Proof. Global class field theory separates the dyadic part of \(\det r\) from its odd conductor part: \[\det r=\delta\,\alpha\circ\varepsilon, \qquad \alpha:\mathbb Z_2^\times\longrightarrow\mathcal O^\times.\] Here \(\delta\) has finite order and odd conductor. Indeed, at an odd prime the pro-\(\ell\) group of principal units has finite image in \(\mathcal O^\times\), whose open subgroup is pro-\(2\); only finitely many primes occur. The remaining unramified global character is trivial by the class field theory of \(\mathbb Q\).

Choose \(w\) with \((-1)^w=\alpha(-1)\). Write \(\mathbb Z_2^\times=\{\pm1\}\times5^{\mathbb Z_2}\). The value \(\alpha(5)\) reduces to one, since a pro-\(2\) group has trivial image in \(k^\times\). After finite coefficient extension choose \[u^2=5^w\alpha(5)^{-1}.\] The element \(u\) is integral and reduces to one. Its powers \(u^{2^n}\) tend to one, so \(a\mapsto u^a\) defines a continuous character of \(\mathbb Z_2\). Define \(\beta(-1)=1\) and \(\beta(5)=u\). Then \(\alpha(z)\beta(z)^2=z^w\) for every \(z\in\mathbb Z_2^\times\). Replacing \(r\) by \(r\otimes(\beta\circ\varepsilon)\) gives the asserted determinant. Since \(\beta(-1)=1\), complex conjugation and oddness are unchanged. Local reciprocity identifies \(\varepsilon\) on dyadic units with the identity character; \(\delta\) is unramified there.

For completeness, the auxiliary modular representation can be chosen without imposing the residual representation of \(r\). Oddness gives \(\delta(-1)=(-1)^{w+1}\). Fix an odd level divisible by the conductor of \(\delta\), and enlarge it to a fine level. The dimension formula (Cohen and Oesterlé 1977, sec. III, Théorème 1) for cusp forms with this character grows linearly with the weight \(w+1\) along the indicated parity class. The CM contribution at this fixed level is bounded independently of the weight. To see the latter assertion, a CM eigenform comes from a Hecke character of an imaginary quadratic field \(K\); its level is \(|\operatorname{disc}K|\operatorname{Norm}(\mathfrak f)\), where \(\mathfrak f\) is the character conductor (Ribet 1977, Theorem (3.4), Corollary (3.5), and Theorem (4.5)). Only finitely many pairs \((K,\mathfrak f)\) occur at the fixed level. For a fixed infinity type the characters with conductor dividing a fixed ideal form, when nonempty, a torsor under a finite ray class character group. Oldform multiplicities are bounded by the fixed level as well. Thus a sufficiently large \(w\) gives a non-CM cuspidal eigenform of determinant \(\delta\varepsilon^w\). Its representation is regular and not virtually solvable. Indeed its restriction to every finite index subgroup is semisimple; if its connected algebraic monodromy were solvable, its unipotent radical would act trivially, so that connected group would be a torus. An open subgroup would then have abelian image, which Ribet’s CM criterion excludes (Ribet 1977, Propositions (4.2) and (4.4)). ◻

Proof. Choose a stable lattice of \(r\) and let \(R^{\square,\chi}_{\bar r,S}\) be its unrestricted framed deformation ring with determinant \(\chi\). The residue here is the actual lattice reduction; no semisimplicity or irreducibility is imposed on it. These rings and their maps to fixed-determinant pseudodeformation rings exist under Mazur’s finiteness condition, which holds for \(G_{\mathbb Q,S}\) (OpenAI 2026a, Proposition 3.1).

Complete at the characteristic-zero point of \(r\). Its real local framed deformation ring is smooth of dimension two: an odd involution has eigenvalues \(1,-1\), and its conjugacy orbit has stabilizer the two-dimensional diagonal torus. For the global ring over this real local ring, the relative relation estimate of (OpenAI 2026a, Proposition 3.6), with no imposed dyadic condition, is \[g-r_{\mathrm{rel}}\ge 3-h^0\bigl((\mathop{\mathrm{ad}}^0r)^\vee(1)\bigr)=3.\] The equality follows from absolute irreducibility. In characteristic zero the trace pairing identifies \(\mathop{\mathrm{ad}}^0r\) with its dual; a nonzero invariant in its cyclotomic twist would give a nonzero map \(r\to r\otimes\varepsilon\). Such a map is an isomorphism, whereas its determinants would require \(\varepsilon^2=1\). Every component of the completed local ring therefore has dimension at least \(2+3=5\). The dimension comparison at a coefficient point in (OpenAI 2026a, Lemma 3.4) adds one on returning to a horizontal integral component. Choose such a component through \(r\) and give it the reduced structure; its ring \(B\) has dimension at least six.

Let \(R^{\mathrm{ps},\chi}_{S}\) be the global pseudodeformation ring of the semisimplified lattice reduction, and set \[D=R^{\mathrm{ps},\chi}_{S}/ \ker\bigl(R^{\mathrm{ps},\chi}_{S}\longrightarrow B\bigr).\] Write \(T(g)\in D\) for its universal trace at \(g\). The ring \(D\) is a horizontal complete local domain. Its image in \(B\) is closed: complete local rings with finite residue field are compact, and their continuous images in Hausdorff rings are closed. The generic representation on \(B\) is absolutely irreducible, since the irreducible locus contains \(r\). Forgetting its single frame loses at most three dimensions, by (OpenAI 2026a, Lemma 3.2). Hence \(\dim D\ge\dim B-3\ge3\).

We justify the last assertion by bounding the trace loci of the virtually solvable representations. The reducible locus is closed and proper on \(D\), because it does not contain \(r\). An absolutely irreducible virtually solvable two-dimensional representation is induced from a character of a quadratic field, or has finite projective image. This follows by considering the connected solvable algebraic subgroup: a noncentral torus has two eigenlines permuted by the whole group, while a central connected subgroup gives finite projective image. A nontrivial connected unipotent subgroup would have a unique invariant line and force reducibility.

Only finitely many quadratic fields \(K\) can occur. The corresponding quadratic self-twist is unramified outside \(S\), so this is Hermite–Minkowski with bounded local degree. For each \(K\), the maximal abelian pro-\(2\) quotient of \(G_{K,S}\) has rank at most two: class field theory bounds its free part by the dyadic unit groups, of total rank \([K:\mathbb Q]=2\); class groups and non-dyadic units contribute only finite groups. The invariant part under the quadratic involution has rank at least one, from the cyclotomic extension. If the inducing character is \(\theta\), fixing the determinant fixes \(\theta\theta^s\) on \(G_K\). Thus at most one free character parameter remains. More explicitly, fixing the character on the image of \(1+s\) fixes a sublattice of rank at least one; the finite index in its saturation contributes an integral, finite extension, not an extra parameter. The resulting complete character rings have integral dimension at most two. There are only finitely many possible residual characters, since their unordered pair is prescribed by \(\bar r|_{G_K}\).

Their images in the trace space are closed of dimension at most two. Indeed, each generating character value satisfies \[X^2-T(h)X+\chi(h)=0\qquad(h\in G_K).\] Finitely many generators of the abelian character group, followed by complete Nakayama, make each character ring finite over its trace image. This is the finite-character argument of (OpenAI 2026a, Proposition 3.1(3)).

For finite projective image, the classification of finite subgroups of \(\operatorname{PGL}_2\) leaves cyclic and dihedral groups, already covered by reducibility and induction, and the three exceptional groups \(A_4,S_4,A_5\). Their exponents divide \(M=60\). For any matrix with projective order dividing \(M\), the ratio \(\zeta\) of its eigenvalues satisfies \(\zeta^M=1\), and \[\frac{T(g)^2}{\chi(g)}=2+\zeta+\zeta^{-1}.\] Consequently \(T(g)\) is a root of the monic polynomial \[\prod_{\zeta^M=1} \bigl(X^2-\chi(g)(2+\zeta+\zeta^{-1})\bigr)\in\mathcal O[X].\] Its coefficients belong to \(\mathcal O\) because the product over roots of unity is Galois invariant and integral. The fixed-determinant pseudo ring is topologically generated by finitely many traces: traces generate it by the determinant-of-a-sum identity, and finitely many suffice by Nakayama on its cotangent space. Imposing these monic equations on such generators gives a finite \(\mathcal O\)-algebra. Thus the exceptional finite-projective trace locus has integral dimension at most one.

The union of these finitely many closed loci cannot contain the generic point of \(D\), whose dimension is at least three. Pull it back to \(B\) and use coefficient-point density (OpenAI 2026a, Lemma 3.3(2)) to obtain a point outside it. Restriction to any finite index subgroup preserves the nonsolvable identity component of its monodromy, so that point remains absolutely irreducible. The same is then true generically on the restricted family and on every component containing that family. ◻

Proof. The determinant and the residual representation enter this argument separately. Lemma 2 supplies a determinant arising from a regular modular representation; the modular seed below will instead lift the residual representation of \(B\). We connect a characteristic-two curve on the seed’s Hecke support to one in an unrestricted locus containing the restricted family. The constraints are used only to select and connect these curves. Once the second curve has potential Hecke support, localized propagation puts the entire unrestricted locus on that support.

We use the following precise propagation input. The localized propagation theorem (OpenAI 2026a, Theorem 5.1) takes a totally real solvable field of even degree, completely split at \(2\), a finite allowed set, a determinant of the form \(\chi\) supplied by a regular target, and, separately, a semisimple residual representation of determinant \(\bar\chi\). If a characteristic-two curve in this pseudodeformation space is potentially pro-modular, has non-virtually-solvable generic representation, and has finite local images at the allowed non-dyadic places, then every irreducible closed locus containing that curve is potentially pro-modular.

Choose a regular cuspidal modular lift of this semisimple residue, using (OpenAI 2026a, Lemma 7.2). That lemma includes character sums and scalar residue. Enlarge the initial allowed set to contain its tame primes and those of \(B\), \(\chi\), and the auxiliary regular representation, before making any field choices. Field preparation (OpenAI 2026a, Proposition 6.1) provides abelian totally real \(2\)-extensions \[\mathbb Q\subset A\subset F_0\] in which \(2\) splits completely. Put \(d_A=[A:\mathbb Q]\) and \(d=[F_0:\mathbb Q]\). The numbers \(u_A,u_0\) of allowed non-dyadic primes are bounded by a constant \(U\) while both degrees grow. Absolutely irreducible residue remains absolutely irreducible. In the reducible case write \(\bar r=\bar\chi_1\oplus\bar\chi_2\) and \(\alpha=\bar\chi_1/\bar\chi_2\). Distinct characters remain distinct, and restriction from \(A\) to \(F_0\) is injective on the residual extension groups for \(\alpha\) and \(\alpha^{-1}\).

When the global residue is reducible and its dyadic characters coincide, the connectedness argument will keep track of chosen local invariant lines at selected dyadic places. These places are the marks. Field preparation lets us make their number large while keeping their proportion among all dyadic places small.

There are no marks if the residue is absolutely irreducible or \(\alpha|_{G_{\mathbb Q_2}}\ne1\). Otherwise field preparation specifies one dyadic place \(v_0\) of \(A\), and we mark all its \(m=[F_0:A]\) extensions to \(F_0\). Thus \(m/d=1/d_A\); put \(m=0\) in the unmarked case. We choose the degrees so that \[ d>u_0+4,\qquad m>u_0+4\ \text{if marked},\qquad 2d-2m-4-u_0>d. \tag{3}\] These are the degree conditions in (OpenAI 2026a, sec. 7.2). Any further fixed lower bound on \(d_A\) is compatible with them.

We now construct two large loci over \(A\). Solvable base change and Jacquet–Langlands transfer the modular seed to definite forms. Regularity keeps this base change cuspidal (OpenAI 2026a, Lemma 4.6). Its central character and the prescribed \(\psi\) agree modulo \(\varpi\). At a sufficiently deep tame level, the integral function description and exact reduction of completed forms therefore give a nonzero residual summand with central character precisely \(\psi\); this is the seed construction in (OpenAI 2026a, sec. 7.4). A component \(C_{\mathrm{seed}}\) of its Hecke support has \[\dim C_{\mathrm{seed}}\ge1+2d_A\] by (OpenAI 2026a, Proposition 4.4).

Independently choose an unrestricted fixed-determinant framed component containing the restricted family \(B|_{G_A}\), by taking a minimal prime below its defining prime. Its generic representation is absolutely irreducible by Lemma 3. The relative presentation over the real local factors gives the component dimension at least \(1+2d_A+3-h^0((\mathop{\mathrm{ad}}^0)^\vee(1))\); the invariant term is at most three. Forgetting the frame loses at most three more dimensions. Thus its trace image \(C_{\mathrm{fam}}\), which contains the restricted trace family, has \[\dim C_{\mathrm{fam}}\ge2d_A-2\] by (OpenAI 2026a, Proposition 3.6, Lemma 3.2, and Section 7.4).

In both loci impose \(\varpi=0\) and the residual trace of one Frobenius element at each allowed non-dyadic place. In the marked case impose also the constant residual local pseudorepresentation at \(v_0\). The latter costs at most \(\kappa\) equations, where \(\kappa\) is the number of generators of the closed-point ideal in one fixed dyadic pseudo ring; it is independent of \(d_A\). Put \(\kappa=0\) when unmarked. Taking \(d_A>U+\kappa+3\), the cut in \(C_{\mathrm{fam}}\) has dimension at least \[2d_A-3-u_A-\kappa>d_A;\] the cut in \(C_{\mathrm{seed}}\) has at least as large a lower bound. Over the abelian field \(A\), the characteristic-two reducible locus has dimension at most one, the dihedral locus at most \(d_A\), and finite-projective-image curves are constant (OpenAI 2026a, Lemma 3.8). Curve avoidance (OpenAI 2026a, Lemma 3.3(1)) therefore gives a non-virtually-solvable characteristic-two curve in each cut. The Frobenius conditions make their allowed non-dyadic local images finite by (OpenAI 2026a, Lemma 3.9). Denote these curves by \(C_s\) and \(C_f\).

It remains to transfer support from \(C_s\) to \(C_f\). Over \(F_0\), consider the characteristic-two framed representation scheme of the fixed semisimple residue, adjoining an invariant projective line at every mark. A closed residual point is called transverse if none of these marked lines is globally invariant; without marks this is an empty condition. We check that both curves enter transverse residual charts. Without marks, any stable lattice works. With marks, the local pseudo at \(v_0\) is constant and scalar after a character twist. The curve representation thus has a local invariant line, after a finite extension of its field. For a nonzero vector \(v\) on that line, over the normalized curve ring \(J=k'[[t]]\) form \[\mathcal V=\sum_{g\in G_A}J\rho(g)v.\] Compactness bounds this module in a lattice, and absolute irreducibility makes it a full stable lattice. The vector \(v\) is primitive: otherwise every translate would belong to \(t\mathcal V\), and Nakayama would give \(\mathcal V=0\). Its reduced orbit spans \(\mathcal V/t\mathcal V\), so its reduction belongs to no global invariant line.

This avoidance survives restriction to \(F_0\). A nonsplit extension of distinct residual characters remains nonsplit by the restriction injectivity, while a split representation retains its two distinct character lines. For globally scalar semisimplification, the orbit condition rules out a split scalar reduction; its nonzero additive extension remains nonzero by the same injectivity. These statements hold after any residual field extension. Finally, \(F_0/A\) is Galois, so transporting the local line to all conjugate marks preserves avoidance of every global invariant line. Both curves therefore pass through transverse residual points, including in the scalar residual case.

We recall what the two connectedness inputs prove, to make the support transfer explicit. At a transverse point, (OpenAI 2026a, Lemma 7.3) gives components of the completed chart of dimension at least \(2d-m\), with successive intersections in a connecting chain of dimension at least \(2d-m-1\). The reducible locus has strictly smaller dimension. For an intersection component of maximal dimension, forgetting its frame and marked lines loses at most \(3+m\) dimensions, and forgetting the characters on those lines is finite. Its trace image therefore has dimension at least \(2d-2m-4\). The \(u_0\) Frobenius trace conditions leave dimension greater than \(d\) by (3), so the same bad-locus avoidance supplies a non-virtually-solvable curve with finite bad local images. Localized propagation carries support across each such intersection. This is (OpenAI 2026a, Lemma 7.4).

The transverse residual fiber is geometrically connected by (OpenAI 2026a, Lemma 7.5). In particular this includes scalar residue: after fixing the unique global invariant line, the nonzero additive extension classes and the transverse marked lines give a connected parameter space. The finite graph of components meeting this fiber therefore joins the seed chart to the family chart. Starting from \(C_s\), which is pro-modular, the preceding chart propagation makes \(C_f|_{G_{F_0}}\) potentially pro-modular. Hence \(C_f\) over \(A\) is potentially pro-modular. Apply localized propagation now over \(A\) to the unrestricted locus \(C_{\mathrm{fam}}\) containing \(C_f\). The entire locus, and therefore the prescribed family, acquires Hecke support. Marks have served only to connect curves in characteristic two; they impose no condition on \(B\).

Only finitely many components, charts, and propagation steps occur. Taking the common Galois compositum of the resulting fields and a common deeper tame level, as in (OpenAI 2026a, Lemma 4.6 and Section 7.4), gives one extension \(F/\mathbb Q\) with all the stated properties. ◻

Proof. Only finiteness and the module assertions remain to be proved. Let \(D_F\) be the closed restricted trace image in \(D\). If \(n=[F:\mathbb Q]\), then \(g^n\in G_F\) for every \(g\in G_\mathbb Q\). The trace-of-powers recurrence \[P_0(X,a)=2,\quad P_1(X,a)=X,\quad P_j(X,a)=XP_{j-1}(X,a)-aP_{j-2}(X,a)\] gives a monic polynomial \(P_n(X,a)\) of degree \(n\), with \[P_n(T(g),\chi(g))=T(g^n)\in D_F.\] Choose finitely many traces topologically generating \(D\) over \(\mathcal O\). Each is integral over \(D_F\), so the subalgebra they generate is finite over \(D_F\). It is compact and hence closed in \(D\); since it also contains a dense subalgebra, it equals \(D\). Thus \(D\) is finite over \(D_F\), and therefore over \(\mathbb T\).

The local-global block comparison (OpenAI 2026a, Proposition 4.2) makes \(\mathbb T\) finite over \(R^{\widehat\otimes b}\), with its actual Galois restriction action in block normalization. On \(D\) this action factors through \[R^{\widehat\otimes b}\longrightarrow R, \qquad a_1\otimes\cdots\otimes a_b\longmapsto a_1\cdots a_b,\] because all dyadic parameters agree. Transitivity of finiteness then proves that \(D\) is finite over the single ring \(R\).

Let \(M\) be the compact definite-forms summand. The same proposition gives \[L_0=\mathop{\mathrm{Hom}}_{G^b,\mathrm{cts}}(P^{\widehat\otimes b},M),\] finite and faithful over \(\mathbb T\), with the asserted compatible product action. The compact module \(M\) is admissible, equivalently finite over a sufficiently deep determinant-one completed group algebra, by (OpenAI 2026a, Proposition 4.1).

At the generic point of \(D\), the base change of \(L_0\) is nonzero. Indeed, if \(\mathfrak q\) is its contraction to \(\mathbb T\), faithfulness and finiteness give \(\operatorname{Supp}_{\mathbb T}L_0=\mathop{\mathrm{Spec}}\mathbb T\); Nakayama then gives \(L_0\otimes_{\mathbb T}\kappa(\mathfrak q)\ne0\). Extending this residue field to \(\mathop{\mathrm{Frac}}(D)\) preserves nonvanishing. Quotienting by \(D\)-torsion consequently leaves a nonzero finite \(D\)-module \(L\). The torsion submodule is stable under every \(\mathcal E\)-action because those actions commute with \(D\). Since \(D\) is horizontal, \(L\) is also \(\mathcal O\)-torsion-free.

Finally, finiteness of \(D\) over \(\mathbb T\) makes \(L_0\otimes_{\mathbb T}D\), and then \(L\), a quotient of a finite direct sum of copies of \(L_0\) as product-block modules. These are maps of compact modules with closed images. Exactness of the product block equivalence makes the inverse object of \(L\) a quotient of the corresponding finite sum of copies of \(M\). It is therefore admissible as asserted. This supplies product admissibility; passing to a single factor is the separate argument of the next section. ◻

Proof. The object \(P/\mathfrak m_RP\) corresponds under the compact equivalence to \(\mathcal E/\mathfrak m_R\mathcal E\), so has finite length. Every smooth simple in the block is admissible. Its compact dual therefore has finite-dimensional \(K\)-coinvariants, with the central character understood. Right exactness of coinvariants now makes \(C_K/\mathfrak m_RC_K\) finite-dimensional. The \(\mathfrak m_R\)-action is topologically nilpotent: on the algebra side this follows from the finite \(R\)-module \(\mathcal E\), and the equivalence transports the inverse limit of its finite-length quotients. Compact Nakayama proves that \(C_K\) is finite over \(R\).

Completed tensor products commute with the cokernels defining coinvariants. After these cokernels, all modules involved are finite over their complete noetherian coefficient rings, so ordinary and completed tensor products agree. For the product formula, apply this argument in each factor. Finite presentations over \(R\), or over its completed tensor powers, then show that the formula commutes with the asserted field base changes. ◻

Proof. We show that \(\mathcal E/\mathcal E e\mathcal E\) is finite over \(\mathcal O\). Under the compact equivalence its modules have only the \(\mathop{\mathrm{SL}}_2(\mathbb Q_2)\)-trivial simple constituents. This is precisely the subcategory of objects with trivial \(\mathop{\mathrm{SL}}_2(\mathbb Q_2)\)-action: that subcategory is thick, and the assertion passes to compact limits (Paškūnas and Tung 2021, sec. 4.2).

Such an action factors through \(\det:G\to\mathbb Q_2^\times\). On squares, the character is prescribed by the central action. Since \(\mathbb Q_2^\times/(\mathbb Q_2^\times)^2\) is finite, the corresponding completed group algebra with these scalar relations is finite over \(\mathcal O\). This remains true if the scalar relations force coefficient torsion. The regular module of \(\mathcal E/\mathcal E e\mathcal E\) corresponds to an object of this subcategory with finite cosocle. Compact Nakayama makes it finite over that finite \(\mathcal O\)-algebra; its endomorphism algebra, and hence \(\mathcal E/\mathcal E e\mathcal E\), is finite over \(\mathcal O\). Its special-fiber support over \(R\) is therefore contained in the closed point. It vanishes at the curve’s generic point, as required. ◻

Proof. The characters take values in \(J^\times\): their values are integral by the characteristic polynomial, and their product is a unit. They are continuous. Indeed, choose an element on which the two characters differ; traces against that element recover their values continuously in \(H\). Their reductions are the residual constituents.

We first construct the families in mixed characteristic. Enlarge the integer coefficients to \(\mathcal O'\), and set \(J_0=\mathcal O'[[X,Y]]\). The pro-\(2\) completion of \(\mathbb Q_2^\times\) has two free generators, represented by \(2\) and \(5\), and torsion generated by \(-1\). Let \(\Gamma_1\) be the universal lift of the first residual character on the two free generators, fixing either lift of its sign on \(-1\), and put \(\Gamma_2=\psi\varepsilon\Gamma_1^{-1}\). There is a continuous map \(J_0\to J\) specializing these characters to \(\gamma_1,\gamma_2\).

For each order, form smooth induction with coefficients in the discrete dual \(J_0^\vee\), and take its compact dual \(N_i\). This defines the family even when the inducing characters become smooth only modulo powers of the coefficient maximal ideal. Local sections on \(B\backslash G=\mathbf P^1(\mathbb Q_2)\) identify its compact model with a pro-free \(J_0\)-module. In particular the model is flat and commutes with coefficient changes. The residual smooth induction has finite length in the specified block. If the residual characters differ it is irreducible; if they coincide it has the character and Steinberg constituents. In either case it has precisely one noncharacter constituent, with multiplicity one (Paškūnas and Tung 2021, sec. 4.1). Filtering the finite coefficient quotients shows that \(N_i\) belongs to the compact block.

Put \(\widetilde W_i=\mathop{\mathrm{Hom}}(P,N_i)\). Compact Nakayama and residual finite length show that this is a finite \(J_0\)-module. Exactness of the equivalence makes it flat: for every finitely generated ideal \(I\subset J_0\), apply the equivalence to the injective map \(I\otimes_{J_0}N_i\to N_i\), using finite presentations to commute these tensors with the equivalence. Thus \(\widetilde W_i\) is finite free. The same argument proves compatibility with coefficient changes. The residual multiplicity calculation gives \[\mathop{\mathrm{rank}}_{J_0}(\widetilde W_i e)=1.\]

At characteristic-zero coefficient points outside the proper loci \(\Gamma_1\Gamma_2^{-1}\in\{1,\varepsilon,\varepsilon^{-1}\}\), Schikhof duality identifies these objects with the two continuous principal series in Theorem 7. Those points are Zariski dense in \(\mathop{\mathrm{Spec}}J_0\). Hence their central \(R\)-action is everywhere the pseudorepresentation \(\Gamma_1+\Gamma_2\): equality may be tested on matrices acting on the finite free modules.

We also need to keep the two character labels distinct upon specialization. The residual block here is nonsupersingular, so Theorem 7 identifies the opposite corner with \(\mathrm{CH}_R^{\mathrm{tf}}\), without a matrix factor. Its Galois action on each free rank-one module \(\widetilde W_i e\) gives a character. At the dense coefficient points the two principal series are distinct simple modules, their nonzero corner modules are distinct, and their corner labels are therefore \(\Gamma_1\) and \(\Gamma_2\). More explicitly, if \(\lambda_i\) is the corner character, set \[I_{ij}=(\lambda_i(g)-\Gamma_j(g):g\in G_{\mathbb Q_2})\subset J_0.\] The dense coefficient points lie in \(V(I_{i1})\cup V(I_{i2})=V(I_{i1}I_{i2})\), so \(I_{i1}I_{i2}=0\). Since \(J_0\) is a domain, one of the two ideals vanishes. The two labels are opposite because the generic specializations are distinct. Thus the labels on the entire family are \(\Gamma_1,\Gamma_2\), up to one fixed permutation, and these identities persist under \(J_0\to J\).

Set \(W_i=\widetilde W_i\otimes_{J_0}J\). By Lemma 9, \(e\) is full over \(H\); its rank-one corner modules therefore make \(W_i\otimes_JH\) simple and distinct. The specialized torsion-free Cayley–Hamilton algebra is a quotient of the Cayley–Hamilton algebra of the specialized parameter. The field structure theorem for a split multiplicity-free determinant allows at most the two character simples (Chenevier 2014, Theorems 2.12 and 2.22). Thus our two modules account for all simples, without assuming that forming the torsion-free quotient commutes with reduction modulo \(2\).

Finally calculate \(K\)-coinvariants directly in the compact induction model. There are finitely many double cosets \(B\backslash G/K\). For a representative \(g\), its contribution is \(J\) modulo the ideal generated by the inducing character minus one on \(B\cap gKg^{-1}\). This follows equally by dualizing the invariant functions with values in \(J^\vee\). Every stabilizer has determinant one. Since \(\varepsilon=1\) over \(J\), the two inducing characters on that stabilizer are inverse. They generate the same ideal, because \(a^{-1}-1=-a^{-1}(a-1)\) for a unit \(a\). The two coinvariant modules consequently vanish simultaneously after tensoring with \(H\). Lemma 8 identifies these modules with the two displayed tensor products. ◻

Proof. The corner is full by Lemma 9. The Cayley–Hamilton comparison and its field structure therefore give at most the simples labelled by the absolute constituents of the parameter. Repeated characters cannot occur on this curve. Indeed the fixed determinant has constant finite values in characteristic two; if it equals \(\gamma^2\), injectivity of squaring in a field makes \(\gamma\), and hence the whole parameter, constant.

If the parameter is absolutely irreducible, the split algebra has only one simple module. A nonzero finite left module has a nonzero semisimple head, so tensoring it with that simple right module is nonzero. If the parameter has two distinct characters, Lemma 10 constructs both simple modules and proves that their tensor tests have identical vanishing. At least one test is nonzero: otherwise the semisimple head of \(C_{K,H}\) would be zero. Both tests are therefore nonzero. ◻

Proof of Proposition 6. The module \(L\) is finite over \(R\), hence over the first \(\mathcal E\). The exact compact equivalence makes \(V\) nonzero and \(\mathcal O\)-torsion-free. It also transports the continuous commuting \(D\)-action. By Lemma 8, \[Q:=V/(\varpi,K-1)=L\otimes_{\mathcal E}C_K\] is finite over \(D/\varpi\). It suffices, by compact Nakayama over \(\mathscr A\), to show that \(Q\) is finite-dimensional over \(k\).

Suppose otherwise. Its closed support contains a one-dimensional complete local domain quotient of \(D/\varpi\). At the generic point of this curve, \(Q\) has nonzero fiber. Since \(D\) is finite over \(R\), its image in \(\mathop{\mathrm{Spec}}R\) is also a nonconstant curve. Extend its fraction field to a field \(H\) as in Lemma 11. With subscripts denoting fibers, we have \[L_H\otimes_{\mathcal E_H}C_{K,H}\ne0.\] This tensor product still has the commuting actions of the remaining \(b-1\) copies of \(\mathcal E_H\).

For any nonzero finite right \(\mathcal E_H\)-module \(U\), choose a simple quotient \(S\). Right exactness of tensor products and Lemma 11 give \(U\otimes_{\mathcal E_H}C_{K,H}\ne0\). Apply this observation successively to the remaining factors. The central parameters are equal, so the same algebra and the same module \(C_{K,H}\) occur each time. We obtain \[L_H\otimes_{\mathcal E_H^{\otimes_H b}}C_{K,H}^{\otimes_H b}\ne0.\] By the product formula in Lemma 8, this is the fiber of \(M/(\varpi,K^b-1)\) on the chosen curve. But product admissibility makes that module finite-dimensional over \(k\). Its continuous \(D\)-action is supported at the closed point, so its fiber on a nonconstant curve is zero. This contradiction proves that \(Q\) is finite-dimensional and hence that \(V\) is finite over \(\mathscr A\). ◻

Proof. The induced filtration on \(fM\) is good. Artin–Rees, followed by the isomorphism \(M\simeq fM\), therefore gives an integer \(c\geq1\) such that \[fM\cap\mathfrak a^nM\subseteq f(\mathfrak a^{n-c}M) \qquad(n\geq c).\] Taking lengths in the induced exact sequence yields \[H_{M/fM}(n) \leq H_M(n)-H_M(n-c)=O(n^{g(M)-1}).\] This proves the first assertion, including the impossibility of a nonzero cokernel when \(g(M)=0\).

The algebra \(\mathscr A\) is a noetherian domain and has an Ore division ring of fractions. If a finite module has rank zero, each element is annihilated by a nonzero element of \(\mathscr A\). A filtration by its finitely many generators has cyclic torsion quotients. Such a quotient is of the form \(\mathscr A/I\), where \(I\) contains a nonzero element \(a\). Its associated graded module is a quotient of the polynomial ring in (4) by the nonzero initial form of \(a\), and hence has growth degree at most three. The same bound holds for a finite extension of these quotients. Thus growth degree four implies positive rank. ◻

Proof. Choose an auxiliary coefficient point \(y\) of \(D\) whose local semisimple parameter, if reducible, has character ratio outside \(\{1,\varepsilon,\varepsilon^{-1}\}\). Such points exist. Indeed, at a reducible exceptional parameter the fixed determinant and the specified ratio determine each character up to a quadratic character. Local class field theory gives only finitely many quadratic characters of \(\mathbb Q_2^\times\). The exceptional parameters therefore lie in a finite union of coefficient-point closures in \(\mathop{\mathrm{Spec}}R\), of integral dimension at most one. Finiteness of \(D\) over \(R\) gives the same bound for their inverse image. Since \(D\) is horizontal of dimension at least three, coefficient-point density (OpenAI 2026a, Lemma 3.3) supplies \(y\) outside that inverse image.

Put \(s=\dim D\). If \(\mathfrak p_y\) is the kernel of the coefficient point, then \(D/\mathfrak p_y\) is finite over \(\mathcal O\) and has dimension one. The complete local domain \(D\) is catenary, so there is a saturated chain \[(0)=\mathfrak p_0\subsetneq\mathfrak p_1 \subsetneq\cdots\subsetneq\mathfrak p_{s-1}=\mathfrak p_y.\] Starting with \(L^{(0)}=L\), define \(L^{(i+1)}\) by tensoring \(L^{(i)}\) with \(D/\mathfrak p_{i+1}\) and removing its torsion over that domain. Each \(L^{(i)}\) is nonzero: the preceding torsion-free finite module has full support, and Nakayama’s lemma at the next prime shows that its fiber there is nonzero. Thus \(L^{(i+1)}\) has nonzero generic fiber. The commuting \(\mathcal E\)-action survives all these operations.

Choose \(a_i\in\mathfrak p_{i+1}\setminus\mathfrak p_i\). Multiplication by \(a_i\) is injective on \(L^{(i)}\). The exact compact block equivalence of Theorem 7 makes it injective on \[V_i=L^{(i)}\widehat\otimes_{\mathcal E}P,\] and \(V_{i+1}\) is a quotient of its cokernel. All these objects are finite over \(\mathscr A\), being successive quotients of \(V\). Lemma 12 gives \[ g(V)\geq s-1+g(V_{s-1}). \tag{5}\]

Finally, \(L^{(s-1)}\) is nonzero and finite torsion-free over \(\mathcal O\). Its associated Banach representation has finite length and, after a finite coefficient extension, every simple factor has the central parameter of a coefficient conjugate of \(y\); see Theorem 7. They are infinite-dimensional: the generic principal-series possibilities are infinite-dimensional, as is the representation attached to an absolutely irreducible two-dimensional Galois parameter. Our choice of \(y\) excludes the exceptional character possibilities. Hence \(V_{s-1}/\varpi V_{s-1}\) is infinite-dimensional over \(k\). Otherwise compact Nakayama would make \(V_{s-1}\) finite over \(\mathcal O\), contradicting this Banach description.

Exactness also makes multiplication by \(\varpi\) injective on \(V_{s-1}\). The infinite-dimensional reduction has growth degree at least one, so Lemma 12 gives \(g(V_{s-1})\geq2\). Together with (5), this proves \[4\geq g(V)\geq\dim D+1\geq4.\] The final assertion of Lemma 12 finishes the proof. ◻

Proof. Positive rank supplies a nonzero \(\mathscr A\)-linear map \(V\to\mathscr A\): take a nonzero linear functional after passing to the Ore division ring and clear right denominators on finitely many generators. Maps between finite \(\mathscr A\)-modules are continuous. Choose \(K\) so that its intersection with the scalar unit subgroup \(Z_0\) is trivial. Extend the action on the target \(\mathscr A\) to \(Z_0K\) by the dual central character \(z^{-d}\). Since \(Z_0K\) has finite index in \(K_0\), Frobenius reciprocity gives a nonzero map from \(V\) to its coinduced compact module. Dualizing gives a nonzero continuous \(K_0\)-map \[ \mathcal C_d\longrightarrow\Pi_V, \qquad \mathcal C_d=\{f\in C(K_0,E):f(kz)=z^{-d}f(k)\}. \tag{6}\] The action on \(\mathcal C_d\) is left translation.

We verify the density needed to use this map. Matrix coefficients of algebraic representations of central degree zero form an algebra separating the points of \(K_0/Z_0\): coefficients of the adjoint representation already separate them. Polynomial approximation on compact subsets of a finite-dimensional \(2\)-adic affine space therefore makes this algebra dense in \(C(K_0/Z_0,E)\). Concretely, embed the compact quotient by the adjoint matrix entries and apply multivariable polynomial approximation on a containing compact box, after extending continuous functions by clopen partitions.

Fix one algebraic representation of central degree \(d\), for example \(\det^{d/2}\) if \(d\) is even and \(E^2\otimes\det^{(d-1)/2}\) if \(d\) is odd. Its matrix coefficients have no common zero on \(K_0\). On a finite clopen cover of \(K_0/Z_0\), divide a function in \(\mathcal C_d\) by a nonvanishing coefficient of this representation evaluated at \(k^{-1}\). Approximate the resulting functions of central degree zero as above, and multiply back. Complete reducibility of algebraic representations shows that the resulting dense span consists of matrix coefficients of irreducible algebraic representations with central degree \(d\). Multiplication by the nowhere-zero function \(\sigma(k^{-1})\) preserves \(\mathcal C_d\) and its topology. Thus the matrix coefficients of the types \(\sigma\otimes U\) in the statement also have dense span. A nonzero continuous map in (6) cannot kill them all.

For finiteness, transpose a type map to a continuous \(K\)-equivariant map from \(V\) to the dual of that finite-dimensional type. A finite set of \(\mathscr A\)-generators of \(V\) determines such a map, so the space of maps is finite-dimensional. Evaluation on those generators also proves continuity of its commuting \(D\)-action. ◻

Proof. Suppose the parameter is reducible. Theorem 7 lists its possible Banach simples. Put \(S_0=\mathop{\mathrm{SL}}_2(\mathbb Z_2)\) and \(u=\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)\). On \(\sigma\otimes U\), the operator \(u\) is minus a unipotent operator. In particular \(u-1\) is invertible over \(E'\). The group \(S_0\) acts transitively on \(\mathbb P^1(\mathbb Q_2)\). An \(S_0\)-map from the type \(\sigma\otimes U\) to a continuous principal series, followed by evaluation at the identity coset, would give a functional invariant under the upper unipotent subgroup, hence under \(u\). This functional is zero, and transitivity makes the whole map zero. The same argument excludes a character, whose restriction to \(S_0\) is trivial.

For a twist of the continuous Steinberg representation, pull back the defining quotient of a continuous induction by a character along a proposed type map. This gives an extension of our finite-dimensional type by a character. Every finite-dimensional continuous representation of \(S_0\) over \(E'\) is semisimple. To see this, continuity and logarithm charts make it analytic on a sufficiently small open subgroup. Complete reducibility for \(\mathfrak{sl}_2(E')\) supplies an equivariant projection onto any invariant subspace. That projection commutes with a sufficiently small open normal subgroup of \(S_0\); averaging its conjugates over the finite quotient gives an \(S_0\)-equivariant projection. The pulled-back extension therefore splits over \(S_0\), producing the already excluded map into continuous induction.

This excludes every reducible parameter in the classification. ◻

Proof. Take a type supplied by Lemma 14. The commuting \(D\)-action on its nonzero finite-dimensional Hom space has a common eigenvector after finite coefficient extension. It gives a continuous character \(x:D\to E'\). Since \(D\) is compact, its image in the endomorphism algebra preserves a lattice; its eigenvalues are integral. Thus \(x\) takes values in \(\mathcal O'\), and the corresponding map \[ \sigma\otimes U\longrightarrow\Pi_V[x] \tag{7}\] is nonzero.

This eigenspace has finite length as a Banach representation. Indeed, extend coefficients to \(\mathcal O'\), impose the scalar relations \(a=x(a)\) for \(a\in D\) on \(L\), and remove \(\mathcal O'\)-torsion. The result is finite over \(\mathcal O'\). The exact compact equivalence identifies its inverse with the corresponding torsion-free scalar quotient of \(V\); continuous maps to \(\mathcal O'\) kill the discarded torsion. Dualizing and inverting \(2\) therefore identifies its Banach representation with \(\Pi_V[x]\). The finite-length assertion is the coefficient-fiber assertion of Theorem 7. A composition series now shows that some Banach simple contains the type in (7): follow a nonzero map through the series, passing to a quotient when its composite is nonzero and otherwise factoring it through the closed subrepresentation. No exactness assertion for the functor of locally algebraic vectors is involved.

Lemma 15 now makes the local block parameter \(q_x\) absolutely irreducible. The global determinant at \(x\) is then absolutely irreducible as well and is represented, after finite coefficient extension, by a continuous \(r_x\); this is the representation theorem for characteristic-zero determinants (Chenevier 2014, Theorem 2.12). Continuity can also be checked by recovering matrix coordinates from traces against a matrix-algebra basis. The normalization is \[ q_x=r_x|_{G_{\mathbb Q_2}}\otimes\varepsilon. \tag{8}\] Since \(2\) splits completely in \(F\), restriction to \(G_F\) is still absolutely irreducible.

We can now transport the type to the classical forms upstairs. The point \(x\) induces a Hecke coefficient point through \(\mathbb T\to D\). By (OpenAI 2026a, Proposition 4.2), its completed definite-forms eigenspace contains a Banach tensor product \[\widehat\bigotimes_{v\mid2}\Pi_v,\] where \(\Pi_v\) is a simple with parameter \(q_x\). All these parameters are identical, and an absolutely irreducible parameter has a unique Banach simple. Thus each \(\Pi_v\) contains the type already found, and their tensor contains \[\bigotimes_{v\mid2}(\sigma\otimes U).\] These are locally algebraic vectors in the eigenspace of completed forms. The classical comparison (OpenAI 2026a, Proposition 4.1) identifies them with definite automorphic forms of algebraic factor \(\bigotimes U\). The weights are regular: the factor \(\mathop{\mathrm{Sym}}^n\) gives classical weight \(n+2\geq2\), while the determinant power only changes the common twist. In the classical decomposition the smooth factor contains \(\sigma\) at every dyadic place. Indeed the Lie algebra identifies the algebraic factor, and the remaining \(K_0\)-action is its smooth multiplicity space. Reduced-norm characters cannot contain this type. Jacquet–Langlands therefore gives a regular cuspidal Hilbert eigensystem over \(F\). Its Galois representation agrees with \(r_x|_{G_F}\) by the Hecke identities and Chebotarev.

The finite-group cuspidality of \(\sigma\), established above, now identifies the local smooth factors. The depth-zero type theorem identifies its characteristic-zero smooth occurrences with depth-zero supercuspidal representations (Henniart 2002, Appendix A, Sections A.3.1–A.3.2). The local Langlands correspondence preserves depth (Aubert et al. 2016, Theorem 2.9); its corresponding Weil parameter therefore has depth zero, which means that it is trivial on wild inertia. The dyadic local–global comparison in (OpenAI 2026a, Proposition 4.1) now gives regular de Rhamness and this wild-inertia assertion for \(r_x|_{G_F}\) at every dyadic place. The splitting of \(2\) in \(F\) gives exactly those properties for \(r_x|_{G_{\mathbb Q_2}}\). The cyclotomic twist in (8) has unramified Weil–Deligne character and does not change the wild-inertia assertion.

Finally, finite ramification is built into \(D\), and its fixed determinant \(\chi\) is odd. All the hypotheses of (OpenAI 2026a, Theorem 1.1) have now been verified for \(r_x\): continuity, absolute irreducibility, oddness, finite ramification, and two distinct de Rham weights at \(2\). That theorem gives the asserted classical modularity up to Tate twist, with no restriction on the residual representation. ◻

Proof. We evaluate all the forms on one ordinary tower. Density of the tame-level forms on that tower will then turn boundedness of the Hecke operators into the continuity required here.

The ordinary tower. Enlarge an integer coefficient ring \(\mathcal O\) as necessary. Over the formal ordinary locus of the compactified tame modular curve of level \(\Gamma_1(N)\), consider the tower of trivializations \[\iota:\widehat{\mathbb G}_m\xrightarrow{\sim}\widehat E\] of the formal group of the universal generalized elliptic curve. The ordinary locus here includes the cusps. Write \(\mathcal I\) for the ring of integral functions on this tower, completed for the coefficient-uniformizer topology, and put \(\mathcal B=\mathcal I[1/2]\), with unit ball \(\mathcal I\). The tower is the inverse system of finite étale trivialization torsors of the ordinary connected \(2\)-divisible group. One can see this by Cartier duality: that group’s dual is étale of height one. This also proves the assertion at a multiplicative cusp. In particular, the tower and \(\mathcal I\) are flat over \(\mathcal O\). We use the ordinary-tower construction of (Katz 1973, sec. 4.2, Lemma 4.2.1 and Theorem 4.2.2). For Drinfeld level structures we use (Katz and Mazur 1985, sec. 3.2 and Lemma 3.5.1); the compactified moduli interpretation, including cusps in bad characteristic, is (Conrad 2007, Theorem 1.2.1 and Definition 2.4.1).

A section of the \(i\)th power of the Hodge line \(\omega\) evaluates to a function on this tower by expressing its pullback using the standard differential \(dT/(1+T)\) on \(\widehat{\mathbb G}_m\).

Density. We claim that the \(\operatorname{Frac}(\mathcal O)\)-linear span of the evaluations of tame-level classical forms of positive weights is dense in \(\mathcal B\). Only rational density is needed.

Here are the details at the prime two. Let \(H\) be the weight-one Hasse invariant on the residue curve. Its evaluation on the tower is \(1\), and the normalized weight-four Eisenstein series \(E_4\) reduces to \(H^4\); see (Katz 1973, sec. 2.1). The open set where \(E_4\) is nonvanishing is affine, since \(\omega\) is ample. Its formal completion is the ordinary base; denote its ring of functions by \(\mathcal I_0\). Serre vanishing for a sufficiently high power of \(\omega\) lifts \(H^{4m+1}\) to an integral form \(G\) of weight \(4m+1\). Consequently \[\eta=G/E_4^m\] is a basis of \(\omega\) on the formal ordinary base. Choose a parameter \(z\) on the formal group whose cotangent is \(\eta\). Such a parameter is obtained by lifting this cotangent successively on the affine base. Express the universal trivialization as \[\iota^*z=a_1T+a_2T^2+\cdots.\] The ring \(\mathcal I\) is topologically generated over \(\mathcal I_0\) by the coefficients \(a_j\) and \(a_1^{-1}\): formal group isomorphisms are represented by their coefficients, the formal group identities, and invertibility of the first coefficient. Over the ordinary base modulo \(\varpi^h\), the ideals cutting out the \(2^n\)-torsion are cofinal with the powers of the parameter ideal: the formal group has height one and \(2\) is nilpotent on this base. Thus an isomorphism of the formal groups is a compatible system of isomorphisms of their finite flat \(2^n\)-torsion groups. Cartier duality identifies this functor with the inverse limit of the finite étale trivialization torsors. The coefficient presentation and the tower therefore have the same coordinate ring modulo every \(\varpi^h\). After inverting \(2\), formal logarithms express each individual \(a_j\) as a polynomial in \(a_1\) with coefficients in \(\mathcal I_0[1/2]\).

Let \(\mathcal C\) be the closed linear span under consideration. It is an algebra, since multiplication adds positive weights. The evaluations of \(E_4\) and \(G\) are \(1\) modulo the uniformizer. Their powers with exponents \(2^n\) tend to \(1\), so \(1\in\mathcal C\); their powers with exponents \(2^n-1\) then show that their inverses also belong to \(\mathcal C\). Ratios of equal-weight sections by powers of \(E_4\) give all functions on the affine base, by the section-ring description of an ample line bundle. Completion gives \(\mathcal I_0[1/2]\subseteq\mathcal C\). Since \(\eta=G/E_4^m\) evaluates to \(a_1\), both \(a_1\) and its inverse belong to \(\mathcal C\). The logarithm identities now give every \(a_j\in\mathcal C\), and topological generation proves \(\mathcal C=\mathcal B\). No uniform bound on the denominators of all logarithm coefficients is required: each coefficient is handled separately before taking the closure.

Dyadic level and Hecke operators. We next evaluate the given form on the same tower and check that its Hecke operators preserve the integral unit ball. For forms of level \(\Gamma_1(2^aN)\), choose a primitive \(2^a\)th root of unity in the coefficient ring and transport its multiplicative torsion point by \(\iota\). This supplies the extra Drinfeld \(\Gamma_1(2^a)\)-structure. The construction works on generalized elliptic curves at cusps: the original tame level is ample, and adding a subgroup in the identity component preserves ampleness. The formal group of a Tate polygon is \(\widehat{\mathbb G}_m\) by (Conrad 2007, Equation (2.5.3)). The integral sections of the Hodge line recover classical forms after inverting \(2\) (Conrad 2007, sec. 4.4, Equation (4.4.2)); hence a fixed scalar multiple of a classical form extends over the integral model and has evaluation in \(\mathcal I\). Evaluation is nonzero for a nonzero form, as its \(q\)-expansion at a compatible multiplicative cusp shows.

The good Hecke operators preserve \(\mathcal I\) and commute with all these evaluations. For \(T_\ell\) the correspondence sums over degree-\(\ell\) isogenies, with normalization \(\ell^{-1}\), a unit in \(\mathcal O\). The isogenies induce isomorphisms of formal groups, and the finite flat correspondence and its trace are integral. The extension over multiplicative cusps, and the normalization of trace–pullback as \(\ell T_\ell\), are (Conrad 2007, Theorem 4.4.3 and Equation (4.5.1)). For \(\ell S_\ell\), the diamond operator and the scalar change of trivialization contribute \(\ell^i\ell^{-1}\) in weight \(i\), which is the required \(\ell^{i-1}\langle\ell\rangle\). This includes the diamond action on the additional \(2\)-power level. Thus every integral polynomial in these operators is a contraction of \(\mathcal B\).

Continuity. Let \(h_\alpha\) be a net of such polynomials that tends to zero at each finite weight stage. On every finite sum of evaluated tame classical forms, \(h_\alpha\) tends to zero. Density and the uniform contraction bound imply \[h_\alpha v\longrightarrow0\qquad(v\in\mathcal B):\] first approximate \(v\) by a finite sum, then use the contraction bound on the error. Apply this to the nonzero evaluation \(v_f\) of \(f\). Since \(h_\alpha v_f=\lambda_f(h_\alpha)v_f\), the eigenvalues tend to zero. The same argument shows that a polynomial relation in \(A(N)\) acts as zero on \(v_f\). The eigenvalue map therefore extends continuously from the dense polynomial algebra to \(A(N)\). Its values are integral, either by the contraction bound or by classical integrality of good Hecke eigenvalues. Compatibility of evaluation with the Hecke action proves the asserted Frobenius identities. ◻

Proof. Fix integral coefficients containing the residual values, and work in the factor \(A\) of the resulting Hecke algebra selected by \(\lambda\). Let \(R^{\mathrm{ps}}\) be the universal global two-dimensional determinant deformation ring for these residual data, with ramification allowed at the primes dividing \(2N\) and with varying determinant. The structural facts above give a continuous surjection \[R^{\mathrm{ps}}\twoheadrightarrow A, \qquad J=\ker(R^{\mathrm{ps}}\longrightarrow A).\]

First suppose \(\theta\) has finite order, and enlarge coefficients to contain its values. Twisting the universal determinant over \(A\) defines a continuous map \(R^{\mathrm{ps}}\to A\) over these enlarged coefficients. There is no change of residual factor: every finite quotient of \(\mathbb Z_2^\times\) is a \(2\)-group, so a finite-order character reduces to \(1\) in characteristic two. At every classical cuspidal point of weight at least three, the twisted determinant belongs to a classical finite-character twist. Its extra conductor is a power of \(2\); Proposition 17 therefore supplies a coefficient point of the same \(A(N)\). For \(j\in J\), its image under the twisted map vanishes at all these points. Their density and reducedness show that this image is zero. Thus twisting factors through \(A\) and preserves all its coefficient points. Equality of the specialized determinants follows from (9) and Chebotarev.

Now fix \(\lambda\), and fix a sign \(s\in\{1,-1\}\). Every \(u\in\mathbb Z_2^\times\) has a unique expression \(u=(-1)^e5^b\), with \(e\in\{0,1\}\) and \(b\in\mathbb Z_2\). Define the universal character \[\Theta_X((-1)^e5^b)=s^e(1+X)^b \quad\text{in }\mathcal O_{E'}[[X]]^\times.\] Twisting the determinant of \(\lambda\) by \(\Theta_X\circ\varepsilon\) gives a continuous map \(R^{\mathrm{ps}}\to\mathcal O_{E'}[[X]]\). For every \(2\)-power root of unity \(\zeta\), evaluation at \(X=\zeta-1\) is a finite-order twist, so the image of every \(j\in J\) vanishes at all such values. A nonzero series in \(\mathcal O_{E'}[[X]]\) cannot do this. Indeed, after dividing by the largest common uniformizer power of its coefficients, Weierstrass preparation expresses it as a distinguished polynomial times a unit. The polynomial has only finitely many roots in the open unit disk. Therefore the image of \(J\) is zero, and the universal twisted map factors continuously through \(A\).

For the given continuous \(\theta\), its image on the pro-\(2\) group \(1+4\mathbb Z_2\) reduces trivially in the odd-order group \(k_{E'}^\times\). Hence \(\theta(5)-1\) lies in the maximal ideal of \(\mathcal O_{E'}\). Specialize \(s=\theta(-1)\) and \(X=\theta(5)-1\). This continuous specialization gives the required point. All ring topologies used here are the completed local topologies, which agree with the prescribed finite-stage topology by (OpenAI 2026b, Proposition 2.3). ◻

Proof. Set \(W=\mathop{\mathrm{ad}}r_x=\mathop{\mathrm{End}}_{E_x}(r_x)\), the full four-dimensional adjoint representation. Tameness of the Weil–Deligne parameter will give the CM qualification for adjoint Selmer vanishing; local irreducibility and distinct de Rham weights will give the local cohomology bound. For a finite place \(v\), the finite local condition is \[H^1_f(\mathbb Q_v,W)= \begin{cases} \ker\bigl(H^1(\mathbb Q_v,W)\to H^1(I_v,W)\bigr),&v\ne2,\\ \ker\bigl(H^1(\mathbb Q_2,W)\to H^1(\mathbb Q_2,W\otimes_{\mathbb Q_2}B_{\mathrm{cris}})\bigr),&v=2. \end{cases}\] Here \(B_{\mathrm{cris}}\) is Fontaine’s crystalline period ring. The global group \(H^1_f(\mathbb Q,W)\) imposes these conditions at all finite places. Positive-degree cohomology at the real place is zero over \(E_x\).

Newton–Thorne’s theorem (Newton and Thorne 2023, Theorem 5.4), specialized to \(\mathop{\mathrm{GL}}_2/\mathbb Q\), states that for a regular algebraic cuspidal automorphic representation \(\pi\) and any prime \(p\), \[H^1_f(\mathbb Q,\mathop{\mathrm{ad}}r_{\pi,p})=0\] provided either \(\pi\) is non-CM, or its CM field \(K\) is not contained in \(\mathbb Q(\zeta_{p^\infty})\). This statement uses the full adjoint and has no residual irreducibility hypothesis. Our classical representation is regular algebraic, including when its classical weight is two, and twists do not change its adjoint. It remains only to verify the CM qualification.

If its CM field were contained in \(\mathbb Q(\zeta_{2^\infty})\), the corresponding quadratic self-twist would restrict to a nontrivial wildly ramified character at \(2\). Indeed the three quadratic subfields are \(\mathbb Q(i)\), \(\mathbb Q(\sqrt2)\), and \(\mathbb Q(\sqrt{-2})\); each ramifies at \(2\), and a quadratic ramified character there is necessarily wild. The Weil–Deligne parameter would then be isomorphic to its twist by that character. On a wild inertia element where the character is \(-1\), these two parameters act as \(I\) and \(-I\), respectively, by the hypothesis on \(x\). This is impossible. Newton–Thorne therefore gives \[ H^1_f(\mathbb Q,W)=0. \tag{11}\]

We next compute the local quotients. The trace pairing identifies \(W^*\) with \(W\), so local Tate duality identifies the dual of \(H^2(\mathbb Q_v,W)\) with \(\mathop{\mathrm{Hom}}_{G_{\mathbb Q_v}}(r_x,r_x(1))\). At \(2\), a nonzero such map is an isomorphism by absolute irreducibility; its determinants would force \(\varepsilon^2=1\) on \(G_{\mathbb Q_2}\). Thus \(H^2(\mathbb Q_2,W)=0\), including in weight two. At \(v\ne2\), local–global compatibility (Carayol 1986, Theorem A) and genericity of each local factor of a cuspidal automorphic representation give the same vanishing: the genericity criterion (Allen 2016, Lemma 1.1.3) says that its Frobenius-semisimple Weil–Deligne parameter \(U\) satisfies \(\mathop{\mathrm{Hom}}_{\mathrm{WD}}(U,U(1))=0\). A Galois morphism would induce such a morphism, also after Frobenius semisimplification. Tate twists do not affect these adjoint computations. This is the non-dyadic argument of (OpenAI 2026b, Lemma 5.2).

Write \(h^j_v=\dim_{E_x}H^j(\mathbb Q_v,W)\). For \(v\ne2\), local Euler characteristic gives \(h^1_v=h^0_v\), whereas \[H^1_f(\mathbb Q_v,W)=W^{I_v}/(\operatorname{Frob}_v-1)W^{I_v}\] also has dimension \(h^0_v\). Hence the local quotient is zero. At \(2\), Schur’s lemma gives \(h^0_2=1\), and the Euler characteristic formula gives \(h^1_2=1+\dim W=5\). The filtration degrees of the filtered de Rham module \(D_{\mathrm{dR}}(W)\) are \(0,0,h,-h\) for some nonzero integer \(h\). The Bloch–Kato dimension formula (Bloch and Kato 1990, Corollary 3.8.4) gives \[\dim H^1_f(\mathbb Q_2,W) =h^0_2+ \dim D_{\mathrm{dR}}(W)/\operatorname{Fil}^0=1+1=2.\] We have proved \[ \dim H^1(\mathbb Q_v,W)/H^1_f(\mathbb Q_v,W) =\begin{cases}0,&v\ne2,\\3,&v=2.\end{cases} \tag{12}\] The kernel of the restriction map \[H^1(G_{\mathbb Q,S},W)\longrightarrow \bigoplus_{v\in S} H^1(\mathbb Q_v,W)/H^1_f(\mathbb Q_v,W)\] lies in \(H^1_f(\mathbb Q,W)\), because classes in its source are already unramified outside \(S\). Equations (11) and (12) now give the bound three.

For the embedding-dimension assertion, apply the characteristic-zero tangent comparison of (OpenAI 2026b, Proposition 4.1) to \(R_{\mathrm{glob}}\). Its trace and determinant values topologically generate the ring, and the specialization at \(\mathfrak x\) is absolutely irreducible. The comparison bounds the embedding dimension by \(\dim_{E_x}H^1(G_{\mathbb Q,S},W)\), with no condition on the semisimplified residue. The continuity needed here follows because \(R_{\mathrm{glob}}/\mathfrak x^2\) is finite over \(\mathcal O\): \(R_{\mathrm{glob}}/\mathfrak x\) is finite over \(\mathcal O\), and \(\mathfrak x/\mathfrak x^2\) is finite over that quotient. Finite extension of \(\mathcal O\) adds no tangent directions, since characteristic-zero derivations kill its fraction field by separability over \(\mathbb Q_2\). The preceding cohomological bound therefore gives the asserted embedding dimension. ◻

Proof. Choose a minimal prime \(P\) of \(A_0\) contained in its prime \(\mathfrak x_0\). The complete local domain \(A_0/P\) is catenary. Moreover \(A_0/\mathfrak x_0\) is an \(\mathcal O\)-submodule of \(\mathcal O'\), hence finite over \(\mathcal O\), and it contains \(\mathcal O\). It therefore has dimension one. The dimension formula gives \[\dim(A_0/P)_{\mathfrak x_0} =\dim A_0/P-\dim A_0/\mathfrak x_0=4-1=3.\] As \(R_0\) surjects onto \(A_0\), we obtain \[3\leq\dim (A_0)_{\mathfrak x_0} \leq\dim (R_0)_{\mathfrak x_0} \leq\mathop{\mathrm{edim}}(R_0)_{\mathfrak x_0}\leq3.\] Thus \((R_0)_{\mathfrak x_0}\) is a regular local ring of dimension three, in particular a domain. Its quotient \((A_0)_{\mathfrak x_0}\) has the same dimension, so the localized kernel \(\ker(R_0\to A_0)_{\mathfrak x_0}\) is zero: a nonzero ideal in this regular local domain has quotient of smaller dimension.

If \(a\in\ker(R_0\to A_0)\), there is consequently \(s\notin\mathfrak x_0\) with \(sa=0\) in \(R_0\). The kernel of \(R_0\to D_0\) is contained in \(\mathfrak x_0\), so the image of \(s\) in \(D_0\) is nonzero. Since \(D_0\) is a domain, the image of \(a\) must vanish. This proves the factorization. ◻

Proof of Theorem 1. Apply Proposition 5 to the original representation, after its indicated continuous cyclotomic normalization and finite coefficient extension. It supplies the horizontal trace domain \(D\) with the normalized original coefficient point. Propositions 6, 13, and 16 supply a coefficient point \(x\) of \(D\) satisfying all the hypotheses of Proposition 19.

Choose an odd \(N\geq5\) sufficiently divisible that it contains every odd ramified prime of the original family and that the classical representation underlying \(r_x\) has level dividing \(2^aN\) for some \(a\). Increase the allowed global ramification set to the primes dividing \(2N\); the existing family remains unramified at any newly added primes. By Propositions 17 and 18, \(r_x\) gives a continuous point of \(\mathbb T_2(N)\) with the required Frobenius values.

Fix the coefficient ring \(\mathcal O\) of \(D\) and the residual embedding selected by \(x\). Let \(A\) be the corresponding residue factor of \(\mathcal O\otimes_{\mathbb Z_2}\mathbb T_2(N)\), and let \(R_{\mathrm{glob}}\) be the universal global determinant deformation ring for this residual determinant with varying determinant. Let \(\mathcal O_x\) be a coefficient ring for \(x\). The determinant on \(D\) and the Hecke determinant give a commutative diagram whose top arrows are continuous surjections: \[\begin{array}{ccc} &R_{\mathrm{glob}}&\\[-2pt] \swarrow&&\searrow\\[-2pt] D&&A\\[-2pt] \searrow&&\swarrow\\[-2pt] &\mathcal O_x.& \end{array}\] The lower arrows are the coefficient point \(x\); equality follows from its Frobenius identities and Chebotarev. The quotient \(D\) has fixed determinant \(\chi\), whereas \(R_{\mathrm{glob}}\) and \(A\) retain determinant variation. Surjectivity onto \(A\) follows from topological generation by the Hecke elements and compactness, as in (OpenAI 2026b, Proposition 2.3); surjectivity onto \(D\) is its definition as a trace-image quotient.

Every component of \(A\) has dimension four by (OpenAI 2026b, Theorem 1.1) and finite coefficient extension. Proposition 19 gives \(\mathop{\mathrm{edim}}(R_{\mathrm{glob}})_{\mathfrak x}\leq3\). Lemma 20 therefore makes \(R_{\mathrm{glob}}\to D\) factor through \(A\). The normalized original coefficient point of \(D\) is consequently a point of the required Hecke algebra over \(\mathbb Q\).

Undo the normalization with Proposition 18. We obtain, over a finite extension \(E'/E\), a continuous map \[\lambda:\mathbb T_2(N)\longrightarrow\mathcal O_{E'}\] whose values on \(T_\ell\) and \(\ell S_\ell\) are respectively \(\mathop{\mathrm{tr}}r(\operatorname{Frob}_\ell)\) and \(\det r(\operatorname{Frob}_\ell)\) for \(\ell\nmid2N\). The compact image of \(r\) preserves an \(\mathcal O_E\)-lattice, so all these values lie in \(\mathcal O_E\). Since the Hecke generators topologically generate \(\mathbb T_2(N)\) and \(\mathcal O_E\) is closed in \(\mathcal O_{E'}\), the entire image of \(\lambda\) lies in \(\mathcal O_E\). The induced map to \(\mathcal O_E\) is continuous for its \(2\)-adic topology. Our choice of \(N\) includes every odd ramified prime of the original \(r\), which completes the proof. ◻