Proof. Fourier inversion in characteristic zero expresses every group-basis coefficient of \(u\) as a sum of the \(u_x\) divided by \(|G|\). Thus it belongs to \(\Lambda[1/2]\). The corresponding coefficient of \(2^C u\) also belongs to \(\mathcal O\). For a power series with bounded \(2\)-denominator, nonnegative \(\mathcal O\)-valuation means that every coefficient is integral. Consequently \(\mathcal O\cap\Lambda[1/2]=\Lambda\), which proves the assertion with the original exponent \(C\). ◻

Proof. Choose generators of \(G\). Every character-value function \(z(x)\) has an expansion \[z(x)=a_\varnothing+ \sum_{\varnothing\ne I\subseteq\{1,\ldots,b\}} 2^{|I|}a_I\prod_{i\in I}x_i, \qquad a_I\in\mathbb Z _2.\] This follows by expanding each sign character as a product of factors \(1-2x_i\). The \(j\)th binary digit of a \(2\)-adic integer \(a\) is \(\binom a{2^j}\bmod2\), equivalently the coefficient of \(Y^{2^j}\) in \((1+Y)^a\) over \(\mathbb F _2\). On the Boolean cube, the contribution of the monomial \(x_I=\prod_{i\in I}x_i\) to this generating function is \[1+x_I\bigl((1+Y)^{2^{|I|}a_I}-1\bigr).\] Its nonconstant terms have \(Y\)-degree at least \(2^{|I|}\). In a contribution to \(Y^{2^j}\), the sum of the quantities \(2^{|I|}\) used is at most \(2^j\). Since \(|I|\leq2^{|I|}\), the resulting Boolean polynomial has degree at most \(2^j\). These formulas for \(2\)-adic exponents follow either by continuity from nonnegative integers or in each finite coefficient quotient.

Express all \(rM\) digit equalities as Boolean equations \(P_{ij}(x)=0\). Their total degree is at most \(r(2^M-1)\). The indicator of their common zero set is \(\prod_{i,j}(1+P_{ij})\). The remaining count is the Chevalley–Warning degree argument (Chevalley 1935; Warning 1935). A Boolean polynomial of degree less than \(b\) has even sum over \(\mathbb F _2^b\): every monomial omits a variable, and summing over that variable gives zero in \(\mathbb F _2\). Hence the number of common zeros is even. Since \(0\) is a common zero, another one exists. ◻

Proof. Here are the finiteness and coefficient issues in the use of global cochains. For a finite coefficient quotient of \(R\), continuous cochains are filtered unions of finite products of free coefficient modules. Refinement of finite partitions makes these modules flat; over an Artin local coefficient ring they are free. For finite residual coefficients, global cohomology is finite in each degree, and restriction to the real places is an isomorphism in degrees greater than two. The latter is the real-place clause of global duality. Thus the residual fiber has bounded amplitude and finite-dimensional cohomology. The same amplitude bound holds for every finite coefficient module, by a composition series and the cohomology long exact sequence.

This gives a bounded finite free model as follows. Over the residue field split off the acyclic disks, leaving its finitely many cohomology spaces. Lift each invertible disk differential over an Artin coefficient quotient and cancel it. Invertibility of a lifted inverse follows from nilpotence of the maximal ideal. Equivalently one can perform this on a free resolution; a minimal free model cannot have an additional term whose reduction has zero cohomology, because its residual differential is zero. Hence the remaining ranks are the finite residual ranks and are bounded independently of the coefficient quotient. Minimal models identified by a homotopy equivalence over an Artin local ring are isomorphic. Lifting their changes of basis therefore makes these models compatible along \(R/\mathfrak m^n\). Taking the inverse limit gives a bounded finite free model over \(R\). The cochain construction with finite free coefficients and the same cancellation also proves derived base change. This is a fixed-\(S\) argument; it does not assert uniform bounds on the number of generators as \(S\) varies.

All the characters of the family have the same real sign, so the action at the real place is the fixed action on \(T\chi_{h_0}\) extended to \(R\). Choose a nonzero vector in its integral invariant line. Its connecting image in the fiber is \(a\). There are no global invariant vectors in the rational elliptic representation, and consequently this connecting map is injective rationally. The choice and its index are fixed. Notice also that \(H^1(\mathbb R,\mathbb T)\) is killed by \(2\); thus twice any ordinary cohomology class lifts to the positive complex. These observations use compact global duality, including the real-place extension (Nekovář 2006, sec. 5.7 and 6.9.1) of the complex formulation in (Nekovář 2006, Theorem 6.3.4). ◻

Proof. The ring \(\mathcal O[G]\) is local, with residue field \(\kappa\) obtained by setting \(2=0\) and every \(g=1\). Thus the quadratic characters disappear. Since \(E[2]\) has a filtration with trivial factors, it is enough first to compute the positive residual complex for the trivial representation.

Let \(K_n\) be the real cyclotomic layer of degree \(d_n=2^n\). The number of finite primes of \(K_n\) above a fixed \(S\) is bounded with \(n\): at \(2\) there is total ramification, and the Frobenius of each fixed odd prime has nonzero image in \(\Gamma\). The \(2\)-ranks of the class groups of \(K_n\) are also bounded. Indeed the cyclotomic \(\mu\)-invariant is zero for this abelian tower (Ferrero and Washington 1979); the structure theorem and class-group control identify their \(2\)-ranks, up to bounded kernels and cokernels, with those of the coinvariants of a finitely generated \(\mathbb Z _2\)-module. The same bound holds for \(S\)-class groups.

The signature map of units in \(K_n\) is surjective. For \(n\geq1\), let \(\zeta\) be a primitive \(2^{n+2}\)th root and put \(\epsilon=1+\zeta+\zeta^{-1}\). If \(f_1(X)=X^2-2\) and \(f_{n+1}(X)=f_n(X^2-2)\), then \(f_n\) is the minimal polynomial of \(\zeta+\zeta^{-1}\) and \(f_n(-1)=-1\). Hence \(N_{K_n/\mathbb Q}(\epsilon)=-1\). Its signature vector has odd augmentation. The embeddings form a regular set for the cyclic \(2\)-group \(\operatorname{Gal}(K_n/\mathbb Q)\), whose group algebra over \(\mathbb F _2\) is local. A vector with odd augmentation is a unit in this group algebra, so its translates span every signature. For \(n=0\) the assertion follows from the unit \(-1\).

We now use Kummer theory to isolate the one dimension that survives after inverting \(t\). Writing \(s_n\) for the number of finite support places and \(c_n\) for the \(2\)-rank of the \(S\)-class group, one has \[\dim H^1(G_{K_n,S},\mu_2)=d_n+s_n+c_n.\] The real degree-one restriction is surjective by the signature result. In degree two, the Kummer sequence has class-group term of dimension \(c_n\) and Brauer term. The Brauer invariants at the real and finite support places have their single sum relation. Since a finite support place above \(2\) is present, restriction of the Brauer term to the real places is surjective and has kernel of dimension \(s_n-1\). Thus the degree-two kernel of restriction to infinity is bounded. Restriction is an isomorphism in higher degrees by global duality. The degree-zero restriction is injective. The cohomology of the positive fiber is therefore bounded in every degree except one; its degree-one dimension is \(d_n+O_S(1)\).

Apply Lemma 8 modulo \(2\). Derived specialization to the \(n\)th layer is specialization modulo \(t^{2^n}\). For a perfect complex over \(\mathbb F _2[[t]]\), the universal-coefficient exact sequence shows that a rank-\(r\) cohomology module contributes \(r2^n+O(1)\) to these dimensions; the torsion terms contribute only \(O(1)\). It follows that after inverting \(t\) the trivial-constituent fiber has dimension one in degree one and vanishes in every other degree. Extensions preserve this concentration, so the fiber for \(E[2]\) has dimension two in degree one. A minimal finite free complex over the local ring \(\mathcal O[G]\) has zero residual differentials. The concentration just proved forces all its terms except the rank-two term in degree one to vanish. No denominator is introduced by this exact freeness statement. ◻

Proof. Abbreviate \(T_h=T_2E^{(h_x)}\) and \(A_h=E^{(h_x)}[2^\infty]\). Use the integral Kummer subgroup \[K_v=\operatorname{im}\bigl(E^{(h_x)}(\mathbb Q _v)^{\wedge}_2 \longrightarrow H^1(\mathbb Q _v,T_h)\bigr)\] at every finite place. For \(v\ne2\) it is all of \(H^1(\mathbb Q _v,T_h)\). One way to see this integrally is the inverse-limit Kummer exact sequence: its quotient is \(T_2H^1(\mathbb Q _v,E^{(h_x)})\), which is zero because local Tate duality identifies the \(2\)-primary local Weil–Châtelet group with the dual of the finite \(2\)-completion of the local point group. At \(2\) the quotient \[L_2=H^1(\mathbb Q _2,T_h)/K_2\] is a free rank-one \(\mathbb Z _2\)-module. The same Kummer sequence and local Tate duality identify it with the \(\mathbb Z _2\)-dual of the free part of \(E^{(h_x)}(\mathbb Q _2)^{\wedge}_2\).

The dual exponential identifies \(L_2\otimes\mathbb Q _2\) with the one-dimensional de Rham target. Integrally its index depends only on the local curve and the chosen differential. This follows from the compatibility of the Kummer map and local Tate duality with the formal elliptic logarithm: on a sufficiently small open formal subgroup that logarithm is an isomorphism, and its adjoint is the dual exponential. Every elliptic curve over \(\mathbb Q _2\) has such a formal subgroup, irrespective of reduction type. Since only finitely many local twists occur, the indices are uniformly bounded.

The assumption on the Selmer corank implies that the compact finite Selmer group is torsion and that the discrete finite Selmer group has length \(s(h_x)\). Rational Poitou–Tate consequently identifies ordinary global \(H^1\) with the singular line at \(2\), and gives ordinary global \(H^2=0\) rationally. Adding the real invariant line gives the claimed dimension two in degree one. This equals the generic rank, so derived base change of the perfect complex specializes its determinant without a change of rational cohomological rank.

The integral calculation explains the two subtractions in \(D(h_x)\). The index of global localization will cancel; the remaining local terms will cancel the unused Euler factors and contribute \(2w(h_x)\) at the active primes. Let \(J\) be the image of the free part of global \(H^1(G_{\mathbb Q,S},T_h)\) in \(L_2\), and put \(i=\operatorname{length}(L_2/J)\). The compact/discrete Poitou–Tate sequence for the mutually orthogonal Kummer conditions contains \[\begin{aligned} 0\longrightarrow L_2/J &\longrightarrow \operatorname{Sel}_{2^\infty}(E^{(h_x)}/\mathbb Q)^\vee \longrightarrow H^2(G_{\mathbb Q,S},T_h)\\ &\longrightarrow \bigoplus_{q\in S}H^2(\mathbb Q _q,T_h) \longrightarrow H^0(\mathbb Q,A_h)^\vee\longrightarrow0, \end{aligned}\] with the usual real Tate-cohomology terms inserted. Those terms are killed by \(2\) and have dimensions bounded by the fixed dimension of \(T\). Taking lengths and using local duality gives \[ \operatorname{length}H^2(G_{\mathbb Q,S},T_h) =s(h_x)-i+\sum_{q\in S}\operatorname{length}H^0(\mathbb Q _q,A_h) +O_E(1). \tag{14}\] This sequence is the cohomology consequence of integral Poitou–Tate duality with the finite local conditions; see (Nekovář 2006, Theorem 6.3.4 and Section 6.9.1), with the real Tate-cochain convention of (Nekovář 2006, sec. 5.7.1.6). No unramified condition replaces the Kummer condition at a bad prime. Global torsion is bounded by its injection into \(E^{(h_x)}(\mathbb Q _2)[2^\infty]\), whose order is bounded by the fixed local twisting classes. The finite terms at \(2\) and at fixed bad primes are bounded for the same reason. Passing between ordinary and positive cohomology changes only these bounded real and global torsion terms.

Suppose first that the central value is nonzero. The coordinate of \(z_{S,x}(0)\) in the free global line has valuation \[v_2(\exp_2^*z_{S,x}(0))-i+O_E(1).\] For the inverse determinant, torsion in degree two subtracts its length and torsion in degree one adds its length. This sign can be checked on the complex \([\mathbb Z _2\xrightarrow{2^a}\mathbb Z _2]\) in degrees one and two: its inverse determinant lattice maps to \(2^a\mathbb Z _2\) in the rational determinant. A vector’s coordinate thus loses \(a\). Using (14), the two appearances of \(i\) cancel, leaving \[v_2(u_x(0))= v_2(\exp_2^*z_{S,x}(0))-s(h_x) -\sum_{q\in S}\operatorname{length}H^0(\mathbb Q _q,A_h)+O_E(1).\]

At an active good prime, quadratic inertia acts by \(-1\) on the Tate module. Its invariants on \(A_h\) are therefore exactly \(E[2]\), and taking Frobenius invariants gives length \(t_q\). At an unused good prime, reduction identifies the local \(2\)-primary torsion with \(E^{(h_x)}(\mathbb F _q)[2^\infty]\), of length \(v_2(\#E^{(h_x)}(\mathbb F _q))\). These unused terms cancel exactly those in (4). The active terms, including the primes of \(h_0\), sum to \(2w(h_x)\). All other terms have already been bounded independently of the growing support. This proves (13).

If the central value is zero, reciprocity makes the dual exponential zero. The rational ordinary specialization then satisfies the finite condition at \(2\), and it satisfies the finite conditions elsewhere automatically. It lies in the rational finite Selmer group, which is zero. Its wedge with the real-place class is therefore zero. The constant-rank determinant specialization proves \(u_x(0)=0\). ◻

Proof of Theorem 4. Use the preceding construction for the single discriminant \(h\). The forward analytic rank-zero theorem gives \(s_2(E^{(h)})=0\) and finiteness of its Shafarevich–Tate group. By (12), \(v_2(u_h(0))\geq-C_{\mathrm{int}}\). Proposition 10 then gives \(D(h)\geq-C_{\mathrm{int}}-C_{\mathrm{loc}}\). Every constant in the construction came from the fixed form, a fixed real period line, or the finite set of local twists at \(2N\). ◻

Proof of Theorem 5. At every nonzero address the forward rank-zero theorem gives Selmer corank zero. Proposition 10 gives \[v_2(u_x(0))\leq B+C_{\mathrm{loc}}\qquad(x\ne0).\] Choose an integer \(M\) such that \(M-C_{\mathrm{int}}>B+C_{\mathrm{loc}}\). By (12), the constant coefficient of \(2^{C_{\mathrm{int}}}u\) belongs to \(\mathbb Z _2[G]\). For sufficiently large \(b\), Lemma 7 gives an \(x\ne0\) such that \[v_2(u_x(0)-u_0(0))\geq M-C_{\mathrm{int}}.\] It follows that \(u_0(0)\) is nonzero and \(v_2(u_0(0))\leq B+C_{\mathrm{loc}}\): either failure would contradict the displayed upper bound at this \(x\). The corank-zero hypothesis at the base and Proposition 10 imply \(L(E^{(h_0)},1)\ne0\) and \[D(h_0)\leq B+2C_{\mathrm{loc}}.\] Taking \(C_E\) large enough proves both theorems with a common constant. In particular no loss proportional to \(b\), \(|Q|\), or the prime count of \(h_0\) has entered this argument. ◻

Proof. Flat modules over an Artin local ring are free, including modules of infinite rank. Here is a useful version of the argument. Lift a basis of \(M/\mathfrak mM\) to a flat module \(M\), and map the corresponding free module to \(M\). Its cokernel is zero because \(\mathfrak m\) is nilpotent. Flatness shows that the kernel has zero reduction, and nilpotence again makes the kernel zero.

Split the residual complex as its cohomology, with zero differential, plus contractible two-term disks. Lift all the graded bases. These lifts are graded isomorphisms: an inverse modulo \(\mathfrak m\) lifts to an inverse by a finite geometric series. Let \(d_0\) be the lifted disk differential and write \(d=d_0+\epsilon\). All coefficients of \(\epsilon\) belong to \(\mathfrak m\). For the disk contraction \((i_0,p_0,h_0)\), arrange \(h_0^2=h_0i_0=p_0h_0=0\) and \(1-i_0p_0=d_0h_0+h_0d_0\). The operators \(1+h_0\epsilon\) and \(1+\epsilon h_0\) have finite geometric inverses. The elementary perturbation identities give the differential \[d_C=p_0\epsilon(1+h_0\epsilon)^{-1}i_0\] on the lifted residual cohomology, and give \[i=(1+h_0\epsilon)^{-1}i_0,\quad p=p_0(1+\epsilon h_0)^{-1},\quad h=h_0(1+\epsilon h_0)^{-1}.\] Substitution, using \((d_0+\epsilon)^2=0\), proves the asserted contraction identities. These calculations are valid for infinite disk sums: each operator is a module homomorphism and every geometric series terminates at the nilpotence exponent of \(\mathfrak m\).

For a map between original complexes use \(p'fi\); insert the contraction homotopies when composing maps. A quasi-isomorphism between bounded free complexes has a contractible cone and hence a homotopy inverse. This gives the stated finite-diagram assertion, including the homotopies expressing triangles and perfect duality. ◻

Proof. There are finitely many possible graded ranks. Fix them on an ultrafilter-large set. At a fixed precision \(j\), all matrix entries belong to the finite ring \(A_j\) (or \(A_j[G]\)), so have unique ultrafilter limits. These limits are compatible as \(j\) varies. Matrix identities, including the homotopy-inverse identities, are finite polynomial identities and pass to the limit. New models and maps may be mapped to old models by their already fixed contractions. Passing to any slower cofinal precision has the same old limits. For the all-sequence evaluation statements below, fix this ultrafilter once and for all, and keep the original contraction at every stage. Later arithmetic diagrams must extend these same stages. Restriction to an ultrafilter-large set and passage to slower cofinal coefficient precision preserve every old evaluation on every old group sequence. An arbitrary later subsequence or a different ultrafilter is not allowed in this assertion. Notice that this argument preserves specified arithmetic maps; it does not create an arithmetic map from a formal description of its expected properties. ◻

Proof. The local analytic description of \(B(F)\) shows that \(M_F\) is a finitely generated \(\mathbf Z_2\)-module. Its torsion is the finite group \(B(F)[2^\infty]\). Consequently \(M_F\) has a free resolution of length at most one. Also \(H^0(F,T_2B)=0\), since \(B(F)[2^\infty]\) has bounded exponent. The integral Kummer injection into \(H^1(F,T_2B)\) therefore defines the displayed morphism using the truncation in degrees at most one of \(R\Gamma(F,T_2B)\).

Tensor the free resolution of \(M_F\) with \(\mathbf Z/2^n\). Its two cohomology groups are \(M_F[2^n]\) and \(M_F/2^nM_F\). The inverse limits of the ordinary Kummer exact sequences identify their maps with the two maps in the statement. In particular finite torsion invariants have not been discarded.

For orthogonality, compose cup product and the Weil pairing with the local invariant map. A morphism \[M_F[-1]\otimes^{\mathbf L}M_F[-1] \longrightarrow\mathbf Z_2[-2]\] is determined by its homomorphism \(M_F\otimes M_F\to\mathbf Z_2\): the possible \(\operatorname{Tor}_1\) is in degree one and gives no extra degree-zero morphism to this target. The indicated homomorphism vanishes, because the finite Kummer image annihilates itself at every level by local Tate duality (Milne 2006, I, Theorem 3.2 and Corollary 3.4). Choose a nullhomotopy in free models. The induced morphism \[R\Gamma(F,T_2B)/U_F \longrightarrow R\operatorname{Hom}_{\mathbf Z_2} (U_F,\mathbf Z_2)[-2]\] is a quasi-isomorphism. This can be checked after reduction modulo \(2\): degree one is the perfect pairing of the finite Kummer quotient with the Kummer image, and degree two is dual to \(B(F)[2]\). Its perfect cone is therefore zero by derived Nakayama. This proves exact orthogonality and also its compatibility with derived reduction. The derived duality of the ambient local cochains is (Nekovář 2006, Proposition 5.2.4); the Kummer condition and its orthogonality have been constructed above. No part of the proof uses an ordinary filtration or a classification of reduction types. ◻

Proof. We give the compatibility argument before imposing the Tate condition. Choose prime ideals \(\mathfrak p_1,\ldots,\mathfrak p_s\) with pairwise disjoint conjugate pairs, including the oriented prescribed primes and the primes above \(2N\), and enough additional split prime ideals to generate \(\operatorname{Cl}(K)\). Such additional ideals may avoid any fixed finite set. Put weight \(w_j=1\) at a prescribed odd prime and \(w_j=0\) at every additional prime and every prime above \(2N\). The conjugate weight is \(-w_j\). Let \[R=\ker\bigl(\mathbf Z^s\longrightarrow\operatorname{Cl}(K)\bigr),\] choose a basis \(m^{(a)}\) of \(R\), and choose \(\beta_a\in K^\times\) with \((\beta_a)=\prod_j\mathfrak p_j^{m_j^{(a)}}\).

Work first over \(F_\infty=K(\mu_{2^\infty})\) and adjoin \(2^n\)-th roots of the finitely many \(\beta_a/\overline\beta_a\). Prescribe on these roots the multipliers \[ \zeta_{2^n}^{\,-\sum_j m_j^{(a)}w_j}. \tag{23}\] The minus sign fixes the choice of connecting map in the ring-class sequence and could be reversed throughout. These prescriptions respect every Kummer relation. Indeed, if \[\prod_a(\beta_a/\overline\beta_a)^{b_a} \in(F_\infty^\times)^{2^n},\] take its valuation at a prime above each \(\mathfrak p_j\) of nonzero weight. These odd primes are unramified in \(F_\infty/K\). Thus \(2^n\) divides \(\sum_a b_am_j^{(a)}\) for each such \(j\), proving the required compatibility of (23). Primes over \(2\) have weight zero and impose no division by a ramification index. Units impose no additional weighted relation. Kummer duality supplies the prescribed automorphism of the radical extension over \(F_\infty\).

At a rational prime \(r\) split in a sufficiently large finite cyclotomic subfield, the power-residue character of \(\mathbf F_r^\times\) is surjective onto \(\mu_{2^n}\). The split ring-class exact sequence is \[1\longrightarrow (\mathcal O_K/r\mathcal O_K)^\times/(\mathbf Z/r\mathbf Z)^\times \longrightarrow\operatorname{Pic}(\mathbf Z+r\mathcal O_K) \longrightarrow\operatorname{Cl}(K)\longrightarrow1.\] There is no unit correction under our exclusion of the extra-unit fields. The first group is \(\mathbf F_r^\times\), with a principal relation measured by \(\beta/\overline\beta\) at the oriented prime over \(r\). Consequently (23) says exactly that its primitive power-residue character extends to the class group with the specified values \(w_j\). Since the ring-class extension is unramified at \(2N\), value zero on Frobenius means that the resulting local character is trivial there. Conjugation acts by inversion.

We now show that imposing these finitely many radical conditions cannot force either forbidden Tate trace. In the non-CM case the image of \(G_{K(\mu_{2^\infty})}\) is an open subgroup of \(\operatorname{SL}_2(\mathbf Z_2)\), by the open-image theorem (Serre 1972). Its index in the corresponding fixed-\(E\) image changes by at most two when \(K\) changes. Its commutator subgroup contains a fixed open subgroup \(U_E\) of \(\operatorname{SL}_2(\mathbf Z_2)\). One may see uniformity by starting with a principal congruence subgroup in the fixed image: every index-two subgroup contains its squares, and the commutators of the resulting fixed congruence subgroup are open. Because the radical extension over \(F_\infty\) is abelian, these commutators act trivially on it. Hence every permitted Tate coset contains a translate of \(U_E\).

For completeness, in the CM case let \(M\) be the CM field. The image over \(KM(\mu_{2^\infty})\) contains a fixed open subgroup of the norm-one Cartan; the index changes by a bounded factor, as \([KM:\mathbf Q]\leq4\). Choose \(\gamma\in G_{KM}\) with \(\chi_{\rm cyc}(\gamma)=5^4=625\). This is possible uniformly: the cyclotomic image of \(G_{KM}\) has index at most four in \(\mathbf Z_2^\times\), so contains fourth powers. On the radical Galois group, conjugation by \(\gamma\) is raising to \(625\), because all the radicands lie in \(K\). On the Tate image over \(KM\), conjugation is trivial. Therefore \[\gamma u\gamma^{-1}u^{-625},\qquad u\in G_{KM(\mu_{2^\infty})},\] fixes every prescribed radical and acts on \(T\) as \(\rho_T(u)^{-624}\). Raising to the fixed nonzero power \(624\) carries an open subgroup of the one-dimensional norm-one Cartan onto an open subgroup. This proves the required uniform kernel statement also in the CM case. The open Cartan image used here is the CM part of the usual \(\ell\)-adic image theorem (Serre 1968, IV, §2.2, Remark).

On a coset of an open subgroup of \(\operatorname{SL}_2\), trace is not identically \(2\) or \(-2\). The same holds on a coset of an open norm-one Cartan subgroup; on the other Cartan-normalizer component trace is zero. The relevant determinant-one sets are compact and have finitely many cosets modulo the fixed open subgroup. Choose in each coset an element whose trace is neither forbidden value. The largest of the finitely many resulting valuations is a constant depending only on \(E\).

Finally restrict the prescribed radical automorphism and such a Tate lift to a finite Galois extension, including as many roots of unity as required. Chebotarev gives infinitely many rational primes with this Frobenius class; choose the place over the prime to realize the chosen representative. They split in \(K\) and have arbitrarily high prescribed congruence to \(1\). Since \(\det(1\pm\operatorname{Fr}_r)=1\pm a_r+r\), a congruence precision larger than the fixed trace bound gives all of (22). This is a finite Chebotarev condition at each stage, not a demand that one prime split in an infinite extension. ◻

Proof. Use tame inertia and unramified Frobenius, or the Hochschild–Serre complex for these two procyclic directions. At \(r\), inertia minus one is multiplication by \(t\) times a unit. It is invertible on all the indicated fields except the characteristic-zero fiber \(t=0\). There, \(r\to1\), and the two remaining Frobenius differentials have nonzero determinant by (22), including either sign of the quadratic twist. The bound also ensures that the integral local torsion at this central fiber has bounded length.

At an active odd conductor prime, inertia of the quadratic character acts by \(-1\). Its differential is thus \(-2\) and is invertible in characteristic zero. In characteristic two the quadratic character disappears. At a split prime the Frobenius action is \(zB\), where \(B\in\operatorname{GL}_2(\mathbf F_2)\) and \(z=(1+t)^a\) has odd \(a\in\mathbf Z_2\). Its coefficient of \(t\) is one, so \(z\) is nonconstant. No nonzero polynomial over the finite constant field vanishes at \(z\). Thus both Frobenius complexes, with operators \(zB-1\) and \(q^{-1}zB-1\), are invertible over \(\Omega\); here \(q=1\) in the residue field. This argument applies also to a reducible residual representation.

The orthogonal full and strict conditions agree when full local cochains are acyclic. Equation (21) then gives self-duality. The global degree-zero assertion will follow from the evaluation maps in Lemma 19. ◻

Proof. Reduce first over \(\mathbf F_2[[t]]\). The quadratic character is now trivial. Removing the active conductor primes from the permitted ramification set changes the global complex by their singular local complexes. These, and their unramified local complexes, are acyclic after inverting \(t\), by Lemma 17. The resulting smaller-support complex, including its comparison maps, can be constructed at the original Artin stages before taking the limit. Its support consists of \(2N,r\), and the retained derivative primes. Its \(\mathbf F_2((t))\) cohomology dimensions are bounded by those at \(t=0\): choose a minimal finite free model over \(\mathbf F_2[[t]]\).

Here is a uniform bound for the latter residual cohomology. Put \(F=\mathbf Q(E[2])\), a fixed field. Then \(FK/F\) is at most quadratic. Enlarge the allowed set of primes of \(F\) by those ramified in \(K\). Its cardinality is bounded by a constant depending on \(E,A\) and the number of derivative primes. Kummer theory bounds the number of generators of the maximal pro-\(2\) Galois group over \(F\) unramified outside this set, using the fixed unit group, the fixed \(2\)-class group, and the allowed valuations. The subgroup defining \(FK\) has index at most two; the elementary Schreier bound bounds its generator number. Every finite Galois \(2\)-extension of \(FK\) with this ramification has Galois closure of \(2\)-power degree over \(F\), so this subgroup indeed controls the required pro-\(2\) extensions of \(FK\).

Over \(FK\) the residual Tate module is trivial. Restriction and inflation–restriction therefore bound the residual \(H^1\) over \(K\). The \(H^0\) bound is immediate, and the global Euler characteristic or global duality bounds \(H^2\). The local terms at the remaining bounded set have bounded dimensions, with degrees of the local fields at most two over the fixed rational local fields. At \(2N\) there are only finitely many local quadratic twisting classes. The same bounds apply to the Selmer cone. Lemma 13 now bounds the minimal ranks. No bound on a torsion exponent is inferred from these rank bounds. ◻

Proof. Let \(C_i\) be a finite model and let \(\iota_i\) be its chosen map to the original global cochains, using the projection from the Selmer cone when appropriate. For every \(g=(g_i)\) and \(h=(h_i)\) define matrices \[e^0_i:C_i^0\to A_i^2,\quad e^1_{g,i}:C_i^1\to A_i^2,\quad e^2_{g,h,i}:C_i^2\to A_i^2\] by evaluating \(\iota_i\) in degrees zero, one, and two. Their source ranks are fixed and their entries belong to \(A_i\). Thus each has a coefficientwise ultrafilter limit, for every choice of \(g,h\). No diagonal selection over the set of group elements is necessary: one fixed ultrafilter defines all these limits. Likewise \(\rho(g)=\lim_i\rho_i(g_i)\) is a representation. The original cochain identities imply \[\begin{align*} e^1_g d&=(\rho(g)-1)e^0,\tag{24}\\ e^2_{g,h}d&=\rho(g)e^1_h-e^1_{gh}+e^1_g. \tag{25}\end{align*}\] All identities pass to the limit because their matrix expressions have finitely many terms. A cycle \(x\in C^1\otimes L\) consequently gives the crossed homomorphism \(g\mapsto e^1_g(x)\), and a boundary gives a coboundary.

We prove injectivity in a way that also controls base change. At each finite stage let \(k=\mathbf F_2\) be the residual field. Choose a space of representatives for residual \(H^1\). Evaluate pairs \((c,m)\), where \(c\) is in this space and \(m\in k^2\), by \(b_c(g)-(\rho(g)-1)m\). The intersection of the kernels of all these evaluations consists precisely of pairs \((0,m)\) with \(m\) an invariant. Since the vector space is finite dimensional, a list of at most \(\dim H^1+2\) elements already has this intersection. For these elements evaluation is a chain map \[C_i\longrightarrow D_i= [\,A_i^2\xrightarrow{((\rho_i(g_j)-1))_j}(A_i^2)^d\,]\] in degrees zero and one, inducing an isomorphism on residual \(H^0\) and an injection on residual \(H^1\). For Selmer complexes this uses the exact sequence of (20): the indicated local degree-zero isomorphisms give \(H^0(C_i)=H^0_{\rm glob}\) and \(H^1(C_i)\hookrightarrow H^1_{\rm glob}\) residually.

Retain these matrices and their chosen group-element sequences. The cone of the limiting map \(C\to D\) has zero residual cohomology in degrees at most zero. Successively cancel its unit differential blocks. The resulting bounded free cone starts in degree one. After every coefficient extension it still starts in degree one; the long exact sequence therefore makes \(H^1(C\otimes L)\to H^1(D\otimes L)\) injective. A crossed cocycle that is a coboundary on all of \(\Gamma_K\) is in particular a coboundary on the detecting list, proving the desired injection. The same cone gives an isomorphism on \(H^0\) after coefficient extension. Since \(H^0(D\otimes L)\) is a subspace of \(L^2\), degree-zero evaluation is injective; by (24), its image is fixed by every element of \(\Gamma_K\), not merely by the detecting list. For induction the same proof has coefficient rank four and the detecting bound is \(\dim H^1+4\). ◻

Proof. The Tate image contains an open subgroup of scalar homotheties, both in the CM and non-CM cases (Bogomolov 1981, sec. 2, Theorem 3 and paragraph 7). Choose once and for all \(g\in G_{\mathbf Q}\) with Tate action \(u\,1\), where \(u\in1+4\mathbf Z_2\) and \(u\ne1\). Its square belongs to every quadratic subgroup. Put \[z=g^2\tau g^2\tau^{-1}.\] The generalized dihedral property of every ring-class extension implies that \(z\) is ring-class trivial. It is also trivial on every quadratic character, and its Tate action is the fixed scalar \(u^4\). For every sequence in \(\Gamma_K\), its commutator with the constant sequence \(z\) is in the exact group \(H\), since its Tate commutator and all its abelian character images are trivial. Thus \(z\) is central in \(\Gamma_K/H\) and acts on the coefficients by \(u^4\).

Here the elementary central-scalar argument suffices. If a crossed cocycle \(b\) is defined on a group with such a central element, comparison of \(b(zg)\) and \(b(gz)\) gives \[(u^4-1)b(g)=(g-1)b(z).\] In characteristic zero this makes \(b\) a coboundary. Inflation–restriction then gives injectivity on \(H\). The valuation of \(u^4-1\) is fixed.

In characteristic two use instead, at the \(i\)-th stage, an inertia element at \(r_i\) mapping to the chosen generator of \(\xi_i\). Its Tate action is trivial and its ring-class image commutes with all ring-class images. The associated sequence is therefore central modulo \(H\), and its coefficient action is \(1+t\). Since \(t\) is invertible in \(\mathbf F_2((t))\), the same calculation applies. It is essential here that \(H\) is defined by the exact actions at each stage, not merely by an action converging to the identity. ◻

Proof. A nonzero element of the quotient has a representative whose valuations are bounded on an ultrafilter-large set. A finite partition then makes its valuation equal to a single integer \(n\) on such a set. It is \(2^n\) times a unit. Products add these finite valuations, so the quotient is a domain, and every nonzero ideal has an element of smallest valuation. It is consequently a DVR. The integer \(2^n\) is nonzero for every \(n\), so its fraction field has characteristic zero.

For each \(j\), all-sequence evaluation maps originally have entries in \(\Lambda\). Extension to \(\mathcal O_j\) and composition with an integral contraction still give integral matrices. Thus evaluation on any sequence of inner group elements has an entrywise image in \(\mathcal O_\infty\). Equations (24) and (25) survive for all such sequences. In particular, if an outer cycle is represented by vectors \(x_j\) for which \(d_jx_j\) has valuation tending to infinity, the evaluation of \(d_jx_j\) does too: the degree-two evaluation matrix is integral. This proves the crossed identity for an outer cycle without replacing the \(x_j\) by genuine inner cycles.

Over the inner residue field \(\Omega_j\), choose a detecting list as in Lemma 19; its length is at most \(\dim_{\Omega_j}H^1(C_j\otimes\Omega_j)+\operatorname{rank}M\). These lengths are bounded. Include in the integral diagrams the evaluation map to its two-term complex and a contraction of its cone to a complex starting in degree one. Such a contraction uses only unit pivots over \(\mathcal O_j\) and has bounded graded sizes. All its matrices, including inverses and homotopies, are integral and pass to the quotient. Its cone still starts in degree one after extension to \(L_\infty\). This proves injectivity just as in Lemma 19.

Let \(H_j\) be the inner exact kernel. The product of the central elements of Lemma 20 is central modulo \(\prod_jH_j\). Its scalar is the same fixed \(u^4\) in every component, and \(u^4-1\) remains nonzero in \(\mathcal O_\infty\). The explicit central-scalar calculation and inflation–restriction give the claimed restriction injection. Additional fixed finite diagrams are treated by the same entrywise operations. ◻

Proof. The two induced summands are distinguished by \(r\)-inertia, whose scalars are \(\Psi\) and \(\Psi^{-1}\). On either summand, multiplying the Tate matrices by scalars does not alter invariant subspaces. The Tate representation restricted to \(G_K\) is absolutely irreducible under the exclusions already specified. These facts give absolute irreducibility of the induction when the two inertia characters differ.

For the outer argument this assertion requires a uniform finite witness. The images of the quadratic subgroups have index at most two in the fixed compact \(2\)-adic Tate image. A compact \(2\)-adic analytic group is topologically finitely generated and hence has only finitely many closed subgroups of index at most two. Choose one on an ultrafilter-large set. Its absolutely irreducible representation has finitely many matrices spanning \(M_2(\mathbf Q_2)\); the determinant expressing this spanning property is a fixed nonzero \(2\)-adic number. Every one of these Tate matrices is realized by choices of Galois elements in each component. Their accompanying character scalars are units and do not change the spanning assertion. Its determinant remains nonzero in \(L_\infty\). This proves absolute irreducibility in that field. Also \(1+t\) has infinite order already in every \(\Omega_j\): for a nonzero integer \(n\), the first nonzero term of \((1+t)^n-1\) occurs in finite degree. It remains a unit modulo \(2\) in every component, so no such power becomes one in the outer field.

If \(\Psi^2=1\), extension across conjugation gives the two quadratic twists. They are inequivalent, since a quadratic self-twist of the Tate representation here would make its restriction to \(G_K\) reducible over an algebraic closure. In the CM case this would force \(K\) to be the CM field, which was excluded. ◻

Proof. At one Artin stage, a coordinate of a cochain in the image of a finite model is a locally constant function. The finitely many coordinates being used factor through one finite quotient of the absolute Galois group. Enlarge it to contain the Tate action at the required precision, the finitely many ring-class fields and quadratic characters currently in use, and all old finite data. Chebotarev realizes the conjugacy class of \(\tau u\) on this finite quotient, with a chosen place realizing the desired representative. It gives a rational prime inert in \(K\), outside the old finite set. Because \(u\) is exactly Tate-trivial, its Frobenius tends to \(\rho_T(\tau)\); in particular \(a_\ell\to0\) and \(\ell\to-1\). Because \(u\) is exactly ring-class trivial, the square Frobenius on the \(K\)-prime is the identity on every retained ring-class field. The unramified local coordinate is therefore the stated Frobenius-square evaluation.

Repeat at each finite stage with increasing precision. All conditions remain finite, and no density bound or prime-size bound is required. For an outer sequence do this separately for each inner sequence, retaining the relevant finite evaluation matrices and degree-two identities before taking the outer quotient. Approximate inner cycles cause no problem, by the integrality calculation in Proposition 21. Previously imposed finite diagrams are unchanged. ◻

Proof. Combine Lemmas 13 and 14, Lemma 15, Proposition 16, Lemmas 17 and 18, Lemmas 19 and 20, Proposition 21, and Lemma 23. Every passage uses actual integral matrices of bounded size; arbitrary evaluations are retained by their original contraction maps, not by an inference from a finite list of identities. ◻

Proof. The prime-to-\(\ell\) local cochains are computed by the tame inertia and residue-field directions. In these two directions the differentials are formed from inertia minus one and Frobenius minus one, with the usual Frobenius action on tame inertia. On \(M\), inertia is trivial at the derivative prime. The square of a rational Frobenius is trivial on every ring-class character. In the limit it is also the identity on \(T\), while \(\ell^2\) tends to one. Thus both differentials become zero, giving (28).

Here is a finite-precision check of the pairing, needed at \(2\). At a retained quotient killed by \(2^a\), impose also \(q=\ell^2\equiv1\pmod{2^{a+1}}\), as well as trivial Frobenius-square action on the coefficient module at this quotient. The tame two-generator complex then has zero differentials, with the coordinates in (29). Its degree-one scalar generators are the unramified and tame characters. More explicitly, use the presentation \(\phi\sigma\phi^{-1}\sigma^{-q}=1\). After augmentation with trivial coefficient action its Fox resolution has terms of ranks \(1,2,1\), with \(d^0=0\) and \(d^1=(0,1-q)\), in the ordered finite and tame coordinates. The diagonal from the same resolution has zero pure-finite term, unit mixed terms, and pure-tame coefficient \(\pm\binom q2\). The latter coefficient is the augmented second derivative of \(\sigma^{-q}\) together with its adjacent \(\sigma\) in the relation. Both \(1-q\) and \(\binom q2\) vanish modulo \(2^a\) under the imposed congruence. Thus this is an actual split complex and an actual pure-plane zero pairing at the retained precision. The field contains the \(2^{a+1}\)-st roots of unity, so \(-1\) is a \(2^a\)-th power. The norm-residue identity \((x,x)=(x,-1)\) for the \(2^a\)-Hilbert symbol therefore makes the square of each scalar generator zero. The two generators form a basis; local Tate duality makes their cross pairing a unit. The same calculation applies coefficientwise to the free Artin coefficient module, including its nilpotent \(t\) and quadratic variables. Thus the pure plane restrictions are already zero at every retained quotient. All these congruences can be imposed simultaneously with the derivative Frobenius conditions by increasing their precision. Taking the compatible limit proves integral vanishing of the pure squares. It is not deduced from skew symmetry in characteristic two.

Equivalently, in the limiting tame presentation \(\phi\sigma\phi^{-1}=\sigma^q\) with \(q\to1\), the rank-\((1,2,1)\) resolution specializes to the exterior resolution for two procyclic directions. The finite Hilbert-symbol calculation specifies the cup product on its split cohomology and rules out a hidden square term. Conjugation transports the finite Tate coordinate by \(J\) and the singular coordinate by \(-J\), with possible scalar units coming from the orientations. Consequently the cross pairing is \[ \beta(x,y)=\epsilon\,\langle x,Jy\rangle,\qquad \epsilon\in\Lambda^\times,\quad x\in F,\ y\in S, \tag{30}\] where \(\langle\, ,\,\rangle\) is the perfect alternating Weil form. This proves the assertions about orthogonal complements. Since the restrictions of this explicit pairing to either pure plane are zero as maps of the split complexes, their isotropy homotopies may be chosen to be zero. Only a product of two degree-one terms could map to the degree-two local invariant target; the degree-zero terms introduce no additional restriction. These zero homotopies agree on the strict subcomplex. These are the actual orthogonality data required in the Selmer duality construction (Nekovář 2006, Theorem 6.3.4); the compatibility assertion is therefore an assertion about these displayed complexes and maps. Lemma  14 is used to retain this finite diagram together with the unchanged local data. ◻

Proof. We use Kolyvagin’s derivative operators (Kolyvagin 1990), keeping their integral descent explicit at \(2\). Choose the orientations coherently. The relative ring-class group at an inert prime \(\ell\) is cyclic of order \(\ell+1\). For distinct new primes these groups form a product: the ring-class exact sequence has kernel the product of the local residue-unit quotients, and the only global units here are \(1,-1\), whose images in these quotients are trivial. Choose a generator \(\sigma_\ell\) compatibly with the tame orientation in Lemma 25; its pro-\(2\) component gives the coordinate \(s\) at the retained precision. Set \[D_\ell=\sum_{i=1}^{\ell} i\sigma_\ell^i,\qquad D_I=\prod_{\ell\in I}D_\ell.\] Direct multiplication gives \[ (\sigma_\ell-1)D_\ell=(\ell+1)- \sum_{i=0}^{\ell}\sigma_\ell^i. \tag{32}\] The Hecke correspondence and reduction of its CM isogenies give, for every class-group translate, \[ \operatorname{Tr}(y_{c r n})=a_\ell y_{c r n/\ell}, \qquad \widetilde y_{c r n}=\operatorname{Fr}_\ell \widetilde y_{c r n/\ell}. \tag{33}\] These are the point relations, not an application of a residual irreducibility theorem. A fixed multiple of the parametrization removes cuspidal terms.

There is a uniform bound on the \(2\)-primary torsion over the union of the ring-class fields in use. To see this, choose a fixed nonidentity scalar homothety in the \(2\)-adic Tate image. Square a lift so that it belongs to every quadratic subgroup under consideration. Multiply it by its conjugate under complex conjugation. Its action on each ring-class field is trivial, because conjugation inverts the abelian ring-class action, while its action on \(T\) is still a fixed nonidentity scalar. Write that scalar minus one as \(2^e\) times a unit. Then \(2^e\) kills the torsion in question. The existence of this fixed scalar, including the CM case, is the Tate-image input used in Lemma 20.

Work first with coefficients modulo \(2^a\). Require \(a_\ell\) and \(\ell+1\) to vanish to a precision substantially larger than \(a\). Equations (32) and (33) then make the derivative Kummer class invariant under the relative ring-class group. In the inflation–restriction sequence, both the obstruction to descent and the ambiguity of descent are killed by \(2^e\), since their coefficient module is killed by \(2^e\). Multiplication by \(2^{2e}\) therefore gives a descent independent of this ambiguity: descend \(2^e\) times the class and multiply its lift by \(2^e\). There is no repetition of this factor for individual primes; descent is performed once from the field with the complete conductor.

The target coefficient module is obtained by an integral transfer. For a finite Artin coefficient ring \(A\), let \(\Delta=\operatorname{Gal}(H_{cr}/K)\), through which the finite scalar and quadratic characters factor. Shapiro’s cochain map and the coefficient homomorphism \[A[\Delta]\longrightarrow A,\qquad [\sigma]\longmapsto\chi_h(\sigma)\Psi(\sigma)\] send the descended class to coefficients \(T\chi_h\Psi\). For a universal quadratic family, replace the quadratic value by its group element in \(A[G]\), also including the fixed base character. Choose inverse actions on both sides when using that Shapiro convention. Explicitly, these maps are finite coset sums with the indicated scalar weights; there is no division by \(\#\Delta\). These chain maps, in all degrees, are used at this finite stage. Naturality of restriction and Kummer corestriction supplies their local lifts and comparison homotopies. Apply the maps to the classes and these witnesses before passing to bounded target models. Neither the complex over \(H_{cr}\) nor its induced coefficient module \(A[\Delta]\) is retained in the limiting diagram. We check the required target local conditions next.

At a place over \(2N\) these descent extensions are unramified. For the identity component of a Neron model, the obstruction to unramified descent vanishes. Indeed, Lang’s theorem gives the assertion on the connected special fiber, and the successive formal-group quotients are additive residue-field modules, whose unramified cyclic cohomology vanishes by the normal-basis and trace calculation. Passing to the inverse limit of these quotients preserves the assertion. The remaining obstruction is killed by the geometric component-group order. Those orders have a uniform bound over unramified extensions of the finitely many fixed local twisting types. Include their product, the descent factor, and the parametrization factor in \(L\). Thus the descended class has the integral Kummer condition at \(2N\). No good-ordinary or semistability condition was used. Transfer with the scalar character preserves this condition because \(\Psi=1\) there.

Here is the local calculation, including the division by two. Apply the derivatives at all primes other than \(\ell\), and write \(X,Y\) for the resulting lower and upper points. Put \(F_\ell=\operatorname{Fr}_\ell\). The lower field splits at the \(K\)-prime \((\ell)\), since that ideal is principal. Reduction therefore gives \(F_\ell^2\widetilde X=\widetilde X\). Good reduction identifies the prime-to-\(\ell\) torsion with the torsion of the reduced curve. In the following coordinate formulas, \(F_\ell\) acts on reduced points, where it satisfies its Frobenius polynomial. If \(Q\) is a \(2^a\)-division point of \(LX\), the finite coordinate of its Kummer class is \[ f=(F_\ell^2-1)\widetilde Q= \bigl(a_\ell F_\ell-(\ell+1)\bigr)\widetilde Q. \tag{34}\] We give the descent-cocycle calculation of the other coordinate. Choose \(Z\) with \(2^aZ=LD_\ell Y\), and choose the descended cocycle \(b\) so that on the upper-field Galois group it is \(b(g)=(g-1)Z\). A change by the already bounded descent ambiguity has no effect on the eventual limit. The crossed identity gives \[U=(\sigma_\ell-1)Z-b(\sigma_\ell) \in E(H_{crn}).\] For \(g\) in the upper-field Galois group, compare the two expansions of \(b(g\sigma_\ell)\); they give \(gU=U\). Moreover, \[2^aU=L\bigl((\ell+1)Y-a_\ell X\bigr).\] Both integers \(\ell+1,a_\ell\) are divisible by \(2^a\). Consequently \[U_0=L\frac{\ell+1}{2^a}Y- L\frac{a_\ell}{2^a}X\] is an actual upper-field point with the same \(2^a\)-multiple as \(U\). Thus \(U-U_0\) is upper-field torsion of bounded exponent. This comparison with an actual integer quotient is what bounds the ambiguity; arbitrary division points would not give this bound. Reduction makes \(\sigma_\ell\) act trivially and \(\widetilde Y=F_\ell\widetilde X\). Therefore reduction of \(b(\sigma_\ell)=-U\) gives its singular coordinate \[ s=\bigl(a_\ell-(\ell+1)F_\ell\bigr)\widetilde Q \tag{35}\] up to the uniformly bounded descent torsion already described. The Frobenius polynomial yields the exact operator identity \[ (F_\ell-a_\ell) \bigl(a_\ell F_\ell-(\ell+1)\bigr) =a_\ell-(\ell+1)F_\ell. \tag{36}\] Thus \(s=(F_\ell-a_\ell)f\); when \(a_\ell\) vanishes at the retained torsion precision, this is \(s=F_\ell f\). Taking the Frobenius limit gives \(s=Jf\).

For the new finite coordinate, reduction of the derivative point is \[ \sum_{i=1}^{\ell}iF_\ell\widetilde X =\frac{\ell(\ell+1)}2F_\ell\widetilde X. \tag{37}\] The conductor extension at \(\ell\) is totally ramified, so a Frobenius lift may be chosen in the upper-field local Galois group. There the descended cocycle restricts to the Kummer cocycle of the derivative point. If \(v_2(\ell+1)\ge a+1\), its displayed reduction is \(2^a\) times an \(F_\ell^2\)-fixed point. Its finite Kummer coordinate is therefore zero modulo \(2^a\). The additional \(1\) is essential at \(2\). It is imposed in the choice of the next precision, and does not multiply the class by a new denominator. Bounded torsion ambiguities disappear on passing from \(E[2^a]\) to \(E[2^m]\) with \(a-m\ge e\). Choose \(a\) and the Frobenius precision successively faster than \(m\).

The calculation is compatible with every translate in the transfer: transport the place and its Frobenius before applying that translate. The scalar weight on the lower and upper class is identical. Applying the same calculation at each \(\ell\in I\) proves the earlier singular conditions as well.

Finally complete the weighted transfer at each stage and retain the resulting target Selmer-cone cycles and their local lifts. An equality of the transferred local classes gives a boundary witness in the target local complex; retain that witness after contraction as well. For instance, retain the degree-zero vectors whose boundaries are the differences in the two switch identities. Include the split local Fox complexes, their zero isotropy homotopies, and the target comparison matrices. The target-side construction in Section 4 proves that this is a bounded diagram for each fixed active support and finite switched set. Its bound is independent of the degrees of the ring-class fields. When a further derivative prime is selected, extend the same original finite-stage diagram and its contraction maps, using the same inner ultrafilter. Do not construct an unrelated cohomologically equivalent old complex. The earlier evaluations, including all degree-two cocycle identities, are consequently retained by the same contraction maps after each new switch. Lemma  14 applies to these extended diagrams. Their limiting classes give (31). The displayed local formulas agree with the calculation underlying (Howard 2004, Proposition 1.7.4); Howard’s odd-prime global hypotheses are not being asserted at \(2\). ◻

Proof. On \(H\) a crossed homomorphism is an additive homomorphism, and its image is equivariant for \(\mathcal G\). A proper submodule of \(W\oplus W\) dominating both factors would impose a nonzero scalar linear relation on the two evaluations: the direct sum of two simple modules is semisimple, and \(\operatorname{End}_{\mathcal G}(W)=F_O\). That relation, followed by injectivity of restriction, would contradict independence of the classes. Thus the joint span is the whole direct sum. Projecting by \(1+\tau\) gives all pairs in the two-dimensional finite plane.

It remains to pass from a span to a single element of the additive evaluation image \(U\). If the determinant quadratic polynomial \(Q\) vanished on \(U\), then its polar form \(Q(u+v)-Q(u)-Q(v)\) would vanish for \(u,v\in U\). Consequently \[Q\Bigl(\sum_i a_i u_i\Bigr) =\sum_i a_i^2 Q(u_i) +\sum_{i<j}a_i a_j \bigl(Q(u_i+u_j)-Q(u_i)-Q(u_j)\bigr) =0.\] It would vanish on the entire field span, where it is not the zero polynomial. This contradiction proves the assertion. This polarization argument is valid in characteristic two as well. ◻

Proof. The zero coordinate needs no argument. Otherwise the generic rank is one and the class spans it. Localize at a height-one prime \(\mathfrak p\ne(2)\), and denote the residue field by \(F_O\). Its characteristic is zero. Lemma 17, the degree-zero local conditions, and duality give zero fiber cohomology in degrees zero and three. Canceling unit blocks therefore gives a free DVR model in degrees one and two. A primitive vector in its generic kernel has nonzero image in its \(\mathfrak p\)-fiber: the kernel of a map of free modules over a DVR is saturated. Include this vector in each localization test. The two-sign reduction below has the same pattern as (Howard 2004, Lemmas 1.5.3 and 1.6.4); the integral switch and the evaluation hypotheses needed here at \(2\) have been constructed above.

Suppose first that \(\Psi^2\ne1\) on \(F_O\). Inertia at \(r\) distinguishes the two summands of the induced representation. The Tate representation over \(K\) is absolutely irreducible in characteristic zero: this follows from the open Tate image, or from its CM normalizer when \(K\) is not the CM field. Thus the induced representation is absolutely irreducible. If the test Selmer dimension is \(s>1\), Lemma 28 and the finite Chebotarev realization in Lemma 23 give a derivative prime with rank-two finite localization, nonzero on the chosen primitive vector.

Poitou–Tate duality now determines the effect on the fiber. The relaxed localization image is a self-annihilating plane. It contains the whole finite plane, and hence equals that plane. The strict kernel has dimension \(s-2\), and its replacement by the singular condition adds no new class. The fiber dimension therefore drops by two. On the generic field, the localization is nonzero, so Lemmas 26 and 27 preserve generic rank one, the nonzero derivative class, and the determinant valuation.

If \(\Psi^2=1\), use conjugation, including its action \((1+t)\mapsto(1+t)^{-1}\), to split the fiber into two signs. All prior plane conditions are stable under this involution in the split local model. The two Galois summands are nonisomorphic absolutely irreducible representations; the only CM quadratic self-twist has been excluded. Each sign has one finite and one singular coordinate. The local pairing in a sign is a nondegenerate symmetric hyperbolic pairing, whose only isotropic lines in characteristic zero are the finite and singular axes. If both signs occur globally, choose one class from each, including a nonzero component of the primitive vector, to obtain the same rank-two reduction.

If only one sign occurs and \(s>1\), choose a rank-one finite localization nonzero on that vector. In this sign the relaxed image is the finite line. In the other sign its intersection with the finite line is zero, and self-duality forces it to be the singular line. A switch retains the old strict kernel of dimension \(s-1\ge1\), introduces the other sign, and preserves the total dimension. The next rank-two switch reduces it. After finitely many such operations the fiber dimension is one. It cannot fall below one, since a primitive vector from the generic rank-one kernel survives after each switch.

At fiber dimension one, a minimal self-dual model over the DVR has one free term in each of degrees one and two and zero differential. The class and paired functional are integral at this height-one prime. Their determinant coordinate is therefore integral. Reversing (38) proves nonnegative valuation at \(\mathfrak p\). Applying this at every height-one prime other than \((2)\), and using that the normal ring \(\Lambda\) is the intersection of its height-one valuation rings inside its fraction field, gives membership in \(\Lambda[1/2]\).

For the stated arithmetic instances, the rational central specialization has Selmer dimension one and no invariants. Upper semicontinuity bounds the generic dimension by one. If the generic class is nonzero, it spans this space. This uses Selmer corank at an unknown base and does not replace that corank by Mordell–Weil rank. ◻

Proof. Permute rows and columns so that the minor is the upper-left block \(A_i\). Write \(D_i=\left(\begin{smallmatrix}A_i&b_i\\c_i&d_i \end{smallmatrix}\right)\). Since its rank is \(n-1\), its Schur complement is zero. Over the fraction field, use the triangular basis changes with entries \(A_i^{-1}b_i,c_iA_i^{-1}\) to leave \(A_i\) and a zero one-by-one block. These triangular changes have determinant one. If \(\alpha_i\) is the last coordinate of \(z_i\) and \(\beta_i\) the value of \(\lambda_i\) on the last target basis vector, cancellation gives exactly \[ u_i=\pm\frac{\alpha_i\beta_i}{\det A_i}. \tag{43}\] Here \(\alpha_i\in O_i\) and \(\beta_i\in2^{-a}O_i\). This proves (42). If \(n=1\), the empty minor is \(1\) and the same formula applies. ◻

Proof. The preceding paired-diagram construction supplies the common clearing exponent in the arithmetic application. The bound on minimal sizes is Lemma 18; in particular, active conductor primes do not contribute to that size after residual localization.

Suppose that \(v_{(2)}(u_i)\to-\infty\). All these coordinates are nonzero. First put the complexes into square two-term form. At the residual \(\Omega\)-test, moving-prime inertia excludes global invariants, and the local degree-zero conditions identify Selmer invariants with those global invariants. Thus \(H^0=0\). The retained self-duality gives \(H^3=0\) and equal dimensions in degrees one and two. A minimal \(\mathcal O\)-model therefore has only two free terms, of equal rank, in those degrees; all other terms cancel in integral disks. Fix a nonprincipal ultrafilter \(\mathcal U\) and fix the bounded rank on an ultrafilter-large set. Write these models as \[C_i=[\,O_i^n\xrightarrow{D_i}O_i^n\,].\]

Transport the class, the uniformly cleared functional, all evaluation matrices, and the finitely many retained comparison maps and boundary witnesses through these integral disk contractions. Their entries remain integral and their identities remain exact. The individual active-place complexes, whose number may grow with the outer instance, are not retained as an outer list of objects. Their effect has already been incorporated in the target cone and its determinant; only the bounded minimal models and the fixed number of switch diagrams enter this limit.

Apply Proposition 21 to these models and their retained integral diagrams. It gives a DVR \(O_\infty\) with uniformizer \(2\), characteristic-zero fraction field \(F_\infty\), and injective all-sequence evaluation, including restriction to the product of the exact inner kernels. The induced representation uses one simultaneous complex conjugation, so the sequence group over the quadratic fields has index two in the rational sequence group. Lemma  22 gives absolute irreducibility over \(F_\infty\): the moving inertia character has infinite order, and the finite Tate-matrix witness survives the outer quotient.

The outer complex may have more cohomology than any inner generic complex. Its new cycles can be represented by vectors \(w_i\) for which \(v(D_iw_i)\to+\infty\), without exact inner lifts. The retained degree-two evaluation matrices are integral, so the evaluations of \(D_iw_i\) also tend to zero. This is the part of Proposition 21 that makes these new classes available for the following rank reduction.

Choose in each inner generic kernel a primitive vector \(v_i\). Some coordinate is a unit on a set in \(\mathcal U\), so its outer limit is nonzero. Thus the outer kernel has dimension at least one. If that dimension is \(s>1\), apply Lemma 28 to two outer classes, one the surviving primitive vector. Realize their rank-two evaluation by inner derivative primes. This realization only asks for finite quotients at each stage. For an approximate inner cocycle one chooses the retained precision below the valuation of its coboundary; no exact inner lift of the extra outer class is being assumed.

More explicitly, retain the original inner index \(i\) for outer instance \(j\), its ultrafilter \(\mathcal U_j\), and the outer ultrafilter \(\mathcal U\). Represent the selected kernel element by \(u_j=(u_{j,i})_i\). At stage \((j,i)\), apply finite Chebotarev to this actual element and the old finite data to choose \(\ell_{j,i}\). Adjoin its local complexes, the derivative cocycle and the pairing homotopies to the same original stage diagram with its old contractions. Take the same \(\mathcal U_j\) and then the same \(\mathcal U\). The retained coefficient precision may increase more slowly, but cofinally; the earlier maps and their arbitrary evaluations are not reconstructed by an independent choice of subsequence or model.

The localization of \(v_i\) is generically nonzero on a set in \(\mathcal U\): its outer image is nonzero. Therefore the original inner spanning class also localizes nontrivially, even if its multiple of \(v_i\) has large valuation. Lemma 27 preserves each inner determinant valuation exactly. On the outer field, the relaxed image is the finite plane and the singular switch lowers \(s\) by two, by the same localization exact sequence as in Proposition 29. The new inner generic kernels again supply primitive vectors, so the outer rank remains at least one.

Only finitely many switches are necessary: the initial outer rank is bounded and drops by two. After each fixed number of switches the required model-size bound remains available. At the terminal outer rank one, the outer differential has a nonzero maximal boundary minor. Its inner determinants \(\Delta_i\) have one finite valuation on a set in \(\mathcal U\), by the definition of \(O_\infty\). Lemma 30 then gives a fixed lower bound on \(v(u_i)\), a contradiction.

This proof uses the minor only after rank reduction. Bounded matrix size alone would not suffice: the matrices \([2^i]\) have bounded size and unbounded torsion length. In that example the entire matrix becomes zero in the outer quotient; the rank-reduction and evaluation hypotheses would have to supply additional arithmetic operations before a nonzero terminal minor could be used. ◻

Proof. Represent \(C\) by a bounded free complex. Derived reduction is its termwise reduction, so the determinant line itself commutes with specialization. For clarity, the cohomology base-change sequence is \[ 0\longrightarrow H^i(C)/tH^i(C) \longrightarrow H^i(C_0) \longrightarrow H^{i+1}(C)[t]\longrightarrow0. \tag{45}\] These last terms are the relevant \(\operatorname{Tor}_1\) terms. Diagonalize the complex over the DVR \(A\), or over its completion \(\mathbb Q_2[[t]]\). A nonunit pivot would create an additional central cohomology class in its two adjacent degrees. The assumed constant dimensions exclude such pivots. Thus over \(\mathbb Q_2[[t]]\) there are only invertible disks and the two free cohomology lines. The rational class-dual tensor consequently specializes to the stated central tensor. Over \(\mathbb Z_2\), all elementary divisors of \(C_0\), including the torsion supplied by (45), remain in its determinant isomorphism. This proves the assertion. ◻

Proof. The pro-\(2\) tame inertia generator acts by \(-1\). Its complex has differential \(-2\), so its cohomology is zero in degree zero and \(T'/2T'\) in degree one. Taking residue-field cohomology gives \[H^1(F_v,T')\simeq E[2]^{\operatorname{Fr}_v},\qquad H^2(F_v,T')\simeq E[2]/(\operatorname{Fr}_v-1)E[2].\] The possible residue-cardinality factor on inertia cohomology is one modulo \(2\). Kernel and cokernel of an endomorphism of a finite vector space have the same dimension, proving the length assertion.

The integral local Kummer exact sequence identifies the \(2\)-adic completion of the points with all of \(H^1(F_v,T')\). Indeed the inverse Tate module of the local Weil–Chatelet group is zero here: local duality identifies its \(2\)-primary part with the dual of the finite completion of the point group. The point completion is finite because the residue characteristic is odd. Equivalently, inertia \(-1\) implies that the local \(2\)-primary torsion is exactly the Frobenius-fixed \(2\)-torsion. The Kummer complex includes the derived invariants of this finite module on reduction, as in Lemma 15. The quotient therefore has exactly the degree-two cohomology displayed above. ◻

Proof. The two rational twists have analytic ranks zero and one. The forward low-analytic-rank theorems give finiteness of their whole Shafarevich–Tate groups; equivalently the Heegner simple-zero theorem over \(K\) gives rank one and finite Shafarevich–Tate group there. Thus all finite lengths in this proof refer to known finite groups.

The specialization comparison is made at each finite stage, before the target data are retained. Put \(c=|h|\). At the split auxiliary prime \(r_i\), the point distribution relation is \[\operatorname{Tr}_{H_{cr_i}/H_c}(y_{cr_i}) =(a_{r_i}-\sigma_{r_i}-\sigma_{\bar r_i})y_c,\] where the two Artin symbols have inverse ring-class actions. At \(t=0\), the scalar character is trivial, while the rational quadratic character has value \(\chi_h(r_i)\) on both symbols. The completed weighted sum therefore gives exactly \(L m_i P_h\), where \(m_i=a_{r_i}-2\chi_h(r_i)\).

This point identity is retained on bounded target complexes as follows. Let \(C_h^{\rm Kum}\) be a fixed free model of the ordinary Kummer Selmer complex over \(K\), and let \(b_h\) represent the Kummer class of \(P_h\). At the retained coefficient precision, enlargement of the ramification set and relaxation at the active and auxiliary primes give a target comparison \(j_i:C_h^{\rm Kum}/2^i\to C_i/(t)\). Each retained local condition is an isomorphism on degree zero and an injection on degree one, so the Selmer-cone exact sequence injects its degree-one cohomology into global cohomology. Kummer naturality and the point relation therefore give, after contraction, a target degree-zero vector \(v_i\) with \[z_i(0)-L m_i j_i(b_h)=d v_i.\] Include this vector, the comparison matrix, and the associated local localization triangles in the same target diagram. Their ranks are bounded for this fixed \(h,K\), independently of \(r_i\) and of \([H_{cr_i}:K]\). All statements are reduced to the same slower cofinal precision if needed. Lemma 14 now retains the displayed boundary identity. The trace bound gives \(v_2(m_i)\le c_E\); hence their limit \(m\in\mathbf Z_2\) on the same ultrafilter is nonzero and has \(v_2(m)\le c_E\). The active local quotients specialize to Lemma 33, and the Kummer conditions at \(2N\), including their differential lattices, are unchanged. The central point is therefore nonzero, so generic and central rational cohomology both have rank one.

To specialize the paired functional, choose its representative over \(\Lambda_{(t)}\). At an active ramified prime, inertia minus one is \(-2\), a unit in this ring. At \(r\), the central Frobenius determinants are nonzero by the trace bound. Their local complexes thus contract over \(\Lambda_{(t)}\). Composing these contractions with retained global duality gives the same generic functional as the contractions over \(\mathcal O\), and specializes to rational central Kummer duality. Lemma 32 applies to this representative; no regularity is claimed for an arbitrary \(\mathcal O\)-cochain representing the functional.

The ordinary integral Kummer complex has compact Selmer cohomology in degree one. Its degree-two torsion has length equal to that of the finite quotient of the discrete Selmer group by its maximal divisible subgroup. This follows by derived base change to \(\mathbb Q_2/\mathbb Z_2\) and the exact local Kummer conditions; the base-change terms must be retained. Here that quotient is \(\operatorname{Sha}(E^{(h)}/K)[2^\infty]\). The free parts of degrees one and two are paired unimodularly by integral duality. Torsion in degree one and its dual contribution in degree three have bounded length, by the fixed-scalar bound on global torsion.

For completeness, an elementary divisor of size \(2^d\) whose cokernel is in degree \(i\) contributes \((-1)^{i+1}d\) to the coordinate in the inverse determinant. Consequently the degree-two Shafarevich–Tate torsion subtracts its length. On the free line, a point of index \(2^{j_h}\) and its functional each contribute \(j_h\), giving \(2j_h\).

At an active prime \(p\mid h\), both primes of \(K\) above \(p\) are governed by Lemma 33, each with length \(t_p\). Relaxation therefore subtracts exactly \(2t_p\). There is no bound hiding an error at every such prime. At \(r\), the central scalar character is trivial; the local degree-two determinant has the valuation of \(r+1-a_r\chi_h(r)\) at each split place. The trace condition \[v_2(2-a_r),\ v_2(2+a_r)\le c_E,\qquad v_2(r-1)>c_E\] makes both valuations at most \(c_E\). It also bounds the trace multiplier \(a_r-2\chi_h(r)\) on the central point. At primes over \(2N\) the integral Kummer condition is retained. The local twisting types form a fixed finite set and \(K\) is split there; all comparison indices, including those of Neron differential lattices at \(2\), are therefore bounded in terms of \(E\). This proves the first equality, using Lemma 32 to compare the generic tensor with the central tensor.

We spell out the quadratic comparison needed for the second equality. Set \[A_h=\operatorname{Res}_{K/\mathbb Q}(E^{(h)}_K), \qquad B_h=E^{(h)}\times E^{(hk)}.\] There are isogenies in both directions whose composites are multiplication by \(2\). The isogeny-invariance identity for the BSD quotient, with finite Shafarevich–Tate groups, is \[ \frac{\#\operatorname{Sha}(A_h)\operatorname{Reg}(A_h) \operatorname{Per}(A_h)\prod_p c_p(A_h)} {\#A_h(\mathbb Q)_{\rm tors} \#A_h^\vee(\mathbb Q)_{\rm tors}} = \frac{\#\operatorname{Sha}(B_h)\operatorname{Reg}(B_h) \operatorname{Per}(B_h)\prod_p c_p(B_h)} {\#B_h(\mathbb Q)_{\rm tors} \#B_h^\vee(\mathbb Q)_{\rm tors}} . \tag{48}\] Regulators and periods are taken with compatible dual and Neron normalizations. This is the Cassels–Tate isogeny identity. The elliptic-curve local-measure formula is due to Cassels (Cassels 1965, Theorem 1.1). For the general abelian-variety identity used here, see the proof of (Milne 2006, I, Theorem 7.3, especially (7.3.1), pp. 97–100) and (Dokchitser and Dokchitser 2010, sec. 4.1). This identity requires finite Shafarevich–Tate groups, as already established at these noncentral twists, but does not assume the BSD conjecture. The fixed-degree composites bound the rank-one regulator indices, the torsion ratios, and the differential and period indices by fixed powers of \(2\). The identity is essential: a statement that the maps on Shafarevich–Tate groups have kernel and cokernel killed by \(2\) would not bound their lengths uniformly.

At every split prime of \(K/\mathbb Q\) the component factors of \(A_h\) and \(B_h\) agree. In particular this cancels all primes dividing \(h\). At an odd prime \(p\mid k\) outside \(2N\), the curve over \(K\) has good reduction. Its Weil restriction has connected special fiber: the reduction of Weil restriction over the ramified quadratic extension is an extension of the good special fiber by a connected additive group. Thus its component factor is \(1\). Of the two rational twists, one is good and the other is ramified quadratic with good original curve. The latter has \(2\)-part of its component number equal to \(2^{t_p}\). This follows as well from the ramified tame description in Lemma 33: its connected reduction is unipotent, and its local \(2\)-primary torsion is the fixed \(2\)-torsion. At other good nonsplit places the component factors are \(1\); at \(2N\) there are only the fixed local comparisons. Taking valuations in (48) therefore gives \[ \operatorname{length}\operatorname{Sha}(E^{(h)}/K)[2^\infty] =s(h)+s(hk)+\sum_{p\mid k}t_p+O_E(1). \tag{49}\] Since \(0\le t_p\le2\), the sum over \(k\) is at most \(2A\). Substituting this equality and \(2\sum_{p\mid h}t_p=4w(h)\) proves the result. ◻

Proof of Corollary 11. The arithmetic diagrams constructed with Lemma 26, their paired functionals, and the model bound of Lemma 18 give the hypotheses of Propositions 29 and 31. Those propositions give nonnegative height-one valuations away from \((2)\) and a lower bound \(-C(E,A)\) at \((2)\). Normality of \(\Lambda\) gives \(2^{C(E,A)}u_h\in\Lambda\). Its constant coefficient has valuation at least \(-C(E,A)\). Apply (47). ◻

Proof. At an unused split prime \(q\), write \(z\) for the full scalar Frobenius at one orientation. Conjugation gives \(z^{-1}\) at the other orientation. The point trace contributes \[e_q=a_q-z-z^{-1},\] whereas relaxation of the singular quotient contributes \[ l_q= \det(1-q^{-1}z\operatorname{Fr}_q\mid T) \det(1-q^{-1}z^{-1}\operatorname{Fr}_q\mid T). \tag{50}\] Using reciprocal Frobenius conventions changes only the corresponding unit conventions in the paired determinant. The localization triangle gives \(u_{\rm rel}=e_q^2u_{\rm prim}/l_q\) at this unused prime. The square is required because the point and its dual both occur.

Both factors are units in the local ring \(\mathcal O[G]\). To check this it suffices to reduce modulo its maximal ideal, generated by \(2\) and the augmentation ideal. There the scalar is \((1+t)^m\) with \(m\) odd, and the residual Frobenius polynomial is nonzero at this nonconstant scalar. More precisely, before character evaluation the following identity holds modulo \(2\): \[ (1+a_qz+z^2)(1+a_qz^{-1}+z^{-2}) =a_q^2+z^2+z^{-2} =(a_q+z+z^{-1})^2 . \tag{51}\] Thus \(l_q/e_q^2-1\in2\mathcal O[G]\), and \[ \eta= \prod_{q\in Q} \left(1+ \frac{l_q/e_q^2-1}{2}(1+g_q)\right) \in\mathcal O[G]. \tag{52}\] Its factor evaluates to \(l_q/e_q^2\) when \(q\) is unused and to \(1\) when \(q\) is active. This proves the claim. The apparent inertia projector denominator is canceled inside each factor before taking the product. ◻

Proof. For a denominator \(s\in\Lambda\setminus(2)\), write \(s=t^d v+2w\), with \(v\) a power-series unit. Its inverse modulo \(2^M\) is its finite geometric expansion about \(t^dv\), which has finite negative support. Conversely all Laurent series of finite negative support occur after inverting \(t\). This proves (53). It also proves that an element of \(\mathcal O\) has a unique series \(\sum a_nt^n\), \(a_n\in\mathbb Z_2\), with \(v(a_n)\to+\infty\) as \(n\to-\infty\), and that its valuation is \(\min_n v(a_n)\).

Write \(\det A_j=t^{d_j}v_j+2w_j\), with \(d_j\le h\), and put \[C_j=\frac{\operatorname{adj}(A_j)}{t^{d_j}v_j}, \qquad C=\operatorname{diag}(C_j).\] Each entry of \(C\) has pole at most \(h\), and \(BC=1+2F\), where entries of \(F\) have the same pole bound. Therefore \[ B^{-1}\equiv C\sum_{j=0}^{M-1}(-2F)^j \pmod{2^M}. \tag{54}\] The \(j\)-th summand has pole at most \((j+1)h\). Adding more entries in a matrix product cannot worsen this pole bound. Multiplication by power-series matrices proves the Schur assertion.

For a block over \(R\) whose augmentation is \(A\), the block is invertible over \(\mathcal O[G]\): its residual determinant in the local ring is a unit. Each sign evaluation is \(A+2E_x\) with power-series entries, so the same estimate holds at all characters. If \(S_0=0\), its Schur complement has augmentation exactly \(P_0\). ◻

Proof. Put the central integral complex into elementary-divisor bases over \(\mathbb Z_2\). Its positive pivot valuations sum to at most \(h_1\). Lift these constant integral basis changes. Formally cancel the same pivot positions upstairs, recording the pivot numerators and all denominators in the resulting rational basis changes and determinant comparison. There are at most \(r_0\) such operations. A product \(P\) of sufficiently large, but bounded, powers of their numerators clears every denominator. Its degree in the differential entries is bounded in terms of \(r_0\); its augmentation belongs to \(\Lambda\) and has central valuation bounded in terms of \(r_0,h_1\).

On constituents where \(P_x\ne0\), the remaining complex is one-by-one in degrees one and two. If its differential is nonzero, the cycle identity forces the projected cycle to be zero, so the same rational determinant expression gives the stipulated zero coordinate. Otherwise the expression is the product of the projected cycle and functional, with the canceled-pivot factor. Clearing denominators gives \[u_x=2^{-a_0}\eta_x Q_x/P_x,\qquad Q\in\mathcal O[G].\] Take \(f=2^{a_0}P^2\) and \(d=\eta P Q\). On a constituent where a pivot first vanishes, \(P_x=0\), so both sides of \(f_xu_x=d_x\) are zero. Thus the identity holds on every character. Since \(f\) is a polynomial of bounded degree in the bounded differential entries, its central valuation and pole properties follow. All unused-prime corrections and eliminated-block units occur in \(d\); none is inserted into \(f\). ◻

Proof. Use the integral universal class and paired functional just constructed. To apply Lemma 37, it remains to reduce the complex to bounded size without losing uniform control of its negative Laurent powers.

At augmentation, the localization triangles present the complex as the old complex extended by a direct sum of singular local blocks in degrees one and two, one block for each new prime. Its differential is upper triangular, with the old complex a subcomplex. This graded model lifts over \(R\): split disks over the augmentation local ring, identify the minimal parts, and lift the invertible basis changes. This is an application of the actual finite diagram construction, rather than just an equality of Euler characteristics.

Each new block has determinant \(l_{q,0}\). Its residual \(t\)-order is uniformly bounded. In the reducible residual range the residual Frobenius polynomial is \((X+1)^2\), and an odd scalar exponent gives order two at each split place; thus order four suffices for the two-place block. Eliminate the entire new block over \(\mathcal O[G]\). By Lemma 36, the resulting complex \(D\) has the bounded old size, has augmentation equal to the old complex, and has a uniform pole bound at every fixed precision.

The central cohomology of the old complex is the fixed base Kummer cohomology, modified only by the bounded auxiliary-prime terms. Its rational dimension is one in degrees one and two. The quotient of the discrete Selmer group by its maximal divisible subgroup is finite because its Pontryagin dual is a finitely generated \(\mathbb Z_2\)-module. For this fixed \(K,E\), its length is a finite constant, even when the Shafarevich–Tate group itself is not known to be finite. Together with global torsion and the trace bound at \(r\), this bounds all central torsion lengths of \(D_0\).

Apply Lemma 37. The determinant of the eliminated block is a unit over \(\mathcal O[G]\), so its inverse is an allowed integral multiplier. Include it and \(\eta\) from (52) in the numerator multiplier of that lemma. This proves the assertion.

The dependence here is on fixed \(K,E\); it is different from the uniform \(C(E,A)\) of Proposition 31. No uniform bound on the finite Selmer quotient as \(K\) varies is asserted or needed for the missing-vertex argument. ◻

Proof. The paired arithmetic diagram constructed after Lemma 26 contains the actual weighted target classes, local Kummer lifts, target comparison and boundary witnesses, and simultaneous duality triangles. The target-side construction gives its inner rank bound for each fixed finite conductor support; no ring-class source complex is passed to the limit. Theorem 24 retains their evaluations and exact-kernel restrictions. The same-stage extension in Proposition 31 shows explicitly that these data remain compatible after each further switch.

Lemma 18 bounds the minimal models in terms of \(E,A\) and the number of retained derivative primes, not the active conductor support. Outer rank reduction uses only a bounded number of switches, hence only finitely many of these bounds. The paired functional has the common clearing exponent established above. These are the three hypotheses of Proposition 31; the characteristic-zero evaluations and local acyclicity give those of Proposition 29. The actual central trace and \(\Lambda_{(t)}\)-representative, together with the local quotients of Lemma 33, were verified in the proof of Proposition 34.

For a binary family, Section 5.4 constructs the universal trace and its dual functional over the group ring before evaluating characters. Lemma 35 then removes the unused-prime factors integrally. The bounded complex in Proposition 38 comes from the actual localization triangles and simultaneous Schur elimination, with the order-four residual local bound. Its central torsion is controlled by the finite quotient of the fixed base Selmer group and the bounded auxiliary-prime terms. Together with the stated constituent-rank assumption, these give all the hypotheses of Lemma 37.

For varying \(K\), only its bounded ramification count enters the residual-model and local-comparison constants. For the final binary argument \(K\) is fixed, so its finite base Selmer quotient may enter the clearing constants. At \(2N\) the original Kummer conditions have been retained throughout; the construction imposes no ordinary or semistable hypothesis. ◻

Proof. Write the element supplying the \(i\)-th test as \(a_i=\sum_{g\in G}a_{i,g}(t)g\), and let \(n_i\) be the Laurent degree being tested. Lemma 36 gives integral Laurent coefficients, so \[z_i=\sum_{g\in G}[t^{n_i}]a_{i,g}(t)\,g\in\mathbb Z_2[G].\] The group \(G\) is finite, hence coefficient extraction commutes with character evaluation: \((z_i)_x=A_i(x)\). Apply Lemma 7 to these \(q\) integral group-ring elements. Its bound is exactly (56); no bound on the Laurent degrees or on the coefficients enters this argument. ◻

Proof. Apply Lemma 40 to the coefficients of \(f_x\) in degrees \(-L(M),\ldots,0\) and the constant coefficient of \(d_x\). At its nonzero solution, all negative coefficients of \(f_x\) are divisible by \(2^M\), and its constant coefficient is congruent to \(f_0(0)\). Hence the latter has valuation \(c\), and \(v_{\mathcal O}(f_x)\le c\). Since \(d_x\) is integral over \(\mathcal O\), \(v_{\mathcal O}(u_x)\ge-c\), and the same holds for \(u_0\). The Laurent expansion in Lemma 36 then shows that all their coefficients lie in \(2^{-c}\mathbb Z_2\). They have no negative coefficients because they belong to \(\Lambda[1/2]\).

The constant-term convolution converges \(2\)-adically, and gives \[[t^0]d_x\equiv [t^0]f_x\,u_x(0) \pmod{2^{M-c}}, \qquad [t^0]d_0=f_0(0)u_0(0).\] Use the imposed congruences and divide by \(f_0(0)\). This proves (57). The possibly large negative support of \(d_x\) was never tested; only its constant coefficient was used. ◻

Proof of Theorem 12. For every character the primitive generic cohomology has rank at most one. At nonzero vertices this follows from the simple-zero theorem; at zero it follows from (17) and rational specialization. There are no generic invariants, and duality pairs degrees one and two; thus each primitive constituent is acyclic or has exactly the rank-one pattern. The unused-prime local blocks are acyclic over the fraction field, by the nonzero factors (50). The universal Selmer complex has the same generic pattern, as required by Proposition 38. With this hypothesis verified, Proposition 39 supplies the compatible universal arithmetic construction. By Proposition 29 and Lemma 35, the corrected coordinates belong to \(\Lambda[1/2]\). Proposition 34 gives \[v(u_x(0))\le B_1\quad(x\ne0)\] for a constant \(B_1\) depending only on \(B,E,K\) and the fixed local data.

If the base derivative vanished, the Gross–Zagier formula would make the central Heegner sum torsion: its canonical height is a fixed nonzero multiple of that derivative. Here only the conductor-one formula at the fixed base \(h=1\) is used, with the corrected primitive sum and the split Heegner data of (Cai et al. 2014, Theorem 1.1). At the base this implies \(u_0(0)=0\). Indeed, if the generic rank is zero the coordinate was defined to be zero. If the generic rank is one, its central rank is also one by (17), so Lemma 32 identifies its specialized class with the zero rational Kummer class of that torsion point.

Proposition 38 supplies \(f,d\) with a common bound \(c_*\) on \(v(f_0(0))\) and common pole bounds \(L(M)\). Choose \[M>\max\{c_*,B_1+2c_*\}\] and then take \(b\) above the threshold of Lemma 41. That lemma gives a nonzero vertex with \(v(u_x(0))\ge M-2c_*>B_1\), a contradiction. Thus the base derivative is nonzero. The classical split Heegner sign is odd, so the base product has a simple zero. ◻

Proof. Use the Shimura packet of \(E^{(-\sigma)}\). Its positive index \(D\) tests the twist by \(\chi_{-D}\), which is \(E^{(\sigma D)}\). The nonvanishing at \(D_0\) selects a globally occurring member admitting that Fourier functional. Take its spherical vector outside \(S\) and its holomorphic vector of weight \(3/2\) at infinity.

Here is the support-local vector selection. Start with a vector on which the \(D_0\) Fourier functional is nonzero. Projection under a sufficiently large compact group of upper unipotents to additive frequency \(D_0\) restricts the potentially nonzero unit translates to arbitrarily small neighborhoods of the two solutions of \(u^2=1\). Choose a unit character agreeing with the forced action at \(-1\). It can have \(2\)-power order: the image of \(-1\) is detected in the \(2\)-primary quotient of the unit group. The neighborhoods of \(1\) and \(-1\) then contribute with the same sign, so this projection preserves the nonzero functional. Upper integral translations act trivially, and smoothness gives sufficiently deep lower-unipotent invariance. The vector therefore defines a classical form at a level supported on \(S\), allowing arbitrary depth at \(2\) and at the bad primes. The unit-squareclass and coprimality sieves commute with good \(T(p^2)\), since their conditions are unchanged by multiplication of the index by \(p^2\). The classical Hecke formula (Shimura 1973) gives (62), with \(\eta=\psi\chi_{-\sigma}\).

At the level just chosen, let \(V\) be the full space of weight-\(3/2\) cusp forms orthogonal to unary theta series, with scalar character \(\psi\) and good \(T(p^2)\) eigenvalues \(\eta(p)a_p(E)\). It contains the constructed form, whose coefficient at \(D_0\) is nonzero. For \(D/D_0\) a unit square at every support prime, Waldspurger’s comparison (Waldspurger 1981, Corollary 2, pp. 483–484) gives, for every \(f\in V\), \[a_f(D)^2= \psi(D/D_0)\frac{L_*(\sigma D)}{L_*(\sigma D_0)}a_f(D_0)^2.\] Its hypothesis (H1) follows from the nonzero eigenspace (Waldspurger 1981, sec. I.2, Proposition 2); the final paragraph of the comparison proof removes (H2) for these local squareclasses. Factoring the displayed quadratic identity in the two coefficient functionals shows that they are proportional on all of \(V\), with one scalar independent of \(f\). This includes oldvectors and their linear combinations; if the central value at \(D\) vanishes, the first functional is simply zero.

Choose an algebraic form \(g\) in the filtered subspace of \(V\) with \(a(D_0)\ne0\). To justify this choice, multiplication by ordinary theta embeds the space into an integral-weight space, and divisibility by theta is specified by algebraic cusp conditions (Ramsey 2006, secs. 3–4). The character, good-Hecke and filter conditions are algebraic linear conditions as well. Since evaluation at \(D_0\) is nonzero after extension to \(\mathbb C\), it is nonzero on some algebraic form.

Let \(u\) be an automorphism of \(\overline{\mathbb Q}\) acting on the character values by an odd power \(e\), and put \(\nu=\eta^{(e-1)/2}\). The forms \(g^u\) and \(g\otimes\nu\) have the same scalar character \(\psi^e=\psi\nu^2\) and the same good eigenvalues \(\eta(p)^e a_p(E)=\nu(p)^2\eta(p)a_p(E)\); here \(\psi/\eta\) is quadratic. They belong to one good-Hecke eigenspace at a common level still supported on \(S\). Applying the preceding proportionality in that nonzero space and normalizing at \(D_0\) gives \[u(\xi)=\nu(D/D_0)\xi,\qquad \xi=a(D)/a(D_0).\] Local square roots of \(D/D_0\) at the conductor of \(\eta\) give a value \(z\) of \(\eta\) with \(\eta(D/D_0)=z^2\). Then \(u(z)=z^e\) and \(\nu(D/D_0)=z^{e-1}\), so \(\xi/z\) is fixed by every algebraic automorphism and is rational. Moreover the same local-square condition gives \(\psi(D/D_0)=\eta(D/D_0)=z^2\). The comparison therefore reads \[(\xi/z)^2=\frac{L_*(\sigma D)}{L_*(\sigma D_0)},\] which proves (63) since \(v(z)=0\). There is no product over the new prime factors: the unramified squarefree formula includes both local valuations zero and one. No exceptional local representation at \(2\) is excluded by the same-local-class comparison. The vanishing assertion and (64) follow. ◻

Proof. For every elliptic curve \(A/\mathbb Q\), the rational Tate module \(V_2A\) is irreducible. Indeed Faltings’ semisimplicity and endomorphism theorem (Faltings 1983, sec. 1, assertions (a),(b)) would turn a proper summand into a nontrivial idempotent in \(\operatorname{End}_{\mathbb Q}(A)\otimes\mathbb Q_2=\mathbb Q_2\). The equality of this endomorphism ring with \(\mathbb Z\) includes CM curves: a nonreal CM endomorphism is not defined over \(\mathbb Q\). The Tate image contains a nonidentity central scalar, by the homothety theorem (Bogomolov 1981, sec. 2, Theorem 3 and paragraph 7).

Let \(Q\in A(\mathbb Q)\) be nontorsion and let \(c_Q\) be its division cocycle. On the kernel of the Tate action its image is a closed \(\mathbb Z_2\)-submodule of \(T_2A\), stable under the full Galois image. If its rational span were zero, the rational Kummer class would factor through the Tate image. A central scalar \(z\neq1\) annihilates its first cohomology: for a cocycle \(c\) the equality of \(c(zg)\) and \(c(gz)\) gives \((z-1)c(g)=(g-1)c(z)\), so \(c\) is a coboundary over \(\mathbb Q_2\). This contradicts Kummer injectivity and the finite generation of \(A(\mathbb Q)\). Irreducibility consequently makes the translation image contain \(2^aT_2A\) for some \(a\). Intersecting with the kernel of additional finite data retains a full lattice.

Apply this to the rational twist point supplied by \(LP_{D_w}\) on \(A=E^{(D_w)}\). Since \(D_w>0\), conjugation acts on its Tate module as \(C\). Modify a conjugation lift by an element of the simultaneous kernel whose translation is \(v\). The square of the modified element has translation \((1+C)v\), and \(b(1+C)v=2b(v)\). The full-lattice property allows a choice with a fixed finite nonzero valuation. Conjugation acts trivially on \(\mathbb Q(\sqrt{D_w})\) and nontrivially on \(\mathbb Q(\sqrt{h_*})\). Chebotarev in each finite torsion, division, and character extension gives (65) and the chosen finite nonzero detection.

For good \(\ell\), Lang’s theorem and finite-field Kummer theory give \[E(\mathbb F_{\ell^2})/2^M E(\mathbb F_{\ell^2}) \simeq T/\bigl(2^MT+(\operatorname{Fr}_\ell^2-1)T\bigr)=T/2^MT.\] This is evaluation of a cocycle at arithmetic Frobenius squared. Its equivariance and \(bC=b\) prove the sign assertion. Reduction of the division cocycle gives precisely the detection already arranged. Finally \(\det C=-1\) implies \(\ell\equiv-1\pmod {2^M}\). ◻

Proof. Let \(B\) be ramified at \(\ell\) and infinity, \(R\) an Eichler order of level \(N\), and \(V=B^0\) with \(Q(x)=-x^2=\operatorname{Nrd}(x)\). The root characters at the fixed level primes below are chosen once. Let \(\mathcal O\) be the valuation ring at a place above \(2\) of the field generated by their values. The other local weights are quadratic, so this field is independent of \(M\) and \(\ell\). In this construction integrality means membership in \(\mathcal O\), and congruences mean congruences modulo \(2^M\mathcal O\).

The supersingular coefficient function. The supersingular pairs on \(X_0(N)\) are the finite set \[X=B^\times\backslash\widehat B^\times/\widehat R^\times.\] This description permits every \(N\) prime to \(\ell\), including arbitrary depth at \(2\); see (Jetchev and Kane 2011, secs. 1.1–1.2,2.4–2.6). For \(z\in X\) put \(R_z=B\cap z\widehat Rz^{-1}\). Choose \(L\) killing the finite subgroup generated by the images of the cuspidal constants in the Atkin–Lehner relations, and also all the bounded-degree torsion used below. Then \[P_M(z)=\lambda_M(L\varphi_\ell(z)),\] where \(\varphi_\ell\) is the specialization of \(\varphi\) to the good-reduction special fiber. This function has Brandt eigenvalues \(a_p(E)\) away from \(N\ell\), Atkin–Lehner eigenvalue \(w_{p^e}\) at \(p^e\Vert N\), and eigenvalue \(+1\) under the right \(\ell\)-uniformizer. These identities are modulo \(2^M\). Choose any integer lifts of this finite list of values.

The local Schwartz weights. We now define the local functions, rather than appealing to an unspecified weighted theta theorem. Off \(2Nh_*\ell\) take the characteristic function of \(R_p^0\). At \(p^e\Vert N\), use a model \[R_p=\left\{\begin{pmatrix}a&b\\c&d\end{pmatrix} :a,b,d\in\mathbb Z_p, c\in p^e\mathbb Z_p\right\}.\] Require the split fundamental unit squareclass and an integral embedding. Its action on the cyclic level quotient selects one root \(r\in\mathbb Z_p^\times\) of \(x^2\). Give \(x\) weight \(\alpha_p(r)\), where \(\alpha_p\) is a finite unit character of \(2\)-power order with \[ \alpha_p(-1)=w_{p^e}\chi_{h_*}(p)^e. \tag{67}\] Such a character exists for either prescribed sign. The root is a locally constant function to any fixed character precision: the two idempotents of the split integral quadratic algebra distinguish the two summands of the cyclic quotient. Unit scaling \(x\mapsto ux\) multiplies this weight by \(\alpha_p(u)\), and order-unit conjugation preserves it.

At \(2\) require \((1+x)/2\in R_2\) and \(x^2\in\mathbb Z_2^{\times2}\). If \(2\mid N\), the preceding selected-root rule is imposed using \(e_x=(1+x)/2\). Its two eigenvalues \((1+r)/2,(1-r)/2\) differ by the odd unit \(r\); hence the two idempotents remain distinct modulo \(2\). This verifies the construction at every exponent of \(2\) in \(N\). Scaling by an odd \(u\) preserves integrality, since \((1+ux)/2=u(1+x)/2+(1-u)/2\). This is an invertible affine change of the operator \(e_x\), so it preserves its two summands individually. The selected summand is selected by the cyclic level quotient, not by declaring its eigenvalue to be \(0\) or \(1\) modulo \(2\). On that same summand \(x\) acts as \(r\) and \(ux\) acts as \(ur\). Even when \(u\equiv-1\pmod4\) interchanges the two eigenvalue parities, the root weight therefore changes by exactly \(\alpha_2(u)\). If \(2\nmid N\) there is no root weight at \(2\).

For \(p\mid h_*\) require \(x\in M_2(\mathbb Z_p)^0\) and \(v_p(Q(x))=1\). Its reduction is a nonzero nilpotent. Fix \(X_0=\left(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right)\) and define \[\epsilon_p(gX_0g^{-1})=\left(\frac{\det g}{p}\right).\] The centralizer consists of \(\left(\begin{smallmatrix}a&b\\0&a\end{smallmatrix}\right)\), with determinant \(a^2\), so this is well-defined on the entire nonzero nilpotent orbit. It obeys \[\epsilon_p(uyu^{-1})=(\det u/p)\epsilon_p(y),\qquad \epsilon_p(ay)=(a/p)\epsilon_p(y).\] The second equality follows by conjugating \(X_0\) by \(\operatorname{diag}(a,1)\). Set \(f_p(x)=\epsilon_p(\bar x)\) on the specified support. Thus the right conjugation character is precisely the local component of \(\chi_{h_*}\circ\operatorname{Nrd}\).

At \(\ell\), let \(\mathfrak P\) be the maximal ideal of the maximal order. Its residue field is \(\mathbb F_{\ell^2}\), and the reduction of an integral trace-zero element lies on its one-dimensional trace-zero line. Fix a nonzero basis vector \(\beta\) on that line and put \[f_\ell(x)=\left(\frac{t}{\ell}\right) \quad\text{if }\bar x=t\beta\neq0, \qquad f_\ell(x)=0\quad\text{otherwise}.\] Order units act trivially by conjugation on the residue field. A uniformizer acts by its nontrivial automorphism and sends \(\beta\) to \(-\beta\). Therefore its multiplier is \(-1\), since \(\ell\equiv-1\pmod4\); scaling has character \((u/\ell)\). Moreover \(\beta^2\) is a nonsquare and \(-1\) is a nonsquare modulo \(\ell\), so \(Q(x)=-t^2\beta^2\) is a nonzero square. Because \((|h_*|/\ell)=1\), the extracted indices automatically satisfy \((D/\ell)=1\). No average over \(\mathbb F_\ell^\times\) is used.

All these functions have compact support and are locally constant. Let \(f=\prod f_p\), incorporating any additional fixed unit-squareclass conditions by their characteristic functions. Those conditions are stable under unit squares and change none of the scaling characters. For \(u\in\widehat R^\times\), \[ f(u^{-1}xu)=\chi_{h_*}(\operatorname{Nrd}u)^{-1}f(x). \tag{68}\] The product scaling character is even. Indeed its value at \(-1\) is \[\left(\prod_{p^e\Vert N}w_{p^e}\chi_{h_*}(p)^e\right) \chi_{h_*}(-1)(-1)=1;\] the parenthesized factor is the even functional sign of \(E^{(h_*)}\). Denote its associated classical character by \(\eta_B\). In the inverse-idele convention fixed above, if \(\alpha\) is this finite scaling character, then \(\eta_B(d)=\alpha(d)\) for positive integers \(d\) prime to the support: the inverse idele character becomes the product of the finite unit values at the lower-right entry.

The scalar transformation law. Here are all the Weil-representation checks needed for scalar modularity. For an additive character of conductor \(\mathbb Z_p\), write \(n(b)=\left(\begin{smallmatrix}1&b\\0&1\end{smallmatrix}\right)\), \(m(u)=\operatorname{diag}(u,u^{-1})\), and \(n^-(c)=\left(\begin{smallmatrix}1&0\\c&1\end{smallmatrix}\right)\). Use a Weyl lift \(w\), its compatible self-dual Fourier transform, and the Weil factors \(\gamma_p(u),\gamma_p\) for this additive character. The defining formulas are \[\omega(n(b))f_p(x)=\mathbf e_p(bQ(x))f_p(x),\quad \omega(m(u))f_p(x)=\gamma_p(u)|u|_p^{3/2}f_p(ux),\quad \omega(w)f_p=\gamma_p\widehat f_p.\] They follow from the Weil construction and Poisson summation (Weil 1964, I, no. 13). Integral \(b\) fixes \(f_p\), since \(Q\) is integral on its support. If \(f_p\) is invariant under translations by a lattice \(L_p'\), its Fourier transform is supported on the dual lattice \((L_p')^\vee\). Choose \(A_p\) so that \(p^{A_p}Q((L_p')^\vee)\subset\mathbb Z_p\). Then \(n(p^{A_p}b)\) fixes \(\widehat f_p\), and conjugating by \(w\) proves that every \(n^-(p^{A_p}b)\) fixes \(f_p\). This argument works without change at \(2\) and for all finite character depths.

Relative to the ordinary theta multiplier cubed, the diagonal Weil factor is \((\det Q,u)_p\). Indeed diagonalizing \(Q\) and multiplying the one-dimensional Weil-index identities (Weil 1964, II, nos. 25, 28) gives the Hilbert symbol of the product of its three coefficients. In a presentation \(B=(-a,-b)\) that product is the determinant of \(aX^2+bY^2+abZ^2\), namely \((ab)^2\). Thus this additional factor is \(1\). The remaining diagonal action is exactly the scaling character already computed. Increasing \(A_2\) to the depth needed for the standard metaplectic splitting, the factorization \[\begin{pmatrix}a&b\\c&d\end{pmatrix} =n^-(c/a)m(a)n(b/a),\qquad c\in p^{A_p}\mathbb Z_p, a\in\mathbb Z_p^\times,\] proves scalar transformation on the required local \(\Gamma_0\). Conjugation by \(z\) commutes with these Weil actions and preserves \(Q\). Consequently the same level and scalar character work for every summand. The Gaussian at infinity gives holomorphic weight \(3/2\); positive definiteness gives nonnegative exponents at every cusp.

Choose an integer \(c_0\) divisible by every possible \(|R_z^\times|\). Such a uniform choice exists because these unit groups embed in \(\operatorname{GL}_4(\mathbb Z)\) and their orders are bounded. The theta series is \[ H_M(n)=c_0\sum_{z\in X} \frac{P_M(z)\chi_{h_*}(\operatorname{Nrd}z)}{|R_z^\times|} \sum_{\substack{x\in B^0\\Q(x)=n}} f(z^{-1}xz). \tag{69}\] Equation (68) proves independence of right representatives; rational reduced norms are positive, so the norm character is trivial on the left rational action. The preceding Weil calculation proves modularity (Weil 1964, III, Theorem 6), and the choice of \(c_0\) proves integral Fourier coefficients. More precisely, each \(c_0/|R_z^\times|\) is an integer, so changing a lift of \(P_M(z)\) by \(2^M\) changes every coefficient by an element of \(2^M\mathcal O\). No precision is lost to a stabilizer denominator. The chosen integer lifts need satisfy no characteristic-zero eigenrelation.

The good-prime Hecke relation. For completeness, the good-prime relation can be checked in a matrix order. Neighbors correspond to the lines in \(\mathbb F_p^2\). A nonzero primitive nilpotent reduction has exactly one invariant line. For an integral element of nonzero reduction and norm \(n\), the number of invariant lines is \(1+(-n/p)\); zero reduction contributes the additional \(p\) lines. If a primitive \(y\) has \(p^2\mid Q(y)\), choose a basis in which its reduction is \(X_0\). Its lower-left entry is divisible by \(p^2\), and exactly the neighbor for that invariant line makes \(y/p\) integral. These counts partition the three terms of the theta Hecke operator. Scaling by \(p\) at the support contributes \(\eta_B(p)\), and the reduced norm of a Brandt neighbor contributes \(\chi_{h_*}(p)\). Thus, with \(A(n)=H_M(n)\), \[A(p^2n)+\eta_B(p)(-n/p)A(n)+p\eta_B(p)^2A(n/p^2) =\eta_B(p)\chi_{h_*}(p)a_p(E)A(n)\pmod {2^M}.\] The orbit-stabilizer weights in (69) are exactly the weights in this neighbor count. Extracting \(a_M(n)=H_M(|h_*|n)\) gives \(\eta=\eta_B\chi_{h_*}\) and \(\psi=\eta_B(|h_*|/\cdot)\), proving (62). Fixed-support coefficient sieves are exact zero-or-retain operations; they preserve coefficient integrality and commute with this relation.

The weighted coefficient comparison. It remains to prove the weighted comparison; modularity alone does not supply it. At \(n=|h_*|D\) the counted elements are precisely the optimal embeddings \(\mathcal O_{K_D}\hookrightarrow R_z\), sending \(\sqrt{Dh_*}\) to \(x\). The condition at \(2\) is exactly the condition on the integral generator \((1+\sqrt{Dh_*})/2\). At a level prime there are two orientations. Indeed a stable lattice splits into its two integral idempotent summands, and a cyclic quotient of order \(p^e\) lies wholly in one summand. At \(\ell\) there are two orientations, the two residue-field embeddings. At all remaining primes there is a single local optimal-embedding orbit.

Fix a rational embedding \(x_0\) and an initial orientation tuple \(z_0\). The orbits with these orientations are indexed by \(\operatorname{Pic}(\mathcal O_{K_D})\), with representatives \(az_0\) for \(a\in\widehat K_D^\times\). The equality \[(az_0)^{-1}x_0(az_0)=z_0^{-1}x_0z_0\] shows that the Schwartz weight is constant along that orbit. Its remaining relative weight is \(\chi_{h_*}(\operatorname{Norm}_{K_D/\mathbb Q}a)\), exactly the unramified genus character. The reduction correspondence commutes with this ideal-class action and remembers the cyclic level subgroup (Jetchev and Kane 2011, sec. 2); a Galois-equivariant description of the lift, class action and reduction is given in (Cornut and Jetchev 2013, sec. 3.2, §§4.6, 4.9–4.10 and Corollary 4.1). We use these correspondences only for their geometry; the weight identity has just been computed, and the multiplicity is computed next.

An embedding stabilizer is \(\mathcal O_{K_D}^\times=\{\pm1\}\). An orbit in \(R_z\) therefore contains \(|R_z^\times|/2\) elements, giving the factor \(1/2\) in (69). At \(p^e\Vert N\), reversing orientation multiplies its Schwartz factor by \(\alpha_p(-1)=w_{p^e}\chi_{h_*}(p)^e\), and its norm-weighted parametrization value by the same sign. At \(\ell\) both multipliers are \(-1\): one is the residue-line calculation and the other is \(\chi_{h_*}(\ell)\) times the Frobenius eigenvalue \(+1\). Every one of the \(2^{\omega(N)+1}\) orientation tuples consequently has the same contribution. The initial tuple has a \(2\)-power unit phase. Combining this count with its factor \(1/2\) gives exactly (66). There is no division by a class number, and none of its rational multipliers grows with \(\ell\) or the number of factors of \(D\). ◻

Proof. Gross–Zagier identifies the height of the genus sum with the product derivative (Gross and Zagier 1986). Since \(L(E^{(h_*)},1)\neq0\), vanishing of the derivative makes \(P_D\) torsion, and a nontorsion \(P_D\) is equivalent to a simple zero of \(E^{(D)}\). The genus sum lies in a biquadratic field, so a fixed integer kills all possible torsion: reduce prime-to-residue-characteristic torsion at two fixed good primes of different residue characteristic; the residue degrees are at most four. Include that integer in \(L\). On the free twist line, divisibility by \(2^{j(P_D)}\) remains divisibility after the integral maps of reduction and \(\lambda_M\). Restriction and the two quadratic projections have composite multiplication by \(2\), so their integral indices have a fixed bound. Equation (66) therefore proves both valuation assertions, with fixed losses only. Lemma 47 gives the witness. ◻

Proof. For the even source normalize \(a(D_0)=1\); the phase-removed squarefree coefficients are rational by Proposition 46. For the odd source, (66) makes every visible squarefree coefficient an integer times a root of unity modulo \(2^M\). Their finite valuations are therefore integers in both cases, not arbitrary elements of a ramified coefficient field’s value group. The fractional part of \(w(D)\) is fixed on the filter, because the parity of the number of simple primes is its fixed quadratic discriminant character. The lower bounds and a nonzero witness therefore give a minimum for the even source.

For the odd source choose a nonprincipal ultrafilter on the increasing precisions. Minimize \(v(a_M(D_M))-w(D_M)\) over sequences with bounded weight and visible coefficient on an ultrafilter-large set. The uniform lower bound, discrete values, and fixed witness give a finite attained minimum. At each fixed weight cutoff \(W\), failure of its divisibility on an ultrafilter-large set would allow a choice of a violating \(D_M\) with \(w(D_M)\leq W\), contrary to the minimum. Hence every prescribed finite list of these divisibilities holds at a sufficiently advanced actual stage. If larger weights arise in a coefficient-integrality check, the coarse bound \(\min\{M,w(D)-C\}\) already pays any fixed normalization once both \(M\) and \(w(D)\) exceed its fixed threshold. This also explains why no infinite-precision modular lift is asserted or needed. The last conclusions follow from the two coefficient comparisons. ◻

Proof. After removing the powers of \(\eta(p)\), the square-index multipliers in (62) are \[b_0=1,\quad b_1=a_p-\kappa,\quad b_{s+1}=a_pb_s-pb_{s-1},\qquad \kappa=(\psi/\eta)(p)(-D/p).\] If \(p\mid D\), then \(\kappa=0\) and \(b_s\) alternates between \(1\) and \(0\) modulo \(2\), because \(a_p\) is even. Thus the difference from \(p\in U/J\) is odd. If \(p\nmid D\), all \(b_s\) are odd and the difference from \(p\in J\) is even. Subtracting the normalization in (71) gives exactly \[\delta+w((U/J)r)+\#J-\lceil\delta+w(U)\rceil =w(r)+\tfrac12\#\{p\in J:p\text{ simple}\}-e_U.\] This is an integer on the allowed filter. Its only potential negative value before imposing parity is \(-1/2\), so it is nonnegative.

For the odd source this calculation first applies on a fixed bounded range of squarefree weights, with enough precision to pay the fixed normalizing power of \(2\). To obtain the global truncated assertion, fix \(R\) and a bound \(W\) for \(w(U)\). The normalizing exponents \(\lceil\delta+w(U)\rceil\) are uniformly bounded. Above a sufficiently large weight threshold, the coarse bound \(\min\{M,w(D)-C\}\) from Lemma 50 pays that normalizing power and at least \(R\) further factors of \(2\), once \(M\) is large enough. Below the threshold use the stabilized minimum and increase the precision again if needed. The recurrence has integral multipliers, so errors modulo \(2^M\) remain divisible by \(2^M\) at every square shell before normalization. Thus at the chosen stage the truncated bound holds at every coefficient for every \(U\) with \(w(U)\le W\), giving integrality without asserting an unbounded valuation estimate for a fixed finite-precision lift.

For split \(p\), rationality of \(E[2]\) over \(\mathbb F_p\) gives \(a_p\equiv p+1\pmod4\). With \(\Delta_s=b_{s+1}-b_s\), every \(\Delta_s\) is even and \(\Delta_{s+1}=a_p\Delta_s-p\Delta_{s-1}\), so \(\Delta_{s+1}\equiv\Delta_{s-1}\pmod4\). Direct calculation gives \[\Delta_1-\Delta_0 =a_p^2-a_p\kappa-2a_p+2\kappa+1-p\equiv0\pmod4.\] Consequently every \(\Delta_s/2\) has the residue in (73). At a simple prime only \(\Delta_0\) is used.

In matching parity the right side of (72) can vanish only when \(r=1\) and \(J\) is split only. The same recurrence, retaining the character phases, gives \(B_U[Jm^2]\equiv B_U[J]\eta(m)\pmod {2}\). The corresponding coefficient of \(\Theta_J\) has exactly that residue, and different squarefree \(J\) have disjoint supports. The numerator in (74) is therefore divisible by the integer \(2\), not just by the maximal ideal of a ramified character field. The ordinary odd-character unary theta transformation and dilation by \(J\) give the asserted character and level. For the odd source these calculations use the same sufficiently advanced stage, uniformly for \(w(U)\le W\), with precision paying the fixed divisions as in Lemma 50. ◻

Proof. We first check integrality of the generators. Diamonds act by scalars on \(B_U\) and on each unary term. For (74), the only apparent denominator is removed by \[\langle u\rangle H_U=\psi_U(u)H_U+ \frac12\sum_J B_U[J] \bigl(\psi_U(u)-\eta(u)\chi_{-4}(u)^i(-J/u)\bigr)\Theta_J;\] the difference of the two characters is zero or twice a unit. The same formula applies after multiplication by theta. Good Hecke operators commute with diamonds, and their coefficient formula preserves integrality. Thus all generators of \(\mathcal M\) are integral. Finitely many Fourier coefficients inject the ambient finite-dimensional space into a finite-dimensional coefficient space, so \(\mathcal M\) embeds in \(\mathcal O^r\) for some finite \(r\). It is consequently a finite, closed \(\mathcal O\)-module. Continuity and simultaneous Chebotarev approximation on the finitely many eigensystems approximate \(t(u)\) and \(d(u)\) by \(T_q\) and \(q\langle q\rangle\) at every prescribed precision. These operators preserve \(\mathcal M\), so their limits do as well. They act by scalars on each good eigensystem and by the identity on its old multiplicity space, which retains the asserted commutation with level operators.

The formula \((T_qf)[n]=f[qn]+q(\langle q\rangle f)[n/q]\) has no second term when \(q\nmid n\). Successive distinct fresh primes therefore give the coefficient assertion. On a single character system, determinants reduce to \(1\) because all characters have \(2\)-power order and the cyclotomic character is odd. On a subtracted vector the displayed diamond formula leaves unary corrections. A fresh trace kills each correction modulo the maximal ideal: the tested index \(nq\) cannot have its prescribed squarefree part \(J\) when \(q\nmid nJ\). This continues to hold after further fresh tests. Thus \(d(u)t(v)=t(v)\) on the reduced trace orbit. Apply the matrix identity \[\operatorname{tr}(A)\operatorname{tr}(B) =\operatorname{tr}(AB)+\det(B)\operatorname{tr}(AB^{-1})\] to each characteristic-zero system and use that determinant identity. Finally \(t(1)=2\) before reduction. ◻

Proof. A constituent old from a level omitting \(p\) has trivial inertia, so both differences are zero, whatever \(U_p^{(2)}\) does on its old multiplicity space. Every other constituent has conductor exponent one and primitive ramified quadratic determinant. The conductor formula excludes monodromy and identifies it as principal series with one unramified and one ramified character. Local compatibility (Carayol 1986, Theorem (A)), after converting its geometric Frobenius convention to the arithmetic convention fixed above, identifies its \(U_p^{(2)}\) eigenvalue with the Frobenius eigenvalue \(\lambda\) on the unramified line. Write \[\rho(\tau_p)=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\qquad \rho(\phi_p)=\begin{pmatrix}\lambda&0\\0&\mu\end{pmatrix}.\] If \(z\) is the lower-right entry of \(\rho(g)\), the two trace differences are \(-2z\mu\) and \(-2z\). Both sides of (76) equal \(-2z\lambda\mu\). The Eisenstein character pair has the same calculation. A unary term is old if \(p\notin J\), and has precisely this ramified quadratic character if \(p\in J\). This exhausts the constituents. ◻

Proof. For an odd source choose the stage in Lemma 51 with cutoff \(R\ge2\); only these fixed divisibilities and finitely many bounded-weight coefficient tests are needed below. Two fresh traces in matching parity, at least one split, add weight at least \(3/2\). The allowed valuation in (72) is an integer, so it is at least \(2\) before the unary division and the result vanishes after it. In opposite parity any two fresh traces add weight at least \(1\); subtracting \(e_U=1/2\) and using integral valuation again gives vanishing. The estimates persist under further tests. Hence respectively \(t(g)t(u)=0\) for split \(u\), or \(t(g)t(u)=0\) for every \(u\). Applying (75) to \(t(gu)t(u)\) gives \(t(gu^2)=t(g)\). The quotient by the subgroup generated by squares is elementary abelian, and the Galois systems are unramified off their level. This proves the stated factorization. At coefficient \(q\) the fresh split weight \(1\) pays exactly the division in \(H_U\); the fresh simple weight \(1/2\) pays exactly \(e_U\) in opposite parity. No unary term survives. This proves the symbol-reading assertion.

For contraction, apply (76) at coefficient \(1\), so that its left side is a coefficient-\(p\) test. Replace its trace arguments by fresh primes. The first-shell calculation in Lemma 51 gives, in each of the split–split, split–simple, and simple–split cases, \[c_p(D')A(D'),\qquad D'=(U/p)q',\quad \operatorname{Fr}_{q'}\sim g\phi_p.\] Here the omitted source is exactly \(B_V\) or \(H_V\), with \(V=U\setminus\{p\}\), constructed from the same original modular form and the same \(\eta,\psi,\delta\); no new minimum or normalization is chosen. To check the integer powers explicitly, put \(n_U=\lceil\delta+w(U)\rceil\) and let \(m_U\) be \(1\) in matching parity and \(0\) otherwise. After fresh tests kill the unary terms, the first shell relative to the omitted-source test has multiplier \[2^{1+n_V-n_U-m_U+m_V}c_p(D').\] For split \(p\) the exponent is zero. For simple \(p\) and matching \(U\) it is also zero. For simple \(p\) and opposite \(U\) it is one; this is precisely the simple–simple case. The character change \(\psi_U/\psi_V=(p/\cdot)\) is already the Legendre factor in this shell calculation. The phase is a unit and disappears on reduction. Let \(q_0',q_1'\) realize \(g\phi_p,g\tau_p\phi_p\), respectively, and put \(D_i'=Vq_i'\). Applying inertia changes the Legendre term in (73) by a sign, so \(c_p(D_1')=c_p(D_0')+1\). Although these Frobenius elements may first be specified in the full current elementary extension, their restrictions to the omitted-source extension agree: inertia at either place above \(p\) is killed in that extension. Its original characters are unramified at \(p\), and its only new character factors come from \(V\). Thus it is unramified at the whole rational prime \(p\), giving \(A(D_1')=A(D_0')\). If \(\Delta_{\tau_p}A(Uq)\) denotes the difference of the original tests with Frobenius labels \(g\tau_p\) and \(g\), the reduced local identity is now explicitly \[\Delta_{\tau_p}A(Uq) =c_p(D_1')A(D_1')+c_p(D_0')A(D_0')=A(D_0'),\] which is (77). When both vertices are simple the product Frobenius is split and the first-shell valuation has a positive excess; its reduction is zero. The entire-prime assertion follows from the actual support of the omitted-\(p\) coefficient form, rather than from any convention about a chosen orientation. Isolation is the same symbol-reading test with \(t(1)=0\).

For the final assertion first use two copies of a single simple Frobenius trace on matching \(H_U\). Put \(b=B_U[1]\), \(\epsilon=\chi_{-4}(q)\), and \(\kappa=(\psi_U/\eta)(q)\). Substitution of (62) and of the unary character gives the two terms in the coefficient-\(1\) test of \(T_q^2\), modulo \(\mathfrak m\): \[\begin{align*} H_U[q^2]&=\frac{b\eta(q)}2 (a_q-\kappa\epsilon-q\epsilon^i),\\ q(\langle q\rangle H_U)[1]&=\frac{b\eta(q)}2 q(\kappa-\epsilon^{i+1}). \end{align*}\] Their sum is \[\frac{b\eta(q)}2 \left(a_q+\kappa(q-\epsilon) -q\epsilon^i(1+\epsilon)\right).\] For \(q\equiv3\pmod4\) the parenthesized half has residue \(a_q/2\); for \(q\equiv1\pmod4\) it has residue \(a_q/2+1\). In both cases it is \(\#E(\mathbb F_q)/2\) modulo \(2\). Since \(b\) reduces to \(A(U)\), the squared trace has the required coefficient. Fresh-prime testing replaces its two copies by distinct primes. To pass to any pair with the specified labels, vary one prime at a time. After that simple prime is omitted the opposite-parity symbol depends only on rational elementary labels; simple–simple changes vanish by the proved contraction rule. Thus only the stated support and split-vertex labels matter. ◻

Proof. First note that the listed generators account for all the relevant radicals. If \(U\) is a finite set of new primes and a squareclass has even valuations outside \(S_F\cup U_F\), divide by the generators at the primes of \(U_F\) where its valuation is odd. The result belongs to \(V_S(F)\). At an inert rational prime its rational generator has valuation one at the unique place of \(F\). Thus fixed-support labels and the displayed residue bits determine every Frobenius test in an extension with this allowed ramification.

We describe the prime selection carefully. For a new split vertex take a template \(\pi_0\) of its prescribed type, with prime ideal \(\mathfrak p_0\) away from the old vertices. Weak approximation gives \(a\in F^\times\) which is sufficiently close to \(1\) at \(S_F\), positive at the prescribed real places, and has any prescribed nonzero residues at the old prime ideals. Choose those residues so that \(a\pi_0\) has the desired quadratic residues there. Use the ray modulus consisting of sufficiently deep powers of the primes of \(S_F\), all old prime ideals, and the real places. Choose a prime ideal \(\mathfrak q\) in the ray class of \((a)\mathfrak p_0\). Then \[\mathfrak q=(a\beta)\mathfrak p_0, \qquad \beta\equiv1\pmod{\text{ray modulus}},\] and \(\pi_{\mathfrak q}=a\beta\pi_0\) is the required generator outside \(S_F\). There are infinitely many such degree-one prime ideals. Indeed, the prime-ideal theorem in a ray class gives order \(X/\log X\) prime ideals of norm at most \(X\), whereas prime ideals of residue degree greater than one contribute only \(O(X^{1/2})\) in a quadratic field. Degree one gives a split rational prime, and finite exclusions remove only finitely many choices.

At an old inert prime with residue field \(\mathbf F_{p^2}\), prescribe the norm of the nonzero residue. The norm onto \(\mathbf F_p^\times\) is surjective, so either rational residue bit is possible. For a new simple vertex the requirements are rational congruences, together with the inert class in \(F\). The Chinese remainder theorem and Dirichlet’s theorem supply fresh primes. Successively adjoining vertices proves simultaneous finite realization. At every step reciprocity determines the reverse bits: the local Hilbert symbols at the support are fixed, and the only other contributions are the two endpoint residues. Taking norms gives exactly (78), so it imposes no additional independent bit.

The same argument applies to a product template. Its valuations and local squareclasses at \(S_F\) are those of the product, and its residues at the remaining vertices multiply. The ray-class construction replaces that template by a degree-one prime with those data. In a simple cluster we instead use rational congruences for the norm labels. No residue at a prime inside the cluster is imposed on its replacement.

In particular this preserves the rational filter, including its norm correction. Write \(\pi=\prod_i\pi_i\) for the oriented product template and \(\pi'=a\beta\pi\) for its fresh representative; \(a\beta\) is a square unit at \(S_F\) and positive at its real places. The support valuations of \(\pi'\) and \(\pi\) agree exactly. Hence \[N\pi=(\prod_i p_i)u,\qquad N\pi'=q u\] with the same signed rational \(S\)-unit \(u\): all its valuations and its sign agree. The quotient \(q/\prod_i p_i=N(a\beta)\) is a local square at the rational support. Its squareclass at \(2\) also identifies \(q^*\) with \(\prod_i p_i^*\) at those fixed tests. Thus no free choice of an untracked \(S\)-unit has entered a product replacement.

Finally, compare two disjoint realizations with identical graph data. Introduce fresh bridge vertices with matching bits to both realizations, and replace vertices one at a time. At each replacement the omitted-vertex elementary Frobenius tests agree, by the generator description in the first paragraph. The symbol interface gives equal values. A preliminary bridge removes any overlap between the original realizations. The same reasoning applies to the product replacements. ◻

Proof. The one-edge assertion is precisely the arithmetic contraction rule, with product replacements justified by Lemma 55. Induction proves the tree assertion: a split–split merger is split, a split–simple merger is simple, and the first merger of two components containing simple vertices has zero derivative. Along a tree there is a unique choice of relative orientations once one root orientation is fixed. Its product template depends on support types and external bits, not on the internal bits being differentiated.

For clarity, consider first the two variables on a split pair \(p,q\). Differentiation in the inertia bit at one place over \(p\) gives the symbol formed with the entire rational prime \(p\) omitted. Its elementary extension is unramified at both places over \(p\). Differentiating in the other oriented bit is therefore zero. Repetition of the same variable is zero by \(\Delta_z^2=0\). This argument would not follow from omission of only one oriented prime ideal.

For a longer cycle, first contract a spanning path. By the product-label rule the closing variable has become internal to the merged cluster. Its change affects neither the product’s support data nor any incidence to a remaining vertex. Equivalently it is an inertia change at an omitted endpoint, covered by the preceding unramifiedness argument. Thus its derivative vanishes. Possible conjugation of the product generator only changes the induced orientation and does not reintroduce that endpoint. Since there is no self-residue among the inputs to the replacement symbol, there is no extra closing-edge datum to retain. Additional derivatives commute with this zero derivative.

For example, contract the aligned edge \(12\) in a triangle. With \(\pi_{12}=\pi_1\pi_2\), the two bits toward vertex \(3\) are \(t_0=z_{13}^0+z_{23}^0\) and \(t_1=z_{13}^1+z_{23}^1\), up to fixed type corrections. An aligned edge \(23\) then differentiates \(t_0\); a conjugately aligned closing edge \(31\) differentiates \(t_1\). Their mixed derivative is the omitted-endpoint zero just proved. If the closing edge is aligned instead, both become the same difference and its square is zero. Contracting a longer spanning path reduces to exactly these two alternatives. ◻

Proof. Expand the evaluation at the programmed incidences from the graph with all primary incidences zero. For a set \(E_0\) of programmed bits this is the elementary identity \[f(\mathbf1)=\sum_{T\subseteq E_0}(\Delta_T f)(\mathbf0).\] Lemma 58 kills cyclic \(T\). A tree component meeting no terminal contracts to an isolated primary vertex, and the isolation rule kills it. This includes an untouched primary vertex. In any remaining component all primary labels are trivial and all external primary bits at the expansion baseline are zero. Its contracted label and its incidences are exactly those obtained by contracting its terminal vertices in the original graph. Choose any terminal tree inducing its relative orientations; the corresponding terminal differences therefore give the same value. Its degree is the number of lost terminal vertices, namely \(t-1\). This establishes the operator interpretation, rather than merely a count of possible contraction degrees.

Expanding two disjoint networks at once gives the composition of these operators: the product-label rule gives the same final labels in either order, and any cycles created between networks vanish by Lemma 58. Ordinary polynomial differentiation now proves the asserted degree bound.

This argument also explains the compatible projections. Taking norms commutes with every oriented product in a forest; primary factors have neutral norm corrections, and each programmed oriented toggle projects to its one rational toggle. Thus the projected forest acts by the corresponding differences on the projected original terminal polynomial at every degree. If projected differences repeat an edge or merge two simple clusters, that term is zero. Forgetting fixed-label coordinates plainly commutes with the same products. Each projected network therefore acts on the original projected terminal polynomial.

The determinant counts are instances of the all-minors matrix-tree theorem (Chaiken 1982, sec. 1). We give the Cauchy–Binet proof in the form needed here, including singular \(Q\). Let \(I\) be the internal vertex set and \(A\) the terminal set. Write the full Laplacian as \(L=DD^{\mathsf t}\), where a column of the incidence matrix \(D\) has ones at its edge’s endpoints. Cauchy–Binet applied to \(L_{I,I}=Q\) counts edge sets whose incidence matrix with the terminal rows deleted is invertible. Such an edge set is a forest with exactly one terminal in each component: a cycle gives dependent columns; a component with no deleted row gives dependent rows; and for a tree with one row deleted, leaf removal proves invertibility. These are exactly the degree-zero terms.

For the pair coefficient apply Cauchy–Binet with row set \(I\cup\{a\}\) and column set \(I\cup\{b\}\). Each counted forest has one root from \(A\setminus\{a\}\) and one from \(A\setminus\{b\}\) in each component. This requires that \(a,b\) lie together and that every other terminal lie in its own component. The corresponding Laplacian minor is \[\det\begin{pmatrix}Q&c_b\\c_a^{\mathsf t}&0\end{pmatrix} =c_a^{\mathsf t}\operatorname{adj}(Q)c_b.\] The corner is zero because the network has no direct terminal edge; signs disappear in characteristic two. The determinant identity uses the adjugate and so remains valid when \(Q\) is singular. ◻

Proof. For the first program take \(T=I+P\), where \(P\) is the cyclic permutation matrix on \(b\) coordinates, and \(B\) diagonal with exactly one nonzero entry. A proper principal group restriction of \(P\) is a union of directed paths, hence nilpotent. Thus \(T_J\) is invertible for every proper \(J\), and so is \(H(T_J,B_J)\): its kernel equations first give \(v=0\) and then \(u=0\). This includes the empty restriction, whose determinant is \(1\).

On the full group set, both \(\ker T\) and \(\ker T^{\mathsf t}\) are spanned by the all-one vector. A radical vector of \(H\) has \(v=t\mathbf1\); summing the equation \(T^{\mathsf t}u+Bv=0\) gives \(t=0\) because \(\mathbf1^{\mathsf t}B\mathbf1=1\). Thus the full \(Q\) has one-dimensional radical spanned by \(r=\sum_i u_i\). In characteristic two its adjugate is \(rr^{\mathsf t}\). Indeed a rank-\(n-1\) matrix has a nonzero rank-one adjugate with image its kernel; symmetry identifies the two factors, and the only nonzero scalar is \(1\). Choose \(c\) to be the covector dual to \(u_1\), so \(c^{\mathsf t}r=1\). All anchors pair to zero with \(r\). At \(x_*\) the only columns pairing nontrivially with \(r\) are consequently those of \(a_0,a_1\). Formula (79) gives zero constant and exactly \(\Delta_z\) in degree one. At every other address the group subset is proper and the constant is \(1\). When \(b=1\), \(I+P=0\) and \(B=(1)\), so the same radical calculation applies without an exception.

For the second program let \(T=I+L\), where \(L\) is the consecutive lower shift, and take \(B_\varepsilon=\varepsilon E_{bb}\). Every group restriction of \(T\) is invertible, and block multiplication gives \[ H(T,B)^{-1}= \begin{pmatrix}T^{-\mathsf t}BT^{-1}&T^{-\mathsf t}\\ T^{-1}&0\end{pmatrix}. \tag{81}\] Again take \(c\) dual to \(u_1\). On a group subset \(J\), the change of \(c^{\mathsf t}Q_J^{-1}c\) is \[\bigl((T_J^{-1})_{b1}\bigr)^2,\] interpreted as zero if \(1\) or \(b\) is absent. Since \((I+L_J)^{-1}=I+L_J+L_J^2+\cdots\), that entry is \(1\) precisely when the entire path \(1,2,\ldots,b\) is present. This is exactly the target address. Pairings involving an anchor do not change: \(Q_J^{-1}h_i=e_i\), and the vectors \(e_i,h_i,c\) do not depend on \(\varepsilon\). At the target the balancing terminal is absent, so the only changed pair coefficient is the one for \(a_0,a_1\). Every constant is \(1\). Lemma 59 proves the claims, including \(b=1\), when the path consists of its single vertex. ◻

Proof of Theorem 57. Use the prescribed realization of the terminal and stationary vertices as a common baseline for all terminal bits. Regard their toggles as common variables, omitting variables invisible to a given symbol. A nonzero iterated difference implies that the polynomial is nonzero. Conversely, the unique multilinear polynomial of a nonzero function has a top monomial with nonzero coefficient. These observations also explain the last assertion using Lemma 58. Every nonzero address has an active terminal, and contractions never remove the last vertex. Thus only nonempty configurations are used.

First consider one symbol. At each \(x\ne0\) let \(P_x\) have degree \(d_x\) and choose a monomial with coefficient \(1\) in its top homogeneous part. For each variable of this monomial add a separate singular program \(\mathcal S\) targeted at \(x\). Give all these networks disjoint internal vertices. At address \(x\), its own \(d_x\) networks each have zero constant, so every surviving term spends at least \(d_x\) contraction degrees. An extra degree from any network kills \(P_x\). Thus only the designated degree-one contractions of its own networks and the constant terms of all other networks survive. Their value is the chosen top coefficient, namely \(1\). A nonzero constant polynomial needs no network and already has value \(1\). This argument applies equally to always active terminals, since Lemma 60 supplies their stationary anchors.

For two symbols write their polynomials at address \(x\ne0\) as \(P_x,R_x\), of degrees \(d_x,e_x\), and let \(P_x^{\rm top},R_x^{\rm top}\) be their top homogeneous parts. Multiply these parts in the ordinary polynomial ring \(\mathbf F_2[z_1,\ldots,z_m]\), before imposing any Boolean relations. That ring is an integral domain, so choose a coefficient equal to \(1\) at a monomial \[ M_x=\prod_z z^{\nu_{xz}},\qquad \nu_{xz}\in\{0,1,2\},\qquad \sum_z\nu_{xz}=d_x+e_x. \tag{82}\] For \(\nu_{xz}=2\) add one singular program for \(z\) targeted at \(x\). For \(\nu_{xz}=1\) add one nonsingular program for \(z\) targeted at \(x\), with its own choice \(\varepsilon_{xz}\in\mathbf F_2\). All internal vertices of different programs are disjoint. Let \[\Phi(\varepsilon)=\prod_{x\ne0} \bigl(\text{final evaluation of }P_x\bigr) \bigl(\text{final evaluation of }R_x\bigr).\] We prove that the sum of \(\Phi\) over all choices of \(\varepsilon\) is \(1\). This proves that some single choice has all the required evaluations equal to \(1\).

Expand every program in contraction degree using Lemma 59. A singular program targeted at \(x\) has zero constant in both factors there, so it costs at least one degree in \(P_x\) and one in \(R_x\). A nonsingular program has constant \(1\) independent of its choice. In the sum over that choice, a term using only its constants in all factors cancels. Every remaining term must therefore spend at least one degree somewhere for this program. The sum of these minimum costs is exactly \[ D=\sum_{x\ne0}(d_x+e_x), \tag{83}\] by (82). A nonzero term cannot exceed this total: if it did, some factor would receive more differences than its degree. Hence every surviving term uses every minimum cost exactly.

It follows that a singular program spends exactly one degree in each factor at its target and zero elsewhere. Its degree-one term there is precisely \(\Delta_z\). Each nonsingular program spends exactly one degree in exactly one factor. After summing its choice, that degree-one contribution is zero away from its target, and is \(\Delta_z\) at its target. All other occurrences of these programs contribute their common constant \(1\). Thus no unknown higher forest coefficient enters the surviving calculation.

At a fixed address, let \(Z_2\) be the variables with exponent two and \(Z_1\) those with exponent one. The local allocation sum is \[\sum_{J\subseteq Z_1} [z_{Z_2}z_J]P_x^{\rm top}\, [z_{Z_2}z_{Z_1\setminus J}]R_x^{\rm top} =[M_x](P_x^{\rm top}R_x^{\rm top})=1.\] Coefficients of the wrong degree in this display are zero. Every factor must in fact receive its full degree, since the total is \(D\) and none can exceed its own degree. Therefore its final derivative is exactly the displayed top coefficient, independent of the chosen baseline values. Allocations at different addresses are independent, so their product is \(1\), as asserted. This also covers a constant factor, whose degree is zero. For example, when both top forms are \(z\), their ordinary product is \(z^2\) and the singular program differentiates each factor once; Boolean reduction would lose this allocation.

In the two-symbol case, choose the program parameters for which \(\Phi=1\). In either case, apply Lemma 55 to adjoin the resulting finite networks to the fixed original vertices. Only new vertices and incidences involving them have been prescribed. The primary types preserve the filters. All activations are nonzero linear forms, so no new vertex appears at \(0\), and the stipulated neutral and splitting conditions persist. ◻

Proof. These are the cited results, with the arithmetic ring-class hypotheses verified by Proposition 39. The even coefficient minimum uses the cyclotomic lower bound; the odd detector of Proposition 49 still needs the arithmetic lower bound for Lemma 50. Neither interpolation statement nor the even symbol uses that odd normalization. ◻

Proof. These are Lemmas 3 and 2. In particular, the quadratic identity is an identity of coranks, not an integral direct-sum decomposition. ◻

Proof. For the first assertion, a derivative in an edge of an \(r\)-vertex graph is a symbol on \(r-1\) vertices. If \(r\ge2\), this graph remains nonempty and has value zero by minimality. If \(r=1\), there is no edge. The omitted-vertex description then leaves exactly the fixed-support labels.

For the second assertion let \(r(s)\) be the minimum split count at simple count \(s\), with value \(+\infty\) when no witness exists. Once a finite value occurs, good-twin insertion gives \(r(s+2)\le r(s)\). A nonincreasing sequence of nonnegative integers stabilizes. Choose \(s_1\ge1\) in that range. A derivative of an edge with a split endpoint reduces the split count and preserves the simple count; it is therefore zero at the minimum. A simple–simple derivative is zero by the symbol rule. Twin insertion gives the last assertion. ◻

Proof. In the first case retain the first \(r-1\) seed labels and merge all other vertices to the last cluster. Sufficient copies ensure that this cluster is nonempty. Its label is forced by the total, hence is the last seed label. Lemma 63 makes its unknown edges immaterial.

For the second case, the rational labels of split attachments form the subgroup with trivial \(F\)-coordinate. They act transitively on the simple coset, so every prescribed change of a simple label has a realizable split shift. Reserve sufficiently many representatives of every such shift. Remove these reserves and the anchors from the donor pool. If \(F\ne F'\), (90), with \(C_0\) exceeding the number removed, leaves a distinct donor for each bad simple vertex. Attach these donors; every simple vertex is now good. If \(F=F'\), this operation was unnecessary.

Pair equal enriched good labels. By withholding a bounded number of pairs, leave between \(s_1\) and \(s_1+C_1\) unpaired vertices, with the correct parity; \(C_1\) depends only on the number of enriched labels. Use reserved shifts to turn the excess unpaired vertices into good pairs and to set the first \(s_1-1\) base positions to the desired seed labels. The seed labels themselves need not all be good. Attach every remaining nonanchor split vertex to the last base position. Its projected label is forced by the total, the anchors, the other base positions, and the twin pairs. It is the required last seed label. Each cluster contains one simple vertex, so every contraction is permitted. Only a bounded number of reserve shifts was used.

When incidences to a retained split anchor are needed only to remove twins, they can be prescribed after the contracted labels are fixed, by the independent terminal-edge realization. Fixed stationary vertices instead enter the finite label enrichment from the outset. These two uses do not increase the number of label types with the original list length. ◻

Proof. Choose ample copies of every required label, including anchors and shifts. In the quadratic case with \(F\ne F'\), add enough donors to exceed all the losses from the other labels and the prescribed reserve margin. Duplicate every full label in the resulting list. Duplication gives total zero in every rational elementary coordinate and preserves any strict donor surplus. Choose the margins so that after reserving the finitely many anchors in any one packet, the union of any positive number of packets still meets the reserve requirements. All labels are realized by distinct fresh primes; duplication never means repeating an actual prime factor. ◻

Proof. If the parameter has no prime factors, a contraction certificate means that the empty symbol itself is a unit. Equation (85) then proves the assertion directly. Otherwise the stationary terminal set is nonempty. Keep the prime factors of \(h\) stationary. Their terminal polynomial is nonzero by its contraction certificate. The stationary one-symbol case of Theorem 57 gives arbitrarily large binary cubes based at \(h\), with only primary new primes, whose other addresses have unit symbol. Equation (85) and \(s(h_x)\ge0\) give the uniform upper bound required by the cyclotomic transfer. The base corank is zero, so that transfer gives (91). A base-dependent dimension threshold does not affect the asserted constant. ◻

Proof. Lemma 3 gives even functional sign. Fix an even symbol on the filter of \(1\), including all auxiliary class-group tests. If its value at \(1\) is a unit, analytic nonvanishing is already detected. Otherwise use a nonempty minimum seed in the full-torsion case, or a stabilized quadratic seed in the other case.

For each \(b\), assign a neutral packet from Lemma 65 to every nonzero linear form on \(\mathbb F_2^b\), and activate its primes by that form. At every nonzero address at least one packet is active; their union satisfies the reserve problem and has a nonzero terminal polynomial. At zero no terminal is active. The single-symbol case of Theorem 57 adds primary auxiliaries and makes the symbol a unit at every nonzero address. All local classes remain those of \(1\). Equation (85), followed by cyclotomic transfer at the original corank-zero base, gives \(a(E)=0\). ◻

Proof. At the fixed class-group support and the other fixed label tests, prepare an even negative-index symbol for each possible total type at this support, and for both isogeny orientations if needed. There are finitely many such preparations. The positive index \(|hk'|\) has nonzero total rational type: its negative-discriminant parity is already nontrivial among the tests at \(2\). Thus in the full-torsion case a unit seed has positive count. In the quadratic case use the stabilized seed of Lemma 63.

When \(F\ne F'\), interchanging the isogeny orientation interchanges the two counts in (90). Select the orientation with initial balance at least zero. The existing donors can then pay for arbitrarily many existing bad simples. They do not have to be supplied by new primes.

Let \(R\) be the absolute prime-count bound in Proposition 44. Insert in \(k'\) a finite dummy supply that meets the anchors, minimum simple count, and reserve requirements for every one of the finitely many prepared symbols. When \(F\ne F'\), add donor slack paying for every reserved loss, the new dummy bad simples, and a further possible loss of \(R\) from the later rough factor. When \(F=F'\), all simple primes, including the later rough primes, are already good; only the simple and split-reserve supplies are needed. Add a fresh ramified prime if needed to exclude the exceptional imaginary fields. The number of dummy primes is bounded independently of \(h\). Their labels and initial incidences may be chosen by Lemma 55.

Absorb \(h\) and the dummy product into the curve. Apply Proposition 44 to complete \(k'\) by a squarefree factor coprime to all primes already used. Prescribe its sign and its local unit classes so that the final \(k'\) is an odd negative fundamental discriminant, is a square at \(2Nh\), and has the required original local classes at \(2N\). These conditions give a nonempty reduced progression by the Chinese remainder theorem. Its root number is even: since \(E^{(h)}\) has odd sign and \(k'\) is negative and square at every prime of \(2Nh\), the quadratic twist formula multiplies that sign by \(\chi_{k'}(-N_{E^{(h)}})=-1\). Thus this is the positive-root-number case of Proposition 44 for the absorbed curve. Additional fixed primes may be excluded. The proposition makes the central value at \(hk'\) nonzero with at most \(R\) new factors. Its size threshold may depend on \(h\) and all these congruences; \(R\) does not.

No particular resulting unit class at the auxiliary class-group support was assumed in advance: dummy supplies were prepared for every possible total there. For the actual total, Lemma 64 makes \(hk'\) partitionable. Analytic nonvanishing gives Selmer corank zero by the forward theorem. Proposition 66 therefore gives (92), with the isogeny comparison above if the other orientation was used. The finite preparation and the bound on dummy and rough factors give \(A,C\) independent of \(h\). ◻

Proof. Use the character-sum normalization of the explicit Gross–Zagier formula in (Cai et al. 2014, Theorem 1.1). The restriction of \(\chi_h\) to \(\mathbb Q(\sqrt{k})\) has exact ring-class conductor \(|h|\): each odd prime dividing \(h\) splits and retains its ramified quadratic character. The genus character over \(\mathbb Q(\sqrt{hh_*})\) is unramified, so the sum defining \(P_h\) has conductor one. Their canonical heights are fixed modular factors times, respectively, \[|h|\sqrt{|k|}\,L'(E^{(h)},1)L(E^{(hk)},1), \qquad \sqrt{|hh_*|}\,L'(E^{(h)},1)L(E^{(h_*)},1).\] In that theorem the multiplier is \(c\sqrt{|D|}\), the level-discriminant factor is \(2^{\mu(N,D)}\), and the unit factor is \([\mathcal O_c^\times:\mathbb Z^\times]^2\). Here every level prime splits, so \((N,D)=1\) and \(\mu(N,D)=0\); the excluded extra-unit fields make the unit factor \(1\) in both applications. Take their ratio. The derivative and the rank-one regulator cancel; the remaining varying real factor is \(\sqrt{|hk|}\), precisely the factor in \(L_*(hk)\). Comparison of the rational free line with its two quadratic projections has bounded index because their composites are multiplication by \(2\). The modular parametrization and the factor involving \(h_*\) are fixed. The level is split and the discriminant and conductor are prime to it; no growing conductor-prime product occurs in this normalization. Local differential and period comparisons have finitely many possibilities at the fixed support. The extra-unit fields have been excluded, and the number of additional places in the varying \(k\) case is bounded. These observations give the asserted bounded valuation error. ◻

Proof. Choose \(k'\) by Lemma 68. Combine (84), (93), and (92). Since the factors are coprime, \(w(hk')=w(h)+w(k')\), and \(w(k')\le\omega(k')\le A\). Hence \[\begin{align*} 2j(P_h) &\ge 4w(h)+s(h)+s(hk')-v(L_*(hk'))-C\\ &\ge 2w(h)+s(h)-2w(k')-C\\ &\ge 2w(h)+s(h)-C'. \end{align*}\] Only even symbols and independently nonvanishing companions were used. Thus this proof precedes, and does not assume, the odd minimum-depth normalization. ◻

Proof. Modulo \(2\) the recurrence is \(b_{s+1}=b_{s-1}\). Its initial values prove the parity assertions. For \(d_s=b_{s+1}-b_s\), the recurrence for \(d_s\) is the same. Directly, \(d_1-d_0\equiv(p-1)(p-\kappa)\equiv0\pmod4\) when \(\kappa=\pm1\). Thus \(d_s/2\) is constant modulo \(2\). The last assertion follows by substituting the sequence \(p^s\) in the recurrence and checking its first two terms. ◻

Proof. If the ordinary coefficient test already has a nonempty split witness, there is nothing to construct. Otherwise we first obtain an additional trace congruence modulo \(8\). That congruence makes one further division of the coefficient form integral. We verify its contraction identities, then use a prime inert in \(K\) to force a nonzero test at a split prime.

Extra depth if the ordinary test has no witness. Use the ordinary even coefficient form for \(E^{(k)}\), normalized by \(a(1)=1\). The positive filter is chosen so that \(kD\) lies in the original negative filter. Since \(k\) attains its least normalized depth, the exact ratio comparison of Proposition 46 gives \[ v(a(D))\ge\omega(D) \tag{96}\] at every allowed squarefree \(D\). The factor \(|k|\) cancels from this ratio; one is not subtracting unrelated bounded-error inequalities.

If equality holds at a nonempty product of \(K\)-split primes, use the usual even bit, projected from oriented \(K\)-labels to rational labels. It has all the required properties by Proposition 54. We therefore assume for the rest of the proof that equality never occurs there. By the rational-phase assertion in Proposition 46, normalization by \(a(1)=1\) makes each nonzero squarefree coefficient a rational number times a root of unity. Its valuation is an integer, not just an element of the possibly ramified coefficient field’s discrete value group. Thus strict inequality implies \[ v(a(D))\ge\omega(D)+1 \quad\text{for every nonempty allowed all-$K$-split }D. \tag{97}\]

Write \(g=\sum a(n)q^n\) for the form. Its characters satisfy \(\psi/\eta=\chi_{-4}\), so \(\eta\) is odd. For a set \(U\) of \(K\)-split primes put \[B_U=2^{-|U|}\prod_{p\in U}(U_p-\eta(p)V_p)g, \qquad \Theta_J=\sum_{m>0,\ (m,S_0)=1}\eta(m)m q^{Jm^2}.\] All expansions retain the prescribed support sieves. The ordinary divided form is \[H_U=\frac12\left(B_U-\sum_{J\subseteq U}B_U[J]\Theta_J\right).\] Here \(B_U[J]\) denotes its coefficient at the squarefree index \(J\).

The resulting trace congruence. We first prove \[ a_p(E^{(k)})\equiv p+1\pmod8 \quad\text{for every allowed good }p\text{ split in }K. \tag{98}\] Apply the ordinary \(H_{\{p\}}\) trace at a Frobenius lift \(\phi_p\), testing coefficient \(p^2\) after theta multiplication. The form \(H_{\{p\}}\) is already integral, and \(\theta\equiv1\pmod2\); hence theta multiplication does not alter any of its reduced coefficients after these divisions. On the conductor-one ramified principal-series part the identity is \[U_p^2t(\phi_p)=U_p^3+d(\phi_p)U_p.\] On the unraised unary part it is the ordinary good-prime coefficient identity. These are the only two types in this expression. Consequently \[(t(\phi_p)f)[p^2]=f[p^3]+(d(\phi_p)f)[p].\] Replace the trace by a fresh split prime \(p'\) with the same finite data, including every support-unit test. At each rational quadratic character unramified outside \(S_0\), a Frobenius lift at \(p\) has the character value of \(p\). Hence the matched \(p'\) has the same value, and \(pp'\) has value \(1\). Equivalently, \(p'\) has the same unit squareclasses as \(p\) at all places of \(S_0\), so their product is square at all of them, including \(2\). Since \(\phi_p\in G_K\), the new prime is also split in \(K\). Its coefficient on the left tests \(pp'\); by (97) it is zero modulo the maximal ideal after the two divisions. The determinant term on the right is zero: the unary summand at \(p\) has the same character and cancels there.

On the right, the indices \(p\) and \(p^3\) in \(B_{\{p\}}\) have underlying squarefree index \(1\); they involve only \(a(1),a(p^2),a(p^4)\). The \(J=\varnothing\) unary term has no coefficient at either nonsquare index. Thus no other squarefree coefficient contributes there. For \(a=a_p(E^{(k)})\), the phase-free square coefficients are \(1,a-1,a(a-1)-p\). Hence the remaining coefficient is, up to a unit, \[H_{\{p\}}[p^3] =\frac{a^2-2a-p+1-p(a-2)}4 =\frac{(a-1)(a-p-1)}4.\] It is zero modulo \(2\). Full rational two-torsion makes \(a\) even, so \(a-1\) is odd; this proves (98). No individual total filter condition on \(p\) was needed: its matched prime \(p'\) makes \(pp'\) lie on the filter, and the square-index terms on the right come from \(1\).

One further integral division. Now define \[ \widetilde H_U=\frac14 \left(B_U-\sum_{J\subseteq U}B_U[J]\Theta_J\right). \tag{99}\] We verify integrality at every coefficient. Write a test index uniquely as \(Jrm^2\), where \(J\subseteq U\) and \(r\) is squarefree and disjoint from \(U\). The underlying squarefree coefficient is at \(D=(U/J)r\). Removing the unit phases, a prime of \(U/J\) contributes an odd factor \(b_s-b_{s-1}\) with \(\kappa=0\), and a prime of \(J\) contributes \(b_{s+1}-b_s\) with \(\kappa=(D/p)\). Thus \[ v(B_U[Jrm^2])\ge\omega(r) \tag{100}\] by (96), with one additional unit of valuation when \(D\) is nonempty and all its primes split in \(K\).

If \(r\ne1\) and all primes of \(D\) split in \(K\), the last inequality is at least \(\omega(r)+1\ge2\). If \(D\) contains an inert prime, then \(r\) contains at least two: the total rational \(K\)-character is trivial on the filter, and all primes of \(U\) split. Then (100) is again at least \(2\). No unary term has this squarefree part. This proves divisibility by \(4\) for \(r\ne1\).

Suppose \(r=1\) and \(J\ne U\). Then \(D=U/J\) is nonempty and all split, so \(B_U[J]\) is even. This cancellation can be checked without dividing by a leading multiplier, which may be zero. Factor from both shell coefficients the common phase and the scalar \(2^{-|U|+|J|}a(U/J)\), whose valuation is at least \(1\) by (97). The remaining factors are the odd differences at \(U/J\), the integral divided differences at \(J\), and the ordinary square-index factors outside \(U\). By Lemma 71, their product is congruent modulo \(2\) to the product of their leading values, even when one of those values is zero modulo \(2\). Since \(m\) is odd, multiplication by \(m\) preserves this congruence. The common scalar supplies the second factor \(2\). Retaining the phase \(\eta(m)\) therefore gives \[B_U[Jm^2]\equiv B_U[J]\eta(m)m\pmod4.\] This is exactly the cancellation with \(B_U[J]\Theta_J\).

Finally let \(J=U\), so \(D=1\). At a prime outside \(U\) the recurrence gives \(b_s\equiv p^s\pmod4\). At a prime of \(U\), (98) gives \(b_s\equiv p^s\pmod8\), and hence \[\frac{b_{s+1}-b_s}{2} \equiv \frac{a_p-2}{2}\,p^s\pmod4.\] Multiplying these congruences, again retaining the phases, gives \(B_U[Um^2]\equiv B_U[U]\eta(m)m\pmod4\). This proves integrality in the last case and establishes (99).

For the trace argument we specify the integral module being used. Write \(\psi_U\) for the character of \(B_U\). The character of \(\Theta_U\) is also \(\psi_U\), so its diamond correction is zero. For the proper subsets the diamond formula is \[\langle u\rangle\widetilde H_U =\psi_U(u)\widetilde H_U+ \sum_{J\subsetneq U}\frac{B_U[J]}2 \frac{\psi_U(u)-\eta(u)(-J/u)}2\Theta_J.\] Each \(B_U[J]/2\) here is integral, and each divided character difference is zero or a unit. Let \(\widetilde{\mathcal M}_U\) be the \(\mathcal O\)-module generated by \(\theta\widetilde H_U\) and \(\theta(B_U[J]/2)\Theta_J\) for \(J\subsetneq U\), together with all their good-Hecke and diamond translates. Its generators are integral by this formula and the good-Hecke coefficient formula. The finite-module and Chebotarev-continuity argument of Lemma 52 therefore applies to \(\widetilde{\mathcal M}_U\). A fresh trace kills every displayed unary correction after coefficientwise reduction, also after further fresh tests. Thus the same coefficientwise reduced determinant and trace identities hold on the required orbit.

The contraction and isolation identities. On \(\theta\widetilde H_U\), two fresh traces in \(G_K\) give zero after reduction at every coefficient. To see this, start with an arbitrary test index \(Jrm^2\) and choose the two fresh primes outside that index and the level. The filter imposes \(\chi_K(D)=1\); since \(U\) is all split, the number of inert factors in \(r\) is either zero or at least two. The fresh traces replace \(r\) by \(rq_1q_2\), with both \(q_i\) split. In the first case the valuation after division by \(4\) is at least \(\omega(r)+2+1-2=\omega(r)+1\ge1\). In the second case it is at least \(\omega(r)+2-2=\omega(r)\ge2\). The unary terms disappear under these fresh tests. Additional fresh \(G_K\) traces only improve these estimates, so vanishing holds on the entire reduced trace orbit. The trace identity therefore gives \[t(gu^2)=t(g)\qquad(g,u\in G_K)\] on this reduced trace orbit. Thus the tests factor through the elementary extension of \(K\) unramified off the level.

A single fresh \(K\)-split trace at coefficient \(1\) reads the bit \[ \widetilde A(D)= \left(2^{-\omega(D)-1}a(D)\text{ with its phase removed}\right)\bmod2 \tag{101}\] for a nonempty all-split configuration. The contraction rule follows from the same integral local identity as in Lemma 53. For completeness its first shell, with \(p\in U\) and a fresh trace \(q\), has underlying configuration \(D'=(U/p)q\), which is still nonempty, and its reduction is \[c_p(D')\widetilde A(D'),\qquad c_p(D')=\frac{a_p-(D'/p)-1}{2}\bmod2.\] Changing the oriented inertia bit changes \(c_p\) once. The omitted-prime identity gives the product label and removes the entire rational prime \(p\) from the new ramification support. Consequently its repeated contraction has precisely the forest interpretation of Lemma 58. An omitted-vertex label equal to the identity has trace zero, which is primary isolation. No contraction to the empty configuration has occurred.

A nonempty witness for the deeper test. It remains to prove that this deeper test has a witness. Fresh ramification gives \(K\not\subseteq\mathbb Q(E[8])\). By Chebotarev there is a good prime \(p\) inert in \(K\) and trivial on the original \(E[8]\). Then \[p\equiv1\pmod8,\qquad a_p(E^{(k)})\equiv-2\pmod8.\] On \(\widetilde H_\varnothing\), whose coefficient at \(1\) is zero, test \(t(u)^2\) at coefficient \(1\), where \(u=\operatorname{Fr}_p\). The square coefficient and matching unary character give, up to a unit, \[\frac{a_p(E^{(k)})-1-p}{4}\bmod2=1.\] The determinant term at coefficient \(1\) cancels. In characteristic two, \(t(u^2)=t(u)^2\), so a single trace in \(G_K\) is nonzero. A fresh prime realizing this finite trace test is split in \(K\). Moreover \(u^2\) is trivial on every rational quadratic character, so the prime lies on the required total rational filter. It is a nonempty witness for (101). Finally the exact coefficient ratio gives (95), with the additional depth at most one. ◻

Proof. Complex conjugation has order two outside \(G_K\), so \(G_{\mathbb Q}=G_K\rtimes\langle c\rangle\). A \(c\)-invariant quadratic character of \(G_K\) extends by giving \(c\) value zero. At a rational prime outside \(S_0\), inertia lies in \(G_K\) and the character is trivial there; thus its extension remains unramified outside \(S_0\). The assumed total therefore annihilates \((V_0^*)^c\).

For a finite vector space with involution, the annihilator of \(\ker(1+c^*)=(V_0^*)^c\) is \(\operatorname{im}(1+c)\). This proves the first assertion. Flipping a vertex of label \(v\) changes its oriented label from \(v\) to \(cv\), and hence changes the total by \((1+c)v\). Reserve one representative for every preimage needed to express the desired element of the image. Their independent flips give the second assertion. These flips preserve every rational norm label. ◻

Proof. Parity gives odd functional sign. Use the normalized odd limit symbol on the positive filter of \(1\) from Section 9.5, with its fixed support, genus partner \(h_*\), and chosen ultrafilter. If it detects the base, a sufficiently advanced actual stage proves the result. Otherwise choose its nonempty minimum seed. The moving detecting prime \(\ell\) is not an extra fixed support requirement.

Fixing the companion. Choose a separate even symbol for negative discriminants square at \(2N\), excluding the fixed odd support, the relevant torsion ramification, and the exceptional imaginary fields. Choose a unit \(k\) of that symbol. Then \(L(E^{(k)},1)\ne0\). Fix this \(k\) and \(K=\mathbb Q(\sqrt{k})\) for the rest of the proof, before choosing any cube dimension. Its fresh ramification makes it independent of the old fixed label extensions. At sufficiently advanced stages \(\ell\) avoids \(k\). The intrinsic odd label tests continue to omit the primes of \(k\).

We construct common terminal packets, individually split in \(K\), such that the odd terminal polynomial and an even terminal polynomial testing \(hk\) are separately nonzero at each nonzero address. They need not have a common unit evaluation yet; the address lemma will supply that step.

The quadratic residual case. First suppose \(F,F'\) are quadratic. Keep the original even symbol evaluated on \(hk\), treating the vertices of \(k\) as stationary external data. Enrich the rational labels of the \(h\)-vertices by the primes of \(k\). Their total there and at the even support is required to be zero. Let \(V\) be the resulting rational elementary label space, including the \(F,F'\) and \(K\) coordinates and all relations among these characters. The allowed simple labels form the affine set \(\{v\in V:\chi_F(v)=1,\ \chi_K(v)=0\}\), in additive notation. The difference of any two of its elements belongs to \(\ker\chi_F\cap\ker\chi_K\), precisely the realizable rational labels of the permitted split attachments. Translation by this subgroup is therefore transitive. The fields \(F,F'\) are unramified outside the old support, whereas \(K\) has a fresh ramified prime; thus \(K\) is independent of their compositum. When \(F\ne F'\), this realizes both the good-simple pattern \((1,1,0)\) and the donor pattern \((0,1,0)\) in their three coordinates. When \(F=F'\), all simple labels are good.

For the old odd projection, take a normal closure of its finite label extension at any stage with \(\ell\) avoiding \(k\). It is unramified outside the old support and \(\ell\). Once \(\ell\) avoids \(k\), a prime ramified in \(K\) remains outside that set. Hence \(K\) has trivial intersection with this normal closure, and every old odd label can be realized together with splitting in \(K\). This applies also to oriented split anchors over \(F\). The added stationary \(k\) coordinates may have relations, but they are already included in \(V\); they do not impose extra requirements on the omitted odd projection. These compatible lifts let us stabilize the common alphabet and its projection maps on the same ultrafilter, as in Section 9.5. The intrinsic odd minimum and seed are unchanged. Choose the following packets in this fixed abstract alphabet, before selecting an actual stage.

Use packets with sufficiently large simple supply and reserves in this kernel, and with donor slack when \(F\ne F'\). For the even test, take no split anchors. If the fields are distinct, correct all bad simples to good ones; if they coincide, every simple is already good. Pair equal enriched labels, and retain a nonempty good base pair. Use reserves to match the pair and the remaining good pairs, and attach every leftover split vertex to the last base position. Its label is forced by total zero. The output is a collection of good twin pairs, each matching also at every split vertex of \(k\). Twin insertion into the unit at \(k\) proves that this output has even symbol one. Thus the even terminal polynomial is nonzero.

For the odd test, project away the newly added tests at \(k\). The same packets contain enough split anchors, simple vertices and reserve shifts to contract to the stabilized odd seed and good pairs, by Lemma 64. Its total on that projection is the prescribed zero total. Incidences between the retained anchors and the good pairs can be set after contraction, by independent edge choices. The odd polynomial is therefore nonzero as well. This odd test does not include the stationary vertices of \(k\).

Full rational two-torsion. Now suppose the curve has full rational two-torsion. Use the even test of Lemma 72, with \(K\)-oriented labels on its support \(S_0\), and let \(v_1,\ldots,v_r\) be a minimum nonempty witness. Its oriented total \(v_1+\cdots+v_r\) lies in \((1+c)V_0\), by Lemma 73. This even test uses the oriented elementary extension of \(K\), whereas the odd test reads only its rational projection; the even contraction rule is needed only for nonempty outputs. Every odd seed label lifts to a \(K\)-split prime because \(K\) was freshly ramified outside its intrinsic support. Each even label also has a compatible lift to the common alphabet. Stabilize the finite common alphabet and projections on the same ultrafilter, preserving the intrinsic odd symbol as in Section 9.5.

Choose packets containing ample copies of every needed even label, all odd seed labels on their rational projection, and orientation-shifting reserves. Arbitrary pairs of projected labels need not lift: take a union of sufficiently many compatible lifts for each projection separately. Duplicate each full label, so the packet has rational total zero.

For the even contraction, retain anchors with labels \(v_1,\ldots,v_{r-1}\). After reserving these anchors, use orientation flips to make the total equal to \(\sum v_i\); this is possible by Lemma 73. In a contraction certificate these flips are choices of conjugations along the forest, as permitted by Lemma 55; they do not change the common terminal data separately at different addresses. Merge all remaining vertices into the last cluster. Its label is \(v_r\). The cluster is nonempty, and the minimum nonempty count makes the output a unit independently of its mutual edges. For the odd contraction, use its rational projection. Retain all but the last odd seed label and merge the remainder. Total zero gives the last label, so this is also a unit. Orientation corrections for the even test did not change these rational labels.

Simultaneous addressing and an actual stage. Now fix \(b\) and put a sufficient common packet at every nonzero linear form on \(\mathbb F_2^b\). Each nonzero address has two separately nonzero terminal polynomials. In the fixed abstract alphabet, Theorem 57 gives finite primary-auxiliary networks making both symbols one there. Require their primary types to split in \(K\) (and \(F\) in the quadratic case) and to be neutral at the stationary primes of \(k\). They are primary at every support coordinate used by either projection, so both symbol identities apply to the same common graph data.

Let \(W_b\) bound all graph weights required by these finite networks. A finite intersection of ultrafilter-large sets supplies an actual stage with the same truth table through \(W_b\), all required divisibilities and trace identities, and sufficient precision for the prescribed divisions. The moving prime also avoids the fixed companion \(k\). Realize the resulting finite graph at that stage using Lemma 55. All active products \(h\) are positive and square at the fixed tests, and all their primes split in \(K\). No auxiliary appears at address zero. The depth constant and fixed companion remain unchanged as \(b\) grows.

Returning to the original curve. At every nonzero address the analytic orders of \(E^{(h)}\) and \(E^{(hk)}\) are respectively one and zero. The odd unit bound, the even unit bound (or (95)), and Lemma 69 with fixed \(k\) give \[2j_h(k)\le 4w(h)+C.\] The finite Sha lengths at these addresses are nonnegative, so this also bounds \(2j_h(k)-4w(h)-s(h)-s(hk)\) from above.

At address zero, \(L(E^{(k)},1)\ne0\) gives \(s_2(E^{(k)})=0\) by the forward theorem. Lemma 62 therefore gives \(s_2(E/K)=1\). Ring-class missing-vertex transfer now makes the base product \(L(E,s)L(E^{(k)},s)\) have a simple zero. Its second factor is nonzero at \(1\), and hence \(a(E)=1\). ◻

Proof of Theorem 1. Use Lemma 62 to choose the indicated isogeny orientation before constructing either coefficient system. The even coefficient system and cyclotomic transfer give both Proposition 66 and the corank-zero conclusion of Proposition 67. The bounded varying companion, ring-class lower bound, and height comparison then establish Proposition 70. This discharges the arithmetic premise before the odd normalization and finite-graph construction of Section 9.5.

For the corank-one case, Proposition 74 fixes a nonvanishing even companion before the cube dimension, constructs simultaneous unit vertices, and applies ring-class transfer at the original base. This gives \(a(E)=s_2(E)\) in both cases. The established Gross–Zagier–Kolyvagin implication in analytic rank zero or one gives \(r(E)=a(E)\) and finiteness of the whole Shafarevich–Tate group. Equation (88) remains the underlying corank identity throughout. In particular, neither a finite two-Selmer dimension nor Mordell–Weil rank was substituted for the original Selmer corank hypothesis. ◻