{proof}","\\begin{corollary}[Optimality of the rank prime sieve]\\label{cor:rank_sieve_optimal}\nWhen \\(p\\mid n\\), no finite collection of determinant or central character tests obtained from algebraic functors on \\(\\GL_n\\) can distinguish the order \\(p\\) scalar extension twists.\n\\end{corollary}","\\begin{standardfact}[Stable equality of epsilon factors]\\label{fact:epsilon_stability}\nLet \\(V,V'\\) be finite dimensional complex Weil representations with finite inertia image, of equal dimension and determinant. For every sufficiently ramified auxiliary character \\(\\eta\\),\n\\[\n \\epsilon(V\\otimes\\eta,\\Psi)=\\epsilon(V'\\otimes\\eta,\\Psi).\n\\]\nHere \\(\\Psi\\) is a fixed nontrivial additive character and the same normalization is used on both sides. Deligne and Henniart's stability theorem applies to the virtual representation \\(B=V-V'\\), which has dimension zero and determinant one. Their Theorem 4.6 gives the ratio as a determinant evaluation once the slope of \\(\\eta\\) exceeds twice the largest relevant slope of \\(B\\). That ratio is one \\cite{DeligneHenniart1981}. We use this equality and avo","\\begin{standardfact}[Stable equality of epsilon factors]\\label{fact:epsilon_stability}\nLet \\(V,V'\\) be finite dimensional complex Weil representations with finite inertia image, of equal dimension and determinant. For every sufficiently ramified auxiliary character \\(\\eta\\),\n\\[\n \\epsilon(V\\otimes\\eta,\\Psi)=\\epsilon(V'\\otimes\\eta,\\Psi).\n\\]\nHere \\(\\Psi\\) is a fixed nontrivial additive character and the same normalization is used on both sides. Deligne and Henniart's stability theorem applies to the virtual representation \\(B=V-V'\\), which has dimension zero and determinant one. Their Theorem 4.6 gives the ratio as a determinant evaluation once the slope of \\(\\eta\\) exceeds twice the largest relevant slope of \\(B\\). That ratio is one \\cite{DeligneHenniart1981}. We use this equality and avoid an unspecified choice of the auxiliary determinant evaluation element.\n\\end{standardfact}","ex Weil representations with finite inertia image, of equal dimension and determinant. For every sufficiently ramified auxiliary character \\(\\eta\\),\n\\[\n \\epsilon(V\\otimes\\eta,\\Psi)=\\epsilon(V'\\otimes\\eta,\\Psi).\n\\]\nHere \\(\\Psi\\) is a fixed nontrivial additive character and the same normalization is used on both sides. Deligne and Henniart's stability theorem applies to the virtual representation \\(B=V-V'\\), which has dimension zero and determinant one. Their Theorem 4.6 gives the ratio as a determinant evaluation once the slope of \\(\\eta\\) exceeds twice the largest relevant slope of \\(B\\). That ratio is one \\cite{DeligneHenniart1981}. We use this equality and avoid an unspecified choice of the auxiliary determinant evaluation element.\n\\end{standardfact}","\\(\\eta\\),\n\\[\n \\epsilon(V\\otimes\\eta,\\Psi)=\\epsilon(V'\\otimes\\eta,\\Psi).\n\\]\nHere \\(\\Psi\\) is a fixed nontrivial additive character and the same normalization is used on both sides. Deligne and Henniart's stability theorem applies to the virtual representation \\(B=V-V'\\), which has dimension zero and determinant one. Their Theorem 4.6 gives the ratio as a determinant evaluation once the slope of \\(\\eta\\) exceeds twice the largest relevant slope of \\(B\\). That ratio is one \\cite{DeligneHenniart1981}. We use this equality and avoid an unspecified choice of the auxiliary determinant evaluation element.\n\\end{standardfact}","'\\otimes\\eta,\\Psi).\n\\]\nHere \\(\\Psi\\) is a fixed nontrivial additive character and the same normalization is used on both sides. Deligne and Henniart's stability theorem applies to the virtual representation \\(B=V-V'\\), which has dimension zero and determinant one. Their Theorem 4.6 gives the ratio as a determinant evaluation once the slope of \\(\\eta\\) exceeds twice the largest relevant slope of \\(B\\). That ratio is one \\cite{DeligneHenniart1981}. We use this equality and avoid an unspecified choice of the auxiliary determinant evaluation element.\n\\end{standardfact}","\\begin{theorem}[Highly ramified epsilon twists are blind]\\label{thm:epsilon_no_go}\nLet \\(V' = V\\otimes\\chi\\), where \\(\\chi\\) has order \\(p\\), and assume \\(p\\mid\\dim V\\). Then \\(V\\) and \\(V'\\) have the same determinant. For every sufficiently ramified auxiliary character \\(\\eta\\),\n\\[\n\\boxed{\n \\epsilon(V\\otimes\\eta,\\Psi)\n =\n \\epsilon(V'\\otimes\\eta,\\Psi).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nThe determinant changes by \\(\\chi^{\\dim V}=1\\). The two representations have the same dimension and determinant. Apply \\cref{fact:epsilon_stability}.\n\\end{proof}","\\begin{theorem}[The total induced object is torsor blind]\\label{thm:total_induction_blind}\nLet \\(H\\triangleleft G\\) with cyclic quotient character group \\(X\\), and let \\(V\\) be an extension of an \\(H\\) representation. Then\n\\[\n\\boxed{\n \\bigoplus_{\\chi\\in X}\n V\\otimes\\chi\n \\cong\n \\Ind_H^G\\Res_HV.\n}\n\\]\nThe left side is unchanged when \\(V\\) is replaced by another member of its extension torsor. Likewise, for cyclic base change,\n\\[\n\\boxed{\n \\mathop{\\boxplus}_{\\chi\\in X}\n \\pi\\otimes\\chi\n \\cong\n \\AI_{E/F}(\\BC_{E/F}\\pi)\n}\n\\]\ninside the established automorphic induction formalism. Therefore comparison of the total induced objects cannot determine the cyclic attachment offset.\n\\end{theorem}","ced object is torsor blind]\\label{thm:total_induction_blind}\nLet \\(H\\triangleleft G\\) with cyclic quotient character group \\(X\\), and let \\(V\\) be an extension of an \\(H\\) representation. Then\n\\[\n\\boxed{\n \\bigoplus_{\\chi\\in X}\n V\\otimes\\chi\n \\cong\n \\Ind_H^G\\Res_HV.\n}\n\\]\nThe left side is unchanged when \\(V\\) is replaced by another member of its extension torsor. Likewise, for cyclic base change,\n\\[\n\\boxed{\n \\mathop{\\boxplus}_{\\chi\\in X}\n \\pi\\otimes\\chi\n \\cong\n \\AI_{E/F}(\\BC_{E/F}\\pi)\n}\n\\]\ninside the established automorphic induction formalism. Therefore comparison of the total induced objects cannot determine the cyclic attachment offset.\n\\end{theorem}","\\begin{proof}\nThe restriction of every quotient character to \\(H\\) is trivial. The regular representation of \\(G/H\\) decomposes as the direct sum of all characters in \\(X\\). Hence\n\\[\n \\Ind_H^G\\Res_HV\n \\cong\n V\\otimes\\CC[G/H]\n \\cong\n \\bigoplus_{\\chi\\in X}V\\otimes\\chi.\n\\]\nReplacing \\(V\\) by \\(V\\otimes\\chi_0\\) only permutes the summands. The automorphic identity is the cyclic automorphic induction and base change analogue. It has the same permutation invariance.\n\\end{proof}","Let \\(G/H\\cong C_p\\), and choose a quotient character \\(\\xi\\). Let \\(G\\) be finite and let \\(V\\) be a finite dimensional semisimple complex representation of \\(G\\).","\\begin{definition}[Stable and moving types]\\label{def:stable_moving}\nAn irreducible representation \\(\\tau\\in\\Irr(H)\\) is stable when \\(\\tau^g\\cong\\tau\\) for every \\(g\\in G\\). Otherwise its orbit has size \\(p\\) and is called moving.\n\\end{definition}","\\begin{lemma}[Extension of a stable type]\\label{lem:stable_extension}\nEvery stable irreducible complex representation of \\(H\\) extends to \\(G\\).\n\\end{lemma}","\\begin{proof}\nChoose \\(g\\in G\\) whose class generates \\(G/H\\), so \\(g^p\\in H\\). Stability gives an invertible intertwiner \\(A\\) with\n\\[\n A\\tau(h)A^{-1}=\\tau(ghg^{-1})\\qquad(h\\in H).\n\\]\nIterate this identity \\(p\\) times. The operator \\(\\tau(g^p)^{-1}A^p\\) commutes with \\(\\tau(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of","\\begin{proof}\nChoose \\(g\\in G\\) whose class generates \\(G/H\\), so \\(g^p\\in H\\). Stability gives an invertible intertwiner \\(A\\) with\n\\[\n A\\tau(h)A^{-1}=\\tau(ghg^{-1})\\qquad(h\\in H).\n\\]\nIterate this identity \\(p\\) times. The operator \\(\\tau(g^p)^{-1}A^p\\) commutes with \\(\\tau(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its","\\begin{proof}\nChoose \\(g\\in G\\) whose class generates \\(G/H\\), so \\(g^p\\in H\\). Stability gives an invertible intertwiner \\(A\\) with\n\\[\n A\\tau(h)A^{-1}=\\tau(ghg^{-1})\\qquad(h\\in H).\n\\]\nIterate this identity \\(p\\) times. The operator \\(\\tau(g^p)^{-1}A^p\\) commutes with \\(\\tau(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is ir","\\begin{proof}\nChoose \\(g\\in G\\) whose class generates \\(G/H\\), so \\(g^p\\in H\\). Stability gives an invertible intertwiner \\(A\\) with\n\\[\n A\\tau(h)A^{-1}=\\tau(ghg^{-1})\\qquad(h\\in H).\n\\]\nIterate this identity \\(p\\) times. The operator \\(\\tau(g^p)^{-1}A^p\\) commutes with \\(\\tau(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In co","\\begin{proof}\nChoose \\(g\\in G\\) whose class generates \\(G/H\\), so \\(g^p\\in H\\). Stability gives an invertible intertwiner \\(A\\) with\n\\[\n A\\tau(h)A^{-1}=\\tau(ghg^{-1})\\qquad(h\\in H).\n\\]\nIterate this identity \\(p\\) times. The operator \\(\\tau(g^p)^{-1}A^p\\) commutes with \\(\\tau(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In cohomological language the scalar cyclic obstruction vanish","\\begin{proof}\nChoose \\(g\\in G\\) whose class generates \\(G/H\\), so \\(g^p\\in H\\). Stability gives an invertible intertwiner \\(A\\) with\n\\[\n A\\tau(h)A^{-1}=\\tau(ghg^{-1})\\qquad(h\\in H).\n\\]\nIterate this identity \\(p\\) times. The operator \\(\\tau(g^p)^{-1}A^p\\) commutes with \\(\\tau(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In cohomological language the scalar cyclic obstruction vanishes because a nonzero complex","\\begin{proof}\nChoose \\(g\\in G\\) whose class generates \\(G/H\\), so \\(g^p\\in H\\). Stability gives an invertible intertwiner \\(A\\) with\n\\[\n A\\tau(h)A^{-1}=\\tau(ghg^{-1})\\qquad(h\\in H).\n\\]\nIterate this identity \\(p\\) times. The operator \\(\\tau(g^p)^{-1}A^p\\) commutes with \\(\\tau(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In cohomological language the scalar cyclic obstruction vanishes because a nonzero complex number has a \\(p\\)th root. This uses the","\\begin{proof}\nChoose \\(g\\in G\\) whose class generates \\(G/H\\), so \\(g^p\\in H\\). Stability gives an invertible intertwiner \\(A\\) with\n\\[\n A\\tau(h)A^{-1}=\\tau(ghg^{-1})\\qquad(h\\in H).\n\\]\nIterate this identity \\(p\\) times. The operator \\(\\tau(g^p)^{-1}A^p\\) commutes with \\(\\tau(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In cohomological language the scalar cyclic obstruction vanishes because a nonzero complex number has a \\(p\\)th root. This uses the coefficient field, and mus","\\begin{proof}\nChoose \\(g\\in G\\) whose class generates \\(G/H\\), so \\(g^p\\in H\\). Stability gives an invertible intertwiner \\(A\\) with\n\\[\n A\\tau(h)A^{-1}=\\tau(ghg^{-1})\\qquad(h\\in H).\n\\]\nIterate this identity \\(p\\) times. The operator \\(\\tau(g^p)^{-1}A^p\\) commutes with \\(\\tau(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In cohomological language the scalar cyclic obstruction vanishes because a nonzero complex number has a \\(p\\)th root. This uses the coefficient field, and must not be transplanted unchanged into an arbitrary i","\\begin{proof}\nChoose \\(g\\in G\\) whose class generates \\(G/H\\), so \\(g^p\\in H\\). Stability gives an invertible intertwiner \\(A\\) with\n\\[\n A\\tau(h)A^{-1}=\\tau(ghg^{-1})\\qquad(h\\in H).\n\\]\nIterate this identity \\(p\\) times. The operator \\(\\tau(g^p)^{-1}A^p\\) commutes with \\(\\tau(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In cohomological language the scalar cyclic obstruction vanishes because a nonzero complex number has a \\(p\\)th root. This uses the coefficient field, and must not be transplanted unchanged into an arbitrary integral or r","\\begin{proof}\nChoose \\(g\\in G\\) whose class generates \\(G/H\\), so \\(g^p\\in H\\). Stability gives an invertible intertwiner \\(A\\) with\n\\[\n A\\tau(h)A^{-1}=\\tau(ghg^{-1})\\qquad(h\\in H).\n\\]\nIterate this identity \\(p\\) times. The operator \\(\\tau(g^p)^{-1}A^p\\) commutes with \\(\\tau(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In cohomological language the scalar cyclic obstruction vanishes because a nonzero complex number has a \\(p\\)th root. This uses the coefficient field, and must not be transplanted unchanged into an arbitrary integral or residual lifting problem.\n\\en","}\nChoose \\(g\\in G\\) whose class generates \\(G/H\\), so \\(g^p\\in H\\). Stability gives an invertible intertwiner \\(A\\) with\n\\[\n A\\tau(h)A^{-1}=\\tau(ghg^{-1})\\qquad(h\\in H).\n\\]\nIterate this identity \\(p\\) times. The operator \\(\\tau(g^p)^{-1}A^p\\) commutes with \\(\\tau(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In cohomological language the scalar cyclic obstruction vanishes because a nonzero complex number has a \\(p\\)th root. This uses the coefficient field, and must not be transplanted unchanged into an arbitrary integral or residual lifting problem.\n\\end{proof}",") whose class generates \\(G/H\\), so \\(g^p\\in H\\). Stability gives an invertible intertwiner \\(A\\) with\n\\[\n A\\tau(h)A^{-1}=\\tau(ghg^{-1})\\qquad(h\\in H).\n\\]\nIterate this identity \\(p\\) times. The operator \\(\\tau(g^p)^{-1}A^p\\) commutes with \\(\\tau(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In cohomological language the scalar cyclic obstruction vanishes because a nonzero complex number has a \\(p\\)th root. This uses the coefficient field, and must not be transplanted unchanged into an arbitrary integral or residual lifting problem.\n\\end{proof}","\\(G/H\\), so \\(g^p\\in H\\). Stability gives an invertible intertwiner \\(A\\) with\n\\[\n A\\tau(h)A^{-1}=\\tau(ghg^{-1})\\qquad(h\\in H).\n\\]\nIterate this identity \\(p\\) times. The operator \\(\\tau(g^p)^{-1}A^p\\) commutes with \\(\\tau(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In cohomological language the scalar cyclic obstruction vanishes because a nonzero complex number has a \\(p\\)th root. This uses the coefficient field, and must not be transplanted unchanged into an arbitrary integral or residual lifting problem.\n\\end{proof}","). Stability gives an invertible intertwiner \\(A\\) with\n\\[\n A\\tau(h)A^{-1}=\\tau(ghg^{-1})\\qquad(h\\in H).\n\\]\nIterate this identity \\(p\\) times. The operator \\(\\tau(g^p)^{-1}A^p\\) commutes with \\(\\tau(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In cohomological language the scalar cyclic obstruction vanishes because a nonzero complex number has a \\(p\\)th root. This uses the coefficient field, and must not be transplanted unchanged into an arbitrary integral or residual lifting problem.\n\\end{proof}","ves an invertible intertwiner \\(A\\) with\n\\[\n A\\tau(h)A^{-1}=\\tau(ghg^{-1})\\qquad(h\\in H).\n\\]\nIterate this identity \\(p\\) times. The operator \\(\\tau(g^p)^{-1}A^p\\) commutes with \\(\\tau(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In cohomological language the scalar cyclic obstruction vanishes because a nonzero complex number has a \\(p\\)th root. This uses the coefficient field, and must not be transplanted unchanged into an arbitrary integral or residual lifting problem.\n\\end{proof}","au(g^p)^{-1}A^p\\) commutes with \\(\\tau(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In cohomological language the scalar cyclic obstruction vanishes because a nonzero complex number has a \\(p\\)th root. This uses the coefficient field, and must not be transplanted unchanged into an arbitrary integral or residual lifting problem.\n\\end{proof}","commutes with \\(\\tau(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In cohomological language the scalar cyclic obstruction vanishes because a nonzero complex number has a \\(p\\)th root. This uses the coefficient field, and must not be transplanted unchanged into an arbitrary integral or residual lifting problem.\n\\end{proof}","au(H)\\), so Schur's lemma makes it a nonzero scalar \\(cI\\). Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In cohomological language the scalar cyclic obstruction vanishes because a nonzero complex number has a \\(p\\)th root. This uses the coefficient field, and must not be transplanted unchanged into an arbitrary integral or residual lifting problem.\n\\end{proof}",". Also \\(A\\) commutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In cohomological language the scalar cyclic obstruction vanishes because a nonzero complex number has a \\(p\\)th root. This uses the coefficient field, and must not be transplanted unchanged into an arbitrary integral or residual lifting problem.\n\\end{proof}","mutes with \\(\\tau(g^p)\\), by the same intertwining identity at \\(h=g^p\\). Choose \\(b\\in\\CC^\\times\\) with \\(b^p=c^{-1}\\) and put \\(R=bA\\). Then\n\\[\n R\\tau(h)R^{-1}=\\tau(ghg^{-1}),\\qquad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In cohomological language the scalar cyclic obstruction vanishes because a nonzero complex number has a \\(p\\)th root. This uses the coefficient field, and must not be transplanted unchanged into an arbitrary integral or residual lifting problem.\n\\end{proof}","quad R^p=\\tau(g^p).\n\\]\nThese are exactly the conjugation and closing relations for the coset presentation of \\(G\\). Assigning \\(g\\mapsto R\\) and \\(h\\mapsto\\tau(h)\\) therefore defines a representation of \\(G\\) extending \\(\\tau\\). Its restriction is irreducible, so the extension is irreducible. In cohomological language the scalar cyclic obstruction vanishes because a nonzero complex number has a \\(p\\)th root. This uses the coefficient field, and must not be transplanted unchanged into an arbitrary integral or residual lifting problem.\n\\end{proof}","Fix one extension \\(\\widetilde\\tau\\) for every stable \\(\\tau\\).","\\begin{definition}[Extension spectrum polynomial]\\label{def:extension_spectrum}\nFor a stable type \\(\\tau\\), the multiplicity space\n\\[\n M_\\tau(V)=\\Hom_H(\\tau,V)\n\\]\nbecomes a representation of \\(G/H\\) after comparing the action on \\(V\\) with \\(\\widetilde\\tau\\). Write\n\\[\n M_\\tau(V)\n \\cong\n \\bigoplus_{a\\in\\ZZ/p\\ZZ}\n (\\xi^a)^{\\oplus m_{\\tau,a}(V)}.\n\\]\nDefine\n\\[\n \\boxed{\n P_{\\tau,V}(z)\n =\n \\sum_{a\\in\\ZZ/p\\ZZ}\n m_{\\tau,a}(V)z^a\n }\n\\]\nin \\(\\ZZ[z]/(z^p-1)\\).\n\\end{definition}","m polynomial]\\label{def:extension_spectrum}\nFor a stable type \\(\\tau\\), the multiplicity space\n\\[\n M_\\tau(V)=\\Hom_H(\\tau,V)\n\\]\nbecomes a representation of \\(G/H\\) after comparing the action on \\(V\\) with \\(\\widetilde\\tau\\). Write\n\\[\n M_\\tau(V)\n \\cong\n \\bigoplus_{a\\in\\ZZ/p\\ZZ}\n (\\xi^a)^{\\oplus m_{\\tau,a}(V)}.\n\\]\nDefine\n\\[\n \\boxed{\n P_{\\tau,V}(z)\n =\n \\sum_{a\\in\\ZZ/p\\ZZ}\n m_{\\tau,a}(V)z^a\n }\n\\]\nin \\(\\ZZ[z]/(z^p-1)\\).\n\\end{definition}","\\begin{proposition}[Effect of quotient twisting]\\label{prop:spectrum_twist}\nFor every \\(j\\in\\ZZ/p\\ZZ\\),\n\\[\n \\boxed{\n P_{\\tau,V\\otimes\\xi^j}(z)\n =\n z^jP_{\\tau,V}(z).\n }\n\\]\nMoving orbit blocks are unchanged up to isomorphism by quotient twisting.\n\\end{proposition}","\\begin{proof}\nThe quotient character acts only on the multiplicity space, shifting every character label by \\(j\\). For a moving type, its inertia group is \\(H\\), and the corresponding irreducible block is induced from \\(H\\). Tensoring an induced representation by a character trivial on \\(H\\) produces an isomorphic induced representation.\n\\end{proof}","\\begin{theorem}[Cyclic extension spectrum criterion]\\label{thm:extension_spectrum_criterion}\nLet \\(V\\) and \\(W\\) be semisimple representations of \\(G\\) with\n\\[\n \\Res_HV\\cong\\Res_HW.\n\\]\nThen\n\\[\n \\boxed{\n W\\cong V\\otimes\\xi^j\n }\n\\]\nfor some \\(j\\in\\ZZ/p\\ZZ\\) if and only if, for every stable irreducible \\(H\\) type \\(\\tau\\),\n\\[\n \\boxed{\n P_{\\tau,W}(z)\n =\n z^jP_{\\tau,V}(z).\n }\n\\]\nThe moving orbit blocks require no additional condition.\n\\end{theorem}","\\begin{proof}\nDecompose the restriction to \\(H\\) into \\(G\\) orbits of irreducible types. For a moving orbit, Clifford theory gives one irreducible induced block, with multiplicity determined by the restriction. Quotient twisting fixes its isomorphism class.","For a stable type \\(\\tau\\), the full \\(\\tau\\) isotypic contribution is\n\\[\n \\widetilde\\tau\\otimes M_\\tau(V).\n\\]\nThe corresponding contribution for \\(W\\) is determined by \\(M_\\tau(W)\\). Tensoring by \\(\\xi^j\\) shifts every quotient character in every stable multiplicity space by the same \\(j\\). Thus the displayed polynomial identities are necessary and sufficient on each stable block. Direct sum over all blocks gives the result.\n\\end{proof}","\\begin{definition}[Alignment set]\\label{def:alignment_set}\nFor \\(V,W\\) with the same restriction to \\(H\\), define\n\\[\n \\boxed{\n \\calJ_H(V,W)\n =\n \\{j\\in\\ZZ/p\\ZZ:\n P_{\\tau,W}=z^jP_{\\tau,V}\n \\text{ for every stable }\\tau\\}.\n }\n\\]\n\\end{definition}","\\begin{corollary}[Structure of the alignment set]\\label{cor:alignment_coset}\nThe alignment set is empty or a coset of the common rotational stabilizer\n\\[\n \\bigcap_\\tau\n \\{j:z^jP_{\\tau,V}=P_{\\tau,V}\\}.\n\\]\nIn particular, one stable type with aperiodic spectrum makes the alignment index unique whenever it exists.\n\\end{corollary}","\\begin{proof}\nIf \\(j_0\\) is one solution, every other solution differs from it by a rotation stabilizing every spectrum polynomial. The converse is immediate.\n\\end{proof}","\\begin{example}[Irreducible stable restriction]\\label{ex:irreducible_stable}\nIf \\(\\Res_HV\\) is irreducible, its multiplicity polynomial is one monomial. Hence \\(\\calJ_H(V,W)\\) is a singleton. This recovers the ordinary extension torsor of \\cref{prop:galois_torsor}.\n\\end{example}","Fix a primitive \\(p\\)th root \\(\\zeta\\). For\n\\[\n P(z)=\\sum_{a\\in\\ZZ/p\\ZZ}m_az^a,\n\\]\ndefine\n\\[\n \\widehat P(r)\n =\n \\sum_{a\\in\\ZZ/p\\ZZ}m_a\\zeta^{ar},\n \\qquad\n r\\in\\ZZ/p\\ZZ.\n\\]","\\begin{proposition}[Fourier trace interpretation]\\label{prop:fourier_trace}\nLet \\(g\\) be a lift of a generator of \\(G/H\\). For a stable type \\(\\tau\\),\n\\[\n \\boxed{\n \\widehat P_{\\tau,V}(r)\n =\n \\Tr\\left(g^r\\mid M_\\tau(V)\\right).\n }\n\\]\nIf \\(V\\) is twisted by \\(\\xi^j\\), then\n\\[\n \\boxed{\n \\widehat P_{\\tau,V\\otimes\\xi^j}(r)\n =\n \\zeta^{jr}\\widehat P_{\\tau,V}(r).\n }\n\\]\n\\end{proposition}","\\begin{proof}\nThe quotient generator acts on the \\(\\xi^a\\) summand by \\(\\zeta^a\\). Taking the trace of its \\(r\\)th power gives the Fourier sum. The twist multiplies the action of \\(g^r\\) by \\(\\zeta^{jr}\\).\n\\end{proof}","\\begin{corollary}[Fourier attachment test]\\label{cor:fourier_attachment}\nSuppose \\(\\Res_HV\\cong\\Res_HW\\). Then \\(W\\cong V\\otimes\\xi^j\\) if and only if\n\\[\n \\boxed{\n \\widehat P_{\\tau,W}(r)\n =\n \\zeta^{jr}\\widehat P_{\\tau,V}(r)\n }\n\\]\nfor every stable type \\(\\tau\\) and every \\(r\\in\\ZZ/p\\ZZ\\).\n\\end{corollary}","\\begin{proof}\nThe discrete Fourier transform on \\(\\ZZ/p\\ZZ\\) is invertible. Thus the Fourier identities are equivalent to the polynomial shift identities in \\cref{thm:extension_spectrum_criterion}.\n\\end{proof}","\\begin{corollary}[One nonzero Fourier mode selects the shift]\\label{cor:one_fourier_mode}\nLet \\(r\\neq0\\) modulo \\(p\\). If\n\\[\n \\widehat P_{\\tau,V}(r)\\neq0,\n\\]\nthen any possible attachment shift is uniquely determined by the ratio\n\\[\n \\boxed{\n \\frac{\\widehat P_{\\tau,W}(r)}{\\widehat P_{\\tau,V}(r)}\n =\n \\zeta^{jr}.\n }\n\\]\n\\end{corollary}","\\begin{proof}\nSince \\(p\\) is prime, multiplication by the nonzero index \\(r\\) is invertible modulo \\(p\\). Thus \\(\\zeta^{jr}\\) uniquely determines \\(j\\).\n\\end{proof}","\\begin{definition}[Local spectral obstruction]\\label{def:local_spectral_obstruction}\nThe local spectral obstruction vanishes when, for every stable type \\(\\tau\\), the zero Fourier modes of the two multiplicity spectra occur at the same indices and all ratios at the remaining indices satisfy\n\\[\n \\widehat P_{\\tau,W}(r)/\\widehat P_{\\tau,V}(r)=\\zeta^{jr}\n\\]\nfor one common \\(j\\). When \\(\\widehat P_{\\tau,V}(r)=0\\), the condition requires \\(\\widehat P_{\\tau,W}(r)=0\\), rather than omitting the comparison. Equivalently, all the Fourier identities in \\cref{cor:fourier_attachment} hold. If no such \\(j\\) exists, the obstruction is nonzero.\n\\end{definition}","ruction]\\label{def:local_spectral_obstruction}\nThe local spectral obstruction vanishes when, for every stable type \\(\\tau\\), the zero Fourier modes of the two multiplicity spectra occur at the same indices and all ratios at the remaining indices satisfy\n\\[\n \\widehat P_{\\tau,W}(r)/\\widehat P_{\\tau,V}(r)=\\zeta^{jr}\n\\]\nfor one common \\(j\\). When \\(\\widehat P_{\\tau,V}(r)=0\\), the condition requires \\(\\widehat P_{\\tau,W}(r)=0\\), rather than omitting the comparison. Equivalently, all the Fourier identities in \\cref{cor:fourier_attachment} hold. If no such \\(j\\) exists, the obstruction is nonzero.\n\\end{definition}","multiplicity spectra occur at the same indices and all ratios at the remaining indices satisfy\n\\[\n \\widehat P_{\\tau,W}(r)/\\widehat P_{\\tau,V}(r)=\\zeta^{jr}\n\\]\nfor one common \\(j\\). When \\(\\widehat P_{\\tau,V}(r)=0\\), the condition requires \\(\\widehat P_{\\tau,W}(r)=0\\), rather than omitting the comparison. Equivalently, all the Fourier identities in \\cref{cor:fourier_attachment} hold. If no such \\(j\\) exists, the obstruction is nonzero.\n\\end{definition}","\\begin{proof}\nThere are finitely many stable types and \\(p\\) Fourier modes. Apply \\cref{cor:fourier_attachment}.\n\\end{proof}","\\begin{example}[A stable cyclic type]\\label{ex:stable_cyclic_type}\nLet\n\\[\n G=C_3\\times C_2,\n \\qquad\n H=C_3.\n\\]\nFix a nontrivial character \\(\\tau\\) of \\(H\\). It is stable under \\(G\\). After choosing the extension with trivial \\(C_2\\) coordinate, the two one dimensional extensions have spectra\n\\[\n P_{\\tau,V_0}(z)=1,\n \\qquad\n P_{\\tau,V_1}(z)=z.\n\\]\nThe alignment set is the singleton containing the unique quotient twist carrying one extension to the other.\n\\end{example}","\\begin{proof}\nThe quotient \\(C_2\\) has the trivial and sign characters. The multiplicity space is one dimensional, so its quotient character is the complete extension label.\n\\end{proof}","\\begin{example}[A moving type and self twist]\\label{ex:S3_moving_type}\nLet \\(G=S_3\\), let \\(H=A_3\\), and let \\(V\\) be the standard two dimensional representation. Its restriction is\n\\[\n \\Res_HV=\\tau\\oplus\\tau^{-1}\n\\]\nfor the two nontrivial characters of \\(C_3\\). These characters form one moving orbit under \\(G/H\\cong C_2\\). Hence there is no stable extension spectrum, and\n\\[\n V\\otimes\\operatorname{sgn}\\cong V.\n\\]\n\\end{example}","\\begin{proof}\nThe standard representation is induced from either nontrivial character of \\(A_3\\). Quotient twisting is trivial on the inducing subgroup, so it preserves the induced representation.\n\\end{proof}","Let \\(E/F\\) be cyclic of prime degree \\(p\\). Let \\(\\epsilon\\) be the corresponding idele class character. Suppose \\(\\rho\\) is an irreducible finite image representation of \\(G_F\\), its restriction \\(\\tau\\) to \\(G_E\\) is irreducible, and \\(\\Pi\\) is an automorphic representation attached to \\(\\tau\\). Let \\(\\pi_0\\) be one cuspidal descent of \\(\\Pi\\).","For each place \\(v\\) of \\(F\\), put\n\\[\n R_v=\\rho|_{W_{F_v}},\n \\qquad\n A_v=\\rec_{F_v}(\\pi_{0,v}).\n\\]\nCyclic base change compatibility gives\n\\[\n \\Res_{W_{E_w}}R_v\n \\cong\n \\Res_{W_{E_w}}A_v.\n\\]","\\begin{definition}[Local attachment set]\\label{def:local_attachment_set}\nWhen the local quotient has order \\(p\\), define\n\\[\n \\calJ_v\n =\n \\{j\\in\\ZZ/p\\ZZ:\n A_v\\otimes\\epsilon_v^j\\cong R_v\\}.\n\\]\nAt a split place, set \\(\\calJ_v=\\ZZ/p\\ZZ\\).\n\\end{definition}","\\begin{theorem}[Local computability of attachment]\\label{thm:local_attachment_spectrum}\nEvery set \\(\\calJ_v\\) is computed by the extension spectrum criterion of \\cref{thm:extension_spectrum_criterion} applied to the finite quotient through which the two local parameters factor.\n\\end{theorem}","\\begin{proof}\nThe restrictions of \\(A_v\\) and \\(R_v\\) to \\(W_{E_w}\\) are isomorphic. Their common restriction has finite image because \\(R_v\\) does. The parameter \\(A_v\\) also has finite image by \\cref{thm:local_artin_descent} below, so both factor through one finite quotient after replacing it by a common finite quotient. Apply \\cref{thm:extension_spectrum_criterion}.\n\\end{proof}","\\begin{theorem}[Global cyclic attachment criterion]\\label{thm:global_attachment_criterion}\nWith the preceding notation, the descent \\(\\pi_0\\otimes\\epsilon^j\\) is attached to \\(\\rho\\) if and only if\n\\[\n \\boxed{\n j\\in\\bigcap_v\\calJ_v.\n }\n\\]\nConsequently, the cyclic Strong Artin step succeeds exactly when this intersection is nonempty.\n\\end{theorem}","\\begin{proof}\nLocal Langlands is compatible with character twists. Hence\n\\[\n \\rec(\\pi_{0,v}\\otimes\\epsilon_v^j)\n \\cong\n A_v\\otimes\\epsilon_v^j.\n\\]\nThe global descent is attached exactly when this is isomorphic to \\(R_v\\) for every place \\(v\\), which is the displayed condition.\n\\end{proof}","\\begin{corollary}[Determinant chosen candidate]\\label{cor:det_candidate}\nIf \\(p\\nmid n\\), determinant and central character select one candidate \\(j_{\\det}\\). The cyclic Strong Artin step is then equivalent to\n\\[\n \\boxed{\n j_{\\det}\\in\\calJ_v\n \\quad\\text{for every }v.\n }\n\\]\n\\end{corollary}","\\begin{theorem}[Local Artin descent]\\label{thm:local_artin_descent}\nLet \\(E/F\\) be a finite extension, let \\(\\pi\\) be an automorphic representation of \\(\\GL_n(\\Adele_F)\\), and suppose its base change \\(\\Pi\\) to \\(E\\) exists with local compatibility. Assume every local Weil Deligne parameter of \\(\\Pi\\) has zero monodromy and finite Weil image. Then the same is true for every local parameter of \\(\\pi\\).\n\\end{theorem}","\\begin{proof}\nFix \\(v\\) and \\(w\\mid v\\), and write the local Weil Deligne parameter as \\((\\phi_v,N_v)\\). Base change restricts \\(\\phi_v\\) to \\(W_{E_w}\\) and keeps the monodromy operator. Since the parameter of \\(\\Pi_w\\) has zero monodromy, \\(N_v=0\\).","The image of \\(W_{E_w}\\) under \\(\\phi_v\\) is finite. The subgroup \\(W_{E_w}\\) has finite index in \\(W_{F_v}\\). Choose finitely many coset representatives. The full image \\(\\phi_v(W_{F_v})\\) is contained in the union of the corresponding finitely many cosets of the finite subgroup \\(\\phi_v(W_{E_w})\\). Hence it is finite.\n\\end{proof}","\\begin{corollary}[Tempered local descent]\\label{cor:tempered_local_descent}\nUnder the hypotheses of \\cref{thm:local_artin_descent}, every local parameter is bounded. Thus every local component of \\(\\pi\\) is tempered in the normalized local Langlands correspondence.\n\\end{corollary}","\\begin{proof}\nA finite subgroup of \\(\\GL_n(\\CC)\\) is bounded. Local Langlands for general linear groups identifies bounded parameters with tempered representations \\cite{Henniart2000}.\n\\end{proof}","Let \\(E/F\\) be a finite Galois extension of degree \\(d\\). Suppose \\(\\pi\\) base changes to an automorphic representation \\(\\Pi\\) attached to a finite image Artin representation \\(\\tau\\). Let \\(M\\) be the exponent of the finite image of \\(\\tau\\). Fix a finite set \\(S\\) containing the exceptional comparison places and all ramification of the extension and the representations. The unramified conclusions in this section refer to places outside this fixed set. Weak attachment over the extension field suffices for these conclusions. If all local attachment over the extension field is known, the preceding local finiteness theorem supplies the stronger conclusion at every place.","\\begin{theorem}[Finite Satake roots]\\label{thm:finite_satake_roots}\nAt every finite place \\(v\\) where \\(E/F\\), \\(\\pi\\), and \\(\\Pi\\) are unramified and the stated Artin polynomial comparison holds above \\(v\\), every normalized Satake root of \\(\\pi_v\\) belongs to\n\\[\n \\boxed{\n \\mu_{dM}.\n }\n\\]\n\\end{theorem}","\\begin{proof}\nChoose \\(w\\mid v\\) and let \\(f=f(w/v)\\). If the normalized Satake roots of \\(\\pi_v\\) are \\(\\alpha_1,\\ldots,\\alpha_n\\), unramified base change replaces the Satake conjugacy class by its \\(f\\)th power. Hence the roots of \\(\\Pi_w\\) are \\(\\alpha_i^f\\). Since \\(\\Pi_w\\) is attached to a representation of exponent dividing \\(M\\),\n\\[\n (\\alpha_i^f)^M=1.\n\\]\nThus \\(\\alpha_i\\in\\mu_{fM}\\). Since \\(f\\mid d\\), this group is contained in \\(\\mu_{dM}\\).\n\\end{proof}","egin{proof}\nChoose \\(w\\mid v\\) and let \\(f=f(w/v)\\). If the normalized Satake roots of \\(\\pi_v\\) are \\(\\alpha_1,\\ldots,\\alpha_n\\), unramified base change replaces the Satake conjugacy class by its \\(f\\)th power. Hence the roots of \\(\\Pi_w\\) are \\(\\alpha_i^f\\). Since \\(\\Pi_w\\) is attached to a representation of exponent dividing \\(M\\),\n\\[\n (\\alpha_i^f)^M=1.\n\\]\nThus \\(\\alpha_i\\in\\mu_{fM}\\). Since \\(f\\mid d\\), this group is contained in \\(\\mu_{dM}\\).\n\\end{proof}","(f=f(w/v)\\). If the normalized Satake roots of \\(\\pi_v\\) are \\(\\alpha_1,\\ldots,\\alpha_n\\), unramified base change replaces the Satake conjugacy class by its \\(f\\)th power. Hence the roots of \\(\\Pi_w\\) are \\(\\alpha_i^f\\). Since \\(\\Pi_w\\) is attached to a representation of exponent dividing \\(M\\),\n\\[\n (\\alpha_i^f)^M=1.\n\\]\nThus \\(\\alpha_i\\in\\mu_{fM}\\). Since \\(f\\mid d\\), this group is contained in \\(\\mu_{dM}\\).\n\\end{proof}","\\begin{corollary}[Integral Hecke polynomials]\\label{cor:integral_hecke}\nEvery normalized Hecke polynomial of \\(\\pi\\) at a place outside the fixed exceptional set \\(S\\) has coefficients in\n\\[\n \\boxed{\n \\ZZ[\\mu_{dM}].\n }\n\\]\nIn particular, all of its coefficients are algebraic integers.\n\\end{corollary}","\\begin{proof}\nThe coefficients are elementary symmetric polynomials in roots of unity from \\(\\mu_{dM}\\).\n\\end{proof}","\\begin{corollary}[Finite Hecke field]\\label{cor:finite_hecke_field}\nThe field generated by the normalized Hecke coefficients of \\(\\pi\\) outside \\(S\\) is a number field contained in\n\\[\n \\boxed{\n \\QQ(\\mu_{dM}).\n }\n\\]\nThere are only finitely many possible normalized Hecke polynomials of degree \\(n\\) at these places.\n\\end{corollary}","\\begin{proof}\nEvery coefficient belongs to the fixed cyclotomic field by \\cref{cor:integral_hecke}. Every root lies in the finite set \\(\\mu_{dM}\\), so only finitely many multisets of \\(n\\) roots, and therefore only finitely many Hecke polynomials, can occur.\n\\end{proof}","\\begin{corollary}[Finite order central character]\\label{cor:finite_central_char}\nThe central character of \\(\\pi\\) has finite order.\n\\end{corollary}","\\begin{proof}\nOutside the fixed exceptional set, every unramified determinant value lies in \\(\\mu_{dM}\\). The Hecke character \\(\\omega_\\pi^{dM}\\) is therefore trivial at every unramified place outside a finite set. The one dimensional strong multiplicity one statement, or equivalently uniqueness of a Hecke character from its almost everywhere unramified values, makes it the trivial global character. Thus \\(\\omega_\\pi\\) has finite order. This argument does not require a separate local finiteness assertion at an exceptional place.\n\\end{proof}","of}\nOutside the fixed exceptional set, every unramified determinant value lies in \\(\\mu_{dM}\\). The Hecke character \\(\\omega_\\pi^{dM}\\) is therefore trivial at every unramified place outside a finite set. The one dimensional strong multiplicity one statement, or equivalently uniqueness of a Hecke character from its almost everywhere unramified values, makes it the trivial global character. Thus \\(\\omega_\\pi\\) has finite order. This argument does not require a separate local finiteness assertion at an exceptional place.\n\\end{proof}","\\begin{definition}[Finite Satake cusp form]\\label{def:finite_satake_cusp}\nA cuspidal representation of \\(\\GL_n(\\Adele_F)\\) is finite Satake when, outside one finite exceptional set, its normalized unramified Satake roots lie in one finite subgroup of \\(\\CC^\\times\\). The exceptional set is fixed, not allowed to change separately for each coefficient or prime.\n\\end{definition}","\\begin{proposition}[Finite number of unramified Hecke polynomials]\\label{prop:finite_number_hecke}\nUnder the hypotheses of \\cref{thm:finite_satake_roots}, put \\(N=dM\\). The number of possible normalized Hecke polynomials of degree \\(n\\) outside \\(S\\) is at most\n\\[\n \\boxed{\n \\binom{N+n-1}{n}.\n }\n\\]\n\\end{proposition}","\\begin{proof}\nA polynomial is determined by the multiset of its \\(n\\) Satake roots. Those roots lie in the \\(N\\) element set \\(\\mu_N\\). The number of multisets of size \\(n\\) selected from \\(N\\) elements is the displayed binomial coefficient.\n\\end{proof}","\\begin{corollary}[Finite trace alphabet]\\label{cor:finite_trace_alphabet}\nThe normalized Hecke traces of \\(\\pi\\) outside \\(S\\) belong to a finite subset of \\(\\ZZ[\\mu_{dM}]\\).\n\\end{corollary}","\\begin{proof}\nEach trace is a sum of \\(n\\) roots from the fixed finite set \\(\\mu_{dM}\\).\n\\end{proof}","Let \\(K\\) be a number field containing the normalized unramified Hecke coefficients of a finite Satake descent, the character values of the target Artin representation, and the values of the quotient character. Enlarge it to a splitting field in which the target and its restriction have matrix realizations. Let \\(\\OO_K\\) be its ring of integers.","\\begin{definition}[Finite polynomial alphabet]\\label{def:finite_alphabet}\nThe finite polynomial alphabet of the cyclic attachment problem is the union\n\\[\n\\calA\n=\n\\calA_{\\pi}\n\\cup\n\\calA_{\\rho},\n\\]\nwhere \\(\\calA_{\\pi}\\) is the set of normalized Hecke polynomials occurring for \\(\\pi\\) outside the fixed exceptional set, and \\(\\calA_{\\rho}\\) is the set of characteristic polynomials occurring for the finitely many twists\n\\[\n\\rho\\otimes\\epsilon^j,\n\\qquad\nj\\in\\ZZ/p\\ZZ.\n\\]\n\\end{definition}","nite polynomial alphabet]\\label{def:finite_alphabet}\nThe finite polynomial alphabet of the cyclic attachment problem is the union\n\\[\n\\calA\n=\n\\calA_{\\pi}\n\\cup\n\\calA_{\\rho},\n\\]\nwhere \\(\\calA_{\\pi}\\) is the set of normalized Hecke polynomials occurring for \\(\\pi\\) outside the fixed exceptional set, and \\(\\calA_{\\rho}\\) is the set of characteristic polynomials occurring for the finitely many twists\n\\[\n\\rho\\otimes\\epsilon^j,\n\\qquad\nj\\in\\ZZ/p\\ZZ.\n\\]\n\\end{definition}","\\begin{proposition}[Finiteness of the alphabet]\\label{prop:finite_alphabet}\nThe set \\(\\calA\\) is finite and is contained in \\(\\OO_K[T]\\).\n\\end{proposition}","\\begin{definition}[Separating coefficient prime]\\label{def:separating_prime}\nA finite place \\(\\lambda\\) of \\(K\\) is separating for \\(\\calA\\) when reduction modulo \\(\\lambda\\) is injective on \\(\\calA\\).\n\\end{definition}","\\begin{theorem}[Existence of separating primes]\\label{thm:separating_primes}\nAll but finitely many finite places of \\(K\\) are separating for \\(\\calA\\).\n\\end{theorem}","\\begin{proof}\nFor every ordered pair of distinct polynomials \\(P,Q\\in\\calA\\), choose one nonzero coefficient difference. Only finitely many finite places divide one of these finitely many nonzero algebraic integers. Outside their union, no two polynomials have the same reduction.\n\\end{proof}","\\begin{lemma}[Good reduction of an absolutely irreducible finite representation]\\label{lem:good_irreducible_reduction}\nLet \\(\\tau\\) be an absolutely irreducible representation of a finite group over a number field \\(K\\). In a fixed matrix realization, exclude the finitely many coefficient places at which a group matrix has a denominator. The resulting stable local lattices have absolutely irreducible reduction at all but finitely many further places.\n\\end{lemma}","\\begin{proof}\nAbsolute irreducibility and Burnside's theorem give group elements \\(g_1,\\ldots,g_{n^2}\\) whose matrices form a \\(K\\) basis of \\(M_n(K)\\). Vectorize these matrices in the fixed basis. Their determinant is nonzero. Exclude places at which that determinant is zero or has a denominator, together with denominators of the finitely many group matrices. At a remaining place the coordinate lattice is stable, and the reduced matrices span the full matrix algebra over the residue field. They continue to span it after extending the residue field. Any invariant subspace is then invariant under every matrix, so it is zero or the full space. This proves absolute irreducibility of the reduction. The argument is local at the good places and does not require a globally free integral lattice ov","\\begin{proof}\nAbsolute irreducibility and Burnside's theorem give group elements \\(g_1,\\ldots,g_{n^2}\\) whose matrices form a \\(K\\) basis of \\(M_n(K)\\). Vectorize these matrices in the fixed basis. Their determinant is nonzero. Exclude places at which that determinant is zero or has a denominator, together with denominators of the finitely many group matrices. At a remaining place the coordinate lattice is stable, and the reduced matrices span the full matrix algebra over the residue field. They continue to span it after extending the residue field. Any invariant subspace is then invariant under every matrix, so it is zero or the full space. This proves absolute irreducibility of the reduction. The argument is local at the good places and does not require a globally free integral lattice over \\(\\OO_K\\).\n\\end{proof}","is stable, and the reduced matrices span the full matrix algebra over the residue field. They continue to span it after extending the residue field. Any invariant subspace is then invariant under every matrix, so it is zero or the full space. This proves absolute irreducibility of the reduction. The argument is local at the good places and does not require a globally free integral lattice over \\(\\OO_K\\).\n\\end{proof}","\\begin{definition}[Good separating prime]\\label{def:good_separating_prime}\nA finite place \\(\\lambda\\) is good for the cyclic attachment problem when","\\begin{enumerate}\n\\item it is separating for \\(\\calA\\)\n\\item its residue characteristic is different from \\(p\\)\n\\item the reduction of \\(\\tau=\\rho|_{G_E}\\) is absolutely irreducible.\n\\end{enumerate}\n\\end{definition}","\\begin{corollary}[Abundance of good primes]\\label{cor:abundant_good_primes}\nAll but finitely many finite places of \\(K\\) are good.\n\\end{corollary}","\\begin{proof}\nCombine \\cref{thm:separating_primes,lem:good_irreducible_reduction} and exclude the finitely many places above \\(p\\).\n\\end{proof}","\\begin{theorem}[Finite trace image bound]\\label{thm:finite_trace_image_bound}\nLet \\(k\\) be algebraically closed of characteristic zero or characteristic \\(\\ell>n\\). Let \\(\\Gamma\\subseteq\\GL_n(k)\\) be a subgroup, and assume its defining module \\(k^n\\) is semisimple. Suppose its trace set is finite, with\n\\[\n t=\\#\\{\\Tr(\\gamma):\\gamma\\in\\Gamma\\}.\n\\]\nThen \\(\\Gamma\\) is finite and\n\\[\n \\abs\\Gamma\\le t^{\\dim_k A}\\le t^{n^2},\n \\qquad A=\\operatorname{span}_k\\Gamma.\n\\]\nIn particular this bounds a finite semisimple residual matrix group in characteristic greater than its rank.\n\\end{theorem}","\\begin{proof}\nBecause products of group matrices are group matrices, \\(A\\) is a finite dimensional subalgebra of \\(M_n(k)\\). An \\(A\\) submodule is the same thing as a \\(\\Gamma\\) submodule. The given module is semisimple and faithful for \\(A\\). The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form\n\\[\n A\\cong\\bigoplus_iM_{d_i}(k),\\qquad\n k^n\\cong\\bigoplus_i(k^{d_i})^{\\oplus m_i},\n \\qquad\\sum_i d_im_i=n.\n\\]\nFaithfulness gives \\(m_i\\ge1\\) for every block. The trace pairing\n\\[\n B(x,y)=\\Tr_{k^n}(xy)\n\\]\nrestricts on the \\(i\\)th block to \\(m_i\\Tr(XY)\\). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units \\(E_{ab}\\) with \\(E_{ba}\\). Each \\(m_","\\begin{proof}\nBecause products of group matrices are group matrices, \\(A\\) is a finite dimensional subalgebra of \\(M_n(k)\\). An \\(A\\) submodule is the same thing as a \\(\\Gamma\\) submodule. The given module is semisimple and faithful for \\(A\\). The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form\n\\[\n A\\cong\\bigoplus_iM_{d_i}(k),\\qquad\n k^n\\cong\\bigoplus_i(k^{d_i})^{\\oplus m_i},\n \\qquad\\sum_i d_im_i=n.\n\\]\nFaithfulness gives \\(m_i\\ge1\\) for every block. The trace pairing\n\\[\n B(x,y)=\\Tr_{k^n}(xy)\n\\]\nrestricts on the \\(i\\)th block to \\(m_i\\Tr(XY)\\). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units \\(E_{ab}\\) with \\(E_{ba}\\). Each \\(m_i\\) is nonzero in \\(k\\): this is automatic in cha","\\begin{proof}\nBecause products of group matrices are group matrices, \\(A\\) is a finite dimensional subalgebra of \\(M_n(k)\\). An \\(A\\) submodule is the same thing as a \\(\\Gamma\\) submodule. The given module is semisimple and faithful for \\(A\\). The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form\n\\[\n A\\cong\\bigoplus_iM_{d_i}(k),\\qquad\n k^n\\cong\\bigoplus_i(k^{d_i})^{\\oplus m_i},\n \\qquad\\sum_i d_im_i=n.\n\\]\nFaithfulness gives \\(m_i\\ge1\\) for every block. The trace pairing\n\\[\n B(x,y)=\\Tr_{k^n}(xy)\n\\]\nrestricts on the \\(i\\)th block to \\(m_i\\Tr(XY)\\). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units \\(E_{ab}\\) with \\(E_{ba}\\). Each \\(m_i\\) is nonzero in \\(k\\): this is automatic in characteristi","\\begin{proof}\nBecause products of group matrices are group matrices, \\(A\\) is a finite dimensional subalgebra of \\(M_n(k)\\). An \\(A\\) submodule is the same thing as a \\(\\Gamma\\) submodule. The given module is semisimple and faithful for \\(A\\). The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form\n\\[\n A\\cong\\bigoplus_iM_{d_i}(k),\\qquad\n k^n\\cong\\bigoplus_i(k^{d_i})^{\\oplus m_i},\n \\qquad\\sum_i d_im_i=n.\n\\]\nFaithfulness gives \\(m_i\\ge1\\) for every block. The trace pairing\n\\[\n B(x,y)=\\Tr_{k^n}(xy)\n\\]\nrestricts on the \\(i\\)th block to \\(m_i\\Tr(XY)\\). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units \\(E_{ab}\\) with \\(E_{ba}\\). Each \\(m_i\\) is nonzero in \\(k\\): this is automatic in characteristic zero, and follows from \\(1\\le m_i\\le n<\\el","\\begin{proof}\nBecause products of group matrices are group matrices, \\(A\\) is a finite dimensional subalgebra of \\(M_n(k)\\). An \\(A\\) submodule is the same thing as a \\(\\Gamma\\) submodule. The given module is semisimple and faithful for \\(A\\). The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form\n\\[\n A\\cong\\bigoplus_iM_{d_i}(k),\\qquad\n k^n\\cong\\bigoplus_i(k^{d_i})^{\\oplus m_i},\n \\qquad\\sum_i d_im_i=n.\n\\]\nFaithfulness gives \\(m_i\\ge1\\) for every block. The trace pairing\n\\[\n B(x,y)=\\Tr_{k^n}(xy)\n\\]\nrestricts on the \\(i\\)th block to \\(m_i\\Tr(XY)\\). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units \\(E_{ab}\\) with \\(E_{ba}\\). Each \\(m_i\\) is nonzero in \\(k\\): this is automatic in characteristic zero, and follows from \\(1\\le m_i\\le n<\\ell\\) in residual characteristic. Thus \\(B\\) is nondegenerate on \\(","oof}\nBecause products of group matrices are group matrices, \\(A\\) is a finite dimensional subalgebra of \\(M_n(k)\\). An \\(A\\) submodule is the same thing as a \\(\\Gamma\\) submodule. The given module is semisimple and faithful for \\(A\\). The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form\n\\[\n A\\cong\\bigoplus_iM_{d_i}(k),\\qquad\n k^n\\cong\\bigoplus_i(k^{d_i})^{\\oplus m_i},\n \\qquad\\sum_i d_im_i=n.\n\\]\nFaithfulness gives \\(m_i\\ge1\\) for every block. The trace pairing\n\\[\n B(x,y)=\\Tr_{k^n}(xy)\n\\]\nrestricts on the \\(i\\)th block to \\(m_i\\Tr(XY)\\). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units \\(E_{ab}\\) with \\(E_{ba}\\). Each \\(m_i\\) is nonzero in \\(k\\): this is automatic in characteristic zero, and follows from \\(1\\le m_i\\le n<\\ell\\) in residual characteristic. Thus \\(B\\) is nondegenerate on \\(A\\).","odule is the same thing as a \\(\\Gamma\\) submodule. The given module is semisimple and faithful for \\(A\\). The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form\n\\[\n A\\cong\\bigoplus_iM_{d_i}(k),\\qquad\n k^n\\cong\\bigoplus_i(k^{d_i})^{\\oplus m_i},\n \\qquad\\sum_i d_im_i=n.\n\\]\nFaithfulness gives \\(m_i\\ge1\\) for every block. The trace pairing\n\\[\n B(x,y)=\\Tr_{k^n}(xy)\n\\]\nrestricts on the \\(i\\)th block to \\(m_i\\Tr(XY)\\). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units \\(E_{ab}\\) with \\(E_{ba}\\). Each \\(m_i\\) is nonzero in \\(k\\): this is automatic in characteristic zero, and follows from \\(1\\le m_i\\le n<\\ell\\) in residual characteristic. Thus \\(B\\) is nondegenerate on \\(A\\).","le. The given module is semisimple and faithful for \\(A\\). The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form\n\\[\n A\\cong\\bigoplus_iM_{d_i}(k),\\qquad\n k^n\\cong\\bigoplus_i(k^{d_i})^{\\oplus m_i},\n \\qquad\\sum_i d_im_i=n.\n\\]\nFaithfulness gives \\(m_i\\ge1\\) for every block. The trace pairing\n\\[\n B(x,y)=\\Tr_{k^n}(xy)\n\\]\nrestricts on the \\(i\\)th block to \\(m_i\\Tr(XY)\\). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units \\(E_{ab}\\) with \\(E_{ba}\\). Each \\(m_i\\) is nonzero in \\(k\\): this is automatic in characteristic zero, and follows from \\(1\\le m_i\\le n<\\ell\\) in residual characteristic. Thus \\(B\\) is nondegenerate on \\(A\\).","ul for \\(A\\). The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form\n\\[\n A\\cong\\bigoplus_iM_{d_i}(k),\\qquad\n k^n\\cong\\bigoplus_i(k^{d_i})^{\\oplus m_i},\n \\qquad\\sum_i d_im_i=n.\n\\]\nFaithfulness gives \\(m_i\\ge1\\) for every block. The trace pairing\n\\[\n B(x,y)=\\Tr_{k^n}(xy)\n\\]\nrestricts on the \\(i\\)th block to \\(m_i\\Tr(XY)\\). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units \\(E_{ab}\\) with \\(E_{ba}\\). Each \\(m_i\\) is nonzero in \\(k\\): this is automatic in characteristic zero, and follows from \\(1\\le m_i\\le n<\\ell\\) in residual characteristic. Thus \\(B\\) is nondegenerate on \\(A\\).","Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form\n\\[\n A\\cong\\bigoplus_iM_{d_i}(k),\\qquad\n k^n\\cong\\bigoplus_i(k^{d_i})^{\\oplus m_i},\n \\qquad\\sum_i d_im_i=n.\n\\]\nFaithfulness gives \\(m_i\\ge1\\) for every block. The trace pairing\n\\[\n B(x,y)=\\Tr_{k^n}(xy)\n\\]\nrestricts on the \\(i\\)th block to \\(m_i\\Tr(XY)\\). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units \\(E_{ab}\\) with \\(E_{ba}\\). Each \\(m_i\\) is nonzero in \\(k\\): this is automatic in characteristic zero, and follows from \\(1\\le m_i\\le n<\\ell\\) in residual characteristic. Thus \\(B\\) is nondegenerate on \\(A\\).","dule. It is therefore zero. The split semisimple algebra has the form\n\\[\n A\\cong\\bigoplus_iM_{d_i}(k),\\qquad\n k^n\\cong\\bigoplus_i(k^{d_i})^{\\oplus m_i},\n \\qquad\\sum_i d_im_i=n.\n\\]\nFaithfulness gives \\(m_i\\ge1\\) for every block. The trace pairing\n\\[\n B(x,y)=\\Tr_{k^n}(xy)\n\\]\nrestricts on the \\(i\\)th block to \\(m_i\\Tr(XY)\\). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units \\(E_{ab}\\) with \\(E_{ba}\\). Each \\(m_i\\) is nonzero in \\(k\\): this is automatic in characteristic zero, and follows from \\(1\\le m_i\\le n<\\ell\\) in residual characteristic. Thus \\(B\\) is nondegenerate on \\(A\\).","efore zero. The split semisimple algebra has the form\n\\[\n A\\cong\\bigoplus_iM_{d_i}(k),\\qquad\n k^n\\cong\\bigoplus_i(k^{d_i})^{\\oplus m_i},\n \\qquad\\sum_i d_im_i=n.\n\\]\nFaithfulness gives \\(m_i\\ge1\\) for every block. The trace pairing\n\\[\n B(x,y)=\\Tr_{k^n}(xy)\n\\]\nrestricts on the \\(i\\)th block to \\(m_i\\Tr(XY)\\). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units \\(E_{ab}\\) with \\(E_{ba}\\). Each \\(m_i\\) is nonzero in \\(k\\): this is automatic in characteristic zero, and follows from \\(1\\le m_i\\le n<\\ell\\) in residual characteristic. Thus \\(B\\) is nondegenerate on \\(A\\).","plit semisimple algebra has the form\n\\[\n A\\cong\\bigoplus_iM_{d_i}(k),\\qquad\n k^n\\cong\\bigoplus_i(k^{d_i})^{\\oplus m_i},\n \\qquad\\sum_i d_im_i=n.\n\\]\nFaithfulness gives \\(m_i\\ge1\\) for every block. The trace pairing\n\\[\n B(x,y)=\\Tr_{k^n}(xy)\n\\]\nrestricts on the \\(i\\)th block to \\(m_i\\Tr(XY)\\). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units \\(E_{ab}\\) with \\(E_{ba}\\). Each \\(m_i\\) is nonzero in \\(k\\): this is automatic in characteristic zero, and follows from \\(1\\le m_i\\le n<\\ell\\) in residual characteristic. Thus \\(B\\) is nondegenerate on \\(A\\).","a has the form\n\\[\n A\\cong\\bigoplus_iM_{d_i}(k),\\qquad\n k^n\\cong\\bigoplus_i(k^{d_i})^{\\oplus m_i},\n \\qquad\\sum_i d_im_i=n.\n\\]\nFaithfulness gives \\(m_i\\ge1\\) for every block. The trace pairing\n\\[\n B(x,y)=\\Tr_{k^n}(xy)\n\\]\nrestricts on the \\(i\\)th block to \\(m_i\\Tr(XY)\\). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units \\(E_{ab}\\) with \\(E_{ba}\\). Each \\(m_i\\) is nonzero in \\(k\\): this is automatic in characteristic zero, and follows from \\(1\\le m_i\\le n<\\ell\\) in residual characteristic. Thus \\(B\\) is nondegenerate on \\(A\\).","goplus_i(k^{d_i})^{\\oplus m_i},\n \\qquad\\sum_i d_im_i=n.\n\\]\nFaithfulness gives \\(m_i\\ge1\\) for every block. The trace pairing\n\\[\n B(x,y)=\\Tr_{k^n}(xy)\n\\]\nrestricts on the \\(i\\)th block to \\(m_i\\Tr(XY)\\). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units \\(E_{ab}\\) with \\(E_{ba}\\). Each \\(m_i\\) is nonzero in \\(k\\): this is automatic in characteristic zero, and follows from \\(1\\le m_i\\le n<\\ell\\) in residual characteristic. Thus \\(B\\) is nondegenerate on \\(A\\).","\\]\nFaithfulness gives \\(m_i\\ge1\\) for every block. The trace pairing\n\\[\n B(x,y)=\\Tr_{k^n}(xy)\n\\]\nrestricts on the \\(i\\)th block to \\(m_i\\Tr(XY)\\). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units \\(E_{ab}\\) with \\(E_{ba}\\). Each \\(m_i\\) is nonzero in \\(k\\): this is automatic in characteristic zero, and follows from \\(1\\le m_i\\le n<\\ell\\) in residual characteristic. Thus \\(B\\) is nondegenerate on \\(A\\).","i\\ge1\\) for every block. The trace pairing\n\\[\n B(x,y)=\\Tr_{k^n}(xy)\n\\]\nrestricts on the \\(i\\)th block to \\(m_i\\Tr(XY)\\). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units \\(E_{ab}\\) with \\(E_{ba}\\). Each \\(m_i\\) is nonzero in \\(k\\): this is automatic in characteristic zero, and follows from \\(1\\le m_i\\le n<\\ell\\) in residual characteristic. Thus \\(B\\) is nondegenerate on \\(A\\).","Choose a basis \\(\\gamma_1,\\ldots,\\gamma_r\\) of \\(A\\) from the spanning set \\(\\Gamma\\), where \\(r=\\dim_k A\\le n^2\\). Send an image element to\n\\[\n \\Phi(\\gamma)=\\bigl(\\Tr(\\gamma\\gamma_1),\\ldots,\n                         \\Tr(\\gamma\\gamma_r)\\bigr).\n\\]\nAll coordinates lie in the trace set, because \\(\\gamma\\gamma_i\\in\\Gamma\\). If two tuples agree, then \\(\\gamma-\\gamma'\\) pairs to zero with every basis element, hence every element of \\(A\\). Nondegeneracy gives \\(\\gamma=\\gamma'\\). Therefore \\(\\Phi\\) injects \\(\\Gamma\\) into a set of size \\(t^r\\). This proves finiteness as well as the bound. Prior finiteness of \\(\\Gamma\\) was not needed in this argument.\n\\end{proof}",",\\gamma_r\\) of \\(A\\) from the spanning set \\(\\Gamma\\), where \\(r=\\dim_k A\\le n^2\\). Send an image element to\n\\[\n \\Phi(\\gamma)=\\bigl(\\Tr(\\gamma\\gamma_1),\\ldots,\n                         \\Tr(\\gamma\\gamma_r)\\bigr).\n\\]\nAll coordinates lie in the trace set, because \\(\\gamma\\gamma_i\\in\\Gamma\\). If two tuples agree, then \\(\\gamma-\\gamma'\\) pairs to zero with every basis element, hence every element of \\(A\\). Nondegeneracy gives \\(\\gamma=\\gamma'\\). Therefore \\(\\Phi\\) injects \\(\\Gamma\\) into a set of size \\(t^r\\). This proves finiteness as well as the bound. Prior finiteness of \\(\\Gamma\\) was not needed in this argument.\n\\end{proof}","he spanning set \\(\\Gamma\\), where \\(r=\\dim_k A\\le n^2\\). Send an image element to\n\\[\n \\Phi(\\gamma)=\\bigl(\\Tr(\\gamma\\gamma_1),\\ldots,\n                         \\Tr(\\gamma\\gamma_r)\\bigr).\n\\]\nAll coordinates lie in the trace set, because \\(\\gamma\\gamma_i\\in\\Gamma\\). If two tuples agree, then \\(\\gamma-\\gamma'\\) pairs to zero with every basis element, hence every element of \\(A\\). Nondegeneracy gives \\(\\gamma=\\gamma'\\). Therefore \\(\\Phi\\) injects \\(\\Gamma\\) into a set of size \\(t^r\\). This proves finiteness as well as the bound. Prior finiteness of \\(\\Gamma\\) was not needed in this argument.\n\\end{proof}","ere \\(r=\\dim_k A\\le n^2\\). Send an image element to\n\\[\n \\Phi(\\gamma)=\\bigl(\\Tr(\\gamma\\gamma_1),\\ldots,\n                         \\Tr(\\gamma\\gamma_r)\\bigr).\n\\]\nAll coordinates lie in the trace set, because \\(\\gamma\\gamma_i\\in\\Gamma\\). If two tuples agree, then \\(\\gamma-\\gamma'\\) pairs to zero with every basis element, hence every element of \\(A\\). Nondegeneracy gives \\(\\gamma=\\gamma'\\). Therefore \\(\\Phi\\) injects \\(\\Gamma\\) into a set of size \\(t^r\\). This proves finiteness as well as the bound. Prior finiteness of \\(\\Gamma\\) was not needed in this argument.\n\\end{proof}","le n^2\\). Send an image element to\n\\[\n \\Phi(\\gamma)=\\bigl(\\Tr(\\gamma\\gamma_1),\\ldots,\n                         \\Tr(\\gamma\\gamma_r)\\bigr).\n\\]\nAll coordinates lie in the trace set, because \\(\\gamma\\gamma_i\\in\\Gamma\\). If two tuples agree, then \\(\\gamma-\\gamma'\\) pairs to zero with every basis element, hence every element of \\(A\\). Nondegeneracy gives \\(\\gamma=\\gamma'\\). Therefore \\(\\Phi\\) injects \\(\\Gamma\\) into a set of size \\(t^r\\). This proves finiteness as well as the bound. Prior finiteness of \\(\\Gamma\\) was not needed in this argument.\n\\end{proof}","t to\n\\[\n \\Phi(\\gamma)=\\bigl(\\Tr(\\gamma\\gamma_1),\\ldots,\n                         \\Tr(\\gamma\\gamma_r)\\bigr).\n\\]\nAll coordinates lie in the trace set, because \\(\\gamma\\gamma_i\\in\\Gamma\\). If two tuples agree, then \\(\\gamma-\\gamma'\\) pairs to zero with every basis element, hence every element of \\(A\\). Nondegeneracy gives \\(\\gamma=\\gamma'\\). Therefore \\(\\Phi\\) injects \\(\\Gamma\\) into a set of size \\(t^r\\). This proves finiteness as well as the bound. Prior finiteness of \\(\\Gamma\\) was not needed in this argument.\n\\end{proof}","\\Tr(\\gamma\\gamma_r)\\bigr).\n\\]\nAll coordinates lie in the trace set, because \\(\\gamma\\gamma_i\\in\\Gamma\\). If two tuples agree, then \\(\\gamma-\\gamma'\\) pairs to zero with every basis element, hence every element of \\(A\\). Nondegeneracy gives \\(\\gamma=\\gamma'\\). Therefore \\(\\Phi\\) injects \\(\\Gamma\\) into a set of size \\(t^r\\). This proves finiteness as well as the bound. Prior finiteness of \\(\\Gamma\\) was not needed in this argument.\n\\end{proof}","\\begin{corollary}[Uniform bound for a residual Galois image]\\label{cor:residual_image_bound}\nLet\n\\[\n \\overline r:G_F\\longrightarrow\\GL_n(\\overline\\FF_{\\ell})\n\\]\nbe continuous and semisimple, with \\(\\ell>n\\). If its Frobenius traces belong to a set of size at most \\(t\\) on a Chebotarev dense set, then\n\\[\n\\boxed{\n \\abs{\\im\\overline r}\\leq t^{n^2}.\n}\n\\]\n\\end{corollary}","\\begin{standardfact}[Cross characteristic lifting]\\label{fact:cross_characteristic_lifting}\nLet \\(\\Gamma\\) be finite and let \\(\\ell\\nmid\\abs\\Gamma\\). Every semisimple representation of \\(\\Gamma\\) over an algebraically closed field of characteristic \\(\\ell\\) is the reduction of a characteristic zero representation after a finite extension of coefficient fields. Irreducible representations lift to irreducible representations, and characteristic polynomials reduce compatibly. This is the nonmodular case of Brauer character theory \\cite{BrauerNesbitt1941,CurtisReiner1962,Serre1977}.\n\\end{standardfact}","\\begin{definition}[Universal bounded cyclotomic alphabet]\\label{def:bounded_cyclotomic_alphabet}\nFor positive integers \\(n\\) and \\(B\\), let \\(\\calQ_{n,B}\\) be the finite set of monic degree \\(n\\) polynomials whose roots are roots of unity of order at most \\(B\\).\n\\end{definition}","\\begin{theorem}[One large residual realization gives an Artin representation]\\label{thm:large_residual_realization}\nLet \\(K\\) be a number field, let \\(S\\) be a finite set of places of \\(F\\), and let \\(\\calP\\subseteq\\OO_K[T]\\) be a nonempty finite set of monic degree \\(n\\) polynomials. Assume all their roots are roots of unity. For \\(v\\notin S\\), choose \\(P_v\\in\\calP\\).","There is a finite set \\(\\Sigma(\\calP)\\) of rational primes, fixed from the alphabet before any realization is chosen, with the following property. Let \\(\\lambda\\) be a coefficient place whose residue characteristic \\(\\ell\\) is outside \\(\\Sigma(\\calP)\\). Suppose a continuous semisimple representation\n\\[\n \\overline r_\\lambda:G_F\\longrightarrow\\GL_n(\\overline\\FF_\\ell)\n\\]\nis unramified outside \\(S\\) and the places above \\(\\ell\\), and has characteristic polynomial \\(\\overline{P_v}\\) at every remaining unramified place. Then there is a finite image semisimple complex representation \\(r\\) with polynomial \\(P_v\\) at those places. If \\(\\overline r_\\lambda\\) is absolutely irreducible, then \\(r\\) is irreducible.\n\\end{theorem}","\\(\\Sigma(\\calP)\\) of rational primes, fixed from the alphabet before any realization is chosen, with the following property. Let \\(\\lambda\\) be a coefficient place whose residue characteristic \\(\\ell\\) is outside \\(\\Sigma(\\calP)\\). Suppose a continuous semisimple representation\n\\[\n \\overline r_\\lambda:G_F\\longrightarrow\\GL_n(\\overline\\FF_\\ell)\n\\]\nis unramified outside \\(S\\) and the places above \\(\\ell\\), and has characteristic polynomial \\(\\overline{P_v}\\) at every remaining unramified place. Then there is a finite image semisimple complex representation \\(r\\) with polynomial \\(P_v\\) at those places. If \\(\\overline r_\\lambda\\) is absolutely irreducible, then \\(r\\) is irreducible.\n\\end{theorem}","from the alphabet before any realization is chosen, with the following property. Let \\(\\lambda\\) be a coefficient place whose residue characteristic \\(\\ell\\) is outside \\(\\Sigma(\\calP)\\). Suppose a continuous semisimple representation\n\\[\n \\overline r_\\lambda:G_F\\longrightarrow\\GL_n(\\overline\\FF_\\ell)\n\\]\nis unramified outside \\(S\\) and the places above \\(\\ell\\), and has characteristic polynomial \\(\\overline{P_v}\\) at every remaining unramified place. Then there is a finite image semisimple complex representation \\(r\\) with polynomial \\(P_v\\) at those places. If \\(\\overline r_\\lambda\\) is absolutely irreducible, then \\(r\\) is irreducible.\n\\end{theorem}","whose residue characteristic \\(\\ell\\) is outside \\(\\Sigma(\\calP)\\). Suppose a continuous semisimple representation\n\\[\n \\overline r_\\lambda:G_F\\longrightarrow\\GL_n(\\overline\\FF_\\ell)\n\\]\nis unramified outside \\(S\\) and the places above \\(\\ell\\), and has characteristic polynomial \\(\\overline{P_v}\\) at every remaining unramified place. Then there is a finite image semisimple complex representation \\(r\\) with polynomial \\(P_v\\) at those places. If \\(\\overline r_\\lambda\\) is absolutely irreducible, then \\(r\\) is irreducible.\n\\end{theorem}","eristic \\(\\ell\\) is outside \\(\\Sigma(\\calP)\\). Suppose a continuous semisimple representation\n\\[\n \\overline r_\\lambda:G_F\\longrightarrow\\GL_n(\\overline\\FF_\\ell)\n\\]\nis unramified outside \\(S\\) and the places above \\(\\ell\\), and has characteristic polynomial \\(\\overline{P_v}\\) at every remaining unramified place. Then there is a finite image semisimple complex representation \\(r\\) with polynomial \\(P_v\\) at those places. If \\(\\overline r_\\lambda\\) is absolutely irreducible, then \\(r\\) is irreducible.\n\\end{theorem}","\\(\\Sigma(\\calP)\\). Suppose a continuous semisimple representation\n\\[\n \\overline r_\\lambda:G_F\\longrightarrow\\GL_n(\\overline\\FF_\\ell)\n\\]\nis unramified outside \\(S\\) and the places above \\(\\ell\\), and has characteristic polynomial \\(\\overline{P_v}\\) at every remaining unramified place. Then there is a finite image semisimple complex representation \\(r\\) with polynomial \\(P_v\\) at those places. If \\(\\overline r_\\lambda\\) is absolutely irreducible, then \\(r\\) is irreducible.\n\\end{theorem}","sentation\n\\[\n \\overline r_\\lambda:G_F\\longrightarrow\\GL_n(\\overline\\FF_\\ell)\n\\]\nis unramified outside \\(S\\) and the places above \\(\\ell\\), and has characteristic polynomial \\(\\overline{P_v}\\) at every remaining unramified place. Then there is a finite image semisimple complex representation \\(r\\) with polynomial \\(P_v\\) at those places. If \\(\\overline r_\\lambda\\) is absolutely irreducible, then \\(r\\) is irreducible.\n\\end{theorem}","\\begin{proof}\nLet \\(t\\ge1\\) be the number of trace values in the source alphabet and put \\(B=t^{n^2}\\). First fix a finite normal extension \\(L/\\QQ\\) containing \\(K\\) and the roots of unity of orders dividing\n\\[\n D_B=\\operatorname{lcm}(1,2,\\ldots,B).\n\\]\nEvery polynomial in \\(\\calQ_{n,B}\\), the bounded cyclotomic alphabet, has coefficients in \\(L\\). Form the union of all Galois conjugates in \\(L\\) of \\(\\calP\\cup\\calQ_{n,B}\\). This is a fixed finite set of integral polynomials. For each distinct pair choose one nonzero coefficient difference. Exclude the rational primes below the prime divisors of all these differences, and exclude the rational primes \\(\\ell\\le\\max(n,B)\\). This defines a finite exceptional set indepen","\\begin{proof}\nLet \\(t\\ge1\\) be the number of trace values in the source alphabet and put \\(B=t^{n^2}\\). First fix a finite normal extension \\(L/\\QQ\\) containing \\(K\\) and the roots of unity of orders dividing\n\\[\n D_B=\\operatorname{lcm}(1,2,\\ldots,B).\n\\]\nEvery polynomial in \\(\\calQ_{n,B}\\), the bounded cyclotomic alphabet, has coefficients in \\(L\\). Form the union of all Galois conjugates in \\(L\\) of \\(\\calP\\cup\\calQ_{n,B}\\). This is a fixed finite set of integral polynomials. For each distinct pair choose one nonzero coefficient difference. Exclude the rational primes below the prime divisors of all these differences, and exclude the rational primes \\(\\ell\\le\\max(n,B)\\). This defines a finite exceptional set independent of any residual representation or choice of lift. Outside it, reductio","\\begin{proof}\nLet \\(t\\ge1\\) be the number of trace values in the source alphabet and put \\(B=t^{n^2}\\). First fix a finite normal extension \\(L/\\QQ\\) containing \\(K\\) and the roots of unity of orders dividing\n\\[\n D_B=\\operatorname{lcm}(1,2,\\ldots,B).\n\\]\nEvery polynomial in \\(\\calQ_{n,B}\\), the bounded cyclotomic alphabet, has coefficients in \\(L\\). Form the union of all Galois conjugates in \\(L\\) of \\(\\calP\\cup\\calQ_{n,B}\\). This is a fixed finite set of integral polynomials. For each distinct pair choose one nonzero coefficient difference. Exclude the rational primes below the prime divisors of all these differences, and exclude the rational primes \\(\\ell\\le\\max(n,B)\\). This defines a finite exceptional set independent of any residual representation or choice of lift. Outside it, reduction at every place of \\(L\\) above \\(\\ell\\) is inj","\\begin{proof}\nLet \\(t\\ge1\\) be the number of trace values in the source alphabet and put \\(B=t^{n^2}\\). First fix a finite normal extension \\(L/\\QQ\\) containing \\(K\\) and the roots of unity of orders dividing\n\\[\n D_B=\\operatorname{lcm}(1,2,\\ldots,B).\n\\]\nEvery polynomial in \\(\\calQ_{n,B}\\), the bounded cyclotomic alphabet, has coefficients in \\(L\\). Form the union of all Galois conjugates in \\(L\\) of \\(\\calP\\cup\\calQ_{n,B}\\). This is a fixed finite set of integral polynomials. For each distinct pair choose one nonzero coefficient difference. Exclude the rational primes below the prime divisors of all these differences, and exclude the rational primes \\(\\ell\\le\\max(n,B)\\). This defines a finite exceptional set independent of any residual representation or choice of lift. Outside it, reduction at every place of \\(L\\) above \\(\\ell\\) is injective on this en","\\begin{proof}\nLet \\(t\\ge1\\) be the number of trace values in the source alphabet and put \\(B=t^{n^2}\\). First fix a finite normal extension \\(L/\\QQ\\) containing \\(K\\) and the roots of unity of orders dividing\n\\[\n D_B=\\operatorname{lcm}(1,2,\\ldots,B).\n\\]\nEvery polynomial in \\(\\calQ_{n,B}\\), the bounded cyclotomic alphabet, has coefficients in \\(L\\). Form the union of all Galois conjugates in \\(L\\) of \\(\\calP\\cup\\calQ_{n,B}\\). This is a fixed finite set of integral polynomials. For each distinct pair choose one nonzero coefficient difference. Exclude the rational primes below the prime divisors of all these differences, and exclude the rational primes \\(\\ell\\le\\max(n,B)\\). This defines a finite exceptional set independent of any residual representation or choice of lift. Outside it, reduction at every place of \\(L\\) above \\(\\ell\\) is injective on this enlarged polynomial set.","n{proof}\nLet \\(t\\ge1\\) be the number of trace values in the source alphabet and put \\(B=t^{n^2}\\). First fix a finite normal extension \\(L/\\QQ\\) containing \\(K\\) and the roots of unity of orders dividing\n\\[\n D_B=\\operatorname{lcm}(1,2,\\ldots,B).\n\\]\nEvery polynomial in \\(\\calQ_{n,B}\\), the bounded cyclotomic alphabet, has coefficients in \\(L\\). Form the union of all Galois conjugates in \\(L\\) of \\(\\calP\\cup\\calQ_{n,B}\\). This is a fixed finite set of integral polynomials. For each distinct pair choose one nonzero coefficient difference. Exclude the rational primes below the prime divisors of all these differences, and exclude the rational primes \\(\\ell\\le\\max(n,B)\\). This defines a finite exceptional set independent of any residual representation or choice of lift. Outside it, reduction at every place of \\(L\\) above \\(\\ell\\) is injective on this enlarged polynomial set.","polynomial in \\(\\calQ_{n,B}\\), the bounded cyclotomic alphabet, has coefficients in \\(L\\). Form the union of all Galois conjugates in \\(L\\) of \\(\\calP\\cup\\calQ_{n,B}\\). This is a fixed finite set of integral polynomials. For each distinct pair choose one nonzero coefficient difference. Exclude the rational primes below the prime divisors of all these differences, and exclude the rational primes \\(\\ell\\le\\max(n,B)\\). This defines a finite exceptional set independent of any residual representation or choice of lift. Outside it, reduction at every place of \\(L\\) above \\(\\ell\\) is injective on this enlarged polynomial set.",",B}\\). This is a fixed finite set of integral polynomials. For each distinct pair choose one nonzero coefficient difference. Exclude the rational primes below the prime divisors of all these differences, and exclude the rational primes \\(\\ell\\le\\max(n,B)\\). This defines a finite exceptional set independent of any residual representation or choice of lift. Outside it, reduction at every place of \\(L\\) above \\(\\ell\\) is injective on this enlarged polynomial set.","is a fixed finite set of integral polynomials. For each distinct pair choose one nonzero coefficient difference. Exclude the rational primes below the prime divisors of all these differences, and exclude the rational primes \\(\\ell\\le\\max(n,B)\\). This defines a finite exceptional set independent of any residual representation or choice of lift. Outside it, reduction at every place of \\(L\\) above \\(\\ell\\) is injective on this enlarged polynomial set.","Continuity of \\(\\overline r_\\lambda\\) into the discrete residual matrix group gives finite image \\(\\Gamma\\). Every conjugacy class of that image occurs at unramified Frobenius places away from the finite excluded set by Chebotarev. Its trace set has size at most \\(t\\), because the Frobenius polynomials are reductions of the prescribed alphabet. \\Cref{thm:finite_trace_image_bound} gives \\(\\abs\\Gamma\\le B<\\ell\\), so \\(\\ell\\nmid\\abs\\Gamma\\).","rline r_\\lambda\\) into the discrete residual matrix group gives finite image \\(\\Gamma\\). Every conjugacy class of that image occurs at unramified Frobenius places away from the finite excluded set by Chebotarev. Its trace set has size at most \\(t\\), because the Frobenius polynomials are reductions of the prescribed alphabet. \\Cref{thm:finite_trace_image_bound} gives \\(\\abs\\Gamma\\le B<\\ell\\), so \\(\\ell\\nmid\\abs\\Gamma\\).","Use the cross characteristic lifting statement in \\cref{fact:cross_characteristic_lifting}, in a splitting modular system compatible with the chosen coefficient place. It gives an ordinary characteristic zero representation of \\(\\Gamma\\) whose reduction is the prescribed residual representation. Compose it with \\(G_F\\to\\Gamma\\). Each lifted eigenvalue has order dividing the order of an element of \\(\\Gamma\\), hence at most \\(B\\). Every lifted Frobenius polynomial consequently belongs to \\(\\calQ_{n,B}\\).","e the cross characteristic lifting statement in \\cref{fact:cross_characteristic_lifting}, in a splitting modular system compatible with the chosen coefficient place. It gives an ordinary characteristic zero representation of \\(\\Gamma\\) whose reduction is the prescribed residual representation. Compose it with \\(G_F\\to\\Gamma\\). Each lifted eigenvalue has order dividing the order of an element of \\(\\Gamma\\), hence at most \\(B\\). Every lifted Frobenius polynomial consequently belongs to \\(\\calQ_{n,B}\\).","lifting statement in \\cref{fact:cross_characteristic_lifting}, in a splitting modular system compatible with the chosen coefficient place. It gives an ordinary characteristic zero representation of \\(\\Gamma\\) whose reduction is the prescribed residual representation. Compose it with \\(G_F\\to\\Gamma\\). Each lifted eigenvalue has order dividing the order of an element of \\(\\Gamma\\), hence at most \\(B\\). Every lifted Frobenius polynomial consequently belongs to \\(\\calQ_{n,B}\\).","n a splitting modular system compatible with the chosen coefficient place. It gives an ordinary characteristic zero representation of \\(\\Gamma\\) whose reduction is the prescribed residual representation. Compose it with \\(G_F\\to\\Gamma\\). Each lifted eigenvalue has order dividing the order of an element of \\(\\Gamma\\), hence at most \\(B\\). Every lifted Frobenius polynomial consequently belongs to \\(\\calQ_{n,B}\\).","Only characteristic polynomials, not all matrix entries of the lift, are needed in the fixed separation field \\(L\\). The lift can be realized over a further number field and reduced at a compatible place above \\(\\lambda\\). Its Frobenius polynomial and \\(P_v\\) have equal reduction there. On restriction to \\(L\\), injectivity of the fixed enlarged alphabet makes them equal in characteristic zero. The chosen compatible embedding fixes the specified copy of \\(K\\), so the equalities recover the original \\(P_v\\) simultaneously. The nonmodular correspondence preserves irreducibility. This proves the final assertion.\n\\end{proof}","the lift, are needed in the fixed separation field \\(L\\). The lift can be realized over a further number field and reduced at a compatible place above \\(\\lambda\\). Its Frobenius polynomial and \\(P_v\\) have equal reduction there. On restriction to \\(L\\), injectivity of the fixed enlarged alphabet makes them equal in characteristic zero. The chosen compatible embedding fixes the specified copy of \\(K\\), so the equalities recover the original \\(P_v\\) simultaneously. The nonmodular correspondence preserves irreducibility. This proves the final assertion.\n\\end{proof}","ield \\(L\\). The lift can be realized over a further number field and reduced at a compatible place above \\(\\lambda\\). Its Frobenius polynomial and \\(P_v\\) have equal reduction there. On restriction to \\(L\\), injectivity of the fixed enlarged alphabet makes them equal in characteristic zero. The chosen compatible embedding fixes the specified copy of \\(K\\), so the equalities recover the original \\(P_v\\) simultaneously. The nonmodular correspondence preserves irreducibility. This proves the final assertion.\n\\end{proof}","\\begin{corollary}[Large residual realization of a finite Satake cusp form]\\label{cor:large_residual_finite_satake}\nLet \\(\\pi\\) be a finite Satake cuspidal representation whose normalized Hecke polynomials lie in \\(K\\). If one sufficiently large residual Galois realization with the expected unramified polynomials exists, then \\(\\pi\\) has a finite image complex Galois realization. Absolute irreducibility of the residual realization gives an irreducible Artin representation.\n\\end{corollary}","\\begin{proof}\nApply \\cref{thm:large_residual_realization} to the finite Hecke polynomial alphabet of \\(\\pi\\).\n\\end{proof}","\\begin{definition}[Residual realization of a descent]\\label{def:residual_realization}\nLet \\(\\lambda\\) be a finite place of the coefficient field. A residual realization of \\(\\pi\\) at \\(\\lambda\\) is a continuous semisimple representation\n\\[\n\\overline r_{\\pi,\\lambda}:G_F\\longrightarrow\\GL_n(\\overline k_{\\lambda})\n\\]\nwhose characteristic polynomial at every unramified place outside a finite set is the reduction of the normalized Hecke polynomial of \\(\\pi\\).\n\\end{definition}","]\\label{def:residual_realization}\nLet \\(\\lambda\\) be a finite place of the coefficient field. A residual realization of \\(\\pi\\) at \\(\\lambda\\) is a continuous semisimple representation\n\\[\n\\overline r_{\\pi,\\lambda}:G_F\\longrightarrow\\GL_n(\\overline k_{\\lambda})\n\\]\nwhose characteristic polynomial at every unramified place outside a finite set is the reduction of the normalized Hecke polynomial of \\(\\pi\\).\n\\end{definition}","\\begin{theorem}[One prime residual attachment]\\label{thm:one_prime_attachment}\nUse the cyclic Artin setup of \\cref{def:attached_descent}. For this unramified theorem it is enough that the lower cuspidal representation \\(\\Pi\\) is weakly attached to \\(\\tau\\), with \\(\\pi\\) a cuspidal cyclic descent. Let \\(\\lambda\\) be a good separating prime. Suppose \\(\\pi\\) has a residual realization \\(\\overline r_{\\pi,\\lambda}\\). Then there is a unique index\n\\[\nj_{\\lambda}\\in\\ZZ/p\\ZZ\n\\]\nsuch that\n\\[\n\\boxed{\nP_{\\pi,v}(T)\n=\nP_{\\rho\\otimes\\epsilon^{j_{\\lambda}},v}(T)\n}\n\\]\nfor every finite place \\(v\\) outside a finite set. Consequently,\n\\[\n\\pi\\otimes\\epsilon^{-j_{\\lambda}}\n\\]\nis weakly attached to \\(\\rho\\).\n\\end{theorem}","label{thm:one_prime_attachment}\nUse the cyclic Artin setup of \\cref{def:attached_descent}. For this unramified theorem it is enough that the lower cuspidal representation \\(\\Pi\\) is weakly attached to \\(\\tau\\), with \\(\\pi\\) a cuspidal cyclic descent. Let \\(\\lambda\\) be a good separating prime. Suppose \\(\\pi\\) has a residual realization \\(\\overline r_{\\pi,\\lambda}\\). Then there is a unique index\n\\[\nj_{\\lambda}\\in\\ZZ/p\\ZZ\n\\]\nsuch that\n\\[\n\\boxed{\nP_{\\pi,v}(T)\n=\nP_{\\rho\\otimes\\epsilon^{j_{\\lambda}},v}(T)\n}\n\\]\nfor every finite place \\(v\\) outside a finite set. Consequently,\n\\[\n\\pi\\otimes\\epsilon^{-j_{\\lambda}}\n\\]\nis weakly attached to \\(\\rho\\).\n\\end{theorem}","cyclic Artin setup of \\cref{def:attached_descent}. For this unramified theorem it is enough that the lower cuspidal representation \\(\\Pi\\) is weakly attached to \\(\\tau\\), with \\(\\pi\\) a cuspidal cyclic descent. Let \\(\\lambda\\) be a good separating prime. Suppose \\(\\pi\\) has a residual realization \\(\\overline r_{\\pi,\\lambda}\\). Then there is a unique index\n\\[\nj_{\\lambda}\\in\\ZZ/p\\ZZ\n\\]\nsuch that\n\\[\n\\boxed{\nP_{\\pi,v}(T)\n=\nP_{\\rho\\otimes\\epsilon^{j_{\\lambda}},v}(T)\n}\n\\]\nfor every finite place \\(v\\) outside a finite set. Consequently,\n\\[\n\\pi\\otimes\\epsilon^{-j_{\\lambda}}\n\\]\nis weakly attached to \\(\\rho\\).\n\\end{theorem}","the lower cuspidal representation \\(\\Pi\\) is weakly attached to \\(\\tau\\), with \\(\\pi\\) a cuspidal cyclic descent. Let \\(\\lambda\\) be a good separating prime. Suppose \\(\\pi\\) has a residual realization \\(\\overline r_{\\pi,\\lambda}\\). Then there is a unique index\n\\[\nj_{\\lambda}\\in\\ZZ/p\\ZZ\n\\]\nsuch that\n\\[\n\\boxed{\nP_{\\pi,v}(T)\n=\nP_{\\rho\\otimes\\epsilon^{j_{\\lambda}},v}(T)\n}\n\\]\nfor every finite place \\(v\\) outside a finite set. Consequently,\n\\[\n\\pi\\otimes\\epsilon^{-j_{\\lambda}}\n\\]\nis weakly attached to \\(\\rho\\).\n\\end{theorem}","is weakly attached to \\(\\tau\\), with \\(\\pi\\) a cuspidal cyclic descent. Let \\(\\lambda\\) be a good separating prime. Suppose \\(\\pi\\) has a residual realization \\(\\overline r_{\\pi,\\lambda}\\). Then there is a unique index\n\\[\nj_{\\lambda}\\in\\ZZ/p\\ZZ\n\\]\nsuch that\n\\[\n\\boxed{\nP_{\\pi,v}(T)\n=\nP_{\\rho\\otimes\\epsilon^{j_{\\lambda}},v}(T)\n}\n\\]\nfor every finite place \\(v\\) outside a finite set. Consequently,\n\\[\n\\pi\\otimes\\epsilon^{-j_{\\lambda}}\n\\]\nis weakly attached to \\(\\rho\\).\n\\end{theorem}","c descent. Let \\(\\lambda\\) be a good separating prime. Suppose \\(\\pi\\) has a residual realization \\(\\overline r_{\\pi,\\lambda}\\). Then there is a unique index\n\\[\nj_{\\lambda}\\in\\ZZ/p\\ZZ\n\\]\nsuch that\n\\[\n\\boxed{\nP_{\\pi,v}(T)\n=\nP_{\\rho\\otimes\\epsilon^{j_{\\lambda}},v}(T)\n}\n\\]\nfor every finite place \\(v\\) outside a finite set. Consequently,\n\\[\n\\pi\\otimes\\epsilon^{-j_{\\lambda}}\n\\]\nis weakly attached to \\(\\rho\\).\n\\end{theorem}","\\begin{proof}\nRestrict the residual realization to \\(G_E\\). At every unramified place \\(w\\) of \\(E\\) above a good place \\(v\\) of \\(F\\), cyclic base change identifies the characteristic polynomial of the restriction with the reduction of the Hecke polynomial of \\(\\Pi_w\\). Since \\(\\Pi\\) is attached to \\(\\tau\\), this is the characteristic polynomial of \\(\\overline\\tau(\\Frob_w)\\). Chebotarev density and Brauer Nesbitt give\n\\[\n\\left(\n\\overline r_{\\pi,\\lambda}|_{G_E}\n\\right)^{\\mathrm{ss}}\n\\cong\n\\overline\\tau.\n\\]\nThe right side is absolutely irreducible and has the full dimension \\(n\\). Hence the restriction itself has one composition factor and is irreducible, so it is isomorphic to \\(\\overline\\tau\\).","e residual realization to \\(G_E\\). At every unramified place \\(w\\) of \\(E\\) above a good place \\(v\\) of \\(F\\), cyclic base change identifies the characteristic polynomial of the restriction with the reduction of the Hecke polynomial of \\(\\Pi_w\\). Since \\(\\Pi\\) is attached to \\(\\tau\\), this is the characteristic polynomial of \\(\\overline\\tau(\\Frob_w)\\). Chebotarev density and Brauer Nesbitt give\n\\[\n\\left(\n\\overline r_{\\pi,\\lambda}|_{G_E}\n\\right)^{\\mathrm{ss}}\n\\cong\n\\overline\\tau.\n\\]\nThe right side is absolutely irreducible and has the full dimension \\(n\\). Hence the restriction itself has one composition factor and is irreducible, so it is isomorphic to \\(\\overline\\tau\\).","ic polynomial of the restriction with the reduction of the Hecke polynomial of \\(\\Pi_w\\). Since \\(\\Pi\\) is attached to \\(\\tau\\), this is the characteristic polynomial of \\(\\overline\\tau(\\Frob_w)\\). Chebotarev density and Brauer Nesbitt give\n\\[\n\\left(\n\\overline r_{\\pi,\\lambda}|_{G_E}\n\\right)^{\\mathrm{ss}}\n\\cong\n\\overline\\tau.\n\\]\nThe right side is absolutely irreducible and has the full dimension \\(n\\). Hence the restriction itself has one composition factor and is irreducible, so it is isomorphic to \\(\\overline\\tau\\).","attached to \\(\\tau\\), this is the characteristic polynomial of \\(\\overline\\tau(\\Frob_w)\\). Chebotarev density and Brauer Nesbitt give\n\\[\n\\left(\n\\overline r_{\\pi,\\lambda}|_{G_E}\n\\right)^{\\mathrm{ss}}\n\\cong\n\\overline\\tau.\n\\]\nThe right side is absolutely irreducible and has the full dimension \\(n\\). Hence the restriction itself has one composition factor and is irreducible, so it is isomorphic to \\(\\overline\\tau\\).","The representations \\(\\overline r_{\\pi,\\lambda}\\) and \\(\\overline\\rho\\) are therefore two extensions of the same absolutely irreducible representation of \\(G_E\\). Their ratio is a character of \\(G_F/G_E\\). Since the residue characteristic is different from \\(p\\), the reduced quotient character has order \\(p\\), and there is a unique index \\(j_{\\lambda}\\) with\n\\[\n\\overline r_{\\pi,\\lambda}\n\\cong\n\\overline\\rho\\otimes\\overline\\epsilon^{j_{\\lambda}}.\n\\]\nAt every good unramified place,\n\\[\nP_{\\pi,v}\n\\equiv\nP_{\\rho\\otimes\\epsilon^{j_{\\lambda}},v}\n\\pmod\\lambda.\n\\]\nBoth polynomials belong to the finite alphabet \\(\\calA\\). The prime is separating, so the congruence is equality in characteristic zero. Uniqueness follows from the freeness of the exten","The representations \\(\\overline r_{\\pi,\\lambda}\\) and \\(\\overline\\rho\\) are therefore two extensions of the same absolutely irreducible representation of \\(G_E\\). Their ratio is a character of \\(G_F/G_E\\). Since the residue characteristic is different from \\(p\\), the reduced quotient character has order \\(p\\), and there is a unique index \\(j_{\\lambda}\\) with\n\\[\n\\overline r_{\\pi,\\lambda}\n\\cong\n\\overline\\rho\\otimes\\overline\\epsilon^{j_{\\lambda}}.\n\\]\nAt every good unramified place,\n\\[\nP_{\\pi,v}\n\\equiv\nP_{\\rho\\otimes\\epsilon^{j_{\\lambda}},v}\n\\pmod\\lambda.\n\\]\nBoth polynomials belong to the finite alphabet \\(\\calA\\). The prime is separating, so the congruence is equality in characteristic zero. Uniqueness follows from the freeness of the extension torsor.\n\\end{proo","The representations \\(\\overline r_{\\pi,\\lambda}\\) and \\(\\overline\\rho\\) are therefore two extensions of the same absolutely irreducible representation of \\(G_E\\). Their ratio is a character of \\(G_F/G_E\\). Since the residue characteristic is different from \\(p\\), the reduced quotient character has order \\(p\\), and there is a unique index \\(j_{\\lambda}\\) with\n\\[\n\\overline r_{\\pi,\\lambda}\n\\cong\n\\overline\\rho\\otimes\\overline\\epsilon^{j_{\\lambda}}.\n\\]\nAt every good unramified place,\n\\[\nP_{\\pi,v}\n\\equiv\nP_{\\rho\\otimes\\epsilon^{j_{\\lambda}},v}\n\\pmod\\lambda.\n\\]\nBoth polynomials belong to the finite alphabet \\(\\calA\\). The prime is separating, so the congruence is equality in characteristic zero. Uniqueness follows from the freeness of the extension torsor.\n\\end{proof}","two extensions of the same absolutely irreducible representation of \\(G_E\\). Their ratio is a character of \\(G_F/G_E\\). Since the residue characteristic is different from \\(p\\), the reduced quotient character has order \\(p\\), and there is a unique index \\(j_{\\lambda}\\) with\n\\[\n\\overline r_{\\pi,\\lambda}\n\\cong\n\\overline\\rho\\otimes\\overline\\epsilon^{j_{\\lambda}}.\n\\]\nAt every good unramified place,\n\\[\nP_{\\pi,v}\n\\equiv\nP_{\\rho\\otimes\\epsilon^{j_{\\lambda}},v}\n\\pmod\\lambda.\n\\]\nBoth polynomials belong to the finite alphabet \\(\\calA\\). The prime is separating, so the congruence is equality in characteristic zero. Uniqueness follows from the freeness of the extension torsor.\n\\end{proof}","e the residue characteristic is different from \\(p\\), the reduced quotient character has order \\(p\\), and there is a unique index \\(j_{\\lambda}\\) with\n\\[\n\\overline r_{\\pi,\\lambda}\n\\cong\n\\overline\\rho\\otimes\\overline\\epsilon^{j_{\\lambda}}.\n\\]\nAt every good unramified place,\n\\[\nP_{\\pi,v}\n\\equiv\nP_{\\rho\\otimes\\epsilon^{j_{\\lambda}},v}\n\\pmod\\lambda.\n\\]\nBoth polynomials belong to the finite alphabet \\(\\calA\\). The prime is separating, so the congruence is equality in characteristic zero. Uniqueness follows from the freeness of the extension torsor.\n\\end{proof}","\\begin{corollary}[Only finitely many local checks remain]\\label{cor:finitely_many_local_checks}\nUnder \\cref{thm:one_prime_attachment}, all local attachment to \\(\\rho\\) is equivalent to checking the label \\(-j_\\lambda\\) at every exceptional finite place excluded from the unramified comparison and at every archimedean place.\n\\end{corollary}","\\begin{corollary}[Infinitely many residual realizations suffice]\\label{cor:infinitely_many_residual}\nIf residual realizations of \\(\\pi\\) exist at infinitely many coefficient places, then one of them is good, and \\(\\pi\\) is weakly attached after one quotient twist.\n\\end{corollary}","\\begin{corollary}[A congruent cohomological eigensystem suffices]\\label{cor:congruent_cohomological}\nLet \\(\\lambda\\) be good. Suppose the reduction of the Hecke eigensystem of \\(\\pi\\) occurs in a setting that supplies a semisimple residual Galois representation with the expected unramified characteristic polynomials. Then the conclusion of \\cref{thm:one_prime_attachment} holds.\n\\end{corollary}","Let \\(K\\) be a number field containing the values of the order \\(p\\) quotient characters, let \\(\\lambda_p\\) be a place above \\(p\\), and let \\(\\OO\\) be the corresponding valuation ring.","\\begin{theorem}[Semisimple collapse of cyclic extension labels]\\label{thm:semisimple_label_collapse}\nLet \\(G\\) be finite, let \\(H\\triangleleft G\\) with \\(G/H\\cong C_p\\), and let \\(V,W\\) be semisimple representations over a number field \\(K\\) with isomorphic restrictions to \\(H\\). Choose stable lattices at a coefficient place above \\(p\\). Over an algebraic closure of the residue field their semisimplified reductions satisfy\n\\[\n \\overline V^{\\mathrm{ss}}\\cong\\overline W^{\\mathrm{ss}}.\n\\]\nThe same statement applies to finite image representations of an arbitrary group after passing to a common finite quotient that retains the cyclic quotient.\n\\end{theorem}","llapse of cyclic extension labels]\\label{thm:semisimple_label_collapse}\nLet \\(G\\) be finite, let \\(H\\triangleleft G\\) with \\(G/H\\cong C_p\\), and let \\(V,W\\) be semisimple representations over a number field \\(K\\) with isomorphic restrictions to \\(H\\). Choose stable lattices at a coefficient place above \\(p\\). Over an algebraic closure of the residue field their semisimplified reductions satisfy\n\\[\n \\overline V^{\\mathrm{ss}}\\cong\\overline W^{\\mathrm{ss}}.\n\\]\nThe same statement applies to finite image representations of an arbitrary group after passing to a common finite quotient that retains the cyclic quotient.\n\\end{theorem}","\\begin{proof}\nFor \\(g\\in G\\), write its commuting primary decomposition as \\(g=g_p g_s\\), where \\(g_p\\) has \\(p\\) power order and \\(g_s\\) has order prime to \\(p\\). The image of \\(g_s\\) in \\(C_p\\) is trivial, so \\(g_s\\in H\\). In residue characteristic \\(p\\), the matrix of \\(g_s\\) is diagonalizable after extending the field, and the matrix of \\(g_p\\) is unipotent. They commute, so the latter preserves each eigenspace of the former. On an eigenspace with eigenvalue \\(\\alpha\\), the product is \\(\\alpha\\) times a unipotent operator. Its characteristic polynomial has only the root \\(\\alpha\\), with that eigenspace's dimension as multiplicity. Therefore the reduced characteristic polynomials of \\(g\\) and \\(g_s\\) agree.","on as \\(g=g_p g_s\\), where \\(g_p\\) has \\(p\\) power order and \\(g_s\\) has order prime to \\(p\\). The image of \\(g_s\\) in \\(C_p\\) is trivial, so \\(g_s\\in H\\). In residue characteristic \\(p\\), the matrix of \\(g_s\\) is diagonalizable after extending the field, and the matrix of \\(g_p\\) is unipotent. They commute, so the latter preserves each eigenspace of the former. On an eigenspace with eigenvalue \\(\\alpha\\), the product is \\(\\alpha\\) times a unipotent operator. Its characteristic polynomial has only the root \\(\\alpha\\), with that eigenspace's dimension as multiplicity. Therefore the reduced characteristic polynomials of \\(g\\) and \\(g_s\\) agree.","\\(g_p\\) has \\(p\\) power order and \\(g_s\\) has order prime to \\(p\\). The image of \\(g_s\\) in \\(C_p\\) is trivial, so \\(g_s\\in H\\). In residue characteristic \\(p\\), the matrix of \\(g_s\\) is diagonalizable after extending the field, and the matrix of \\(g_p\\) is unipotent. They commute, so the latter preserves each eigenspace of the former. On an eigenspace with eigenvalue \\(\\alpha\\), the product is \\(\\alpha\\) times a unipotent operator. Its characteristic polynomial has only the root \\(\\alpha\\), with that eigenspace's dimension as multiplicity. Therefore the reduced characteristic polynomials of \\(g\\) and \\(g_s\\) agree.",") in \\(C_p\\) is trivial, so \\(g_s\\in H\\). In residue characteristic \\(p\\), the matrix of \\(g_s\\) is diagonalizable after extending the field, and the matrix of \\(g_p\\) is unipotent. They commute, so the latter preserves each eigenspace of the former. On an eigenspace with eigenvalue \\(\\alpha\\), the product is \\(\\alpha\\) times a unipotent operator. Its characteristic polynomial has only the root \\(\\alpha\\), with that eigenspace's dimension as multiplicity. Therefore the reduced characteristic polynomials of \\(g\\) and \\(g_s\\) agree.","after extending the field, and the matrix of \\(g_p\\) is unipotent. They commute, so the latter preserves each eigenspace of the former. On an eigenspace with eigenvalue \\(\\alpha\\), the product is \\(\\alpha\\) times a unipotent operator. Its characteristic polynomial has only the root \\(\\alpha\\), with that eigenspace's dimension as multiplicity. Therefore the reduced characteristic polynomials of \\(g\\) and \\(g_s\\) agree.","ending the field, and the matrix of \\(g_p\\) is unipotent. They commute, so the latter preserves each eigenspace of the former. On an eigenspace with eigenvalue \\(\\alpha\\), the product is \\(\\alpha\\) times a unipotent operator. Its characteristic polynomial has only the root \\(\\alpha\\), with that eigenspace's dimension as multiplicity. Therefore the reduced characteristic polynomials of \\(g\\) and \\(g_s\\) agree.","The restrictions of \\(V,W\\) agree on \\(H\\), so their reduced characteristic polynomials agree at \\(g_s\\), hence at every \\(g\\). The classical modular Brauer Nesbitt theorem identifies the semisimplifications over the algebraic closure of the residue field \\cite{BrauerNesbitt1941,Chenevier2014}. We use its full characteristic polynomial form, valid in arbitrary residual characteristic. Chenevier's checked preprint Theorem A, Lemma 1.12(ii), and Theorem 2.12 give the determinant and uniqueness interface. No restriction \\(p>n\\) is needed for this uniqueness assertion. This proof does not assume that the original number field is a splitting field, or choose stable type extensions over it. For finite image representation","The restrictions of \\(V,W\\) agree on \\(H\\), so their reduced characteristic polynomials agree at \\(g_s\\), hence at every \\(g\\). The classical modular Brauer Nesbitt theorem identifies the semisimplifications over the algebraic closure of the residue field \\cite{BrauerNesbitt1941,Chenevier2014}. We use its full characteristic polynomial form, valid in arbitrary residual characteristic. Chenevier's checked preprint Theorem A, Lemma 1.12(ii), and Theorem 2.12 give the determinant and uniqueness interface. No restriction \\(p>n\\) is needed for this uniqueness assertion. This proof does not assume that the original number field is a splitting field, or choose stable type extensions over it. For finite image representations of a larger g","The restrictions of \\(V,W\\) agree on \\(H\\), so their reduced characteristic polynomials agree at \\(g_s\\), hence at every \\(g\\). The classical modular Brauer Nesbitt theorem identifies the semisimplifications over the algebraic closure of the residue field \\cite{BrauerNesbitt1941,Chenevier2014}. We use its full characteristic polynomial form, valid in arbitrary residual characteristic. Chenevier's checked preprint Theorem A, Lemma 1.12(ii), and Theorem 2.12 give the determinant and uniqueness interface. No restriction \\(p>n\\) is needed for this uniqueness assertion. This proof does not assume that the original number field is a splitting field, or choose stable type extensions over it. For finite image representations of a larger group, quotient by the intersection of their kernels with \\(H\\)","The restrictions of \\(V,W\\) agree on \\(H\\), so their reduced characteristic polynomials agree at \\(g_s\\), hence at every \\(g\\). The classical modular Brauer Nesbitt theorem identifies the semisimplifications over the algebraic closure of the residue field \\cite{BrauerNesbitt1941,Chenevier2014}. We use its full characteristic polynomial form, valid in arbitrary residual characteristic. Chenevier's checked preprint Theorem A, Lemma 1.12(ii), and Theorem 2.12 give the determinant and uniqueness interface. No restriction \\(p>n\\) is needed for this uniqueness assertion. This proof does not assume that the original number field is a splitting field, or choose stable type extensions over it. For finite image representations of a larger group, quotient by the intersection of their kernels with \\(H\\). The resulting finite","). The classical modular Brauer Nesbitt theorem identifies the semisimplifications over the algebraic closure of the residue field \\cite{BrauerNesbitt1941,Chenevier2014}. We use its full characteristic polynomial form, valid in arbitrary residual characteristic. Chenevier's checked preprint Theorem A, Lemma 1.12(ii), and Theorem 2.12 give the determinant and uniqueness interface. No restriction \\(p>n\\) is needed for this uniqueness assertion. This proof does not assume that the original number field is a splitting field, or choose stable type extensions over it. For finite image representations of a larger group, quotient by the intersection of their kernels with \\(H\\). The resulting finite quotient has the same argument.\n\\end{proof}","checked preprint Theorem A, Lemma 1.12(ii), and Theorem 2.12 give the determinant and uniqueness interface. No restriction \\(p>n\\) is needed for this uniqueness assertion. This proof does not assume that the original number field is a splitting field, or choose stable type extensions over it. For finite image representations of a larger group, quotient by the intersection of their kernels with \\(H\\). The resulting finite quotient has the same argument.\n\\end{proof}","\\begin{corollary}[Residual collapse in the global and local fibres]\\label{cor:two_torsor_residual_collapse}\nAt a coefficient prime above the cyclic degree \\(p\\), all global Galois extensions of a fixed irreducible restriction have the same semisimplified reduction. At any one place, local parameters with the same restricted finite image parameter have the same semisimplified reduction after passing to a common finite local quotient and compatible coefficient field.\n\\end{corollary}","Let \\(\\lambda_p\\) be a finite place of the coefficient field above the cyclic degree \\(p\\), chosen after adjoining the values of the quotient character. Then\n\\[\n\\epsilon\\equiv1\\pmod{\\lambda_p}.\n\\]","\\begin{definition}[Frobenius transform of a polynomial]\\label{def:frobenius_transform}\nIf\n\\[\nP(T)=\\prod_{i=1}^{n}(T-\\alpha_i),\n\\]\ndefine\n\\[\nP^{[p]}(T)\n=\n\\prod_{i=1}^{n}(T-\\alpha_i^p).\n\\]\nThis is a polynomial expression in the coefficients of \\(P\\).\n\\end{definition}","\\begin{lemma}[Frobenius transform after reduction]\\label{lem:frobenius_transform}\nLet \\(P\\) have integral coefficients at a place of residue characteristic \\(p\\). Then\n\\[\n\\boxed{\n\\overline{P^{[p]}}(T)\n=\n\\operatorname{Frob}\\bigl(\\overline P(T)\\bigr),\n}\n\\]\nwhere Frobenius raises every coefficient to its \\(p\\)th power. In particular, the map\n\\[\n\\overline P\\longmapsto\\overline{P^{[p]}}\n\\]\nis injective over the residue field.\n\\end{lemma}","\\begin{proof}\nOver an algebraic closure of the residue field,\n\\[\n\\overline{P^{[p]}}(T)\n=\n\\prod_i(T-\\overline\\alpha_i^p).\n\\]\nThe elementary symmetric functions of the \\(p\\)th powers are the \\(p\\)th powers of the elementary symmetric functions in characteristic \\(p\\). Finite residue fields are perfect, so Frobenius is injective.\n\\end{proof}","\\begin{theorem}[Canonical quotient prime congruence]\\label{thm:quotient_prime_congruence}\nUse the cyclic Artin setup of \\cref{def:attached_descent}. For every finite place \\(v\\) outside a finite set,\n\\[\n\\boxed{\nP_{\\pi,v}(T)\n\\equiv\nP_{\\rho,v}(T)\n\\pmod{\\lambda_p}.\n}\n\\]\nThus the semisimplified reduction of \\(\\rho\\) is residually automorphic with the mod \\(\\lambda_p\\) Hecke eigensystem of every cuspidal descent of \\(\\Pi\\).\n\\end{theorem}","Canonical quotient prime congruence]\\label{thm:quotient_prime_congruence}\nUse the cyclic Artin setup of \\cref{def:attached_descent}. For every finite place \\(v\\) outside a finite set,\n\\[\n\\boxed{\nP_{\\pi,v}(T)\n\\equiv\nP_{\\rho,v}(T)\n\\pmod{\\lambda_p}.\n}\n\\]\nThus the semisimplified reduction of \\(\\rho\\) is residually automorphic with the mod \\(\\lambda_p\\) Hecke eigensystem of every cuspidal descent of \\(\\Pi\\).\n\\end{theorem}","\\begin{proof}\nAssume first that \\(v\\) splits in \\(E\\). Local base change is the identity at every place above \\(v\\), so the two polynomials agree before reduction.","Assume that \\(v\\) is unramified and nonsplit. Since the extension has prime degree, its residue degree is \\(p\\). Unramified base change raises the Satake conjugacy class of \\(\\pi_v\\) to its \\(p\\)th power. Restriction of \\(\\rho_v\\) to \\(W_{E_w}\\) does the same to its Frobenius class. Since \\(\\Pi_w\\) is attached to \\(\\rho|_{W_{E_w}}\\),\n\\[\nP_{\\pi,v}^{[p]}(T)\n=\nP_{\\rho,v}^{[p]}(T).\n\\]\nReduce modulo \\(\\lambda_p\\) and use \\cref{lem:frobenius_transform}. Injectivity of residue Frobenius gives the desired congruence. The finitely many ramified and excluded places do not affect residual attachment.\n\\end{proof}","\\begin{corollary}[All cyclic twists collapse residually]\\label{cor:twists_collapse_residually}\nFor every \\(j\\in\\ZZ/p\\ZZ\\),\n\\[\n\\overline{\\rho\\otimes\\epsilon^j}\n\\cong\n\\overline\\rho\n\\]\nat the semisimple character level modulo \\(\\lambda_p\\). Likewise, all automorphic descents have the same mod \\(\\lambda_p\\) unramified Hecke eigensystem.\n\\end{corollary}","\\begin{proof}\nThe values of \\(\\epsilon\\) are \\(p\\)th roots of unity and reduce to one at \\(\\lambda_p\\). Apply \\cref{thm:quotient_prime_congruence} to every descent twist.\n\\end{proof}","\\begin{definition}[Quotient prime lifting gate]\\label{def:quotient_prime_lifting}\nThe quotient prime lifting gate is the assertion that the matching residual unramified eigensystems supplied by \\cref{thm:quotient_prime_congruence}, together with the local deformation conditions of \\(\\rho\\), lift to characteristic zero automorphy of \\(\\rho\\).\n\\end{definition}","\\begin{theorem}[Quotient prime reduction of a cyclic step]\\label{thm:quotient_prime_reduction}\nIf the quotient prime lifting gate holds for the cyclic step, then the target representation \\(\\rho\\) is automorphic. Hence Strong Artin for all finite solvable images follows if this gate holds at every invariant extension step of every solvable Clifford tower.\n\\end{theorem}","\\begin{theorem}[Finite Frobenius trace rigidity]\\label{thm:finite_trace_ladic_rigidity}\nLet \\(L/\\QQ_\\ell\\) be finite and let \\(r:G_F\\to\\GL_n(L)\\) be continuous, semisimple, and unramified outside a finite set. Suppose its traces at a density one set of unramified places belong to one finite set \\(T\\), of size \\(t\\). Then\n\\[\n \\abs{r(G_F)}\\le t^{n^2}.\n\\]\nIn particular \\(r\\) has finite image.\n\\end{theorem}","\\begin{proof}\nThe image is compact. Choose a stable integral lattice after conjugation, so reductions modulo successive powers of a uniformizer give finite quotient images. Fix \\(g\\in G_F\\) and one such quotient. The Chebotarev class of the image of \\(g\\) has positive density. A density one set meets it, even after removing the finite set of ramified places. Thus some selected Frobenius matrix is conjugate to the image of \\(r(g)\\) modulo that power, and their traces are congruent.","ompact. Choose a stable integral lattice after conjugation, so reductions modulo successive powers of a uniformizer give finite quotient images. Fix \\(g\\in G_F\\) and one such quotient. The Chebotarev class of the image of \\(g\\) has positive density. A density one set meets it, even after removing the finite set of ramified places. Thus some selected Frobenius matrix is conjugate to the image of \\(r(g)\\) modulo that power, and their traces are congruent.","For each precision, \\(\\Tr r(g)\\) is therefore arbitrarily close to an element of \\(T\\). The finite set \\(T\\) is closed in \\(L\\), so \\(\\Tr r(g)\\in T\\). Equivalently, one value in \\(T\\) occurs along an unbounded sequence of precisions, and its distance from \\(\\Tr r(g)\\) tends to zero. This uses conjugacy classes, which suffice because trace is conjugacy invariant. It does not require choosing a preferred Frobenius element inside each class.","Extend scalars to an algebraic closure of \\(L\\). Semisimplicity is preserved in characteristic zero: the finite dimensional semisimple image algebra is separable. Apply the characteristic zero case of \\cref{thm:finite_trace_image_bound} to \\(\\Gamma=r(G_F)\\). Its faithful defining representation is semisimple and its trace set has size at most \\(t\\). This gives the stated bound directly, without an additional monodromy or torus argument.\n\\end{proof}","\\begin{corollary}[Cyclotomic Frobenius rigidity]\\label{thm:cyclotomic_ladic_rigidity}\nLet \\(r\\) be as in \\cref{thm:finite_trace_ladic_rigidity}. If every Frobenius eigenvalue on a density one set belongs to one fixed finite group \\(\\mu_N\\), then \\(r\\) has finite image.\n\\end{corollary}","\\begin{proof}\nThe Frobenius traces belong to the finite set of sums of \\(n\\) members of \\(\\mu_N\\). Apply \\cref{thm:finite_trace_ladic_rigidity}.\n\\end{proof}","\\begin{theorem}[One semisimple l adic realization is enough]\\label{thm:one_ladic_realization}\nLet \\(\\pi\\) be a finite Satake cuspidal representation with algebraic unramified Hecke polynomials. Fix an embedding of their number field into \\(\\overline\\QQ_\\ell\\). Suppose there is a continuous semisimple representation \\(r:G_F\\to\\GL_n(L)\\), for a finite extension \\(L/\\QQ_\\ell\\), unramified outside a finite set, whose Frobenius polynomials agree with the embedded Hecke polynomials at almost all places. Then \\(r\\) has finite image and determines a finite image complex representation weakly attached to \\(\\pi\\).\n\\end{theorem}","finite Satake cuspidal representation with algebraic unramified Hecke polynomials. Fix an embedding of their number field into \\(\\overline\\QQ_\\ell\\). Suppose there is a continuous semisimple representation \\(r:G_F\\to\\GL_n(L)\\), for a finite extension \\(L/\\QQ_\\ell\\), unramified outside a finite set, whose Frobenius polynomials agree with the embedded Hecke polynomials at almost all places. Then \\(r\\) has finite image and determines a finite image complex representation weakly attached to \\(\\pi\\).\n\\end{theorem}","dding of their number field into \\(\\overline\\QQ_\\ell\\). Suppose there is a continuous semisimple representation \\(r:G_F\\to\\GL_n(L)\\), for a finite extension \\(L/\\QQ_\\ell\\), unramified outside a finite set, whose Frobenius polynomials agree with the embedded Hecke polynomials at almost all places. Then \\(r\\) has finite image and determines a finite image complex representation weakly attached to \\(\\pi\\).\n\\end{theorem}","of a finite group has cyclotomic character values and can be realized over an algebraic number field after choosing a splitting field. Choose the ordinary complex realization in a modular system compatible with the fixed embedding of the Hecke field. Characteristic polynomials then give the same algebraic polynomials under the two embeddings, so the complex representation is weakly attached to \\(\\pi\\).","t}[A torsion cohomology realization range]\\label{fact:scholze_torsion}\nScholze's Corollary 5.4.3 supplies a unique continuous semisimple residual representation for a Hecke eigensystem actually occurring in the stated cohomology of a locally symmetric space, under the field, level, coefficient, and theorem dependencies of that source \\cite{Scholze2015}. The specified polynomial has coefficients \\(q_v^{i(n+1)/2}T_{i,v}\\), rather than an unspecified normalization of the eigenvalues. A fully stated unconditional subrange uses a CM field containing an imaginary quadratic field and the stated compatible set of rational primes. The more general totally real or CM formulation in the original article records stabilization dependencies. A determinant over a coefficient ring modulo a bounded nilpotent ideal and an actual residual representation are distinct outputs. For the gate here we require the latter and an explicit identification with our reduced normalized Hecke polynomials.\n\\end{standardfact}","\\begin{standardfact}[A holomorphic limit realization range]\\label{fact:goldring_shin}\nGoldring's theorem, with an appendix by Shin, treats a unitary similitude group from a simple unitary Kottwitz datum and a cuspidal representation whose infinity component is an \\(X\\) holomorphic limit of discrete series \\cite{GoldringShin2014}. At a coefficient prime satisfying the good reduction and reflex field conditions, it supplies a semisimple l adic representation over the CM field. Its local compatibility is stated on the density one split prime set with the twist \\(\\lvert\\cdot\\rvert^{(1-n)/2}\\) in the source normalization. This is a concrete realization range. It is not a theorem for every singular infinity type or every finite Satake representation of \\(\\GL_n\\).\n\\end{standardfact}","from a simple unitary Kottwitz datum and a cuspidal representation whose infinity component is an \\(X\\) holomorphic limit of discrete series \\cite{GoldringShin2014}. At a coefficient prime satisfying the good reduction and reflex field conditions, it supplies a semisimple l adic representation over the CM field. Its local compatibility is stated on the density one split prime set with the twist \\(\\lvert\\cdot\\rvert^{(1-n)/2}\\) in the source normalization. This is a concrete realization range. It is not a theorem for every singular infinity type or every finite Satake representation of \\(\\GL_n\\).\n\\end{standardfact}","sfying the good reduction and reflex field conditions, it supplies a semisimple l adic representation over the CM field. Its local compatibility is stated on the density one split prime set with the twist \\(\\lvert\\cdot\\rvert^{(1-n)/2}\\) in the source normalization. This is a concrete realization range. It is not a theorem for every singular infinity type or every finite Satake representation of \\(\\GL_n\\).\n\\end{standardfact}","\\begin{corollary}[A normalized torsion realization closes the weak step]\\label{cor:scholze_attachment}\nUse the cyclic Artin setup and a good separating prime \\(\\lambda\\). Suppose the reduced eigensystem of a descent occurs in the cohomology range of \\cref{fact:scholze_torsion}, with all source hypotheses met, and an explicit normalization comparison identifies the source Frobenius polynomials with \\(\\overline{P_{\\pi,v}}\\). Then the cyclic step is weakly attached after one quotient twist.\n\\end{corollary}","gin{corollary}[A normalized torsion realization closes the weak step]\\label{cor:scholze_attachment}\nUse the cyclic Artin setup and a good separating prime \\(\\lambda\\). Suppose the reduced eigensystem of a descent occurs in the cohomology range of \\cref{fact:scholze_torsion}, with all source hypotheses met, and an explicit normalization comparison identifies the source Frobenius polynomials with \\(\\overline{P_{\\pi,v}}\\). Then the cyclic step is weakly attached after one quotient twist.\n\\end{corollary}","\\begin{proof}\nThe explicit comparison gives a finite trace alphabet on the density one set covered by the source's local compatibility. Apply \\cref{thm:finite_trace_ladic_rigidity}. The normalization and density condition are essential premises. This corollary does not assert all local attachment or an arbitrary transfer from \\(\\GL_n\\) into the unitary setting.\n\\end{proof}","\\begin{standardfact}[Parallel weight one examples]\\label{fact:weight_one_companion}\nGee and Kassaei's companion construction is for a totally real field unramified at \\(p>2\\), with an irreducible modular residual representation satisfying the stated residual image and local distinct eigenvalue hypotheses. Their Artin consequence assumes the relevant modularity statement for every totally odd residual two dimensional representation and every coefficient prime \\cite{GeeKassaei2013}. Dimitrov and Wiese construct a continuous semisimple residual representation from a parallel weight one Hilbert eigenform of prime to \\(p\\) level. Its trace and determinant data require the stated Hecke and diamond character normalization \\cite{DimitrovWiese2020}. These theorems illustrate realization and weight one methods in their precise ranks and hypotheses. They do not supply re","ation satisfying the stated residual image and local distinct eigenvalue hypotheses. Their Artin consequence assumes the relevant modularity statement for every totally odd residual two dimensional representation and every coefficient prime \\cite{GeeKassaei2013}. Dimitrov and Wiese construct a continuous semisimple residual representation from a parallel weight one Hilbert eigenform of prime to \\(p\\) level. Its trace and determinant data require the stated Hecke and diamond character normalization \\cite{DimitrovWiese2020}. These theorems illustrate realization and weight one methods in their precise ranks and hypotheses. They do not supply residual representations or weight raising in arbitrary rank.\n\\end{standardfact}","\\begin{definition}[Pseudocharacter family gate]\\label{def:pseudo_family_gate}\nA pseudocharacter family gate for a finite Satake descent \\(\\pi\\) consists of a reduced p adic analytic space \\(X\\), a closed point \\(x\\) valued in a finite extension of \\(\\QQ_\\ell\\), and a continuous dimension \\(n\\) characteristic zero pseudorepresentation over \\(\\mathcal O(X)\\). It factors through the Galois group unramified outside one finite set, and its unramified characteristic polynomials specialize at \\(x\\) to the embedded Hecke polynomials of \\(\\pi\\). We use a point and coefficient setting in which characteristic zero realization produces a continuous semisimple representation after a finite coefficient extension. The existence of this family and the occurrence of the specified point are part of the gate.\n\\end{definition}","er family gate for a finite Satake descent \\(\\pi\\) consists of a reduced p adic analytic space \\(X\\), a closed point \\(x\\) valued in a finite extension of \\(\\QQ_\\ell\\), and a continuous dimension \\(n\\) characteristic zero pseudorepresentation over \\(\\mathcal O(X)\\). It factors through the Galois group unramified outside one finite set, and its unramified characteristic polynomials specialize at \\(x\\) to the embedded Hecke polynomials of \\(\\pi\\). We use a point and coefficient setting in which characteristic zero realization produces a continuous semisimple representation after a finite coefficient extension. The existence of this family and the occurrence of the specified point are part of the gate.\n\\end{definition}","\\(\\pi\\) consists of a reduced p adic analytic space \\(X\\), a closed point \\(x\\) valued in a finite extension of \\(\\QQ_\\ell\\), and a continuous dimension \\(n\\) characteristic zero pseudorepresentation over \\(\\mathcal O(X)\\). It factors through the Galois group unramified outside one finite set, and its unramified characteristic polynomials specialize at \\(x\\) to the embedded Hecke polynomials of \\(\\pi\\). We use a point and coefficient setting in which characteristic zero realization produces a continuous semisimple representation after a finite coefficient extension. The existence of this family and the occurrence of the specified point are part of the gate.\n\\end{definition}","\\begin{proof}\nSpecialize at \\(x\\) and use characteristic zero pseudorepresentation theory in the declared coefficient setting \\cite{Chenevier2014}. This gives a continuous semisimple representation over a finite extension of \\(\\QQ_\\ell\\), factoring through the stated finitely ramified quotient. Its Frobenius polynomials are the required specialization. Apply \\cref{thm:one_ladic_realization}. Neither specialization nor rigidity constructs the analytic space, the Galois pseudorepresentation on it, or the point corresponding to the descent.\n\\end{proof}","\\begin{theorem}[Realization gates imply solvable Strong Artin]\\label{thm:gates_imply_solvable_artin}\nLet \\(\\rho\\) be an irreducible finite image representation with solvable image. Follow it through a prime cyclic Clifford tower. Suppose every invariant extension step admits a realization gate in the sense of \\cref{def:realization_gate_step}. Then \\(\\rho\\) is weakly automorphic. If the remaining exceptional finite and archimedean local parameters align at every step, then the automorphic representation is attached at every place.\n\\end{theorem}","\\begin{standardfact}[Conditional construction of Kim and Yamauchi]\\label{fact:kim_yamauchi}\nKim and Yamauchi's construction for \\(\\GL_n(\\Adele_\\QQ)\\) uses five declared inputs: a finite Hecke field, integrality with a common denominator, cuspidal Galois conjugates, residual Galois realizations for all sufficiently large coefficient primes, and a Rankin Selberg coefficient bound obtainable from the required exterior power functoriality. Their theorem is a conditional construction of Artin representations, not a theorem producing the residual realizations for every weight zero cusp form \\cite{KimYamauchi2016}. The finite alphabet and cross characteristic recovery below occupy closely related representation theoretic ground. Historical novelty is not inferred from their reformulation.\n\\end{standardfact}","\\begin{theorem}[Automatic finite Satake inputs over the rational field]\\label{thm:automatic_KY_inputs}\nLet \\(E/\\QQ\\) be finite Galois, and let \\(\\pi\\) be a cuspidal representation of \\(\\GL_n(\\Adele_\\QQ)\\) whose base change to \\(E\\) is attached to a finite image Artin representation. Then the following inputs in the Kim and Yamauchi strategy are automatic:","\\begin{lemma}[Complex representations of profinite groups have finite image]\\label{lem:complex_profinite_finite}\nLet \\(G\\) be profinite and let \\(r:G\\to\\GL_n(\\CC)\\) be continuous for the usual complex topology. Then \\(r(G)\\) is finite.\n\\end{lemma}","\\begin{proof}\nThe Lie group \\(\\GL_n(\\CC)\\) has a neighborhood \\(U\\) of the identity that contains no nontrivial subgroup. Continuity gives a neighborhood \\(r^{-1}(U)\\) of the identity in \\(G\\). A profinite group has a basis of open normal subgroups, so one such subgroup \\(N\\) lies inside \\(r^{-1}(U)\\). Its image is a subgroup contained in \\(U\\), hence is trivial. Therefore \\(r\\) factors through the finite quotient \\(G/N\\). This proof uses a group neighborhood theorem. It does not use the false general topological assertion that an arbitrary continuous image of a totally disconnected space is totally disconnected.\n\\end{proof}","e group \\(\\GL_n(\\CC)\\) has a neighborhood \\(U\\) of the identity that contains no nontrivial subgroup. Continuity gives a neighborhood \\(r^{-1}(U)\\) of the identity in \\(G\\). A profinite group has a basis of open normal subgroups, so one such subgroup \\(N\\) lies inside \\(r^{-1}(U)\\). Its image is a subgroup contained in \\(U\\), hence is trivial. Therefore \\(r\\) factors through the finite quotient \\(G/N\\). This proof uses a group neighborhood theorem. It does not use the false general topological assertion that an arbitrary continuous image of a totally disconnected space is totally disconnected.\n\\end{proof}","\\begin{standardfact}[Characteristic zero pseudocharacters]\\label{fact:pseudocharacter_realization}\nOver an algebraically closed field of characteristic zero, a semisimple pseudocharacter of dimension \\(n\\) is the trace of a unique semisimple \\(n\\) dimensional representation up to isomorphism. We use this standard pseudocharacter realization principle in the form needed below \\cite{Chenevier2014}.\n\\end{standardfact}","\\begin{theorem}[Complex pseudocharacter gate]\\label{thm:complex_pseudo_gate}\nLet \\(\\pi\\) be a finite Satake cuspidal representation of \\(\\GL_n(\\Adele_F)\\). Suppose there exists a continuous dimension \\(n\\) complex pseudocharacter \\(T\\) of \\(G_F\\), in the characteristic zero sense of \\cref{fact:pseudocharacter_realization}, with characteristic polynomials at almost all unramified Frobenius classes equal to those of \\(\\pi\\). Then \\(\\pi\\) has a finite image semisimple Galois realization with those almost everywhere polynomials.\n\\end{theorem}","lex pseudocharacter gate]\\label{thm:complex_pseudo_gate}\nLet \\(\\pi\\) be a finite Satake cuspidal representation of \\(\\GL_n(\\Adele_F)\\). Suppose there exists a continuous dimension \\(n\\) complex pseudocharacter \\(T\\) of \\(G_F\\), in the characteristic zero sense of \\cref{fact:pseudocharacter_realization}, with characteristic polynomials at almost all unramified Frobenius classes equal to those of \\(\\pi\\). Then \\(\\pi\\) has a finite image semisimple Galois realization with those almost everywhere polynomials.\n\\end{theorem}","acter gate]\\label{thm:complex_pseudo_gate}\nLet \\(\\pi\\) be a finite Satake cuspidal representation of \\(\\GL_n(\\Adele_F)\\). Suppose there exists a continuous dimension \\(n\\) complex pseudocharacter \\(T\\) of \\(G_F\\), in the characteristic zero sense of \\cref{fact:pseudocharacter_realization}, with characteristic polynomials at almost all unramified Frobenius classes equal to those of \\(\\pi\\). Then \\(\\pi\\) has a finite image semisimple Galois realization with those almost everywhere polynomials.\n\\end{theorem}","\\begin{proof}\nThe algebraic pseudocharacter realization theorem supplies a semisimple representation \\(r\\) with trace \\(T\\). Its continuity deserves a separate argument. Let \\(A\\) be the span of its image. The trace pairing on \\(A\\) is nondegenerate, by the characteristic zero semisimple algebra argument in \\cref{thm:finite_trace_image_bound}. Choose a basis \\(r(g_1),\\ldots,r(g_d)\\) from the image. The functions\n\\[\n g\\longmapsto\\Tr(r(g)r(g_i))=T(gg_i)\n\\]\nare continuous. Nondegeneracy says that these finitely many trace functions determine all linear coordinates of \\(r(g)\\) in \\(A\\). Inverting their fixed coefficient matrix proves that every matrix entry of \\(r(g)\\) is continuous. Now \\cref{lem:complex_profinite_finite} gives finite image. The polynomial equality is the hypothesis. This continu","\\begin{proof}\nThe algebraic pseudocharacter realization theorem supplies a semisimple representation \\(r\\) with trace \\(T\\). Its continuity deserves a separate argument. Let \\(A\\) be the span of its image. The trace pairing on \\(A\\) is nondegenerate, by the characteristic zero semisimple algebra argument in \\cref{thm:finite_trace_image_bound}. Choose a basis \\(r(g_1),\\ldots,r(g_d)\\) from the image. The functions\n\\[\n g\\longmapsto\\Tr(r(g)r(g_i))=T(gg_i)\n\\]\nare continuous. Nondegeneracy says that these finitely many trace functions determine all linear coordinates of \\(r(g)\\) in \\(A\\). Inverting their fixed coefficient matrix proves that every matrix entry of \\(r(g)\\) is continuous. Now \\cref{lem:complex_profinite_finite} gives finite image. The polynomial equality is the hypothesis. This continuity step is not s","\\begin{proof}\nThe algebraic pseudocharacter realization theorem supplies a semisimple representation \\(r\\) with trace \\(T\\). Its continuity deserves a separate argument. Let \\(A\\) be the span of its image. The trace pairing on \\(A\\) is nondegenerate, by the characteristic zero semisimple algebra argument in \\cref{thm:finite_trace_image_bound}. Choose a basis \\(r(g_1),\\ldots,r(g_d)\\) from the image. The functions\n\\[\n g\\longmapsto\\Tr(r(g)r(g_i))=T(gg_i)\n\\]\nare continuous. Nondegeneracy says that these finitely many trace functions determine all linear coordinates of \\(r(g)\\) in \\(A\\). Inverting their fixed coefficient matrix proves that every matrix entry of \\(r(g)\\) is continuous. Now \\cref{lem:complex_profinite_finite} gives finite image. The polynomial equality is the hypothesis. This continuity step is not silently supplied by the algebraic realization statement","\\begin{proof}\nThe algebraic pseudocharacter realization theorem supplies a semisimple representation \\(r\\) with trace \\(T\\). Its continuity deserves a separate argument. Let \\(A\\) be the span of its image. The trace pairing on \\(A\\) is nondegenerate, by the characteristic zero semisimple algebra argument in \\cref{thm:finite_trace_image_bound}. Choose a basis \\(r(g_1),\\ldots,r(g_d)\\) from the image. The functions\n\\[\n g\\longmapsto\\Tr(r(g)r(g_i))=T(gg_i)\n\\]\nare continuous. Nondegeneracy says that these finitely many trace functions determine all linear coordinates of \\(r(g)\\) in \\(A\\). Inverting their fixed coefficient matrix proves that every matrix entry of \\(r(g)\\) is continuous. Now \\cref{lem:complex_profinite_finite} gives finite image. The polynomial equality is the hypothesis. This continuity step is not silently supplied by the algebraic realization statement alone.\n\\end{proof}","algebraic pseudocharacter realization theorem supplies a semisimple representation \\(r\\) with trace \\(T\\). Its continuity deserves a separate argument. Let \\(A\\) be the span of its image. The trace pairing on \\(A\\) is nondegenerate, by the characteristic zero semisimple algebra argument in \\cref{thm:finite_trace_image_bound}. Choose a basis \\(r(g_1),\\ldots,r(g_d)\\) from the image. The functions\n\\[\n g\\longmapsto\\Tr(r(g)r(g_i))=T(gg_i)\n\\]\nare continuous. Nondegeneracy says that these finitely many trace functions determine all linear coordinates of \\(r(g)\\) in \\(A\\). Inverting their fixed coefficient matrix proves that every matrix entry of \\(r(g)\\) is continuous. Now \\cref{lem:complex_profinite_finite} gives finite image. The polynomial equality is the hypothesis. This continuity step is not silently supplied by the algebraic realization statement alone.\n\\end{proof}","\\(A\\) be the span of its image. The trace pairing on \\(A\\) is nondegenerate, by the characteristic zero semisimple algebra argument in \\cref{thm:finite_trace_image_bound}. Choose a basis \\(r(g_1),\\ldots,r(g_d)\\) from the image. The functions\n\\[\n g\\longmapsto\\Tr(r(g)r(g_i))=T(gg_i)\n\\]\nare continuous. Nondegeneracy says that these finitely many trace functions determine all linear coordinates of \\(r(g)\\) in \\(A\\). Inverting their fixed coefficient matrix proves that every matrix entry of \\(r(g)\\) is continuous. Now \\cref{lem:complex_profinite_finite} gives finite image. The polynomial equality is the hypothesis. This continuity step is not silently supplied by the algebraic realization statement alone.\n\\end{proof}","e span of its image. The trace pairing on \\(A\\) is nondegenerate, by the characteristic zero semisimple algebra argument in \\cref{thm:finite_trace_image_bound}. Choose a basis \\(r(g_1),\\ldots,r(g_d)\\) from the image. The functions\n\\[\n g\\longmapsto\\Tr(r(g)r(g_i))=T(gg_i)\n\\]\nare continuous. Nondegeneracy says that these finitely many trace functions determine all linear coordinates of \\(r(g)\\) in \\(A\\). Inverting their fixed coefficient matrix proves that every matrix entry of \\(r(g)\\) is continuous. Now \\cref{lem:complex_profinite_finite} gives finite image. The polynomial equality is the hypothesis. This continuity step is not silently supplied by the algebraic realization statement alone.\n\\end{proof}","istic zero semisimple algebra argument in \\cref{thm:finite_trace_image_bound}. Choose a basis \\(r(g_1),\\ldots,r(g_d)\\) from the image. The functions\n\\[\n g\\longmapsto\\Tr(r(g)r(g_i))=T(gg_i)\n\\]\nare continuous. Nondegeneracy says that these finitely many trace functions determine all linear coordinates of \\(r(g)\\) in \\(A\\). Inverting their fixed coefficient matrix proves that every matrix entry of \\(r(g)\\) is continuous. Now \\cref{lem:complex_profinite_finite} gives finite image. The polynomial equality is the hypothesis. This continuity step is not silently supplied by the algebraic realization statement alone.\n\\end{proof}","\\begin{definition}[Finite Satake realization principle]\\label{def:finite_satake_principle}\nThe finite Satake realization principle asserts that a cuspidal representation of \\(\\GL_n(\\Adele_F)\\) with finite local Langlands image at every place and finite Satake data globally is attached to a finite image complex Galois representation.\n\\end{definition}","Base change of the realized representation has the same unramified Frobenius polynomials as \\(\\tau\\). Chebotarev density and semisimplicity identify its restriction to \\(G_E\\) with \\(\\tau\\). Since \\(\\tau\\) is irreducible, the realized representation is irreducible and lies in the finite extension torsor. By \\cref{prop:one_attachment}, twisting the automorphic descent by a quotient character selects the prescribed extension at that step. Iteration reaches the original representation.\n\\end{proof}","\\begin{theorem}[Trace separator for irreducible extensions]\\label{thm:galois_trace_separator}\nLet \\(H\\triangleleft G\\) with \\(G/H\\cong C_p\\), and let \\(\\rho\\) be an irreducible extension of an irreducible \\(G\\) invariant representation of \\(H\\). Then there exists \\(g\\in G\\setminus H\\) such that\n\\[\n \\boxed{\n \\Tr\\rho(g)\\neq0.\n }\n\\]\nConsequently, the \\(p\\) quotient twists of \\(\\rho\\) are distinguished by their traces at \\(g\\).\n\\end{theorem}","eparator for irreducible extensions]\\label{thm:galois_trace_separator}\nLet \\(H\\triangleleft G\\) with \\(G/H\\cong C_p\\), and let \\(\\rho\\) be an irreducible extension of an irreducible \\(G\\) invariant representation of \\(H\\). Then there exists \\(g\\in G\\setminus H\\) such that\n\\[\n \\boxed{\n \\Tr\\rho(g)\\neq0.\n }\n\\]\nConsequently, the \\(p\\) quotient twists of \\(\\rho\\) are distinguished by their traces at \\(g\\).\n\\end{theorem}","\\begin{proof}\nThe \\(p\\) extensions \\(\\rho\\otimes\\xi^j\\) are pairwise nonisomorphic by \\cref{prop:galois_torsor}. Their characters agree on \\(H\\). If \\(\\Tr\\rho(g)=0\\) for every \\(g\\notin H\\), then all \\(p\\) characters would agree on every element of \\(G\\), contradicting their pairwise nonisomorphism. Thus some \\(g\\notin H\\) has nonzero trace. At that element,\n\\[\n \\Tr(\\rho\\otimes\\xi^j)(g)\n =\n \\xi(g)^j\\Tr\\rho(g),\n\\]\nand these values are distinct.\n\\end{proof}","\\begin{corollary}[Chebotarev trace separator]\\label{cor:chebotarev_separator}\nLet the finite image of \\(\\rho\\) cut out a finite Galois extension. There is an unramified finite place \\(v\\) whose Frobenius element lies outside \\(H\\) and satisfies\n\\[\n \\Tr\\rho(\\Frob_v)\\neq0.\n\\]\nAt that place, the Galois extension torsor is separated by the first Hecke coefficient.\n\\end{corollary}","\\begin{theorem}[Existence of an automorphic coefficient separator]\\label{thm:automorphic_separator}\nLet \\(E/F\\) be cyclic of prime degree \\(p\\). Let \\(\\pi\\) be cuspidal and assume its base change to \\(E\\) is cuspidal. Let \\(\\epsilon\\) be the quotient character. Then there is an unramified finite place \\(v\\) and an index \\(k\\) with \\(p\\nmid k\\) such that the degree \\(k\\) coefficient of the normalized Hecke polynomial of \\(\\pi_v\\) is nonzero and \\(\\epsilon_v(\\Frob_v)\\neq1\\).\n\\end{theorem}","of an automorphic coefficient separator]\\label{thm:automorphic_separator}\nLet \\(E/F\\) be cyclic of prime degree \\(p\\). Let \\(\\pi\\) be cuspidal and assume its base change to \\(E\\) is cuspidal. Let \\(\\epsilon\\) be the quotient character. Then there is an unramified finite place \\(v\\) and an index \\(k\\) with \\(p\\nmid k\\) such that the degree \\(k\\) coefficient of the normalized Hecke polynomial of \\(\\pi_v\\) is nonzero and \\(\\epsilon_v(\\Frob_v)\\neq1\\).\n\\end{theorem}","cient separator]\\label{thm:automorphic_separator}\nLet \\(E/F\\) be cyclic of prime degree \\(p\\). Let \\(\\pi\\) be cuspidal and assume its base change to \\(E\\) is cuspidal. Let \\(\\epsilon\\) be the quotient character. Then there is an unramified finite place \\(v\\) and an index \\(k\\) with \\(p\\nmid k\\) such that the degree \\(k\\) coefficient of the normalized Hecke polynomial of \\(\\pi_v\\) is nonzero and \\(\\epsilon_v(\\Frob_v)\\neq1\\).\n\\end{theorem}","\\begin{proof}\nThe representations \\(\\pi\\) and \\(\\pi\\otimes\\epsilon\\) are distinct by the fibre theorem in \\cref{fact:base_change_fibre}. Strong multiplicity one implies that their unramified local components differ at some finite place \\(v\\). If \\(\\epsilon_v(\\Frob_v)=1\\), the two unramified components are identical, so the selected place has nontrivial quotient value.","Let \\(A_v\\) be the normalized Satake matrix. Twisting scales it by \\(\\zeta=\\epsilon_v(\\Frob_v)\\), a nontrivial \\(p\\)th root. Since the characteristic polynomials differ, \\cref{thm:one_polynomial_separator} gives an active coefficient degree \\(k\\) not divisible by \\(p\\).\n\\end{proof}","Let \\(K/F\\) be finite Galois with group \\(G\\). For a complex character \\(\\theta\\) of a subgroup, write \\(L(s,\\theta)\\) for the corresponding Artin \\(L\\) function.","\\begin{definition}[Entire packet set]\\label{def:entire_packet_set}\nAn entire packet set \\(\\calE\\) is a collection of characters of subgroups of \\(G\\) whose Artin \\(L\\) functions are known to be entire.\n\\end{definition}","\\begin{theorem}[Positive saturation criterion]\\label{thm:positive_saturation}\nLet \\(\\chi\\) be a nontrivial irreducible character of \\(G\\). Suppose there are a positive integer \\(m\\), nonnegative integers \\(a_i\\), subgroups \\(H_i\\), and characters \\(\\theta_i\\in\\calE\\) such that\n\\[\n \\boxed{\n m\\chi\n =\n \\sum_i\n a_i\\Ind_{H_i}^G\\theta_i.\n }\n\\]\nThen \\(L(s,\\chi)\\) is entire.\n\\end{theorem}","\\begin{proof}\nArtin formalism gives\n\\[\n L(s,\\chi)^m\n =\n \\prod_iL(s,\\theta_i)^{a_i}.\n\\]\nThe right side is entire by hypothesis. Brauer induction implies that \\(L(s,\\chi)\\) is meromorphic \\cite{Brauer1947}. If it had a pole at any point, its \\(m\\)th power would have a pole there. Thus it has no poles and is entire.\n\\end{proof}","\\begin{definition}[Entire induction cone]\\label{def:entire_cone}\nFix finitely many known entire induced packets \\(\\Theta_1,\\ldots,\\Theta_r\\) in the character lattice of \\(G\\). Define\n\\[\n C_{\\calE}\n =\n \\left\\{\n \\sum_{i=1}^ra_i\\Theta_i:\n a_i\\in\\ZZ_{\\geq0}\n \\right\\}.\n\\]\nIts saturation is\n\\[\n \\operatorname{Sat}(C_{\\calE})\n =\n \\{x:\\text{some }m>0\\text{ has }mx\\in C_{\\calE}\\}.\n\\]\n\\end{definition}","\\begin{corollary}[Finite cone certificate]\\label{cor:cone_certificate}\nIf an irreducible character \\(\\chi\\) belongs to \\(\\operatorname{Sat}(C_{\\calE})\\), then its Artin \\(L\\) function is entire. Membership is an integer feasibility problem once a character table and the generator decompositions are known.\n\\end{corollary}","\\begin{remark}[Why signed Brauer induction is insufficient]\\label{rem:brauer_signed}\nBrauer induction permits negative coefficients. Those coefficients place Artin \\(L\\) functions in denominators and yield meromorphic continuation, not holomorphy. The positivity in \\cref{thm:positive_saturation} is the essential analytic input.\n\\end{remark}","The automorphic induction of the attached representation over \\(E\\) is already attached to\n\\[\n \\Ind_{G_E}^{G_F}\\tau\n \\cong\n \\bigoplus_{j=0}^{p-1}\n \\rho\\otimes\\epsilon^j.\n\\]\nIt is also the isobaric sum of all descents. By \\cref{thm:total_induction_blind}, this total object is invariant under rotating either torsor. It proves the direct sum correspondence but cannot identify one constituent with the target extension.","Determinant and central character select the candidate when \\(p\\nmid n\\). When \\(p\\mid n\\), \\cref{thm:central_character_no_go} shows that every algebraic central character is invariant. Deligne and Henniart stability gives the analogous blindness for sufficiently ramified character twists. The missing information is therefore carried by bounded depth noncentral extension spectra.","If a continuous l adic realization has finite image, then at each place of \\(F\\) above the coefficient prime its restriction to the local Galois group becomes trivial after a finite extension of that local field. Its Hodge Tate weights are therefore all zero. There are \\(n\\) copies of the same weight. For \\(n>1\\) this is not regular. Regularity is not a stylistic label that can be removed from a theorem statement after finding a convenient residual congruence.","Special nonregular theorems also exist. Calegari and Geraghty treat a nonregular ordinary symplectic setting with weights of the form \\((0,0,j-1,j-1)\\), under their numerical, residual modularity, image, and local conditions \\cite{CalegariGeraghty2020}. This shows that regularity is not required by every conceivable lifting method. It does not supply the all zero, arbitrary rank theorem required by the present cyclic gate. The correct conclusion is a mismatch between the declared target and currently invoked theorem hypotheses, rather than a universal prohibition on nonregular automorphy lifting.","The global gate is now sharply stated, but it has not disappeared. A separating coefficient prime converts a supplied residual Galois realization into an exact cyclic label and weak characteristic zero attachment. The quotient prime forces a different statement: equality of the unramified residual eigensystems, with all order \\(p\\) labels collapsed. A continuous semisimple l adic realization becomes finite image by the trace pairing theorem. Each route begins with its declared existence or lifting input.","Let $C_e=\\langle\\gamma\\rangle$ be a finite cyclic group of order $e$, acting\nlinearly on a module $M$. Write $T$ for the action of $\\gamma$ and set\n\\[\n N=1+T+\\cdots+T^{e-1}.\n\\]\nThe relation $T^e=1$ implies $N(T-1)=(T-1)N=0$. These identities explain why\nthe image of $T-1$ lies in the kernel of $N$, but they do not identify the two\nspaces.","A one-cocycle is a function $a:C_e\\to M$ satisfying\n\\[\n a(gh)=a(g)+g\\cdot a(h).\n\\]\nPutting $g=h=1$ gives $a(1)=0$. If $x=a(\\gamma)$, repeated application of the\ncocycle equation gives\n\\[\n a(\\gamma^r)=\\sum_{j=0}^{r-1}T^j x\\qquad(1\\le r\\le e).\n\\]\nThe group relation $\\gamma^e=1$ therefore forces $Nx=0$. Conversely, if\n$Nx=0$, this formula defines a cocycle. Whenever exponents wrap around $e$,\nthe discrepancy is a translate of $Nx$, which is zero. Thus evaluating at\nthe chosen generator identifies cocycles with $\\ker N$.","$a:C_e\\to M$ satisfying\n\\[\n a(gh)=a(g)+g\\cdot a(h).\n\\]\nPutting $g=h=1$ gives $a(1)=0$. If $x=a(\\gamma)$, repeated application of the\ncocycle equation gives\n\\[\n a(\\gamma^r)=\\sum_{j=0}^{r-1}T^j x\\qquad(1\\le r\\le e).\n\\]\nThe group relation $\\gamma^e=1$ therefore forces $Nx=0$. Conversely, if\n$Nx=0$, this formula defines a cocycle. Whenever exponents wrap around $e$,\nthe discrepancy is a translate of $Nx$, which is zero. Thus evaluating at\nthe chosen generator identifies cocycles with $\\ker N$.","If $x=a(\\gamma)$, repeated application of the\ncocycle equation gives\n\\[\n a(\\gamma^r)=\\sum_{j=0}^{r-1}T^j x\\qquad(1\\le r\\le e).\n\\]\nThe group relation $\\gamma^e=1$ therefore forces $Nx=0$. Conversely, if\n$Nx=0$, this formula defines a cocycle. Whenever exponents wrap around $e$,\nthe discrepancy is a translate of $Nx$, which is zero. Thus evaluating at\nthe chosen generator identifies cocycles with $\\ker N$.","For a zero-cochain $y\\in M$, its coboundary at $\\gamma$ is $(T-1)y$. Taking\nthe quotient by these changes gives the precise presentation\n\\begin{equation}\n H^1(C_e,M)\\simeq \\ker N/\\operatorname{im}(T-1).\n \\label{eq:appendix-cyclic-H1}\n\\end{equation}\nIf the norm operator is identically zero, this reduces to\n$M/\\operatorname{im}(T-1)$. That reduction has a hypothesis. Dropping the\nkernel from the general formula introduces classes that cannot satisfy the\ncyclic relation.","Suppose now that $e$ is invertible in the coefficient ring. Define\n$S=\\sum_{j=0}^{e-1}jT^j$. A direct coefficient calculation gives\n\\[\n (T-1)S=e-N.\n\\]\nFor $x\\in\\ker N$, take $y=e^{-1}Sx$. Then $(T-1)y=x$, so every cocycle is\na coboundary and $H^1(C_e,M)=0$. The averaging argument uses division by\n$e$ at its decisive step. In residue characteristic dividing $e$, the same\ndisplay remains an identity but no longer constructs $y$.","ertible in the coefficient ring. Define\n$S=\\sum_{j=0}^{e-1}jT^j$. A direct coefficient calculation gives\n\\[\n (T-1)S=e-N.\n\\]\nFor $x\\in\\ker N$, take $y=e^{-1}Sx$. Then $(T-1)y=x$, so every cocycle is\na coboundary and $H^1(C_e,M)=0$. The averaging argument uses division by\n$e$ at its decisive step. In residue characteristic dividing $e$, the same\ndisplay remains an identity but no longer constructs $y$.","\\begin{example}[The same generator action gives different cyclic cohomology]\nOver $\\mathbb F_3$, let $M=\\operatorname{Mat}_2(\\mathbb F_3)$ and let $T$\nbe conjugation by $F=\\operatorname{diag}(1,-1)$. The diagonal entries of a\nmatrix form the $+1$ eigenspace of $T$. The off-diagonal entries form its\n$-1$ eigenspace. Both spaces have dimension two.","For $C_2$, the norm $1+T$ is multiplication by $2$ on the diagonal space and\nzero on the off-diagonal space. Hence its kernel is the off-diagonal space.\nThe operator $T-1$ is invertible on that space and zero on the diagonal\nspace. Formula~\\eqref{eq:appendix-cyclic-H1} gives $H^1(C_2,M)=0$.","Let instead the generator of $C_6$ act by the same $T$. On the diagonal\nspace, its norm is multiplication by $6=0$ in $\\mathbb F_3$. On the\noff-diagonal space, the alternating six-term sum is also zero. Now $N=0$\non all of $M$, while the image of $T-1$ is still only the off-diagonal\nspace. Therefore $H^1(C_6,M)$ has dimension two. The generator matrix\nalone does not determine this cohomology without the order and coefficient\ncharacteristic.\n\\end{example}","In Frobenius coordinates the relevant action is often $T(X)=FXF^{-1}$.\nThe images of $T-1$ and $1-T$ coincide, since multiplication of a source\nvector by $-1$ exchanges the signs. This is the bridge to the concrete\nquotient by Frobenius coboundaries. The recovered v0.18 source proves that\nbridge under the explicit zero-norm hypothesis. It does not claim that every\nFrobenius quotient is automatically an arithmetic local cohomology group.\nOne must first identify the acting group and the coefficient module.","Let $q:R'\\to R$ be a ring map with kernel $I$, and assume $I^2=0$. Let\n$\\rho:G\\to R^\\times$ be a group representation. Choose unit-valued lifts\n$r_g\\in (R')^\\times$ with $q(r_g)=\\rho(g)$ and $r_1=1$. Such a choice is\nan input to this calculation. It is not already a multiplicative lift.\nDefine its normalized defect by\n\\begin{equation}\n c(g,h)=(r_gr_h-r_{gh})r_{gh}^{-1}\\in I.\n \\label{eq:appendix-normalized-defect}\n\\end{equation}\nEquivalently, $r_gr_h=(1+c(g,h))r_{gh}$. The coefficients are elements of\nthe actual ring kernel, with its additive group, rather than an unrelated\nvector space of the same dimension.","and assume $I^2=0$. Let\n$\\rho:G\\to R^\\times$ be a group representation. Choose unit-valued lifts\n$r_g\\in (R')^\\times$ with $q(r_g)=\\rho(g)$ and $r_1=1$. Such a choice is\nan input to this calculation. It is not already a multiplicative lift.\nDefine its normalized defect by\n\\begin{equation}\n c(g,h)=(r_gr_h-r_{gh})r_{gh}^{-1}\\in I.\n \\label{eq:appendix-normalized-defect}\n\\end{equation}\nEquivalently, $r_gr_h=(1+c(g,h))r_{gh}$. The coefficients are elements of\nthe actual ring kernel, with its additive group, rather than an unrelated\nvector space of the same dimension.","The action of $G$ on $I$ is $g\\cdot x=r_gxr_g^{-1}$. It is independent of\nthe chosen unit lift. Indeed, any other unit lift has the form $(1+b)r_g$\nwith $b\\in I$. Since $(1+b)^{-1}=1-b$ and products of two elements of $I$\nvanish, conjugating $x\\in I$ by $1+b$ leaves $x$ unchanged. The same\nobservation shows that the defect in the multiplication of the $r_g$ does\nnot prevent these conjugations from forming a genuine group action on $I$.","x=r_gxr_g^{-1}$. It is independent of\nthe chosen unit lift. Indeed, any other unit lift has the form $(1+b)r_g$\nwith $b\\in I$. Since $(1+b)^{-1}=1-b$ and products of two elements of $I$\nvanish, conjugating $x\\in I$ by $1+b$ leaves $x$ unchanged. The same\nobservation shows that the defect in the multiplication of the $r_g$ does\nnot prevent these conjugations from forming a genuine group action on $I$.","Compute $(r_gr_h)r_\\ell$ and $r_g(r_hr_\\ell)$. After expressing both in\nterms of $r_{gh\\ell}$, the first is\n\\[\n (1+c(g,h)+c(gh,\\ell))r_{gh\\ell},\n\\]\nwhile the second is\n\\[\n (1+g\\cdot c(h,\\ell)+c(g,h\\ell))r_{gh\\ell}.\n\\]\nAll omitted cross products belong to $I^2$. Associativity therefore yields\n\\begin{equation}\n g\\cdot c(h,\\ell)-c(gh,\\ell)+c(g,h\\ell)-c(g,h)=0.\n \\label{eq:appendix-cocycle}\n\\end{equation}\nThis is the usual inhomogeneous two-cocycle identity. Normalization at the\nidentity follows from $r_1=1$.","st is\n\\[\n (1+c(g,h)+c(gh,\\ell))r_{gh\\ell},\n\\]\nwhile the second is\n\\[\n (1+g\\cdot c(h,\\ell)+c(g,h\\ell))r_{gh\\ell}.\n\\]\nAll omitted cross products belong to $I^2$. Associativity therefore yields\n\\begin{equation}\n g\\cdot c(h,\\ell)-c(gh,\\ell)+c(g,h\\ell)-c(g,h)=0.\n \\label{eq:appendix-cocycle}\n\\end{equation}\nThis is the usual inhomogeneous two-cocycle identity. Normalization at the\nidentity follows from $r_1=1$.","Change the chosen lifts to $r'_g=(1+b(g))r_g$, where $b:G\\to I$ and\n$b(1)=0$. Expanding without commuting the factors gives\n\\[\n r'_gr'_h\n =(1+b(g)+g\\cdot b(h)+c(g,h))r_{gh}.\n\\]\nTo compare with $r'_{gh}$, multiply by $1-b(gh)$. The new defect is\n\\begin{equation}\n c'(g,h)=c(g,h)+g\\cdot b(h)-b(gh)+b(g).\n \\label{eq:appendix-defect-change}\n\\end{equation}\nThus changing the section adds a two-coboundary. The class $[c]\\in H^2(G,I)$\ndoes not depend on the section.","hosen lifts to $r'_g=(1+b(g))r_g$, where $b:G\\to I$ and\n$b(1)=0$. Expanding without commuting the factors gives\n\\[\n r'_gr'_h\n =(1+b(g)+g\\cdot b(h)+c(g,h))r_{gh}.\n\\]\nTo compare with $r'_{gh}$, multiply by $1-b(gh)$. The new defect is\n\\begin{equation}\n c'(g,h)=c(g,h)+g\\cdot b(h)-b(gh)+b(g).\n \\label{eq:appendix-defect-change}\n\\end{equation}\nThus changing the section adds a two-coboundary. The class $[c]\\in H^2(G,I)$\ndoes not depend on the section.","\\begin{proposition}[Exact lifting criterion]\nWith the hypotheses above, a unit-valued representation lifting $\\rho$\nexists if and only if $[c]=0$.\n\\end{proposition}","\\begin{proof}\nIf $c=\\delta b$, replace $r_g$ by $(1-b(g))r_g$. Equation\n\\eqref{eq:appendix-defect-change} shows that the corrected defect is zero.\nThe corrected units are therefore multiplicative and reduce to $\\rho$.\nThe equality $c(1,1)=0$ forces $b(1)=0$ for a primitive in the displayed\nnormalization. Conversely, a multiplicative unit lift has zero defect.\nIts difference from the original section is a kernel-valued one-cochain,\nso the same change formula proves that the original class is zero.\n\\end{proof}","\\begin{example}[An obstruction that really does not vanish]\nReduce $\\mathbb Z/27\\mathbb Z$ to $\\mathbb Z/9\\mathbb Z$. Its kernel is\n$9\\mathbb Z/27\\mathbb Z$, whose square is zero. Map a generator of $C_3$\nto the unit $4$ modulo $9$. This defines a representation because\n$4^3=1$ modulo $9$.","A lift of the generator must be $4+9t$ modulo $27$, for $t=0,1,2$.\nDirect expansion gives\n\\[\n (4+9t)^3\\equiv 4^3\\equiv 10\\pmod {27}.\n\\]\nNone has cube $1$. Hence no representation lift exists and the obstruction\nclass is nonzero. Conjugation is trivial in this scalar example. The\nkernel is additively $\\mathbb F_3$, and the standard cyclic resolution gives\n$H^2(C_3,\\mathbb F_3)=\\mathbb F_3$ for the trivial action. This is a\nfailure of lifting, rather than a failure to find a convenient section.","The defect can also be seen without an existence argument. Choose the\nsection $r_1=1$, $r_\\gamma=4$, and $r_{\\gamma^2}=16$ modulo $27$. These\nreduce to the required units modulo $9$. The products of consecutive\npowers have no defect until the exponent passes through three. At that\npoint $4^3=10=1+9$ modulo $27$. Consequently the normalized defect is\n$9$ when the exponents wrap around and zero otherwise. Under the\nidentification $I\\simeq\\mathbb F_3$, this is the ordinary carry cocycle.","ut an existence argument. Choose the\nsection $r_1=1$, $r_\\gamma=4$, and $r_{\\gamma^2}=16$ modulo $27$. These\nreduce to the required units modulo $9$. The products of consecutive\npowers have no defect until the exponent passes through three. At that\npoint $4^3=10=1+9$ modulo $27$. Consequently the normalized defect is\n$9$ when the exponents wrap around and zero otherwise. Under the\nidentification $I\\simeq\\mathbb F_3$, this is the ordinary carry cocycle.","For a proposed correction $b$, sum the equations\n$c(\\gamma^i,\\gamma)=b(\\gamma)-b(\\gamma^{i+1})+b(\\gamma^i)$ for\n$i=0,1,2$. The right side telescopes to $3b(\\gamma)=0$, using\n$b(1)=0$ and the trivial action. The left side is $9\\ne0$ in $I$.\nThere is therefore no primitive. This explicit calculation recovers the\nnonzero obstruction class directly from its normalized representative.\n\\end{example}","Suppose an absolute two-cocycle has been cancelled by a continuous\none-cochain $b_0$. Prescribed local lifts need not agree with the resulting\nglobal lift. Their remaining differences are local one-cocycles. Changing\n$b_0$ to another absolute primitive changes these differences by the\nrestriction of a global one-cocycle. Conjugating the local lift at index\n$i$ by a unit $1+x_i$, with $x_i\\in I$, changes the local cocycle by a\nlocal coboundary. The conjugators $x_i$ are independently chosen at each\nlocal index. Requiring one common conjugator is a stronger problem.","s\none-cochain $b_0$. Prescribed local lifts need not agree with the resulting\nglobal lift. Their remaining differences are local one-cocycles. Changing\n$b_0$ to another absolute primitive changes these differences by the\nrestriction of a global one-cocycle. Conjugating the local lift at index\n$i$ by a unit $1+x_i$, with $x_i\\in I$, changes the local cocycle by a\nlocal coboundary. The conjugators $x_i$ are independently chosen at each\nlocal index. Requiring one common conjugator is a stronger problem.","After taking the local coboundary quotients, the remaining class belongs to\n\\begin{equation}\n \\operatorname{coker}\\left(\n H^1_{\\mathrm{cont}}(G,I)\\longrightarrow\n \\prod_i H^1_{\\mathrm{cont}}(G_i,I)\\right).\n \\label{eq:appendix-gauge-cokernel}\n\\end{equation}\nHere the local groups, their maps into $G$, and the precise continuous\ncoefficient action must be supplied. The recovered source constructs the\ncorresponding quotients directly in its square-zero data.","To see the criterion, let $a_i$ be the local mismatch after $b_0$. A match\nup to the allowed local conjugations exists exactly when\n\\[\n a_i=\\operatorname{res}_i(z)+\\delta x_i\n\\]\nfor a global continuous one-cocycle $z$ and kernel elements $x_i$. This is\nexactly the assertion that the image of $(a_i)_i$ in\n\\eqref{eq:appendix-gauge-cokernel} is zero. If $b_1$ is another absolute\nprimitive, then $b_1-b_0$ is a global cocycle. Its restriction disappears in\nthe cokernel, so the remaining class does not depend on which primitive\nwas used.","The v0.29 exact sequence identifies this secondary cokernel with the kernel\nof the map that forgets local gauge data from the relative obstruction\nspace. Exactness at that position does not make the forgetting map\nsurjective. It also does not make the localization map in\n\\eqref{eq:appendix-gauge-cokernel} surjective. These are separate arithmetic\nquestions. In particular, solving the absolute $H^2$ obstruction does not\nby itself solve a family of prescribed local conditions.","Let $J:V\\to W$ be a linear map over a field $k$. The correction equation\n$Ju+e=0$ has a solution precisely when $e\\in\\operatorname{im}J$. A useful\ndescription of this image uses all the dual relations among the equations.\nLet\n\\[\n S=\\ker(J^*:W^*\\to V^*),\\qquad\n C=\\{w\\in W:\\phi(w)=0\\text{ for every }\\phi\\in S\\}.\n\\]\nEvery $Ju$ belongs to $C$. Conversely, if $w$ does not belong to\n$\\operatorname{im}J$, its nonzero class in $W/\\operatorname{im}J$ is\ndetected by an algebraic linear functional. Pulling that functional back\nto $W$ gives an element of $S$ that does not annihilate $w$. Hence\n\\begin{equation}\n \\operatorname{im}J=C.\n \\label{eq:appendix-compatible-target}\n\\end{equation}\nThis algebraic dual argument w","Let $J:V\\to W$ be a linear map over a field $k$. The correction equation\n$Ju+e=0$ has a solution precisely when $e\\in\\operatorname{im}J$. A useful\ndescription of this image uses all the dual relations among the equations.\nLet\n\\[\n S=\\ker(J^*:W^*\\to V^*),\\qquad\n C=\\{w\\in W:\\phi(w)=0\\text{ for every }\\phi\\in S\\}.\n\\]\nEvery $Ju$ belongs to $C$. Conversely, if $w$ does not belong to\n$\\operatorname{im}J$, its nonzero class in $W/\\operatorname{im}J$ is\ndetected by an algebraic linear functional. Pulling that functional back\nto $W$ gives an element of $S$ that does not annihilate $w$. Hence\n\\begin{equation}\n \\operatorname{im}J=C.\n \\label{eq:appendix-compatible-target}\n\\end{equation}\nThis algebraic dual argument works without a finite-dimensional","Let $J:V\\to W$ be a linear map over a field $k$. The correction equation\n$Ju+e=0$ has a solution precisely when $e\\in\\operatorname{im}J$. A useful\ndescription of this image uses all the dual relations among the equations.\nLet\n\\[\n S=\\ker(J^*:W^*\\to V^*),\\qquad\n C=\\{w\\in W:\\phi(w)=0\\text{ for every }\\phi\\in S\\}.\n\\]\nEvery $Ju$ belongs to $C$. Conversely, if $w$ does not belong to\n$\\operatorname{im}J$, its nonzero class in $W/\\operatorname{im}J$ is\ndetected by an algebraic linear functional. Pulling that functional back\nto $W$ gives an element of $S$ that does not annihilate $w$. Hence\n\\begin{equation}\n \\operatorname{im}J=C.\n \\label{eq:appendix-compatible-target}\n\\end{equation}\nThis algebraic dual argument works without a finite-dimensional hypothesis.\nIt makes no assertion","Let $J:V\\to W$ be a linear map over a field $k$. The correction equation\n$Ju+e=0$ has a solution precisely when $e\\in\\operatorname{im}J$. A useful\ndescription of this image uses all the dual relations among the equations.\nLet\n\\[\n S=\\ker(J^*:W^*\\to V^*),\\qquad\n C=\\{w\\in W:\\phi(w)=0\\text{ for every }\\phi\\in S\\}.\n\\]\nEvery $Ju$ belongs to $C$. Conversely, if $w$ does not belong to\n$\\operatorname{im}J$, its nonzero class in $W/\\operatorname{im}J$ is\ndetected by an algebraic linear functional. Pulling that functional back\nto $W$ gives an element of $S$ that does not annihilate $w$. Hence\n\\begin{equation}\n \\operatorname{im}J=C.\n \\label{eq:appendix-compatible-target}\n\\end{equation}\nThis algebraic dual argument works without a finite-dimensional hypothesis.\nIt makes no assertion about continuous duals in a different topology.","ar map over a field $k$. The correction equation\n$Ju+e=0$ has a solution precisely when $e\\in\\operatorname{im}J$. A useful\ndescription of this image uses all the dual relations among the equations.\nLet\n\\[\n S=\\ker(J^*:W^*\\to V^*),\\qquad\n C=\\{w\\in W:\\phi(w)=0\\text{ for every }\\phi\\in S\\}.\n\\]\nEvery $Ju$ belongs to $C$. Conversely, if $w$ does not belong to\n$\\operatorname{im}J$, its nonzero class in $W/\\operatorname{im}J$ is\ndetected by an algebraic linear functional. Pulling that functional back\nto $W$ gives an element of $S$ that does not annihilate $w$. Hence\n\\begin{equation}\n \\operatorname{im}J=C.\n \\label{eq:appendix-compatible-target}\n\\end{equation}\nThis algebraic dual argument works without a finite-dimensional hypothesis.\nIt makes no assertion about continuous duals in a different topology.","name{im}J$. A useful\ndescription of this image uses all the dual relations among the equations.\nLet\n\\[\n S=\\ker(J^*:W^*\\to V^*),\\qquad\n C=\\{w\\in W:\\phi(w)=0\\text{ for every }\\phi\\in S\\}.\n\\]\nEvery $Ju$ belongs to $C$. Conversely, if $w$ does not belong to\n$\\operatorname{im}J$, its nonzero class in $W/\\operatorname{im}J$ is\ndetected by an algebraic linear functional. Pulling that functional back\nto $W$ gives an element of $S$ that does not annihilate $w$. Hence\n\\begin{equation}\n \\operatorname{im}J=C.\n \\label{eq:appendix-compatible-target}\n\\end{equation}\nThis algebraic dual argument works without a finite-dimensional hypothesis.\nIt makes no assertion about continuous duals in a different topology.","cription of this image uses all the dual relations among the equations.\nLet\n\\[\n S=\\ker(J^*:W^*\\to V^*),\\qquad\n C=\\{w\\in W:\\phi(w)=0\\text{ for every }\\phi\\in S\\}.\n\\]\nEvery $Ju$ belongs to $C$. Conversely, if $w$ does not belong to\n$\\operatorname{im}J$, its nonzero class in $W/\\operatorname{im}J$ is\ndetected by an algebraic linear functional. Pulling that functional back\nto $W$ gives an element of $S$ that does not annihilate $w$. Hence\n\\begin{equation}\n \\operatorname{im}J=C.\n \\label{eq:appendix-compatible-target}\n\\end{equation}\nThis algebraic dual argument works without a finite-dimensional hypothesis.\nIt makes no assertion about continuous duals in a different topology.","ll the dual relations among the equations.\nLet\n\\[\n S=\\ker(J^*:W^*\\to V^*),\\qquad\n C=\\{w\\in W:\\phi(w)=0\\text{ for every }\\phi\\in S\\}.\n\\]\nEvery $Ju$ belongs to $C$. Conversely, if $w$ does not belong to\n$\\operatorname{im}J$, its nonzero class in $W/\\operatorname{im}J$ is\ndetected by an algebraic linear functional. Pulling that functional back\nto $W$ gives an element of $S$ that does not annihilate $w$. Hence\n\\begin{equation}\n \\operatorname{im}J=C.\n \\label{eq:appendix-compatible-target}\n\\end{equation}\nThis algebraic dual argument works without a finite-dimensional hypothesis.\nIt makes no assertion about continuous duals in a different topology.","g the equations.\nLet\n\\[\n S=\\ker(J^*:W^*\\to V^*),\\qquad\n C=\\{w\\in W:\\phi(w)=0\\text{ for every }\\phi\\in S\\}.\n\\]\nEvery $Ju$ belongs to $C$. Conversely, if $w$ does not belong to\n$\\operatorname{im}J$, its nonzero class in $W/\\operatorname{im}J$ is\ndetected by an algebraic linear functional. Pulling that functional back\nto $W$ gives an element of $S$ that does not annihilate $w$. Hence\n\\begin{equation}\n \\operatorname{im}J=C.\n \\label{eq:appendix-compatible-target}\n\\end{equation}\nThis algebraic dual argument works without a finite-dimensional hypothesis.\nIt makes no assertion about continuous duals in a different topology.","Corestricting $J$ to $C$ makes it surjective onto its actual reachable\ntarget. The equations have not changed. By contrast, adjoining variables\nthat absorb arbitrary errors changes the system unless their removal is\nproved. A solution of an augmented equation is not automatically a solution\nof the original equation. The original target and the quotient that records\nunreachable errors must be kept separate.","\\begin{example}[A reachable target with a visible failed correction]\nLet $V=k^2$, $W=k^3$, and\n$J(u,v)=(u,v,u+v)$. Its image is the plane $w_3=w_1+w_2$. The dual\nkernel is generated by the functional\n$\\phi(w_1,w_2,w_3)=w_3-w_1-w_2$. The error $(1,2,3)$ is compatible\nin every characteristic and is cancelled by $(u,v)=(-1,-2)$. The error\n$(0,0,1)$ is incompatible in every field, since its value under $\\phi$\nis $1$. No choice of the two correction variables can cancel it.","gin{example}[A reachable target with a visible failed correction]\nLet $V=k^2$, $W=k^3$, and\n$J(u,v)=(u,v,u+v)$. Its image is the plane $w_3=w_1+w_2$. The dual\nkernel is generated by the functional\n$\\phi(w_1,w_2,w_3)=w_3-w_1-w_2$. The error $(1,2,3)$ is compatible\nin every characteristic and is cancelled by $(u,v)=(-1,-2)$. The error\n$(0,0,1)$ is incompatible in every field, since its value under $\\phi$\nis $1$. No choice of the two correction variables can cancel it.","e}[A reachable target with a visible failed correction]\nLet $V=k^2$, $W=k^3$, and\n$J(u,v)=(u,v,u+v)$. Its image is the plane $w_3=w_1+w_2$. The dual\nkernel is generated by the functional\n$\\phi(w_1,w_2,w_3)=w_3-w_1-w_2$. The error $(1,2,3)$ is compatible\nin every characteristic and is cancelled by $(u,v)=(-1,-2)$. The error\n$(0,0,1)$ is incompatible in every field, since its value under $\\phi$\nis $1$. No choice of the two correction variables can cancel it.","If a third variable $t$ is added so that the derivative becomes\n$J'(u,v,t)=(u,v,u+v+t)$, the enlarged map is surjective. It cancels the\nsecond error with $(u,v,t)=(0,0,-1)$. But the nonzero value of $t$ records\nexactly the failed original relation. This enlarged solution cannot be\ndeclared a solution of the original equations by forgetting $t$. One must\nshow that the additional variable is removable, or that the target error\nalready lies in the original compatible plane.\n\\end{example}","For a nonlinear system $F(x)=0$, a relation among the residual Jacobian\nrows is only a first-order identity. Consider\n\\[\n F(x)=(x,x^2),\\qquad J_x(h)=(h,2xh).\n\\]\nAt the origin the derivative has horizontal image. The functional selecting\nthe second coordinate is a residual syzygy, but it does not annihilate\n$F(x)$ for general $x$. The exact relation is the moving relation\n\\[\n (-x,1)F(x)=0.\n\\]\nFurthermore,\n\\[\n F(x+h)-F(x)-J_x(h)=(0,h^2).\n\\]\nModulo $8$, taking $x=0$ and $h=2$ gives remainder $(0,4)$, outside the\nhorizontal target. Keeping only the syzygy at the origin would therefore\nlose a genuine nonlinear error even within the same residual fibre.","inear system $F(x)=0$, a relation among the residual Jacobian\nrows is only a first-order identity. Consider\n\\[\n F(x)=(x,x^2),\\qquad J_x(h)=(h,2xh).\n\\]\nAt the origin the derivative has horizontal image. The functional selecting\nthe second coordinate is a residual syzygy, but it does not annihilate\n$F(x)$ for general $x$. The exact relation is the moving relation\n\\[\n (-x,1)F(x)=0.\n\\]\nFurthermore,\n\\[\n F(x+h)-F(x)-J_x(h)=(0,h^2).\n\\]\nModulo $8$, taking $x=0$ and $h=2$ gives remainder $(0,4)$, outside the\nhorizontal target. Keeping only the syzygy at the origin would therefore\nlose a genuine nonlinear error even within the same residual fibre.","he residual Jacobian\nrows is only a first-order identity. Consider\n\\[\n F(x)=(x,x^2),\\qquad J_x(h)=(h,2xh).\n\\]\nAt the origin the derivative has horizontal image. The functional selecting\nthe second coordinate is a residual syzygy, but it does not annihilate\n$F(x)$ for general $x$. The exact relation is the moving relation\n\\[\n (-x,1)F(x)=0.\n\\]\nFurthermore,\n\\[\n F(x+h)-F(x)-J_x(h)=(0,h^2).\n\\]\nModulo $8$, taking $x=0$ and $h=2$ gives remainder $(0,4)$, outside the\nhorizontal target. Keeping only the syzygy at the origin would therefore\nlose a genuine nonlinear error even within the same residual fibre.","Jacobian\nrows is only a first-order identity. Consider\n\\[\n F(x)=(x,x^2),\\qquad J_x(h)=(h,2xh).\n\\]\nAt the origin the derivative has horizontal image. The functional selecting\nthe second coordinate is a residual syzygy, but it does not annihilate\n$F(x)$ for general $x$. The exact relation is the moving relation\n\\[\n (-x,1)F(x)=0.\n\\]\nFurthermore,\n\\[\n F(x+h)-F(x)-J_x(h)=(0,h^2).\n\\]\nModulo $8$, taking $x=0$ and $h=2$ gives remainder $(0,4)$, outside the\nhorizontal target. Keeping only the syzygy at the origin would therefore\nlose a genuine nonlinear error even within the same residual fibre.","ows is only a first-order identity. Consider\n\\[\n F(x)=(x,x^2),\\qquad J_x(h)=(h,2xh).\n\\]\nAt the origin the derivative has horizontal image. The functional selecting\nthe second coordinate is a residual syzygy, but it does not annihilate\n$F(x)$ for general $x$. The exact relation is the moving relation\n\\[\n (-x,1)F(x)=0.\n\\]\nFurthermore,\n\\[\n F(x+h)-F(x)-J_x(h)=(0,h^2).\n\\]\nModulo $8$, taking $x=0$ and $h=2$ gives remainder $(0,4)$, outside the\nhorizontal target. Keeping only the syzygy at the origin would therefore\nlose a genuine nonlinear error even within the same residual fibre.","der identity. Consider\n\\[\n F(x)=(x,x^2),\\qquad J_x(h)=(h,2xh).\n\\]\nAt the origin the derivative has horizontal image. The functional selecting\nthe second coordinate is a residual syzygy, but it does not annihilate\n$F(x)$ for general $x$. The exact relation is the moving relation\n\\[\n (-x,1)F(x)=0.\n\\]\nFurthermore,\n\\[\n F(x+h)-F(x)-J_x(h)=(0,h^2).\n\\]\nModulo $8$, taking $x=0$ and $h=2$ gives remainder $(0,4)$, outside the\nhorizontal target. Keeping only the syzygy at the origin would therefore\nlose a genuine nonlinear error even within the same residual fibre.","The graded correction argument retains an exact moving relation\n$B_xF(x)=0$ and a residual exactness statement\n\\begin{equation}\n \\operatorname{im}\\overline J=\\ker\\overline B.\n \\label{eq:appendix-residual-exactness}\n\\end{equation}\nIf $F(x)=\\pi^r e$, the moving identity gives $\\pi^r B_xe=0$. A faithful\nleading layer allows this equality to imply\n$\\overline B\\,\\overline e=0$. Equation\n\\eqref{eq:appendix-residual-exactness} then supplies a residual correction\n$\\overline u$ with $\\overline J\\,\\overline u=-\\overline e$. Lift\n$\\overline u$ to $u$ and put $x'=x+\\pi^r u$. The linear part of $F(x')$\nis divisible by $\\pi^{r+1}$. For polynomial systems the remainder contains\nat least two copies of $\\pi^r$, so it is divisible by $\\pi^{2r}$ and hence\nby $\\pi^{r+1}$ when $r\\ge1$","The graded correction argument retains an exact moving relation\n$B_xF(x)=0$ and a residual exactness statement\n\\begin{equation}\n \\operatorname{im}\\overline J=\\ker\\overline B.\n \\label{eq:appendix-residual-exactness}\n\\end{equation}\nIf $F(x)=\\pi^r e$, the moving identity gives $\\pi^r B_xe=0$. A faithful\nleading layer allows this equality to imply\n$\\overline B\\,\\overline e=0$. Equation\n\\eqref{eq:appendix-residual-exactness} then supplies a residual correction\n$\\overline u$ with $\\overline J\\,\\overline u=-\\overline e$. Lift\n$\\overline u$ to $u$ and put $x'=x+\\pi^r u$. The linear part of $F(x')$\nis divisible by $\\pi^{r+1}$. For polynomial systems the remainder contains\nat least two copies of $\\pi^r$, so it is divisible by $\\pi^{2r}$ and hence\nby $\\pi^{r+1}$ when $r\\ge1$. This produces the next layer.","on\n$B_xF(x)=0$ and a residual exactness statement\n\\begin{equation}\n \\operatorname{im}\\overline J=\\ker\\overline B.\n \\label{eq:appendix-residual-exactness}\n\\end{equation}\nIf $F(x)=\\pi^r e$, the moving identity gives $\\pi^r B_xe=0$. A faithful\nleading layer allows this equality to imply\n$\\overline B\\,\\overline e=0$. Equation\n\\eqref{eq:appendix-residual-exactness} then supplies a residual correction\n$\\overline u$ with $\\overline J\\,\\overline u=-\\overline e$. Lift\n$\\overline u$ to $u$ and put $x'=x+\\pi^r u$. The linear part of $F(x')$\nis divisible by $\\pi^{r+1}$. For polynomial systems the remainder contains\nat least two copies of $\\pi^r$, so it is divisible by $\\pi^{2r}$ and hence\nby $\\pi^{r+1}$ when $r\\ge1$. This produces the next layer.","a residual exactness statement\n\\begin{equation}\n \\operatorname{im}\\overline J=\\ker\\overline B.\n \\label{eq:appendix-residual-exactness}\n\\end{equation}\nIf $F(x)=\\pi^r e$, the moving identity gives $\\pi^r B_xe=0$. A faithful\nleading layer allows this equality to imply\n$\\overline B\\,\\overline e=0$. Equation\n\\eqref{eq:appendix-residual-exactness} then supplies a residual correction\n$\\overline u$ with $\\overline J\\,\\overline u=-\\overline e$. Lift\n$\\overline u$ to $u$ and put $x'=x+\\pi^r u$. The linear part of $F(x')$\nis divisible by $\\pi^{r+1}$. For polynomial systems the remainder contains\nat least two copies of $\\pi^r$, so it is divisible by $\\pi^{2r}$ and hence\nby $\\pi^{r+1}$ when $r\\ge1$. This produces the next layer.","}\n \\operatorname{im}\\overline J=\\ker\\overline B.\n \\label{eq:appendix-residual-exactness}\n\\end{equation}\nIf $F(x)=\\pi^r e$, the moving identity gives $\\pi^r B_xe=0$. A faithful\nleading layer allows this equality to imply\n$\\overline B\\,\\overline e=0$. Equation\n\\eqref{eq:appendix-residual-exactness} then supplies a residual correction\n$\\overline u$ with $\\overline J\\,\\overline u=-\\overline e$. Lift\n$\\overline u$ to $u$ and put $x'=x+\\pi^r u$. The linear part of $F(x')$\nis divisible by $\\pi^{r+1}$. For polynomial systems the remainder contains\nat least two copies of $\\pi^r$, so it is divisible by $\\pi^{2r}$ and hence\nby $\\pi^{r+1}$ when $r\\ge1$. This produces the next layer.","im}\\overline J=\\ker\\overline B.\n \\label{eq:appendix-residual-exactness}\n\\end{equation}\nIf $F(x)=\\pi^r e$, the moving identity gives $\\pi^r B_xe=0$. A faithful\nleading layer allows this equality to imply\n$\\overline B\\,\\overline e=0$. Equation\n\\eqref{eq:appendix-residual-exactness} then supplies a residual correction\n$\\overline u$ with $\\overline J\\,\\overline u=-\\overline e$. Lift\n$\\overline u$ to $u$ and put $x'=x+\\pi^r u$. The linear part of $F(x')$\nis divisible by $\\pi^{r+1}$. For polynomial systems the remainder contains\nat least two copies of $\\pi^r$, so it is divisible by $\\pi^{2r}$ and hence\nby $\\pi^{r+1}$ when $r\\ge1$. This produces the next layer.","J=\\ker\\overline B.\n \\label{eq:appendix-residual-exactness}\n\\end{equation}\nIf $F(x)=\\pi^r e$, the moving identity gives $\\pi^r B_xe=0$. A faithful\nleading layer allows this equality to imply\n$\\overline B\\,\\overline e=0$. Equation\n\\eqref{eq:appendix-residual-exactness} then supplies a residual correction\n$\\overline u$ with $\\overline J\\,\\overline u=-\\overline e$. Lift\n$\\overline u$ to $u$ and put $x'=x+\\pi^r u$. The linear part of $F(x')$\nis divisible by $\\pi^{r+1}$. For polynomial systems the remainder contains\nat least two copies of $\\pi^r$, so it is divisible by $\\pi^{2r}$ and hence\nby $\\pi^{r+1}$ when $r\\ge1$. This produces the next layer.","ndix-residual-exactness}\n\\end{equation}\nIf $F(x)=\\pi^r e$, the moving identity gives $\\pi^r B_xe=0$. A faithful\nleading layer allows this equality to imply\n$\\overline B\\,\\overline e=0$. Equation\n\\eqref{eq:appendix-residual-exactness} then supplies a residual correction\n$\\overline u$ with $\\overline J\\,\\overline u=-\\overline e$. Lift\n$\\overline u$ to $u$ and put $x'=x+\\pi^r u$. The linear part of $F(x')$\nis divisible by $\\pi^{r+1}$. For polynomial systems the remainder contains\nat least two copies of $\\pi^r$, so it is divisible by $\\pi^{2r}$ and hence\nby $\\pi^{r+1}$ when $r\\ge1$. This produces the next layer.","\\end{equation}\nIf $F(x)=\\pi^r e$, the moving identity gives $\\pi^r B_xe=0$. A faithful\nleading layer allows this equality to imply\n$\\overline B\\,\\overline e=0$. Equation\n\\eqref{eq:appendix-residual-exactness} then supplies a residual correction\n$\\overline u$ with $\\overline J\\,\\overline u=-\\overline e$. Lift\n$\\overline u$ to $u$ and put $x'=x+\\pi^r u$. The linear part of $F(x')$\nis divisible by $\\pi^{r+1}$. For polynomial systems the remainder contains\nat least two copies of $\\pi^r$, so it is divisible by $\\pi^{2r}$ and hence\nby $\\pi^{r+1}$ when $r\\ge1$. This produces the next layer.","thful\nleading layer allows this equality to imply\n$\\overline B\\,\\overline e=0$. Equation\n\\eqref{eq:appendix-residual-exactness} then supplies a residual correction\n$\\overline u$ with $\\overline J\\,\\overline u=-\\overline e$. Lift\n$\\overline u$ to $u$ and put $x'=x+\\pi^r u$. The linear part of $F(x')$\nis divisible by $\\pi^{r+1}$. For polynomial systems the remainder contains\nat least two copies of $\\pi^r$, so it is divisible by $\\pi^{2r}$ and hence\nby $\\pi^{r+1}$ when $r\\ge1$. This produces the next layer.","equality to imply\n$\\overline B\\,\\overline e=0$. Equation\n\\eqref{eq:appendix-residual-exactness} then supplies a residual correction\n$\\overline u$ with $\\overline J\\,\\overline u=-\\overline e$. Lift\n$\\overline u$ to $u$ and put $x'=x+\\pi^r u$. The linear part of $F(x')$\nis divisible by $\\pi^{r+1}$. For polynomial systems the remainder contains\nat least two copies of $\\pi^r$, so it is divisible by $\\pi^{2r}$ and hence\nby $\\pi^{r+1}$ when $r\\ge1$. This produces the next layer.","rline B\\,\\overline e=0$. Equation\n\\eqref{eq:appendix-residual-exactness} then supplies a residual correction\n$\\overline u$ with $\\overline J\\,\\overline u=-\\overline e$. Lift\n$\\overline u$ to $u$ and put $x'=x+\\pi^r u$. The linear part of $F(x')$\nis divisible by $\\pi^{r+1}$. For polynomial systems the remainder contains\nat least two copies of $\\pi^r$, so it is divisible by $\\pi^{2r}$ and hence\nby $\\pi^{r+1}$ when $r\\ge1$. This produces the next layer.","e e=0$. Equation\n\\eqref{eq:appendix-residual-exactness} then supplies a residual correction\n$\\overline u$ with $\\overline J\\,\\overline u=-\\overline e$. Lift\n$\\overline u$ to $u$ and put $x'=x+\\pi^r u$. The linear part of $F(x')$\nis divisible by $\\pi^{r+1}$. For polynomial systems the remainder contains\nat least two copies of $\\pi^r$, so it is divisible by $\\pi^{2r}$ and hence\nby $\\pi^{r+1}$ when $r\\ge1$. This produces the next layer.","Over a suitable truncated principal coefficient ring, iteration terminates\nat $\\pi^N=0$ and produces a root of the original system. The recovered\nv0.25 modules derive the necessary faithful-layer property under their\nstated principal residue and nilpotence hypotheses. They still require\nthe moving identity and exactness. Equality of selected dimensions alone\ndoes not prove either one. A correct dimension count can help prove\nexactness after an inclusion is established, but cannot replace that\ninclusion.","The normalization by $\\pi^r$ deserves care. Writing\n$F(x)=\\pi^r e$ need not determine $e$ uniquely in a ring with torsion.\nIf $e'$ is another choice, then $\\pi^r(e-e')=0$. The faithful leading\nlayer used by the theorem ensures that $e-e'$ has zero residue below the\nterminal cutoff. Thus the residue correction problem is independent of\nthe chosen normalized error. The same hypothesis converts\n$\\pi^r B_xe=0$ into the residue relation needed for exactness. These are\ntwo concrete reasons why the coefficient-ring condition is part of the\ntheorem. Formal cancellation of $\\pi^r$ in a ring where it is a zero\ndivisor would be invalid.","The faithful leading\nlayer used by the theorem ensures that $e-e'$ has zero residue below the\nterminal cutoff. Thus the residue correction problem is independent of\nthe chosen normalized error. The same hypothesis converts\n$\\pi^r B_xe=0$ into the residue relation needed for exactness. These are\ntwo concrete reasons why the coefficient-ring condition is part of the\ntheorem. Formal cancellation of $\\pi^r$ in a ring where it is a zero\ndivisor would be invalid.","There is also an integral warning. In $R=\\mathbb Z/4\\mathbb Z$, the\nsubmodule $C=2R$ contains $2$, and $2$ is divisible by $2$ in $R$. Yet\n$2$ is not divisible by $2$ inside $C$, since $2C=0$. Thus an ambient\ndivisibility statement need not be an internal divisibility statement in a\ncompatible target. An integral splitting cannot be inferred from the\nresidual vector-space picture. The graded argument avoids silently\nassuming such a splitting by working with the normalized residue of each\nactual error.","Let $E$ be a complete normed ring with $\\norm{1}=1$, a submultiplicative\nnorm, and the ultrametric inequality\n\\[\n \\norm{x+y}\\le\\max(\\norm{x},\\norm{y}).\n\\]\nAssume $\\norm A\\le1$. For a nonnegative integer $d$, define\n\\begin{align}\n D_{A,d}(h)&=\\sum_{i=0}^{d-1}A^ihA^{d-1-i},\n \\label{eq:appendix-ordered-derivative}","Let $E$ be a complete normed ring with $\\norm{1}=1$, a submultiplicative\nnorm, and the ultrametric inequality\n\\[\n \\norm{x+y}\\le\\max(\\norm{x},\\norm{y}).\n\\]\nAssume $\\norm A\\le1$. For a nonnegative integer $d$, define\n\\begin{align}\n D_{A,d}(h)&=\\sum_{i=0}^{d-1}A^ihA^{d-1-i},\n \\label{eq:appendix-ordered-derivative}\\\\\n R_{A,d}(h)&=(A+h)^d-A^d-D_{A,d}(h).\n \\label{eq:appendix-ordered-remainder}\n\\end{align}\nFor $d=0$ the sum is empty and both maps are zero. The order of the\nfactors in \\eqref{eq:appendix-ordered-derivative} matters. Replacing this\noperator by $dA^{d-1}h$ requires $A$ and $h$ to commute. The replacement\nis invalid for general matrices.","R_{A,d}(h)&=(A+h)^d-A^d-D_{A,d}(h).\n \\label{eq:appendix-ordered-remainder}\n\\end{align}\nFor $d=0$ the sum is empty and both maps are zero. The order of the\nfactors in \\eqref{eq:appendix-ordered-derivative} matters. Replacing this\noperator by $dA^{d-1}h$ requires $A$ and $h$ to commute. The replacement\nis invalid for general matrices.","The recursive identities are\n\\begin{align*}\n D_{A,d+1}(h)&=D_{A,d}(h)A+A^dh,\\\\\n R_{A,d+1}(h)&=R_{A,d}(h)(A+h)+D_{A,d}(h)h.\n\\end{align*}\nThey follow by multiplying $(A+h)^d$ on the right by $A+h$ and collecting\nthe terms with zero, one, and at least two occurrences of $h$. This\nderivation does not rearrange any noncommuting factors.","R_{A,d+1}(h)&=R_{A,d}(h)(A+h)+D_{A,d}(h)h.\n\\end{align*}\nThey follow by multiplying $(A+h)^d$ on the right by $A+h$ and collecting\nthe terms with zero, one, and at least two occurrences of $h$. This\nderivation does not rearrange any noncommuting factors.","For degree three, the linear term is\n\\[\n D_{A,3}(h)=A^2h+AhA+hA^2,\n\\]\nand the nonlinear remainder is\n\\[\n R_{A,3}(h)=Ah^2+hAh+h^2A+h^3.\n\\]\nThe middle word $hAh$ cannot be absorbed into $Ah^2$ in a matrix ring.\nThe remainder estimate keeps all four words and bounds each by its\nnorm, so it respects the ordered expansion. This also explains why the\nestimate is not an assertion of commutative binomial calculus in\ndisguise. The scalar bound remains uniform because the norm of a finite\nultrametric sum is controlled by its largest summand, rather than by the\nnumber of summands.","\\begin{lemma}[Uniform remainder estimate]\nFor $0\\le r\\le1$ and $\\norm h,\\norm k\\le r$,\n\\[\n \\norm{D_{A,d}(h)}\\le\\norm h,\n \\qquad\n \\norm{R_{A,d}(h)-R_{A,d}(k)}\\le r\\norm{h-k}.\n\\]\nIn particular $\\norm{R_{A,d}(h)}\\le r\\norm h$.\n\\end{lemma}","\\begin{proof}\nThe recursive formula for $D$ shows its additivity. Each term of its\nordered sum has norm at most $\\norm h$, because every power of $A$ has\nnorm at most one. A finite ultrametric sum has norm at most the largest\nterm, proving the first estimate without a factor $d$.","Induct on $d$ for the second estimate. The case $d=0$ is immediate. Write\n$D_d=D_{A,d}$ and $R_d=R_{A,d}$. Subtracting the remainder recursions gives\n\\begin{align*}\n R_{d+1}(h)-R_{d+1}(k)\n ={}&(R_d(h)-R_d(k))(A+h)","Induct on $d$ for the second estimate. The case $d=0$ is immediate. Write\n$D_d=D_{A,d}$ and $R_d=R_{A,d}$. Subtracting the remainder recursions gives\n\\begin{align*}\n R_{d+1}(h)-R_{d+1}(k)\n ={}&(R_d(h)-R_d(k))(A+h)\\\\\n &+R_d(k)(h-k)\\\\\n &+(D_d(h)-D_d(k))h+D_d(k)(h-k).\n\\end{align*}\nThe first term has norm at most $r\\norm{h-k}$ by induction and\n$\\norm{A+h}\\le1$. Taking $h=k$ and the other argument zero in the\ninductive estimate gives $\\norm{R_d(k)}\\le r\\norm k\\le r$, so the\nsecond term has the same bound. Additivity gives\n$\\norm{D_d(h)-D_d(k)}\\le\\norm{h-k}$, which bounds the third term.\nFinally $\\norm{D_d(k)}\\le\\norm k\\le r$ bounds the fourth term.\nUltrametricity bounds their sum by the same maximum. Since $R_{A,d}(0)=0$,\ntaking $k=0$ gives the final claim.\n\\end{proof}","&+(D_d(h)-D_d(k))h+D_d(k)(h-k).\n\\end{align*}\nThe first term has norm at most $r\\norm{h-k}$ by induction and\n$\\norm{A+h}\\le1$. Taking $h=k$ and the other argument zero in the\ninductive estimate gives $\\norm{R_d(k)}\\le r\\norm k\\le r$, so the\nsecond term has the same bound. Additivity gives\n$\\norm{D_d(h)-D_d(k)}\\le\\norm{h-k}$, which bounds the third term.\nFinally $\\norm{D_d(k)}\\le\\norm k\\le r$ bounds the fourth term.\nUltrametricity bounds their sum by the same maximum. Since $R_{A,d}(0)=0$,\ntaking $k=0$ gives the final claim.\n\\end{proof}","r\\norm{h-k}$ by induction and\n$\\norm{A+h}\\le1$. Taking $h=k$ and the other argument zero in the\ninductive estimate gives $\\norm{R_d(k)}\\le r\\norm k\\le r$, so the\nsecond term has the same bound. Additivity gives\n$\\norm{D_d(h)-D_d(k)}\\le\\norm{h-k}$, which bounds the third term.\nFinally $\\norm{D_d(k)}\\le\\norm k\\le r$ bounds the fourth term.\nUltrametricity bounds their sum by the same maximum. Since $R_{A,d}(0)=0$,\ntaking $k=0$ gives the final claim.\n\\end{proof}","on and\n$\\norm{A+h}\\le1$. Taking $h=k$ and the other argument zero in the\ninductive estimate gives $\\norm{R_d(k)}\\le r\\norm k\\le r$, so the\nsecond term has the same bound. Additivity gives\n$\\norm{D_d(h)-D_d(k)}\\le\\norm{h-k}$, which bounds the third term.\nFinally $\\norm{D_d(k)}\\le\\norm k\\le r$ bounds the fourth term.\nUltrametricity bounds their sum by the same maximum. Since $R_{A,d}(0)=0$,\ntaking $k=0$ gives the final claim.\n\\end{proof}","For $E=\\operatorname{Mat}_n(\\mathbb Q_p)$ use the entrywise supremum norm\nand normalize $\\abs p_p=p^{-1}$. Matrix multiplication satisfies\n\\[\n \\abs{(XY)_{ij}}_p\n =\\left|\\sum_tX_{it}Y_{tj}\\right|_p\n \\le\\max_t\\abs{X_{it}}_p\\abs{Y_{tj}}_p\n \\le\\norm X\\norm Y.\n\\]\nTaking the maximum over the finite set of entries proves\nsubmultiplicativity. The same ultrametric calculation removes a matrix\ndimension factor. Completeness is inherited from the finite product of\ncopies of $\\mathbb Q_p$. These are the hypotheses required by the actual\nmatrix application, rather than properties borrowed from a complex matrix\nnorm.","norm\nand normalize $\\abs p_p=p^{-1}$. Matrix multiplication satisfies\n\\[\n \\abs{(XY)_{ij}}_p\n =\\left|\\sum_tX_{it}Y_{tj}\\right|_p\n \\le\\max_t\\abs{X_{it}}_p\\abs{Y_{tj}}_p\n \\le\\norm X\\norm Y.\n\\]\nTaking the maximum over the finite set of entries proves\nsubmultiplicativity. The same ultrametric calculation removes a matrix\ndimension factor. Completeness is inherited from the finite product of\ncopies of $\\mathbb Q_p$. These are the hypotheses required by the actual\nmatrix application, rather than properties borrowed from a complex matrix\nnorm.","Assume that $D=D_{A,d}$ has an additive two-sided inverse $Q$, and that\n$\\norm{Qz}\\le K\\norm z$, with $K\\ge0$. Choose $0\\le r\\le1$ with $Kr<1$. For a\ntarget $U$, set $\\Delta=U-A^d$. The proposed correction map is\n\\begin{equation}\n \\Phi(h)=Q(\\Delta-R_{A,d}(h)).\n \\label{eq:appendix-contraction-map}\n\\end{equation}\nBoth inverse identities will be used. $DQ=1$ turns a fixed point into a\nroot. $QD=1$ proves the necessary condition from an existing root.","\\begin{theorem}[Exact image of a small power ball]\nUnder these hypotheses, there exists a unique $h$ with $\\norm h\\le r$\nand $(A+h)^d=U$ if and only if $\\norm{Q\\Delta}\\le r$.\n\\label{thm:appendix-local-image}\n\\end{theorem}","\\begin{proof}\nSuppose $\\norm{Q\\Delta}\\le r$. On the closed $r$-ball, the remainder\nestimate and ultrametricity give\n\\[\n \\norm{\\Phi(h)}\n \\le\\max\\bigl(\\norm{Q\\Delta},K\\norm{R_{A,d}(h)}\\bigr)\n \\le\\max(r,Kr\\norm h)\\le r.\n\\]\nFor $h,k$ in that ball,\n\\[\n \\norm{\\Phi(h)-\\Phi(k)}\\le Kr\\norm{h-k}.\n\\]\nThe ball is complete and $Kr<1$. The contraction theorem gives a unique\nfixed point $h$. Apply $D$ to its equation to obtain\n$Dh=\\Delta-R_{A,d}(h)$, which is exactly $(A+h)^d=U$.","pose $\\norm{Q\\Delta}\\le r$. On the closed $r$-ball, the remainder\nestimate and ultrametricity give\n\\[\n \\norm{\\Phi(h)}\n \\le\\max\\bigl(\\norm{Q\\Delta},K\\norm{R_{A,d}(h)}\\bigr)\n \\le\\max(r,Kr\\norm h)\\le r.\n\\]\nFor $h,k$ in that ball,\n\\[\n \\norm{\\Phi(h)-\\Phi(k)}\\le Kr\\norm{h-k}.\n\\]\nThe ball is complete and $Kr<1$. The contraction theorem gives a unique\nfixed point $h$. Apply $D$ to its equation to obtain\n$Dh=\\Delta-R_{A,d}(h)$, which is exactly $(A+h)^d=U$.","$r$-ball, the remainder\nestimate and ultrametricity give\n\\[\n \\norm{\\Phi(h)}\n \\le\\max\\bigl(\\norm{Q\\Delta},K\\norm{R_{A,d}(h)}\\bigr)\n \\le\\max(r,Kr\\norm h)\\le r.\n\\]\nFor $h,k$ in that ball,\n\\[\n \\norm{\\Phi(h)-\\Phi(k)}\\le Kr\\norm{h-k}.\n\\]\nThe ball is complete and $Kr<1$. The contraction theorem gives a unique\nfixed point $h$. Apply $D$ to its equation to obtain\n$Dh=\\Delta-R_{A,d}(h)$, which is exactly $(A+h)^d=U$.","Conversely, a root in the ball satisfies $\\Delta=Dh+R_{A,d}(h)$. Applying\n$Q$ gives\n\\[\n Q\\Delta=h+QR_{A,d}(h).\n\\]\nIts norm is at most $r$, since the second summand has norm at most\n$Kr\\norm h\\le r$. Any such root is a fixed point of\n\\eqref{eq:appendix-contraction-map}, so the contraction also proves\nuniqueness among all roots in this ball.\n\\end{proof}","The proof gives a controlled approximation process. Start at $h_0=0$ and\niterate $h_{j+1}=\\Phi(h_j)$. Then $h_1=Q\\Delta$, and\n\\[\n \\norm{h_{j+1}-h_j}\\le (Kr)^j\\norm{h_1}.\n\\]\nFor $t>j$, ultrametricity bounds $\\norm{h_t-h_j}$ by the largest of\nthe intervening increments, which is at most $(Kr)^j\\norm{h_1}$. The\nlimit therefore satisfies the same bound for its distance from $h_j$.\nWith $r=p^{-m}$ and $K=p^s$, every iteration gains at least $m-s$\ndigits in this estimate. This is a quantitative consequence after the\ninverse and target test have been certified. It is not a search over\nunrelated roots at successively deeper levels.","he proof gives a controlled approximation process. Start at $h_0=0$ and\niterate $h_{j+1}=\\Phi(h_j)$. Then $h_1=Q\\Delta$, and\n\\[\n \\norm{h_{j+1}-h_j}\\le (Kr)^j\\norm{h_1}.\n\\]\nFor $t>j$, ultrametricity bounds $\\norm{h_t-h_j}$ by the largest of\nthe intervening increments, which is at most $(Kr)^j\\norm{h_1}$. The\nlimit therefore satisfies the same bound for its distance from $h_j$.\nWith $r=p^{-m}$ and $K=p^s$, every iteration gains at least $m-s$\ndigits in this estimate. This is a quantitative consequence after the\ninverse and target test have been certified. It is not a search over\nunrelated roots at successively deeper levels.","n process. Start at $h_0=0$ and\niterate $h_{j+1}=\\Phi(h_j)$. Then $h_1=Q\\Delta$, and\n\\[\n \\norm{h_{j+1}-h_j}\\le (Kr)^j\\norm{h_1}.\n\\]\nFor $t>j$, ultrametricity bounds $\\norm{h_t-h_j}$ by the largest of\nthe intervening increments, which is at most $(Kr)^j\\norm{h_1}$. The\nlimit therefore satisfies the same bound for its distance from $h_j$.\nWith $r=p^{-m}$ and $K=p^s$, every iteration gains at least $m-s$\ndigits in this estimate. This is a quantitative consequence after the\ninverse and target test have been certified. It is not a search over\nunrelated roots at successively deeper levels.","For integral matrices suppose the inverse has the certified form\n\\[\n Q=p^{-s}B,\\qquad s\\in\\mathbb N,\n\\]\nwhere $B$ is an integral linear operator on the matrix entries. In a fixed\nordering of the $n^2$ entries it is an $n^2$ by $n^2$ matrix over\n$\\mathbb Z_p$. This is an operator on the matrix space, not ordinary\nleft multiplication by an $n$ by $n$ matrix. Integral coefficients imply\n$\\norm{Bz}\\le\\norm z$, so $K=p^s$ is a valid bound.","atrices suppose the inverse has the certified form\n\\[\n Q=p^{-s}B,\\qquad s\\in\\mathbb N,\n\\]\nwhere $B$ is an integral linear operator on the matrix entries. In a fixed\nordering of the $n^2$ entries it is an $n^2$ by $n^2$ matrix over\n$\\mathbb Z_p$. This is an operator on the matrix space, not ordinary\nleft multiplication by an $n$ by $n$ matrix. Integral coefficients imply\n$\\norm{Bz}\\le\\norm z$, so $K=p^s$ is a valid bound.","There is a finite exact certificate for the inverse. For every matrix unit\n$E_{ij}$ check\n\\[\n QD(E_{ij})=E_{ij},\\qquad DQ(E_{ij})=E_{ij}.\n\\]\nLinearity and the matrix-unit basis extend both identities to every matrix\nover $\\mathbb Q_p$. These equalities are exact in the coefficient field.\nThey are not merely congruences modulo $p$. In particular, an operator\nsingular over $\\mathbb Q_p$ cannot satisfy them.","Take $r=p^{-m}$ with $m>s$. Then $Kr=p^{s-m}<1$, and\n\\begin{align}\n \\norm{Q(U-A^d)}\\le p^{-m}\n &\\Longleftrightarrow \\norm{B(U-A^d)}\\le p^{-(m+s)}\\notag","Take $r=p^{-m}$ with $m>s$. Then $Kr=p^{s-m}<1$, and\n\\begin{align}\n \\norm{Q(U-A^d)}\\le p^{-m}\n &\\Longleftrightarrow \\norm{B(U-A^d)}\\le p^{-(m+s)}\\notag\\\\\n &\\Longleftrightarrow\n B\\operatorname{vec}(U-A^d)=0\\pmod {p^{m+s}}.\n \\label{eq:appendix-finite-criterion}\n\\end{align}\nThe last congruence is entrywise. It depends only on the reductions of the\ntarget, center, and integral operator modulo $p^{m+s}$. The theorem\ntherefore decides the existence and uniqueness of a root in\n$A+p^m\\operatorname{Mat}_n(\\mathbb Z_p)$ by one finite target test, once\nthe exact inverse certificate is supplied.","&\\Longleftrightarrow\n B\\operatorname{vec}(U-A^d)=0\\pmod {p^{m+s}}.\n \\label{eq:appendix-finite-criterion}\n\\end{align}\nThe last congruence is entrywise. It depends only on the reductions of the\ntarget, center, and integral operator modulo $p^{m+s}$. The theorem\ntherefore decides the existence and uniqueness of a root in\n$A+p^m\\operatorname{Mat}_n(\\mathbb Z_p)$ by one finite target test, once\nthe exact inverse certificate is supplied.","The root constructed over $\\mathbb Q_p$ is integral. Its center is\nintegral, and every correction entry has norm at most $p^{-m}\\le1$.\nIf $d>0$ and $U$ is a unit, the root $F$ is also a unit. Indeed, $F$\ncommutes with $U=F^d$ and hence with $U^{-1}$, and\n$F^{d-1}U^{-1}$ is a two-sided inverse of $F$. Thus the unit version has\nan actual output in $\\operatorname{GL}_n(\\mathbb Z_p)$.","The value of $s$ measures denominators of the whole inverse derivative.\nIt is not automatically $v_p(d)$. The proof makes no statement about\nshallow balls with $m\\le s$, where its contraction estimate is insufficient.\nIt also does not provide automatic certificate discovery for arbitrary\n$p$-adic input. Exact rational examples can supply certificates by ordinary\nlinear algebra, but a general certified implementation is an additional\ntask.","Work over $\\mathbb Z_2$ and take\n\\[\n A=\\begin{pmatrix}1&1","Work over $\\mathbb Z_2$ and take\n\\[\n A=\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix},\\qquad d=2.\n\\]\nFor $X=\\begin{pmatrix}a&b","0&1\\end{pmatrix},\\qquad d=2.\n\\]\nFor $X=\\begin{pmatrix}a&b\\\\c&e\\end{pmatrix}$, the ordered derivative is\n\\[\n D_{A,2}(X)=AX+XA\n =\\begin{pmatrix}2a+c&a+2b+e","c&e\\end{pmatrix}$, the ordered derivative is\n\\[\n D_{A,2}(X)=AX+XA\n =\\begin{pmatrix}2a+c&a+2b+e\\\\2c&c+2e\\end{pmatrix}.\n\\]\nUsing the entry order $(a,b,c,e)$, its matrix and a scaled inverse are\n\\[\n J=\\begin{pmatrix}\n 2&0&1&0","2c&c+2e\\end{pmatrix}.\n\\]\nUsing the entry order $(a,b,c,e)$, its matrix and a scaled inverse are\n\\[\n J=\\begin{pmatrix}\n 2&0&1&0","2c&c+2e\\end{pmatrix}.\n\\]\nUsing the entry order $(a,b,c,e)$, its matrix and a scaled inverse are\n\\[\n J=\\begin{pmatrix}\n 2&0&1&0\\\\1&2&0&1\\\\0&0&2&0\\\\0&0&1&2\n \\end{pmatrix},\\qquad\n B=\\begin{pmatrix}\n 2&0&-1&0\\\\-1&2&1&-1\\\\0&0&2&0\\\\0&0&-1&2\n \\end{pmatrix}.\n\\]\nDirect multiplication gives $BJ=JB=4I_4$. Therefore $Q=\\tfrac14 B$\nis the exact inverse and $s=2$. The entry $\\tfrac14$ in this inverse\nprevents clearing its denominators with only one factor of $2$.","0&0&-1&2\n \\end{pmatrix}.\n\\]\nDirect multiplication gives $BJ=JB=4I_4$. Therefore $Q=\\tfrac14 B$\nis the exact inverse and $s=2$. The entry $\\tfrac14$ in this inverse\nprevents clearing its denominators with only one factor of $2$.","The residual derivative is singular. In characteristic two,\n$D_{A,2}(I_2)=2A=0$, while $I_2\\ne0$. Nevertheless $J$ is invertible\nover $\\mathbb Q_2$, with determinant $16$. Residual singularity and\nsingularity over the fraction field are different tests.","Choose $m=3$, so the local root ball is $A+8\\operatorname{Mat}_2(\\mathbb Z_2)$\nand the finite target test is modulo $2^{m+s}=32$. Consider\n\\[\n F=\\begin{pmatrix}1&1","Choose $m=3$, so the local root ball is $A+8\\operatorname{Mat}_2(\\mathbb Z_2)$\nand the finite target test is modulo $2^{m+s}=32$. Consider\n\\[\n F=\\begin{pmatrix}1&1\\\\8&1\\end{pmatrix},\\qquad\n U=F^2=\\begin{pmatrix}9&2\\\\16&9\\end{pmatrix}.\n\\]\nHere\n\\[\n AF=\\begin{pmatrix}9&2","16&9\\end{pmatrix}.\n\\]\nHere\n\\[\n AF=\\begin{pmatrix}9&2\\\\8&1\\end{pmatrix},\\qquad\n FA=\\begin{pmatrix}1&2\\\\8&9\\end{pmatrix},\n\\]\nso the root does not commute with the center. Since\n$A^2=\\begin{pmatrix}1&2","8&9\\end{pmatrix},\n\\]\nso the root does not commute with the center. Since\n$A^2=\\begin{pmatrix}1&2\\\\0&1\\end{pmatrix}$, the target difference has\nentry vector $(8,0,16,8)$. Applying $B$ gives\n\\[\n B\\operatorname{vec}(U-A^2)=(0,0,32,0).\n\\]\nThe congruence modulo $32$ passes. The theorem supplies uniqueness in the\nspecified ball, beyond the direct multiplication that exhibits this root.","0&1\\end{pmatrix}$, the target difference has\nentry vector $(8,0,16,8)$. Applying $B$ gives\n\\[\n B\\operatorname{vec}(U-A^2)=(0,0,32,0).\n\\]\nThe congruence modulo $32$ passes. The theorem supplies uniqueness in the\nspecified ball, beyond the direct multiplication that exhibits this root.","For a negative control, take\n\\[\n U_{\\mathrm{bad}}=\\begin{pmatrix}1&10\\\\0&1\\end{pmatrix}.\n\\]\nThe difference from $A^2$ is $(0,8,0,0)$ in entry coordinates, and $B$\nsends it to $(0,16,0,0)$. This is nonzero modulo $32$, so no square root\nof $U_{\\mathrm{bad}}$ lies in that ball. This is a failure for the\nspecified prefix. It is not a claim that the target has no square root\nanywhere in a larger coefficient field.","0&1\\end{pmatrix}.\n\\]\nThe difference from $A^2$ is $(0,8,0,0)$ in entry coordinates, and $B$\nsends it to $(0,16,0,0)$. This is nonzero modulo $32$, so no square root\nof $U_{\\mathrm{bad}}$ lies in that ball. This is a failure for the\nspecified prefix. It is not a claim that the target has no square root\nanywhere in a larger coefficient field.","The recovered exact checker compares the finite image with the finite\ncriterion. Its unipotent square case tests $256$ corrections at the\nstated horizon and $4096$ corrections one level deeper. Both give the same\n$16$ target residues as the criterion. These are bounded corroborating\ncalculations. Infinite existence and uniqueness follow from the general\ncontraction proof and its exact hypotheses.","There is a genuinely singular contrast. Let\n$A_0=\\operatorname{diag}(1,-1)$ over $\\mathbb Q_p$. Then\n$D_{A_0,2}(E_{12})=0$, although $E_{12}\\ne0$. Thus this derivative has\nno inverse over the fraction field, including when $p$ is odd and $2$ is\na unit. The certificate theorem cannot cover this center. A universal\nclaim that invertible centers always have invertible power derivatives\nwould be false.","Let $X_r$ be finite sets with genuine reduction maps\n$\\pi_{r,t}:X_t\\to X_r$ for $r\\le t$. Assume identity and composition\nlaws for these maps. Let $P_r\\subseteq X_r$ be solution sets, and assume\nreduction preserves solutions. Then\n\\begin{equation}\n \\left(\\forall r,\\ P_r\\ne\\varnothing\\right)\n \\Longleftrightarrow\n \\left(\\exists (x_r)_r,\\ x_r\\in P_r\\text{ and }\n \\pi_{r,t}(x_t)=x_r\\text{ for }r\\le t\\right).\n \\label{eq:appendix-branch-criterion}\n\\end{equation}\nThe nontrivial implication is finite branch compactness. For each depth,\na solution at that depth gives a finite path of solutions at all earlier\ndepths. There are arbitrarily long paths. The resulting tree has finite\nbranching because every level","Let $X_r$ be finite sets with genuine reduction maps\n$\\pi_{r,t}:X_t\\to X_r$ for $r\\le t$. Assume identity and composition\nlaws for these maps. Let $P_r\\subseteq X_r$ be solution sets, and assume\nreduction preserves solutions. Then\n\\begin{equation}\n \\left(\\forall r,\\ P_r\\ne\\varnothing\\right)\n \\Longleftrightarrow\n \\left(\\exists (x_r)_r,\\ x_r\\in P_r\\text{ and }\n \\pi_{r,t}(x_t)=x_r\\text{ for }r\\le t\\right).\n \\label{eq:appendix-branch-criterion}\n\\end{equation}\nThe nontrivial implication is finite branch compactness. For each depth,\na solution at that depth gives a finite path of solutions at all earlier\ndepths. There are arbitrarily long paths. The resulting tree has finite\nbranching because every level is finite. An infinite branch follows from\nthe finite-branch form","Let $X_r$ be finite sets with genuine reduction maps\n$\\pi_{r,t}:X_t\\to X_r$ for $r\\le t$. Assume identity and composition\nlaws for these maps. Let $P_r\\subseteq X_r$ be solution sets, and assume\nreduction preserves solutions. Then\n\\begin{equation}\n \\left(\\forall r,\\ P_r\\ne\\varnothing\\right)\n \\Longleftrightarrow\n \\left(\\exists (x_r)_r,\\ x_r\\in P_r\\text{ and }\n \\pi_{r,t}(x_t)=x_r\\text{ for }r\\le t\\right).\n \\label{eq:appendix-branch-criterion}\n\\end{equation}\nThe nontrivial implication is finite branch compactness. For each depth,\na solution at that depth gives a finite path of solutions at all earlier\ndepths. There are arbitrarily long paths. The resulting tree has finite\nbranching because every level is finite. An infinite branch follows from\nthe finite-branch form of K\\\"onig's","Let $X_r$ be finite sets with genuine reduction maps\n$\\pi_{r,t}:X_t\\to X_r$ for $r\\le t$. Assume identity and composition\nlaws for these maps. Let $P_r\\subseteq X_r$ be solution sets, and assume\nreduction preserves solutions. Then\n\\begin{equation}\n \\left(\\forall r,\\ P_r\\ne\\varnothing\\right)\n \\Longleftrightarrow\n \\left(\\exists (x_r)_r,\\ x_r\\in P_r\\text{ and }\n \\pi_{r,t}(x_t)=x_r\\text{ for }r\\le t\\right).\n \\label{eq:appendix-branch-criterion}\n\\end{equation}\nThe nontrivial implication is finite branch compactness. For each depth,\na solution at that depth gives a finite path of solutions at all earlier\ndepths. There are arbitrarily long paths. The resulting tree has finite\nbranching because every level is finite. An infinite branch follows from\nthe finite-branch form of K\\\"onig's lemma. The reduction maps on the\nsolution sets need not be surjective. Compac","Let $X_r$ be finite sets with genuine reduction maps\n$\\pi_{r,t}:X_t\\to X_r$ for $r\\le t$. Assume identity and composition\nlaws for these maps. Let $P_r\\subseteq X_r$ be solution sets, and assume\nreduction preserves solutions. Then\n\\begin{equation}\n \\left(\\forall r,\\ P_r\\ne\\varnothing\\right)\n \\Longleftrightarrow\n \\left(\\exists (x_r)_r,\\ x_r\\in P_r\\text{ and }\n \\pi_{r,t}(x_t)=x_r\\text{ for }r\\le t\\right).\n \\label{eq:appendix-branch-criterion}\n\\end{equation}\nThe nontrivial implication is finite branch compactness. For each depth,\na solution at that depth gives a finite path of solutions at all earlier\ndepths. There are arbitrarily long paths. The resulting tree has finite\nbranching because every level is finite. An infinite branch follows from\nthe finite-branch form of K\\\"onig's lemma. The reduction maps on the\nsolution sets need not be surjective. Compactness selects persistent\nchoices instead of extending a previously chosen node at every step.","This matters for simultaneous matrix prescriptions. At level $r$, a node\ncan contain a shared matrix $F_r$ together with one local conjugator\n$z_{i,r}$ for each member of a finite family. The equations are\n\\[\n z_{i,r}F_r^{d_i}z_{i,r}^{-1}=U_i\\pmod {p^r},\n\\]\nwith the prescribed prefix of $F_r$ and prescribed residual identity for\neach conjugator. These are a single joint predicate on a finite tuple.\nThe arithmetic equations and prefixes are preserved by reduction.\nCompactness therefore produces compatible towers for the shared matrix\nand for all local conjugators, provided this joint problem is nonempty at\nevery depth.","An individual solution for each prescription is weaker. Even the scalar\nrequirements $F=1$ and $F=-1$ modulo an odd prime have solutions\nseparately, but have no common solution. Likewise, solving every member\nof a family at each depth with a different shared matrix does not establish\nthe premise of \\eqref{eq:appendix-branch-criterion}. The compatible branch\nmust come from one family of jointly satisfiable nodes.","Nor does a successful shallow node necessarily extend. For example, the\nscalar prefix $F=4$ modulo $9$ satisfies $F^3=1$ there. Its lifts modulo\n$27$ are $4,13,22$, all with cube $10$. That branch stops at the next\nlevel. Merely checking a finite number of successful levels gives no\nall-depth conclusion without another theorem.","sely, for a downward-closed tower of finite solution sets indexed\nby all natural-number levels, absence of an infinite branch implies that\nsome finite level is empty. This gives a finite\nobstruction witness in principle. It supplies no uniform computable bound\non the depth of that witness. The matrix inverse theorem above provides\nan explicit horizon only in its certified nonsingular sector with $m>s$.\nIt removes the all-depth premise for one local power problem in that sector.\nIt does not remove the joint all-depth premise for arbitrary simultaneous\nprescriptions.","Finally, compatible integral matrix residues determine an actual matrix\nover $\\mathbb Z_p$ entry by entry. Compatible inverse residues determine\nits inverse when the nodes are units. Equality of all reductions then\nimplies equality of the integral matrices. These reconstruction steps turn\nthe selected branch into a genuine integral solution. The remaining\narithmetic attachment is not part of that reconstruction.","\\begin{proposition}[The two prime cyclic cases]\\label{prop:worked_clifford}\nLet \\(G\\) be finite, let \\(H\\triangleleft G\\) with \\(G/H\\cong C_p\\), and let \\(V\\) be irreducible over \\(\\CC\\). Exactly one of the following occurs:\n\\begin{enumerate}\n\\item \\(V|_H\\) is irreducible and invariant under \\(G\\), and \\(V\\) is one of its \\(p\\) quotient character extensions.\n\\item \\(V|_H\\) is a sum of \\(p\\) pairwise nonisomorphic conjugate irreducible representations, and \\(V\\) is induced from any one of them.\n\\end{enumerate}\n\\end{proposition}","ked_clifford}\nLet \\(G\\) be finite, let \\(H\\triangleleft G\\) with \\(G/H\\cong C_p\\), and let \\(V\\) be irreducible over \\(\\CC\\). Exactly one of the following occurs:\n\\begin{enumerate}\n\\item \\(V|_H\\) is irreducible and invariant under \\(G\\), and \\(V\\) is one of its \\(p\\) quotient character extensions.\n\\item \\(V|_H\\) is a sum of \\(p\\) pairwise nonisomorphic conjugate irreducible representations, and \\(V\\) is induced from any one of them.\n\\end{enumerate}\n\\end{proposition}","\\begin{proof}\nComplete reducibility for the finite subgroup gives an isotypic decomposition of \\(V|_H\\). The group \\(G\\) permutes these isotypic spaces by conjugating their types. More than one orbit would give a proper nonzero \\(G\\) stable direct sum, contrary to irreducibility. Thus there is one orbit. Since conjugation by \\(H\\) fixes isomorphism classes of \\(H\\) representations, the permutation action factors through \\(C_p\\). Its transitive orbit has size one or \\(p\\).","ucibility for the finite subgroup gives an isotypic decomposition of \\(V|_H\\). The group \\(G\\) permutes these isotypic spaces by conjugating their types. More than one orbit would give a proper nonzero \\(G\\) stable direct sum, contrary to irreducibility. Thus there is one orbit. Since conjugation by \\(H\\) fixes isomorphism classes of \\(H\\) representations, the permutation action factors through \\(C_p\\). Its transitive orbit has size one or \\(p\\).","typic decomposition of \\(V|_H\\). The group \\(G\\) permutes these isotypic spaces by conjugating their types. More than one orbit would give a proper nonzero \\(G\\) stable direct sum, contrary to irreducibility. Thus there is one orbit. Since conjugation by \\(H\\) fixes isomorphism classes of \\(H\\) representations, the permutation action factors through \\(C_p\\). Its transitive orbit has size one or \\(p\\).","In the size \\(p\\) case, choose one isotypic space \\(U\\). Its translates are pairwise distinct isotypic spaces, with direct sum \\(V\\). The stabilizer of its type is exactly \\(H\\). If \\(U\\) had a proper nonzero \\(H\\) submodule, the direct sum of its \\(p\\) translates would be a proper nonzero \\(G\\) submodule of \\(V\\). Hence \\(U\\) is irreducible. The natural map from \\(\\Ind_H^G U\\) to \\(V\\), sending each coset copy to its translated space, is an isomorphism. Its restriction is precisely the stated orbit sum.","In the size one case, write \\(V|_H\\cong\\tau\\otimes M\\), where \\(H\\) acts trivially on the multiplicity space \\(M\\). The stable type \\(\\tau\\) extends by \\cref{lem:stable_extension}. Remove that extension action from the action on \\(V\\). The remaining action on \\(M\\) factors through \\(C_p\\). If \\(M\\) had a proper quotient stable subspace, its tensor product with \\(\\tau\\) would give a proper \\(G\\) submodule of \\(V\\). Thus \\(M\\) is irreducible as a complex representation of \\(C_p\\). Every such irreducible is one dimensional. Therefore \\(V|_H\\cong\\tau\\), with multiplicity one, and the remaining quotient character is its extension label. The torsor theorem gives the \\(p\\) possible extensions. This proves both alternatives and excludes an intermedia","In the size one case, write \\(V|_H\\cong\\tau\\otimes M\\), where \\(H\\) acts trivially on the multiplicity space \\(M\\). The stable type \\(\\tau\\) extends by \\cref{lem:stable_extension}. Remove that extension action from the action on \\(V\\). The remaining action on \\(M\\) factors through \\(C_p\\). If \\(M\\) had a proper quotient stable subspace, its tensor product with \\(\\tau\\) would give a proper \\(G\\) submodule of \\(V\\). Thus \\(M\\) is irreducible as a complex representation of \\(C_p\\). Every such irreducible is one dimensional. Therefore \\(V|_H\\cong\\tau\\), with multiplicity one, and the remaining quotient character is its extension label. The torsor theorem gives the \\(p\\) possible extensions. This proves both alternatives and excludes an intermediate multiplici","In the size one case, write \\(V|_H\\cong\\tau\\otimes M\\), where \\(H\\) acts trivially on the multiplicity space \\(M\\). The stable type \\(\\tau\\) extends by \\cref{lem:stable_extension}. Remove that extension action from the action on \\(V\\). The remaining action on \\(M\\) factors through \\(C_p\\). If \\(M\\) had a proper quotient stable subspace, its tensor product with \\(\\tau\\) would give a proper \\(G\\) submodule of \\(V\\). Thus \\(M\\) is irreducible as a complex representation of \\(C_p\\). Every such irreducible is one dimensional. Therefore \\(V|_H\\cong\\tau\\), with multiplicity one, and the remaining quotient character is its extension label. The torsor theorem gives the \\(p\\) possible extensions. This proves both alternatives and excludes an intermediate multiplicity case.\n\\end{proof}","ne case, write \\(V|_H\\cong\\tau\\otimes M\\), where \\(H\\) acts trivially on the multiplicity space \\(M\\). The stable type \\(\\tau\\) extends by \\cref{lem:stable_extension}. Remove that extension action from the action on \\(V\\). The remaining action on \\(M\\) factors through \\(C_p\\). If \\(M\\) had a proper quotient stable subspace, its tensor product with \\(\\tau\\) would give a proper \\(G\\) submodule of \\(V\\). Thus \\(M\\) is irreducible as a complex representation of \\(C_p\\). Every such irreducible is one dimensional. Therefore \\(V|_H\\cong\\tau\\), with multiplicity one, and the remaining quotient character is its extension label. The torsor theorem gives the \\(p\\) possible extensions. This proves both alternatives and excludes an intermediate multiplicity case.\n\\end{proof}","te \\(V|_H\\cong\\tau\\otimes M\\), where \\(H\\) acts trivially on the multiplicity space \\(M\\). The stable type \\(\\tau\\) extends by \\cref{lem:stable_extension}. Remove that extension action from the action on \\(V\\). The remaining action on \\(M\\) factors through \\(C_p\\). If \\(M\\) had a proper quotient stable subspace, its tensor product with \\(\\tau\\) would give a proper \\(G\\) submodule of \\(V\\). Thus \\(M\\) is irreducible as a complex representation of \\(C_p\\). Every such irreducible is one dimensional. Therefore \\(V|_H\\cong\\tau\\), with multiplicity one, and the remaining quotient character is its extension label. The torsor theorem gives the \\(p\\) possible extensions. This proves both alternatives and excludes an intermediate multiplicity case.\n\\end{proof}","ially on the multiplicity space \\(M\\). The stable type \\(\\tau\\) extends by \\cref{lem:stable_extension}. Remove that extension action from the action on \\(V\\). The remaining action on \\(M\\) factors through \\(C_p\\). If \\(M\\) had a proper quotient stable subspace, its tensor product with \\(\\tau\\) would give a proper \\(G\\) submodule of \\(V\\). Thus \\(M\\) is irreducible as a complex representation of \\(C_p\\). Every such irreducible is one dimensional. Therefore \\(V|_H\\cong\\tau\\), with multiplicity one, and the remaining quotient character is its extension label. The torsor theorem gives the \\(p\\) possible extensions. This proves both alternatives and excludes an intermediate multiplicity case.\n\\end{proof}",") extends by \\cref{lem:stable_extension}. Remove that extension action from the action on \\(V\\). The remaining action on \\(M\\) factors through \\(C_p\\). If \\(M\\) had a proper quotient stable subspace, its tensor product with \\(\\tau\\) would give a proper \\(G\\) submodule of \\(V\\). Thus \\(M\\) is irreducible as a complex representation of \\(C_p\\). Every such irreducible is one dimensional. Therefore \\(V|_H\\cong\\tau\\), with multiplicity one, and the remaining quotient character is its extension label. The torsor theorem gives the \\(p\\) possible extensions. This proves both alternatives and excludes an intermediate multiplicity case.\n\\end{proof}","rs through \\(C_p\\). If \\(M\\) had a proper quotient stable subspace, its tensor product with \\(\\tau\\) would give a proper \\(G\\) submodule of \\(V\\). Thus \\(M\\) is irreducible as a complex representation of \\(C_p\\). Every such irreducible is one dimensional. Therefore \\(V|_H\\cong\\tau\\), with multiplicity one, and the remaining quotient character is its extension label. The torsor theorem gives the \\(p\\) possible extensions. This proves both alternatives and excludes an intermediate multiplicity case.\n\\end{proof}","For \\(S_3\\) and \\(H=A_3\\), the standard two dimensional representation lies in the moving case. A three cycle has the two nontrivial cube roots as eigenvalues. Restriction is the sum of the two nontrivial \\(C_3\\) characters. Induction from either one gives the standard representation. Its sign twist is isomorphic to itself. This self twist is allowed precisely because the restricted representation is reducible. It does not contradict freeness of the torsor whose restriction is irreducible.","Let \\(\\omega\\) be a primitive cube root of unity and put\n\\[\n A=\\operatorname{diag}(1,1,\\omega).\n\\]\nExpanding directly gives\n\\[\n \\det(TI-A)\n =T^3-(2+\\omega)T^2+(1+2\\omega)T-\\omega.\n\\]\nThe three elementary symmetric coefficients are\n\\[\n e_1=2+\\omega,\\qquad e_2=1+2\\omega,\\qquad e_3=\\omega.\n\\]\nFor scalar twist \\(\\omega^jA\\), they become\n\\[\n \\omega^j(2+\\omega),\\qquad\n \\omega^{2j}(1+2\\omega),\\qquad\n \\omega^{3j}\\omega=\\omega.\n\\]\nThe determinant is constant for \\(j=0,1,2\\), but the first coefficient distinguishes the three labels because \\(2+\\omega\\ne0\\). Thus rank divisibility causes blindness of the determinant, rather than blindness of every coefficient.","A=\\operatorname{diag}(1,1,\\omega).\n\\]\nExpanding directly gives\n\\[\n \\det(TI-A)\n =T^3-(2+\\omega)T^2+(1+2\\omega)T-\\omega.\n\\]\nThe three elementary symmetric coefficients are\n\\[\n e_1=2+\\omega,\\qquad e_2=1+2\\omega,\\qquad e_3=\\omega.\n\\]\nFor scalar twist \\(\\omega^jA\\), they become\n\\[\n \\omega^j(2+\\omega),\\qquad\n \\omega^{2j}(1+2\\omega),\\qquad\n \\omega^{3j}\\omega=\\omega.\n\\]\nThe determinant is constant for \\(j=0,1,2\\), but the first coefficient distinguishes the three labels because \\(2+\\omega\\ne0\\). Thus rank divisibility causes blindness of the determinant, rather than blindness of every coefficient.","=T^3-(2+\\omega)T^2+(1+2\\omega)T-\\omega.\n\\]\nThe three elementary symmetric coefficients are\n\\[\n e_1=2+\\omega,\\qquad e_2=1+2\\omega,\\qquad e_3=\\omega.\n\\]\nFor scalar twist \\(\\omega^jA\\), they become\n\\[\n \\omega^j(2+\\omega),\\qquad\n \\omega^{2j}(1+2\\omega),\\qquad\n \\omega^{3j}\\omega=\\omega.\n\\]\nThe determinant is constant for \\(j=0,1,2\\), but the first coefficient distinguishes the three labels because \\(2+\\omega\\ne0\\). Thus rank divisibility causes blindness of the determinant, rather than blindness of every coefficient.","Now compare the balanced matrix\n\\[\n A_{\\mathrm{bal}}=\\operatorname{diag}(1,\\omega,\\omega^2).\n\\]\nIts polynomial is \\(T^3-1\\). Multiplying all eigenvalues by \\(\\omega\\) merely permutes their multiset. Every lower coefficient vanishes, and the determinant is fixed. Its full characteristic polynomial is therefore blind to the label. This is an actual local rotational stabilizer. A claim that one characteristic polynomial always detects a quotient twist would fail on this example.","At quotient prime three and rank two, the determinant changes by \\(\\omega^{2j}\\). If a required determinant ratio is \\(\\omega^b\\), the unique candidate satisfies \\(2j=b\\) modulo three, so \\(j=2b\\). This solves a finite selection problem. It still does not show that every other coefficient of the candidate matches the target.","The all local comparison includes infinity. For a real place with complex quadratic extension, the quotient is \\(C_2\\). Let the target finite parameter have sign multiplicities \\((3,0)\\), and let a second parameter have multiplicities \\((1,2)\\). Both have determinant sign positive. Twisting the second by sign gives \\((2,1)\\), which still does not equal \\((3,0)\\). Its exact attachment set is empty. This is a finite local model showing why determinant selection, even with quotient prime coprime to rank, does not discharge the remaining infinity condition.","Take \\(G=C_2\\times C_3\\), let \\(H=C_2\\), and let \\(\\xi\\) be the quotient character taking a generator of \\(C_3\\) to \\(\\omega\\). Let \\(\\tau\\) be the nontrivial character of \\(H\\), extended with trivial quotient action. Consider a semisimple representation with spectrum\n\\[\n P_{\\tau,V}(z)=2+z.\n\\]\nIts quotient multiplicity vector is \\((2,1,0)\\). Twisting by \\(\\xi\\) gives\n\\[\n zP_{\\tau,V}(z)=2z+z^2,\n\\]\nwhose vector is \\((0,2,1)\\). Twisting by \\(\\xi^2\\) gives \\((1,0,2)\\). These are ordered vectors. Knowing that each has total multiplicity three does not retain the extension label.","C_3\\), let \\(H=C_2\\), and let \\(\\xi\\) be the quotient character taking a generator of \\(C_3\\) to \\(\\omega\\). Let \\(\\tau\\) be the nontrivial character of \\(H\\), extended with trivial quotient action. Consider a semisimple representation with spectrum\n\\[\n P_{\\tau,V}(z)=2+z.\n\\]\nIts quotient multiplicity vector is \\((2,1,0)\\). Twisting by \\(\\xi\\) gives\n\\[\n zP_{\\tau,V}(z)=2z+z^2,\n\\]\nwhose vector is \\((0,2,1)\\). Twisting by \\(\\xi^2\\) gives \\((1,0,2)\\). These are ordered vectors. Knowing that each has total multiplicity three does not retain the extension label.","d let \\(\\xi\\) be the quotient character taking a generator of \\(C_3\\) to \\(\\omega\\). Let \\(\\tau\\) be the nontrivial character of \\(H\\), extended with trivial quotient action. Consider a semisimple representation with spectrum\n\\[\n P_{\\tau,V}(z)=2+z.\n\\]\nIts quotient multiplicity vector is \\((2,1,0)\\). Twisting by \\(\\xi\\) gives\n\\[\n zP_{\\tau,V}(z)=2z+z^2,\n\\]\nwhose vector is \\((0,2,1)\\). Twisting by \\(\\xi^2\\) gives \\((1,0,2)\\). These are ordered vectors. Knowing that each has total multiplicity three does not retain the extension label.","the quotient character taking a generator of \\(C_3\\) to \\(\\omega\\). Let \\(\\tau\\) be the nontrivial character of \\(H\\), extended with trivial quotient action. Consider a semisimple representation with spectrum\n\\[\n P_{\\tau,V}(z)=2+z.\n\\]\nIts quotient multiplicity vector is \\((2,1,0)\\). Twisting by \\(\\xi\\) gives\n\\[\n zP_{\\tau,V}(z)=2z+z^2,\n\\]\nwhose vector is \\((0,2,1)\\). Twisting by \\(\\xi^2\\) gives \\((1,0,2)\\). These are ordered vectors. Knowing that each has total multiplicity three does not retain the extension label.","Suppose a second stable \\(H\\) type \\(\\sigma\\) occurs with spectrum one. If the target has spectra \\(2z+z^2\\) for \\(\\tau\\) and \\(z^2\\) for \\(\\sigma\\), the first block demands shift one and the second demands shift two. The two restrictions to \\(H\\) have the same multiplicities. Each block separately is a quotient twist of the matching source block. The full representation is not one quotient twist, because the required shifts disagree. This is the reason the spectrum criterion quantifies over one common \\(j\\), rather than an independent \\(j_\\tau\\) for each type.","or \\(\\tau\\) and \\(z^2\\) for \\(\\sigma\\), the first block demands shift one and the second demands shift two. The two restrictions to \\(H\\) have the same multiplicities. Each block separately is a quotient twist of the matching source block. The full representation is not one quotient twist, because the required shifts disagree. This is the reason the spectrum criterion quantifies over one common \\(j\\), rather than an independent \\(j_\\tau\\) for each type.","ma\\), the first block demands shift one and the second demands shift two. The two restrictions to \\(H\\) have the same multiplicities. Each block separately is a quotient twist of the matching source block. The full representation is not one quotient twist, because the required shifts disagree. This is the reason the spectrum criterion quantifies over one common \\(j\\), rather than an independent \\(j_\\tau\\) for each type.","For a balanced multiplicity spectrum \\(1+z+z^2\\), both nonzero Fourier modes vanish:\n\\[\n 1+\\omega+\\omega^2=0,\\qquad\n 1+\\omega^2+\\omega^4=0.\n\\]\nThe spectrum \\(3\\) has the same total multiplicity, but both nonzero modes equal three. Comparing only ratios for which the source mode is nonzero would omit both decisive comparisons. The zero support clause in \\cref{def:local_spectral_obstruction} is therefore necessary. A zero source mode requires a zero target mode. It is not an instruction to skip that mode.","Changing the reference extension of \\(\\tau\\) relabels both spectra by the same rotation. The attachment set does not change. Likewise changing the chosen generator of the quotient rescales all labels by an invertible element of \\(\\ZZ/p\\ZZ\\). These coordinate choices affect the displayed polynomials, but not the existence of an isomorphism after one common quotient twist.","For prime cyclic quotients a nonempty exact alignment set is either a singleton or all labels. Hence two nonempty exact singleton conditions \\(\\{1\\}\\) and \\(\\{2\\}\\) are incompatible, while \\(\\{1\\}\\) together with all labels is compatible. Coarse coefficient filters may produce other finite subsets, but those are not automatically exact extension spectrum sets. Keeping that distinction prevents a visual or computational filter from being assigned a stronger meaning than its construction supports.","Let\n\\[\n \\Gamma=\\{\\operatorname{diag}(\\omega^a,\\omega^{-a}):a=0,1,2\\}.\n\\]\nThe trace alphabet is \\(\\{2,-1\\}\\). The image algebra has dimension two, with basis the identity and \\(\\gamma_1=\\operatorname{diag}(\\omega,\\omega^2)\\). The trace coordinate map from \\cref{thm:finite_trace_image_bound} gives\n\\[\n \\Phi(\\gamma_0)=(2,-1),\\qquad\n \\Phi(\\gamma_1)=(-1,-1),\\qquad\n \\Phi(\\gamma_2)=(-1,2).\n\\]\nThe tuples are distinct. The sharper algebra bound is \\(3\\le2^2=4\\), while the rank only bound is \\(3\\le2^{2^2}=16\\). The latter is deliberately rough. Its role is a uniform bound independent of the coefficient prime, not an efficient classification of the group.","a,\\omega^{-a}):a=0,1,2\\}.\n\\]\nThe trace alphabet is \\(\\{2,-1\\}\\). The image algebra has dimension two, with basis the identity and \\(\\gamma_1=\\operatorname{diag}(\\omega,\\omega^2)\\). The trace coordinate map from \\cref{thm:finite_trace_image_bound} gives\n\\[\n \\Phi(\\gamma_0)=(2,-1),\\qquad\n \\Phi(\\gamma_1)=(-1,-1),\\qquad\n \\Phi(\\gamma_2)=(-1,2).\n\\]\nThe tuples are distinct. The sharper algebra bound is \\(3\\le2^2=4\\), while the rank only bound is \\(3\\le2^{2^2}=16\\). The latter is deliberately rough. Its role is a uniform bound independent of the coefficient prime, not an efficient classification of the group.","bet is \\(\\{2,-1\\}\\). The image algebra has dimension two, with basis the identity and \\(\\gamma_1=\\operatorname{diag}(\\omega,\\omega^2)\\). The trace coordinate map from \\cref{thm:finite_trace_image_bound} gives\n\\[\n \\Phi(\\gamma_0)=(2,-1),\\qquad\n \\Phi(\\gamma_1)=(-1,-1),\\qquad\n \\Phi(\\gamma_2)=(-1,2).\n\\]\nThe tuples are distinct. The sharper algebra bound is \\(3\\le2^2=4\\), while the rank only bound is \\(3\\le2^{2^2}=16\\). The latter is deliberately rough. Its role is a uniform bound independent of the coefficient prime, not an efficient classification of the group.","Semisimplicity cannot be omitted. Over \\(\\CC\\), the matrices\n\\[\n \\begin{pmatrix}1&a","Semisimplicity cannot be omitted. Over \\(\\CC\\), the matrices\n\\[\n \\begin{pmatrix}1&a\\\\0&1\\end{pmatrix},\\qquad a\\in\\ZZ,\n\\]\nform an infinite group with the single trace value two. The invariant first coordinate line has no invariant complementary line. The trace pairing does not see the upper right nilpotent direction. This is exactly the step that fails without semisimplicity.","The characteristic restriction also has a visible purpose. In characteristic two, choose a primitive cube root \\(\\omega\\) and the scalar group\n\\[\n \\{I_2,\\omega I_2,\\omega^2 I_2\\}.\n\\]\nIt is a faithful semisimple representation, as a direct sum of two identical one dimensional characters. Every trace is zero because its multiplicity two vanishes in the coefficient field. The alphabet has size one, but the group has size three. The would be bound \\(3\\le1^{4}\\) is false. In the proof, the block pairing multiplier \\(m_i=2\\) is zero. Requiring \\(\\ell>n\\) prevents precisely this multiplicity cancellation.","purpose. In characteristic two, choose a primitive cube root \\(\\omega\\) and the scalar group\n\\[\n \\{I_2,\\omega I_2,\\omega^2 I_2\\}.\n\\]\nIt is a faithful semisimple representation, as a direct sum of two identical one dimensional characters. Every trace is zero because its multiplicity two vanishes in the coefficient field. The alphabet has size one, but the group has size three. The would be bound \\(3\\le1^{4}\\) is false. In the proof, the block pairing multiplier \\(m_i=2\\) is zero. Requiring \\(\\ell>n\\) prevents precisely this multiplicity cancellation.","ga\\) and the scalar group\n\\[\n \\{I_2,\\omega I_2,\\omega^2 I_2\\}.\n\\]\nIt is a faithful semisimple representation, as a direct sum of two identical one dimensional characters. Every trace is zero because its multiplicity two vanishes in the coefficient field. The alphabet has size one, but the group has size three. The would be bound \\(3\\le1^{4}\\) is false. In the proof, the block pairing multiplier \\(m_i=2\\) is zero. Requiring \\(\\ell>n\\) prevents precisely this multiplicity cancellation.","\\{I_2,\\omega I_2,\\omega^2 I_2\\}.\n\\]\nIt is a faithful semisimple representation, as a direct sum of two identical one dimensional characters. Every trace is zero because its multiplicity two vanishes in the coefficient field. The alphabet has size one, but the group has size three. The would be bound \\(3\\le1^{4}\\) is false. In the proof, the block pairing multiplier \\(m_i=2\\) is zero. Requiring \\(\\ell>n\\) prevents precisely this multiplicity cancellation.","For an elementary example, compare the integral polynomials\n\\[\n P(T)=T^2-2T+1,\\qquad Q(T)=T^2+2T+1.\n\\]\nTheir middle coefficients differ by four. Reduction modulo two makes them equal. Reduction modulo any odd rational prime keeps them distinct. A separating prime is a coefficient prime with this latter injectivity property. It does not say which polynomial is realized by a global Galois representation. It says that, once a realization gives one of the alphabet polynomials modulo that prime, a different alphabet entry cannot imitate it.","The large residual theorem uses two alphabets. One is the supplied automorphic alphabet. The other contains every degree \\(n\\) polynomial whose roots have orders at most \\(B=t^{n^2}\\). A field containing \\(\\mu_{\\operatorname{lcm}(1,\\ldots,B)}\\) contains all their coefficients. Both the field and the exceptional coefficient primes are fixed before choosing a residual realization. This order matters. A proof that chose a separation field from the coefficients of an as yet unknown lift and then called the excluded primes a predetermined set would leave a circular quantifier.","The resulting bound is not claimed to be practical. Even \\(t\\) and \\(n\\) of moderate size make \\(B\\) and the universal cyclotomic field enormous. The theorem establishes an existence implication with a finite exceptional set. A useful computational implementation for a particular finite image may exploit its actual exponent and much smaller image algebra, but that is an additional optimization and must retain the same separation logic.","Certified matrix Hensel sector & Unique actual local root from one finite test when \\(m>s\\) and the two sided inverse certificate holds & A certificate, singular sector analysis, and arithmetic or global realization"]}