Proof. The cofiber sequence for \(M\), together with \(\pi_1\mathbb S=\pi_2\mathbb S=\mathbb Z/2\), gives \(\pi_1M=\pi_2M=0\). Applying maps into \(M\) to this cofiber sequence shows that restriction along the unit \(\eta:\mathbb S\to M\) is injective on \[[M,M]\longrightarrow[\mathbb S,M]=\mathbb Z/5.\] Thus \(5\operatorname{id}_M=0\). The cofiber sequence \[M\xrightarrow{5}M\xrightarrow{\eta\wedge1}M\wedge M \longrightarrow\Sigma M\] therefore permits an extension of \(\operatorname{id}_M\) to a pairing \(\mu\). Its restriction along \(1\wedge\eta\) is also the identity: these endomorphisms of \(M\) agree after restriction along \(\eta\). Moreover, \([\Sigma M,M]=0\), by the same cofiber sequence and the two vanishing homotopy groups. The extension is consequently unique. Smashing \(5\operatorname{id}_M=0\) with any spectrum proves the assertion about coefficients. The products are induced by the usual tensor pairings of \(K\)-theory spectra and this fixed pairing on \(M\). ◻

Proof. Tensoring with a finitely generated projective \(R\)-module is exact on coherent modules and preserves every specified support. Tensor product therefore gives compatible pairings by \(K(R)\) on the supported, whole, and localized categories. These pairings respect the null composite in the localization diagram and hence its cofiber sequence. On a dévissage category, tensoring is tensoring with the restriction to the residue field. The projection formula for a finite map and the usual flat base-change identity are identities of these tensor functors. For each term \(B\) of this localization triangle, use the pairing \[(K(R)\wedge M)\wedge(B\wedge M) \longrightarrow K(R)\wedge B\wedge M\wedge M \longrightarrow B\wedge M,\] where the last arrow uses the integral tensor action and the fixed pairing on \(M\). These pairings commute with all arrows of the triangle. Precomposing with the representative \(\beta:\mathbb S^2\to K(R)\wedge M\) gives the asserted diagrams for \(b\); no integral lift of \(\beta\) is required. Its degree is even, so moving it across a boundary contributes no sign. Passage to a generic point follows by filtered localization. These assertions concern localization boundaries; the coefficient connecting map \(\partial_{\mathrm{coef}}\) is a different map. ◻

Proof of Lemma 4. We use the corrected, strongly convergent motivic spectral sequence with its finite-coefficient multiplicative structure: \[ E_2^{p,q}=H^{p-q}(H,\mathbb Z/5(-q)) \Longrightarrow K_{-p-q}(H;5). \tag{5}\] This form of the spectral sequence is recorded in (Weibel 2013, VI.4.2, Addendum VI.4.2.1, and VI.4.4). For the corrected construction, see (Suslin 2003, Introduction and Theorem 6.1) (the locators use the Trudy pagination). The product is the usual \(K\)-theory product on the abutment and the motivic cup product on \(E_2\) (Friedlander and Suslin 2002, Proposition 16.1 and Theorem 16.2). The Moore-space comultiplication used there corresponds by stable duality to a unital pairing on \(M\). Its coefficient product agrees with ours by the uniqueness in Lemma 2.

Write an \(E_2\)-term as \(H^s(H,\mathbb Z/5(j))\), of total \(K\)-degree \(2j-s\). Terms of negative weight vanish. For \(j\geq0\), the defining motivic complex \(C_*\mathbb Z_{\mathrm{tr}}(\mathbb G_m^{\wedge j})[-j]\) is concentrated in cohomological degrees at most \(j\). Global sections on the Nisnevich site of a field are exact, since every covering of the field point has a splitting, and derived reduction modulo five does not increase this upper degree. Thus the field motivic group vanishes when \(s>j\). For \(s\leq j\), the norm-residue and Beilinson–Lichtenbaum comparison theorems identify it with \[ H^s_{\mathrm{et}}(H,\mu_5^{\otimes j}) \tag{6}\] (Voevodsky 2011, Theorems 6.16–6.17). These are the Rost–Voevodsky norm-residue theorem and its motivic comparison consequence, with the completion of the norm-residue argument in (Weibel 2009); the earlier motivic formulation is (Suslin and Voevodsky 2000). In particular the terms of negative cohomological degree vanish. Under the dimension bounds below, only finitely many terms contribute to each total degree, so the resulting filtrations are finite. In these coordinates a differential \(d_\rho\), for \(\rho\geq2\), goes from \((s,j)\) to \((s+2\rho-1,j+\rho-1)\).

We first derive the algebraically closed case from this same spectral sequence. If \(\overline H\) is an algebraic closure of \(H\), only the terms with \(s=0\) survive. In each degree \(2j\geq0\) there is exactly one term, \(H^0(\overline H,\mu_5^{\otimes j})\cong\mathbb F_5\), and in odd degree there is none. There are no possible differentials. Thus the edge map is an isomorphism in every even degree, and \[K_{2j}(\overline H;5)\cong\mathbb F_5,\qquad K_{2j+1}(\overline H;5)=0\qquad(j\geq0).\] The coefficient sequence surjects \(K_2(\overline H;5)\) onto \(\overline H^\times[5]=\mu_5(\overline H)\). Both groups have dimension one, so this boundary is an isomorphism. Consequently the restriction of \(\beta_H\) is nonzero, and so is its edge image. Multiplicativity now shows that each \(b^j(1)\) over \(\overline H\), computed by iterated left multiplication, has nonzero edge image and generates the corresponding even group. This proves the last assertion of the lemma without a separate rigidity input.

By naturality, the edge image \(\alpha\in H^0(H,\mu_5)\) of \(\beta_H\) is nonzero: restriction on this \(H^0\) group is injective, and its geometric image is the nonzero class just computed. The edge image of \(b^r(1)\) is \(\alpha^{\otimes r}\). Since \(H\) contains \(\mu_5\), this generates the one-dimensional group \(H^0(H,\mu_5^{\otimes r})\). Hence this whole line consists of permanent cycles for every \(r\geq1\). Only \(\alpha\ne0\) is needed; no choice of an identification of \(\alpha\) with the specified primitive root is required.

Suppose first that \(\operatorname{cd}_5H\leq3\). In total degree \(2r\), the only possible terms are \[H^0(H,\mu_5^{\otimes r}),\qquad H^2(H,\mu_5^{\otimes(r+1)}).\] There is no incoming differential to either term. The only possible outgoing differential from the first is \(d_2\) to an \(H^3\)-term, and it vanishes by permanence of the Bott power. Every outgoing differential from the second has cohomological degree at least \(5\) and is zero. We obtain a two-step filtration, expressed as \[ 0\longrightarrow H^2(H,\mu_5^{\otimes(r+1)}) \longrightarrow K_{2r}(H;5) \longrightarrow H^0(H,\mu_5^{\otimes r}) \longrightarrow0. \tag{7}\] Multiplication by \(\beta_H\) preserves these filtrations and acts on both graded pieces by tensoring with \(\alpha\). Each such map is an isomorphism, proving (1).

If \(\operatorname{cd}_5H\leq2\), the sole term in total degree \(2r+1\) is \(H^1(H,\mu_5^{\otimes(r+1)})\), with no possible nonzero differential. Multiplication by \(\beta_H\) again acts by tensoring with \(\alpha\). This proves (2). For a finite field, the same calculation applies in every positive degree since its \(5\)-cohomological dimension is one. In degree zero, the class \(1\) generates \(K_0(H;5)\) and \(b(1)=\beta_H\) generates \(K_2(H;5)=H^0(H,\mu_5)\). This proves (3). ◻

Proof. The sum in (13) is zero when \(m=0\), so its nonvanishing implies \(m>0\). Suppose, for a contradiction, that the sum in (14) is zero. We first turn this assumed relation into prescribed ordinary tame symbols on the generic curve.

Localization and Bott untwisting. Put \(J=L(C)\) and \(A_U=\Gamma(U,\mathcal O_U)\). The ring \(A_U\) is a localization of the coordinate ring of a smooth affine curve. It is therefore a Noetherian regular domain of dimension at most one, with fraction field \(J\). Since one of the \(p_i\) is retained, its dimension is one. Every finitely generated torsion \(A_U\)-module is killed by a nonzero element; the corresponding quotient of \(A_U\) has dimension at most zero and is Noetherian, hence Artinian. Such a module has finite length, with simple factors the residue fields of the retained closed points. This argument does not require \(U\) to be semilocal or the multiplicative set to be finite.

Quillen localization for the torsion category, followed by dévissage and the coefficient convention in Section 2, gives the exact sequence \[ K_6(J;5)\longrightarrow \bigoplus_{q\in U^{(1)}}K_5(\kappa(q);5) \longrightarrow G_5(U;5). \tag{15}\] The second arrow is the sum of coherent pushforwards from the points; the first is the vector of localization residues. These are ordinary localization and dévissage for a multiplicative set (Quillen 1973b, sec. 4, Theorem 3 and Corollary 2; §5, Theorems 4–5; §7, Proposition 3.2 and paragraph 3.4). The index set of this direct sum can be infinite, but each vector has finite support. Every \(q\in U^{(1)}\) is a closed point of \(C\), so \(\kappa(q)\) is a finite extension of \(L\).

Exactness now supplies \(\xi\in K_6(J;5)\) whose residue at \(p_i\) is \(e_i b^2[a_i]_5\) and whose other residues in \(U\) are zero. The field \(J\) has \(\operatorname{cd}_5J\leq3\), and each \(\kappa(q)\) has \(\operatorname{cd}_5\kappa(q)=2\). All these fields are perfect and contain the same primitive root by restriction from \(L\). Thus Lemma 4 applies to the actual multiplication operations used here: it makes \[\begin{aligned} b^2 &:K_2(J;5)\longrightarrow K_6(J;5) &&\text{surjective},\\ b^2 &:K_1(\kappa(q);5)\longrightarrow K_5(\kappa(q);5) &&\text{injective}. \end{aligned}\] Choose \(\xi_0\in K_2(J;5)\) with \(\xi=b^2\xi_0\). Write \(\partial_q^{(5)}\) for a coefficient localization residue. The compatibility in Lemma 3 gives \(\partial_q^{(5)}(b^2\xi_0)=b^2\partial_q^{(5)}\xi_0\), with no sign because \(b\) has even degree. Injectivity on the residue fields then yields \[ \partial_q^{(5)}\xi_0= \begin{cases} e_i[a_i]_5,&q=p_i,\\ 0,&q\in U^{(1)}\setminus\{p_1,\ldots,p_m\}. \end{cases} \tag{16}\]

There is still a coefficient obstruction to representing \(\xi_0\) integrally. Its coefficient boundary belongs to \[K_1(J)[5]=\mu_5(J)=\langle\zeta\rangle.\] The last equality holds because \(L\) already contains all fifth roots of unity. Hence there is \(\lambda\in\mathbb F_5\) with \(\partial_{\mathrm{coef}}\xi_0=\lambda[\zeta]\). Let \(\beta_U\) and \(\beta_J\) be the restrictions of \(\beta_L\). Naturality gives \(\partial_{\mathrm{coef}}\beta_J=[\zeta]\), while all localization residues of \(\beta_J\) on \(U\) vanish because it extends as \(\beta_U\). This uses an extension over the \(L\)-scheme \(U\), not an extension over \(R\). The class \[\eta=\xi_0-\lambda\beta_J\in K_2(J;5)\] therefore has zero coefficient boundary and the same residues as \(\xi_0\). The coefficient exact sequence in Section 2 supplies an integral representative \(z\in K_2(J)\) of \(\eta\). Write \(\partial_q z\in K_1(\kappa(q))=\kappa(q)^\times\) for its ordinary tame symbol. Reduction of localization residues commutes with coefficient reduction, so (16) becomes \[ [\partial_q z]_5= \begin{cases} e_i[a_i]_5,&q=p_i,\\ 0,&q\in U^{(1)}\setminus\{p_1,\ldots,p_m\}. \end{cases} \tag{17}\] In particular, \(\partial_q z\) is a fifth power in \(\kappa(q)^\times\) at every retained point other than the \(p_i\). We have obtained an ordinary \(K_2\)-class with the required control of its tame symbols. It remains to derive the relation that these symbols must satisfy on \(D\).

The horizontal divisor. For each closed point \(q\) of \(C\), let \(R_q\) be the valuation ring of the unique extension of the valuation of \(L\) to \(\kappa(q)\), let \(k_q\) be its residue field, and let \(v_q\) be its normalized valuation. The extension \(\kappa(q)/L\) is finite, so \(R_q\) is a DVR finite over \(R\). Properness gives the extension \[j_q:\mathop{\mathrm{Spec}}R_q\longrightarrow\mathcal X\] used to define \(r(q)\). This map is finite. Indeed, its graph in \(\mathop{\mathrm{Spec}}R_q\times_R\mathcal X\) is a closed immersion because \(\mathcal X/R\) is separated, and the projection from this fiber product to \(\mathcal X\) is finite, being the base change of \(\mathop{\mathrm{Spec}}R_q\to\mathop{\mathrm{Spec}}R\). Set \[f_q=[k_q:\kappa(r(q))].\]

Smoothness over the regular DVR \(R\) makes \(\mathcal X\) a regular surface, and its function field is \(J\). The codimension filtration for coherent \(K\)-theory gives the following row of the coniveau complex: \[ K_2(J)\longrightarrow \bigoplus_{s\in\mathcal X^{(1)}}\kappa(s)^\times \longrightarrow \bigoplus_{r\in\mathcal X^{(2)}}\mathbb Z. \tag{18}\] The assertion needed here is only that the composite is zero: the first differential \(d_1\) of the coniveau spectral sequence satisfies \(d_1^2=0\) (Quillen 1973b, sec. 7, Theorem 5.4 and equation (5.5)). There is exactly one vertical point in \(\mathcal X^{(1)}\), the generic point of the integral curve \(D\). Its residue \(\partial_Dz\in k(D)^\times\) contributes the principal divisor \(\operatorname{div}_D(\partial_Dz)\). The other codimension-one points are the closed points \(q\) of \(C\).

We compute their contributions without assuming that a horizontal closure is normal. On \(\mathop{\mathrm{Spec}}R_q\), the localization boundary \[K_1(\kappa(q))\longrightarrow K_0(k_q)=\mathbb Z\] is the normalized valuation \(v_q\): a unit has boundary zero, and a uniformizer \(\pi_q\) has boundary \([R_q/\pi_qR_q]=[k_q]\) (Weibel 2013, 6.1.2 and V.(6.6)). Functoriality of coherent localization for the finite map \(j_q\) identifies the horizontal boundary on \(\mathcal X\) with the pushforward of this DVR boundary (Quillen 1973b, sec. 7, paragraphs 2.7–2.8, Proposition 3.2 and paragraph 3.4). At the generic point this pushforward is the identity on \(\kappa(q)\). At the closed point it sends \([k_q]\) to \([k_q:\kappa(r(q))]\,[\kappa(r(q))]\). Thus the contribution at \(r(q)\) is exactly \(f_qv_q(\partial_qz)\). There is no additional ramification factor: the boundary already uses the normalized valuation on \(\kappa(q)\), and the closed-point pushforward contributes only the residue degree \(f_q\).

Define the horizontal divisor \[ H(z)=\sum_{q\in C^{(1)}}f_qv_q(\partial_qz)\,(r(q)). \tag{19}\] The sum is finite because the first arrow of (18) takes values in a direct sum. All codimension-two points of the proper surface lie on \(D\). The identity \(d_1^2z=0\) therefore says, as an equality of divisors on \(D\), \[H(z)+\operatorname{div}_D(\partial_Dz)=0.\] Consequently \[ [H(z)]=0\quad\text{in }\mathop{\mathrm{Pic}}(D). \tag{20}\]

The Picard test. Apply \(\psi\) to (20). If \(r(q)\in Z\), the corresponding summand vanishes because \(\psi([r(q)])=0\). If \(r(q)\notin Z\), the hypothesis on \(U\) retains \(q\). For such a point other than the \(p_i\), (17) makes \(v_q(\partial_qz)\) divisible by five, so its contribution also vanishes in \(B\). Finally, \(\kappa(p_i)=L\), \(k_{p_i}=k=\kappa(r_i)\), and hence \(f_{p_i}=1\) and \(v_{p_i}=v_L\). The remaining congruence in (17) gives \[v_{p_i}(\partial_{p_i}z)\equiv e_i v_L(a_i)\pmod5.\] The identity \(\psi([H(z)])=0\) is therefore \[\sum_{i=1}^m e_i v_L(a_i)\,\psi([r_i])=0\quad\text{in }B,\] contrary to (13). This proves (14). ◻

Proof. The underlying curve \(E\) has four points over \(\mathbb F_5\). At \(s=0\) there are two affine points, while at each nonzero \(s\) the value \(s^4+1=2\) is a nonsquare. There are also two rational points at infinity, since the leading coefficient of the quartic is a square. Choosing any rational point as origin over \(\mathbb F_5\), the Frobenius polynomial is therefore \(T^2-2T+5\), and its roots compute extension-field point counts (Silverman 2009, Theorem V.2.3.1). If \(a_m\) is the sum of the \(m\)th powers of its roots, then \[a_0=a_1=2,\qquad a_m=2a_{m-1}-5a_{m-2}.\] It follows that \(a_4=-14\) and \[\#E(k_0)=5^4+1-a_4=640.\] This count is independent of the choice of origin. Hence, with origin \(o\), the finite group \(E(k_0)\) has a point \(Q\notin5E(k_0)\).

Such a point remains nondivisible by \(5\) over \(k\). In fact, if \(Q=5R\) with \(R\in E(k)\), the trace for the quadratic extension \(k/k_0\), using the group law defined over \(k_0\), would give \[2Q=5\bigl(R+\operatorname{Frob}_{k_0}(R)\bigr) \quad\text{in }E(k_0).\] Multiplication by \(2\) is invertible on \(E(k_0)/5E(k_0)\), a contradiction. Every branch point is \(2\)-torsion and therefore divisible by \(5\), so \(Q\) is not a branch point. A fiber of the degree-two morphism \(h\) has a point over an extension of \(k_0\) of degree at most two. Since \(k\) is its quadratic extension, we may choose \(P\in D(k)\) over \(Q\). The description of \(h\) on \(Z=0\) gives \(Z(P)\ne0\).

Finally, on the dense chart \(Y\ne0\) the morphism \(h\circ g_a\) fixes \(s\) and multiplies \(w\) by \(a^{-2}\). It therefore agrees with \(h\) when \(a^2=1\) and with \([-1]\circ h\) when \(a^2=-1\). These are identities of morphisms on all of \(D\): they extend from the dense chart because \(D\) is integral and the target is separated. This also covers the possibility that \(Q\) is a point at infinity. Thus \[h(g_aP)= \begin{cases} Q,&a^2=1,\\ -Q,&a^2=-1. \end{cases}\] Each weighted summand in (24) equals \(Q\) modulo \(5\). Their sum is \(4Q\), which is nonzero because multiplication by \(4\) is invertible on \(E(k)/5E(k)\). ◻

Proof. The regular local ring \(V[x,y]_{(5,x,y)}\) has dimension three. The equation \(5+x^4+y^4\) is congruent to \(5\) modulo the square of its maximal ideal, so it is part of a regular system of parameters. Its quotient is regular local of dimension two and is a domain (The Stacks Project Authors 2026, Tags 00NQ and 00NP).

Equations (25) and (27) show that the assignment respects the defining equation. Every denominator inverted in (28) has nonzero constant residue at \(x=y=0\), hence maps to a unit of \(\mathcal O\). Thus the homomorphism is defined and local. At least one of \(X_0,Y_0\) is a unit of \(V\), so its image contains \(t\). Surjectivity follows from (26).

The element \(Y_0x-X_0y\) belongs to the kernel and has nonzero image in \(\mathfrak m_A/\mathfrak m_A^2\). In particular the kernel is nonzero. It is a nonmaximal prime, since the quotient \(\mathcal O\) is a DVR, and therefore has height one in the two-dimensional local domain \(A\). Since \(Y_0x-X_0y\) is a regular parameter, it generates a height-one prime and hence generates this kernel. ◻

Proof. The \(V\)-module \(L\) is flat, since \(V\to H_0\) is a localization and \(L\) is a vector space over \(H_0\). The substitution \(x=tX\), \(y=tY\) transforms \(5+x^4+y^4=0\) into \(X^4+Y^4=1\). Tensoring and localizing therefore give the asserted description of \(T\). Its coordinate ring is a localization of the coordinate ring of a smooth geometrically integral affine curve over \(L\); it is regular, integral, and Noetherian, of dimension at most one.

Since \(5\) is invertible in the second tensor factor, the coordinate ring of the base-changed divisor is \[ \mathcal O\otimes_V L =L\otimes_{H_0}L \xrightarrow{\sim}\prod_{\delta\in\Delta}L, \qquad a\otimes b\longmapsto(\delta(a)b)_\delta. \tag{31}\] Here \(\mathcal O[1/5]=L\), and the last isomorphism is the usual splitting of a finite Galois extension. The images of \(x\) and \(y\) on the \(\delta\)-factor are \(\delta(t)X_0\) and \(\delta(t)Y_0\), giving (30). Every inverted denominator evaluates to a unit of \(\mathcal O\), so these factors indeed define points of \(T\). They are distinct: the ratios \(\delta(t)/t\) are the four different fourth roots of unity, and at least one of \(X_0,Y_0\) is a unit. Thus \(T\) has closed points and has dimension one.

The coordinates in (30) are integral over \(\mathcal O\) and satisfy the Fermat equation, so they define sections of \(\mathcal D\). Reducing (25) gives their stated specializations. The last assertion follows directly from the factor maps in (31). ◻

Proof of Theorem 1. Let \(c\in K_5(S)\) be the class from Proposition 7, and write \(c_L\in K_5(L)\) for its restriction along \(S\to L\). There is no map \(S\to\mathcal O\) in this construction, since \(5\) is invertible in \(S\). Instead, integral localization for the DVR \(\mathcal O\) gives \[K_5(\mathcal O)\longrightarrow K_5(L)\longrightarrow K_4(k)=0,\] where the last equality is Quillen’s computation for finite fields (Quillen 1972); see also (Weibel 2013, IV.1.13). The residue field here has characteristic five; the equality is the integral finite-field calculation. Choose an integral lift \(\widetilde c\in K_5(\mathcal O)\). The closed immersion \(i:\mathop{\mathrm{Spec}}\mathcal O\hookrightarrow\mathop{\mathrm{Spec}}A\) of Lemma 11 defines \[\gamma=i_*\widetilde c\in G_5(A)=K_5(A).\] The equality uses the canonical resolution isomorphism for the regular ring \(A\). Restriction to \(\operatorname{Frac}A\) annihilates every coherent module supported on \(\mathop{\mathrm{Spec}}\mathcal O\). Thus \(\gamma\) is in the kernel of the canonical augmentation, integrally.

To show that it is nonzero, make the flat base change \(V\to L\). The square \[\begin{tikzcd}[column sep=large,row sep=large] \displaystyle\coprod_{\delta\in\Delta}\mathop{\mathrm{Spec}}L \arrow[r,"\coprod p_\delta"] \arrow[d] & T \arrow[d] \\ \mathop{\mathrm{Spec}}\mathcal O\arrow[r,"i"] & \mathop{\mathrm{Spec}}A \end{tikzcd}\] is Cartesian by Lemma 12. To check flat base change with the point indices, put \(B_T=A\otimes_VL\). For every finitely generated \(\mathcal O\)-module \(M\), scalar extension gives natural isomorphisms of \(B_T\)-modules \[ \begin{aligned} B_T\otimes_A i_*M &\simeq(\mathcal O\otimes_VL)\otimes_{\mathcal O}M\\ &\simeq\bigoplus_{\delta\in\Delta}L\otimes_{\mathcal O,\delta}M. \end{aligned} \tag{32}\] The \(\delta\)-summand is a \(B_T\)-module through the quotient defining \(p_\delta\). These are exact functors: \(B_T\) is flat over \(A\), and each map \(\mathcal O\to L\) is a localization followed by an automorphism. Thus (32) proves the required identity on integral coherent \(K\)-theory.

Write \(\iota:S\to L\) for the fixed embedding. Since \(\widetilde c\) restricts to \(c_L=\operatorname{res}_\iota c\), its image on the \(\delta\)-factor is \[\operatorname{res}_{\delta:\mathcal O\to L}\widetilde c =\operatorname{res}_{\iota\circ\delta}c =\operatorname{res}_{\iota}(\delta c).\] The equality uses the specified identification \(\Delta=\mathop{\mathrm{Gal}}(L/H_0)\) and requires no equivariance of the chosen DVR preimage. Consequently, still integrally, \[\gamma|_T=\sum_{\delta\in\Delta}(p_\delta)_* \operatorname{res}_\iota(\delta c).\] Naturality of coefficient reduction and Proposition 7 now give \[ \overline{\gamma|_T} =\sum_{\delta\in\Delta}(p_\delta)_* \bigl(\chi(\delta)^2b^2[\delta u]_5\bigr) \quad\text{in }G_5(T;5). \tag{33}\] Here \(b\) uses the fixed embedding \(S\to L\), and the character factors come from Proposition 7. No choice of the integral lift \(\widetilde c\) affects this formula.

Apply Proposition 8 to the model \(\mathcal D/\mathcal O\), to \(U=T\), and to the set \(Z_\infty\) and homomorphism \(\psi\) constructed in Section 5. The required specialization condition follows directly from the definition of \(A\): let \(q\) be any closed point of \(C\) and let \(\mathcal O_q\) be the valuation ring of its finite residue-field extension \(\kappa(q)/L\). If \(q\) specializes in the chart \(Z\ne0\), the proper extension \(\mathop{\mathrm{Spec}}\mathcal O_q\to\mathcal D\) lies in that chart, since an open set containing the closed point of a DVR contains its generic point. Hence \(X/Z\) and \(Y/Z\) belong to \(\mathcal O_q\). The original coordinates \(x=tX/Z\) and \(y=tY/Z\) have positive valuation. Every denominator inverted outside \((5,x,y)\) has nonzero constant residue and hence is a unit at \(q\). Thus \(q\) belongs to \(T\).

All the hypotheses of Proposition 8 are now verified: the smooth model and localized chart come from Section 6, the retained-point condition holds for every closed point, and (23) says that \(\psi\) kills the entire set \(Z_\infty\). The points \(p_\delta\) specialize to \(g_{\chi(\delta)}P\) and are \(L\)-rational. Every \(\delta u\) is a uniformizer, so \(v_L(\delta u)=1\). The sum tested by Proposition 8 for (33) is therefore \[\sum_{\delta\in\Delta}\chi(\delta)^2 \psi([g_{\chi(\delta)}P]) =4Q\ne0\quad\text{in }E(k)/5E(k),\] by Lemma 10. It follows that \(\overline{\gamma|_T}\ne0\). Hence \(\gamma\ne0\) in the original integral group, as required. ◻