Proof. First construct the resolution in profinite modules. A compact \(p\)-adic analytic group has a noetherian completed group ring; if it has no \(p\)-torsion, then \(\operatorname{pd}_{\Lambda_G}\mathbb Z_p=\operatorname{cd}_p(G)=\dim G\). Here projective dimension is in profinite modules. These results for compact groups, including the agreement with abstract finitely generated modules, are recorded in (Venjakob 2002, secs. 1.1–1.2, pp. 275–276). The noetherianity input for analytic pro-\(p\) groups is (Lazard 1965, 2.2.4); the compact-group statement uses the finite-index passage recorded by Venjakob. Choose successive finite free presentations. Noetherianity keeps their kernels finitely generated, and the projective-dimension bound makes the last syzygy projective. Their natural compact topologies give the asserted profinite resolution.

To apply the resolution to the geometric coefficient complexes, we pass from continuous actions to solid modules over the completed group ring. Write \(A=\underline{\mathbb Z_p}[\underline G]\) for the free condensed group ring. A continuous action on a complex means an object of \(D(A\text{-}\mathrm{Cond})\) with derived-solid underlying complex. By (Anschütz 2020, Lemma 1.3), the group-ring analytic structure has underived solidification; its normalized ring is \[A^\square\simeq\underline{\mathbb Z_p[[G]]} =\underline{\varprojlim_{m,V}(\mathbb Z/p^m)[G/V]},\] where \(V\) runs through open normal subgroups. The ring identification, including multiplication, is (Tang 2026, sec. 3.6, preceding Proposition 3.21). Analyticity gives fully faithful embeddings of the unbounded derived category of solid \(G\)-modules into both \(D(A\text{-}\mathrm{Cond})\) and \(D(A^\square\text{-}\mathrm{Cond})\), as stated in (Anschütz 2020, sec. 1, pp. 3–4). The solid heart is characterized by solidity of the underlying condensed module. The essential-image criterion of Clausen–Scholze’s analytic theory extends this characterization to unbounded complexes: membership is detected on cohomology objects (Scholze 2019, Proposition 12.4 and Remark 12.5). Thus its objects are precisely the complexes with derived-solid underlying coefficients. Consequently the external derived Hom complexes satisfy \[R\Gamma(G,K)(*) \simeq\mathop{\mathrm{RHom}}_{A\text{-}\mathrm{Cond}}(\underline{\mathbb Z_p},K) \simeq\mathop{\mathrm{RHom}}_{G\text{-}\mathrm{Solid}}(\underline{\mathbb Z_p},K).\] The first identification is the external derived Hom for the condensed classifying topos \(BG_{\mathrm{pro\acute et}}\) (Anschütz 2020, sec. 2, pp. 5 and 8); the last category is that of solid \(\mathbb Z_p[[G]]\)-modules by (Tang 2026, Proposition 3.21). They apply to the given complex \(K\) itself, including when its coefficients are neither profinite nor derived \(p\)-complete.

The profinite resolution stays exact and projective in solid modules (Tang 2026, Theorems 3.2 and 3.14(i)). Exactness can be seen by testing on extremally disconnected profinite sets: maps from such a set lift through a surjection of profinite spaces. Each \(P_i\) is a retract of \(\Lambda_G^{r_i}\), and \[\mathop{\mathrm{Hom}}_{\Lambda_G}(\Lambda_G,M)=M(*).\] Evaluation at the point is exact, since the point is extremally disconnected. This proves projectivity against an arbitrary solid coefficient module \(M\), even when \(M\) is not profinite. A bounded projective resolution computes derived Hom against any complex, which proves the displayed calculation and the amplitude assertion.

Finally, \(\mathop{\mathrm{Hom}}_{\Lambda_G}(P_i,H^j(K))\) is a retract of a finite sum of \(H^j(K)(*)\). The retract commutes with \(\gamma\), because \(\gamma\) commutes with the \(G\)-action. After inverting \(p\), the entire finite complex computing the \(j\)th row therefore has scalar action \(\lambda_j\). Its cohomology has the same action. This uses a scalar identity on point values only; it does not deduce an identity of condensed sheaves from their point values. Boundedness of the resolution gives a finite filtration in each total degree, so exact rationalization preserves the spectral sequence and its abutment. ◻

Proof. For an inclusion \(K_1\subset K_2\) of compact open subgroups of \(\mathop{\mathrm{GL}}_n(\mathbb Q_p)\), choose a \(K_2\)-stable lattice \(\Lambda\) and then \(r\) large enough that \[U=1+p^r\mathop{\mathrm{End}}_{\mathbb Z_p}(\Lambda)\subset K_1.\] This is a uniform subgroup normal in \(K_2\). In the division-algebra case one uses a sufficiently deep congruence subgroup of \(\mathcal O_{D_n}^{\times}\), which is normal in the relevant compact group. For these small groups, Lazard’s comparison is (Huber et al. 2011, Theorem 3.3.3 and Remark 3.3.4) with trivial \(\mathbb Z_p\)-coefficients, followed by inversion of \(p\). Its coefficient naturality, rational agreement, and group naturality are (Huber et al. 2011, Theorem 3.1.1(1)–(3)), with numbering in the arXiv revision specified in the bibliography. Equip the uniform groups with their lower \(p\)-series. At these odd primes the proof of Theorem 3.3.3 gives \(e=1\), and the associated graded target is generated in degree one, as required by the group-naturality clause. Inclusions and the block homomorphisms preserve these filtrations. The common subgroup \(U\) therefore identifies restriction with the identity on the same ambient Lie algebra cohomology.

Here is why passing back to a compact group is valid even for an index divisible by \(p\). For \(U\triangleleft K\), the augmentation and norm of the finite permutation module \(\mathbb Z[K/U]\) induce, by continuous Shapiro, the identities \[\operatorname{cor}\operatorname{res}=[K:U],\qquad \operatorname{res}\operatorname{cor} =\sum_{q\in K/U}q^*\] at each finite coefficient level, compatibly with reduction in \(m\). Shapiro here is the derived induction–restriction adjunction: induction for an open subgroup is a direct sum on underlying condensed modules and is exact by (AB4) (Tang 2026, Theorems 2.2 and 2.4); restriction is also exact. Their adjunction therefore descends directly to derived categories. Its naturality identifies the permutation maps with the actual restrictions and conjugations in the finite continuous bar complexes. Apply \(R\!\varprojlim_m\) to these identities before taking cohomology and inverting \(p\). Only then divide by \([K:U]\). This gives \(H^*_{\mathrm b}(K)\xrightarrow{\sim}H^*_{\mathrm b}(U)^{K/U}\) by restriction, with the compact continuous-cochain interpretation of (5).

The finite quotient acts trivially in every Lie degree. The Chevalley–Eilenberg complex is a finite-dimensional algebraic \(\mathop{\mathrm{GL}}_n\)-representation after a splitting-field extension. Cartan’s formula \(\mathcal L_X=d\iota_X+\iota_Xd\) makes the derivative of its action on cohomology zero. In characteristic zero, an algebraic action of the connected group \(\mathop{\mathrm{GL}}_n\) with zero derivative is trivial. Thus all inner automorphisms act trivially, in every degree; the same conclusion descends for the division algebra. This proves the compact restriction identification without an all-degree numerical calculation.

After a finite splitting-field extension, either Lie algebra is \(\mathfrak{gl}_n\). The finite Chevalley–Eilenberg calculation in 9 gives its asserted groups in degrees one through three, with degree-three representative \[(X,Y,Z)% \BeginAccSupp{method=hex,unicode,ActualText=27FC}% \longmapsto\EndAccSupp{}\mathop{\mathrm{tr}}\bigl(X[Y,Z]\bigr),\] or its reduced-trace version for \(D_n\). This is the invariant-form cocycle of (Chevalley and Eilenberg 1948, sec. 21) and (Koszul 1950, sec. 11); the appendix records the normalization and restriction used here. These representatives are defined over \(\mathbb Q_p\) and invariant under inner automorphisms, so the assertions descend from the splitting field. Restriction of the displayed form to the upper-left or lower-right \(2\times2\) block is the corresponding form for \(\mathfrak{gl}_2\), whose class is nonzero. This proves the compact calculations and their compatibility.

To pass to \(J_n\), let \(\mathcal B_n\) be the reduced affine building of \(\mathop{\mathrm{GL}}_n(\mathbb Q_p)\). It is contractible of dimension \(n-1\). The group \(J_n\) preserves vertex types, its face stabilizers \(K_\sigma\) are compact open, and a chamber \(\Delta\) is a fundamental domain, including its faces. A proof of these building properties by lattice norms is included in 9. Write its bounded augmented cellular resolution as \[P_r=\bigoplus_{\substack{\sigma\subset\Delta\\\dim\sigma=r}} \mathbb Z[\underline{J_n/K_\sigma}].\] It is exact also as a complex of condensed modules. Indeed, on a profinite parameter set a continuous section of a discrete cellular chain group has finite image. Contractibility lifts each of the finitely many cycles in that image; choosing those lifts on its clopen fibers gives a continuous lift. The same exact derived induction–restriction adjunction therefore supplies continuous Shapiro for each summand. At coefficient level \(\Lambda_m\), applying derived Hom gives a finite filtration with graded terms \(\bigoplus_{\dim\sigma=r}C_{\mathrm{cts}}^{\bullet}(K_\sigma,\Lambda_m)[-r]\) in \(D(\mathrm{Ab})\). A cellular boundary is the literal signed projection of the corresponding coset modules, so naturality of that adjunction makes each face map the signed restriction between stabilizers, with no index multiplier. All of these statements are compatible with coefficient reduction.

Take \(R\!\varprojlim_m\) of this finite filtration. There are finitely many orbit terms in each of finitely many cellular degrees, so this step uses only finite sums and cones in the cellular direction. Taking cohomology of the resulting finite filtration gives the integral spectral sequence \[E_1^{r,s}= \bigoplus_{\substack{\sigma\subset\Delta\\\dim\sigma=r}} H^s_{\mathrm{int}}(K_\sigma) \ \Longrightarrow\ H^{r+s}_{\mathrm{int}}(J_n),\] with precisely those signed restriction faces. It does not interchange an infinite group-cochain totalization with the derived inverse limit. Only after this construction do we invert \(p\); exact rationalization preserves the finite filtration and its abutment.

Under the compact comparison, every restriction between face stabilizers is the identity on the same Lie algebra cohomology; any change of representatives contributes an inner conjugation, which also acts trivially. Thus each rationalized row is the cellular cochain complex of the simplex \(\Delta\) with constant coefficients \(H^s(\mathfrak{gl}_n,\mathbb Q_p)\). It has cohomology only in cellular degree zero. The resulting edge map is restriction to a vertex stabilizer. Restriction further to compact open subgroups, using intersections and the compact comparison, proves the asserted canonical identifications. The block restriction can now be tested on compact open subgroups, where it is the trace calculation above. ◻

Proof. The kernel of \(D\to\mathbb Z_p^\times\times J_2\) is \(N=\mathbb Q_p^2\), with its additive topology. Fix \(m\) and exhaust it by the compact open subgroups \(N_r=p^{-r}\mathbb Z_p^2\) for \(r\geq0\). Their finite-coefficient cohomology, computed by the two-variable Koszul resolution, is \[H^q(N_r,\mathbb Z/p^m)=\bigwedge^q(\mathbb Z/p^m)^2.\] The standard bases identify restriction from \(N_{r+1}\) to \(N_r\) with multiplication by \(p^q\) in degree \(q\): restriction multiplies each degree-one homomorphism by \(p\), and is compatible with cup products. For \(q>0\) this inverse system is pro-zero; for \(q=0\) it is constant with identity transition maps.

This calculation computes the cohomology of \(N\) without a missing derived-limit term. Indeed a continuous cochain on \(N\) is exactly a compatible family of cochains on the \(N_r\). Restriction of cochains is surjective, since \(N_r^a\) is clopen in \(N_{r+1}^a\) and a function extends by zero on its complement. Thus the ordinary inverse limit of the cochain complexes realizes their derived inverse limit. The Milnor exact sequence has \(\varprojlim^1_r=0\) in every degree: this holds both for a constant tower with identity maps and for a pro-zero tower. It follows that the constant map is an equivalence \[\Lambda_m\xrightarrow{\ \sim\ }C_{\mathrm{cts}}^{\bullet}(N,\Lambda_m).\] The same proof works with coefficients \(C_{\mathrm{cts}}(S,\mathbb Z/p^m)\) for every profinite parameter space \(S\). The finite Koszul complex gives finite sums of this module, the positive-degree restrictions are again \(p^q\), and extension by zero is continuous on \(N_r^a\times S\). The condensed standard resolution evaluated on each such parameter identifies this calculation with the internal equivalence \[\underline{\Lambda_m}\xrightarrow{\ \sim\ } R\Gamma(N,\underline{\Lambda_m}).\] Thus it proves an equivalence of coefficient objects for extension descent. The equivalence is the map of constants, so it is equivariant under the Levi action.

Now take \(R\!\varprojlim_m\). In degree zero the parameter modules have surjective transition maps, by lifting locally constant values, and their limit is \(C_{\mathrm{cts}}(S,\mathbb Z_p)\). Hence the internal coefficient limit satisfies \[\underline{\mathbb Z_p}\xrightarrow{\ \sim\ } R\!\varprojlim_m R\Gamma(N,\underline{\Lambda_m}).\] At each finite level, internal descent for the split extension identifies \(R\Gamma(D,\underline{\Lambda_m})\) with \(R\Gamma(\mathbb Z_p^\times\times J_2, R\Gamma(N,\underline{\Lambda_m}))\). The Levi-equivariant constant equivalence therefore induces inflation. After point evaluation, the standard-resolution comparison identifies this map with finite continuous-cochain inflation. Taking \(R\!\varprojlim_m\) and using (4) gives the claimed isomorphism integrally before rationalization.

Finally, the compact resolution for \(\mathbb Z_p^\times\) and the finite building resolution for \(J_2\) give the usual rational product calculation. Since \(H^*_{\mathrm b}(\mathbb Z_p^\times)\) is exterior on one degree-one generator and \(H^2_{\mathrm b}(J_2)=0\), degree three of the Levi is exactly \(H^3_{\mathrm b}(J_2)\). Thus \(\rho_3^*e_{J_3}\) is a scalar multiple of \(\rho_2^*e_{J_2}\). Restricting to the section \(A% \BeginAccSupp{method=hex,unicode,ActualText=21A6}% \mapsto\EndAccSupp{}\operatorname{diag}(1,A)\) shows that this scalar is nonzero, by 4. ◻

Proof. The imported result is (Barthel et al. 2026, Proposition 2.7.1(1)–(5)), whose hypotheses are \(h\ge2\), a perfect field containing \(\mathbb F_{p^{h(h-1)}}\), and the coheight-one Gross–Torii extension with the specified determinant identifications. Here \(h=3\) and \(k=\mathbb F_{p^6}\) satisfy those hypotheses exactly.

We recall why its formulation gives all surjections and the indicated structure groups. The infinite-level two-tower theorem (Barthel et al. 2026, Theorem 2.6.3), followed by forgetting the étale framing, identifies \(\mathop{\mathrm{Spd}}\widehat L\) with exact sequences \[0\longrightarrow\mathcal O\longrightarrow\mathcal E_3^0 \longrightarrow\mathcal E_2^0\longrightarrow0\] whose determinant identification is \(\iota\). Forgetting that framing is the \(\mathbb Z_p^{\times}\)-torsor in the proof of (Barthel et al. 2026, Proposition 2.7.1(2)). Given a surjection, its kernel is the line \(\det\mathcal E_3^0\otimes(\det\mathcal E_2^0)^{-1}\); the fixed \(\iota\) trivializes this line and recovers the sequence. Thus no additional condition is imposed on the surjection. Descent along the two framing torsors gives exactly the asserted description of \(Y\), also in families.

Quotienting by \(S_3\) instead allows the source bundle to vary while retaining its determinant trivialization. It therefore classifies extensions \[0\longrightarrow\mathcal O\longrightarrow\mathcal E \longrightarrow\mathcal E_2^0\longrightarrow0\] with middle term of slope \(1/3\). By (Barthel et al. 2026, Proposition 2.7.1(5)), these are precisely the extensions that are nonsplit at every geometric point, which form \(\operatorname{BC}(-1/2)^*\).

To specify the joint action, let \(i_f:\mathcal O\to\mathcal E_3^0\) be the kernel injection normalized by \(\iota\). If \(f'=bfg^{-1}\), the determinant isomorphisms for the two exact sequences give \[g i_f=i_{f'}\,N_3(g)N_2(b)^{-1}.\] Indeed, \(g\) acts on the source determinant by \(N_3(g)\) and \(b\) acts on the target determinant by \(N_2(b)\). The induced diagram of extensions therefore has kernel map \(N_3(g)N_2(b)^{-1}\) and quotient map \(b\). Functoriality of extensions by pushout and pullback proves the displayed \(P_2\times T_3\)-action. This calculation is an identity of bundle maps on every test object and commutes with base change. For \(b=1\), it is pushout by \(a=N_3(g)\). On the extension group, \(a_*=(aI_2)^*\), so this is the action by inverse pullback of the central scalar \(a^{-1}I_2\). The underlying quotient and this \(T_3\) restriction are the parts of (Barthel et al. 2026, Proposition 2.7.1(5)) used below; the joint action includes the target determinant factor just computed. The field assertions are parts (1) and (4) of the same proposition. ◻

Proof. Apply the underlying cohomology calculation of (Barthel et al. 2026, Proposition 3.6.9) to the quotient in 7, retaining only its \(T_3\)-action. The map of constants and the vanishing in the other degrees give a fiber sequence of derived solid \(k\)-modules with \(T_3\)-action \[k\longrightarrow R\Gamma(S_3,\widehat L) \longrightarrow k_{\chi_T}[-3].\] Here \(\chi_T\) denotes the actual character on the line in degree three. This is the restriction of the argument combining quotient descent and cohomology in (Barthel et al. 2026, Theorem 3.6.10) that is needed here. For \(b=1\), the action formula in 7 is pushout by \(a\in T_3\) on the kernel. On the extension group this equals pullback by \(aI_2\), or the action by inverse pullback of \(a^{-1}I_2\). The reduced norm of \(aI_2\) in \(D_2^\times\) is \(a^2\). Hence the subgroup \[P_3^1/S_3=\mu_{p-1}\subset T_3\] acts on the last term by \(\alpha% \BeginAccSupp{method=hex,unicode,ActualText=21A6}% \mapsto\EndAccSupp{}\alpha^{2}\) or \(\alpha% \BeginAccSupp{method=hex,unicode,ActualText=21A6}% \mapsto\EndAccSupp{}\alpha^{-2}\); either convention gives the same vanishing used below. The displayed exponent is also checked in (Barthel et al. 2026, Lemma 5.2.5). This is a subgroup of the norm quotient; it must not be replaced by the central \(\mu_{p-1}\) in \(P_3\). Since \(p\ge5\), this character is nontrivial. The averaging idempotent for the finite group \(\mu_{p-1}\) is defined in characteristic \(p\), so its derived invariants are exact and annihilate \(k_{\chi_T}\). Taking these invariants proves that the underlying map in (11) is an equivalence. The map itself is \(H\)-equivariant: it is induced by constants from the original \(G\)-action on \(\widehat L\), followed by \(P_3^1\)-descent, and \(G\) fixes \(k\). The forgetful functor from \(H\)-equivariant objects detects equivalences, so this gives the required equivalence with its full residual action. No splitting of the fiber sequence or joint character formula is needed.

For completeness, the finite-field hypothesis in this import is substantive. Its proof uses the solid Frobenius comparison (Barthel et al. 2026, Lemma 3.6.3 and Corollary 3.6.8) for \(\bigl(\bigcup_{r\ge0}k((z^{1/p^r}))\bigr)^{\wedge}\) over a finite field \(k\). The geometric base change of this point is a punctured perfectoid ball; the proof solves coefficient Frobenius minus identity on its convergent functions and then requires surjectivity in solid modules. For this precise function ring, 27 supplies a continuous section and hence proves surjectivity on every profinite parameter space. This justifies the solid enhancement directly, without applying a linear open-mapping theorem to Frobenius minus identity. The resulting natural comparison commutes with the \(S_3\)-action. Thus 7 verifies the required hypothesis before finite-field descent is applied.

For the geometric statement, take a quadratic extension \(F/\mathbb Q_p\) that is a maximal subfield of \(D_2\). Proposition 3.1.2 of (Barthel et al. 2026) identifies \[\bigl(\operatorname{BC}(-1/2)^*\bigr)_C \simeq[\mathcal H^1_{F,C}/F],\] where \(\mathcal H^1_{F,C}\) is the one-dimensional Drinfeld upper half-plane for \(F\) and the additive group \(F\) acts by translations. This identification is equivariant for the central \(\mathbb Q_p^{\times}\). It is distinct from the rank-\(n\) Drinfeld space \(\Omega_C^{n-1}\) used in the modification argument. Proposition 3.5.6 of (Barthel et al. 2026) computes the solid \(\mathcal O^{\flat}\)-cohomology of this quotient: it is \(C^{\flat}\) in degree zero, a character line in degree \([F:\mathbb Q_p]+1=3\), and zero otherwise. By the \(b=1\) action formula in 7, this norm-quotient \(\mu_{p-1}\) acts by kernel scalars. Expressing those as central pullbacks on the target gives reduced-norm character \(\alpha% \BeginAccSupp{method=hex,unicode,ActualText=21A6}% \mapsto\EndAccSupp{}\alpha^2\) or its inverse. Both are nontrivial, so the top cohomology has no \(\mu_{p-1}\)-invariants; the choice of action or dual convention does not affect the conclusion. The degree-zero identification is the map of constants. Applying the same averaging argument proves (12).

Only central equivariance of the Drinfeld description was needed to prove this vanishing. The constant map on \(X_C\) itself has the full residual \(H\)-equivariance, and an equivariant map is an equivalence when its underlying map is. This proves the full assertion, without assuming a \(P_2\)-equivariant splitting or a larger equivariance of that chart. Finally, the actions of \(P_2\) and \(\overline Z\) are over the fixed base, so they fix \(k\) and \(C^{\flat}\) pointwise. All arrows used above are induced by structure maps and group descent; their stated naturality follows. ◻

Construction; full proof in 11. Fix a parameter set \(T\). Write \(\mathscr D_\Gamma^{\mathrm{or}}\) and \(\mathscr D_C^{\mathrm{or}}\) for oriented deformations with the specified residue tags of the full and connected reference groups. The connected–étale sequence is uniquely split over the perfect field \(F\). Let \(Q_T\) be the base change of its étale quotient, so \(\Gamma_T=C_T\oplus Q_T\). An oriented deformation \(C_A\) over a complete adic test ring \(A\) has a contractible space of lifts of the tagged étale quotient \(Q_T\). With such a marked lift \(Q_A\), send it to \(C_A\oplus Q_A\), retaining its connected orientation. This constructs \(\mathfrak s\) in the left diagram: \[ \begin{tikzcd}[column sep=large] \mathscr D_C^{\mathrm{or}} \arrow[r,bend left=18,"\mathfrak s"] &\mathscr D_\Gamma^{\mathrm{or}} \arrow[l,bend left=18,"\mathfrak f"] \end{tikzcd} \qquad \begin{tikzcd}[column sep=large] E(R_T,C_T)\arrow[r,bend left=18,"i_T"] &E(R_T,\Gamma_T)\arrow[l,bend left=18,"r_T"] . \end{tikzcd} \tag{15}\] Here \(\mathfrak f\) takes the connected deformation, and \(\mathfrak f\mathfrak s\simeq\mathrm{id}\). Corepresentability reverses the arrows and gives \(r_Ti_T\simeq\mathrm{id}\). In particular the required map is \(r_T\), supplied by the split section. The forgetful map alone supplies \(i_T\) in the other direction.

The tensor-factor identity keeps track of which connected map results. The tensor construction compares the two oriented formal groups through their common Quillen formal group. On the split deformation its comparison is the inverse of the connected inclusion. The resulting connected tag and orientation are therefore exactly those defining \(c_T\), so \(r_Tb_T\simeq c_T\). The appendix verifies this cancellation on the actual tags and proves its compatibility with adic refinement and every continuous group-family map. It then upgrades the identity to the continuous solid maps in the statement. This identifies the right-hand tensor composite with the canonical map used in coefficient descent. ◻

Proof of 9. The connected endpoint calculation gives the coefficients and ordinary locality of every evaluation. The tensor construction and the oriented section supply the left equivariant ring map; the right map is the canonical connected base change already constructed. The constants calculation and finite Koszul argument prove both fixed-point descent and descent for each coefficient module, with their residual actions and maps from Witt constants. Taking derived \(H\)-invariants in the extension \(P_3^1\to G\to H\) gives (14); \(\overline Z\) acts trivially on the \(E_2\) source. The preceding point-evaluation argument proves the ordinary stabilizer-descent identification with its sphere unit. ◻

Proof. The kernel is a vector bundle: locally at the Cartier divisor, the quotient is a coordinate modulo a local equation of the divisor, and the kernel has a basis obtained by multiplying that coordinate by the local equation. At a geometric point, \(\mathop{\mathrm{rank}}(\mathcal V_n)=n\) and \(\deg(\mathcal V_n)=0\). If it had a positive-slope Harder–Narasimhan piece, its maximal destabilizing subbundle \(\mathcal F\) would have rank \(b<n\) and integral degree \(d\geq1\). The injection \(\mathcal F% \BeginAccSupp{method=hex,unicode,ActualText=21AA}% \hookrightarrow\EndAccSupp{}\mathcal E_n\) contradicts semistability of \(\mathcal E_n\), because \[\mu(\mathcal F)=\frac d b\geq\frac1b>\frac1n =\mu(\mathcal E_n).\] One may saturate its image in \(\mathcal E_n\) without decreasing degree. Hence \(\mathcal V_n\) is semistable of degree zero at every geometric point. The relative slope-zero equivalence stated above now applies. ◻

Proof. If \(\ell\subset W_C\) for a proper rational subspace \(W\), modify \(W\otimes_{\mathbb Q_p}\mathcal O\) along \(\ell\). The resulting rank-\(\dim W\), degree-one bundle embeds as a subbundle of \(\mathcal E_\ell\); the quotient is the unchanged trivial bundle associated to \(\mathbb Q_p^n/W\). Its slope is greater than \(1/n\), so \(\mathcal E_\ell\) is not semistable.

For the converse, take a saturated destabilizing subbundle \(\mathcal F\subset\mathcal E_\ell\) of rank \(b<n\). Its integral degree \(d\) is at least one. Set \(\mathcal F'=\mathcal F\cap\mathcal O^n\). The quotient \(\mathcal F/\mathcal F'\) embeds in \(\mathcal E_\ell/\mathcal O^n\) and has length \(\epsilon\leq1\). Since \(\mathcal F'\) embeds in \(\mathcal O^n\), semistability of the latter gives \[0\geq\deg(\mathcal F')=d-\epsilon\geq0.\] Thus \(d=\epsilon=1\) and \(\deg(\mathcal F')=0\). The saturation of \(\mathcal F'\) in \(\mathcal O^n\) cannot increase its degree: a positive increase would contradict the same semistability inequality. It is therefore already saturated. All its subbundles have nonpositive degree, so it is semistable of slope zero. The classification of bundles at a geometric point identifies it with \(W\otimes_{\mathbb Q_p}\mathcal O\) for a \(b\)-dimensional rational subspace \(W\); here an inclusion between trivial bundles is a matrix over \(H^0(\mathcal X_C^{\mathrm{FF}},\mathcal O)=\mathbb Q_p\). The equality \(\epsilon=1\) in the displayed local lattice description then says precisely that \(\ell\subset W_C\).

This proves the criterion at every geometric point. The classification in (Fargues and Scholze 2024, Theorem II.2.14) gives \(\mathcal O(1/n)\) for a semistable rank-\(n\), degree-one bundle, and the relative basic-stratum theorem gives the claimed family of forms. ◻

Proof. Fixing a quotient line of \(i_\infty^*\mathcal E_n^0\) fixes its kernel \(\mathcal V_n\). By 12 its frames exist pro-étale locally. Requiring a frame to preserve the determinant lattice cuts its \(\mathop{\mathrm{GL}}_n(\mathbb Q_p)\)-torsor down to a \(J_n\)-torsor. Thus \(\mathcal Z_n/J_n=\mathbb P(i_\infty^*\mathcal E_n^0)\) as v-sheaves.

In the other direction, a framed lower bundle gives an upper modification \(\mathcal O^n% \BeginAccSupp{method=hex,unicode,ActualText=21AA}% \hookrightarrow\EndAccSupp{}\mathcal E_\ell\). By 13 its allowed directions are exactly \(\Omega_C^{n-1}\). Its determinant lattice is induced by the standard one on \(\mathcal O^n\). The isomorphisms \(\mathcal E_\ell\simeq\mathcal E_n^0\) form a \(D_n^\times\)-torsor, by the relative basic-stratum theorem. The norm valuation \(D_n^\times\to\mathbb Z\) is surjective, so this torsor has, v-locally, isomorphisms carrying the induced lattice onto \(\Lambda_n\). Those isomorphisms form precisely a \(P_n\)-torsor. It follows that \(\mathcal Z_n/P_n=\Omega_C^{n-1}\).

Both identifications commute with base change and the remaining group action. Taking the further quotients proves [eq:two-realizations]. They are Galois compatible because every operation used—forming the kernel, determinant, lattice condition, and sheaf of frames—is compatible with the descent just constructed. More explicitly, an upper direction is a line in \(\mathbb Q_p^n\otimes_{\mathbb Q_p}(\mathcal O(\infty)/\mathcal O)|_\infty\). The second factor is common to all coordinates, so its semilinear Galois scalar cancels upon projectivization. The action on \(\Omega_C^{n-1}\) is consequently the standard arithmetic action, commuting with \(J_n\). The lower frames form a Galois-equivariant \(J_n\)-torsor, which gives the equivariant map to \(B_CJ_n\). Here Galois acts on the \(\mathop{\mathrm{Spd}}C\) base of \(B_CJ_n\) and trivially on the group \(J_n\); the analogous statement holds for \(B_CP_n\). This assertion does not require the torsor itself to have a Galois-fixed frame. ◻

Proof. If \(q:\mathcal E_2\twoheadrightarrow i_{\infty *}\mathcal L\) is the universal quotient, then \(\mathcal V_2=\ker(q)\) and \(\mathcal V_3=\ker(qf)\). Thus \(\mathcal V_3=f^{-1}\mathcal V_2\), proving the exact sequence. The two lower modifications are slope zero by 12. The kernel of \(f\) has rank one and degree \(\deg(\mathcal E_3)-\deg(\mathcal E_2)=0\), so it too has slope zero.

To check exactness on local systems, pass to a pro-étale cover trivializing the three slope-zero bundles. Full faithfulness of the relative correspondence identifies the maps with matrices of continuous \(\mathbb Q_p\)-valued functions. The matrices have the indicated ranks on every geometric fiber; on the open loci of invertible minors they give exact, locally split sequences of rational local systems. These loci cover the base, proving exactness before descent.

Transport \(\Lambda_3\) and \(\Lambda_2\) to the determinant local systems of \(\mathcal V_3\) and \(\mathcal V_2\). The determinant isomorphism in [eq:filtered-modification] defines the lattice on the kernel as their tensor quotient. Locally choose a lattice-compatible basis of the kernel and determinant-compatible frames of the quotient, and lift the latter to the middle local system. Such adapted frames form a torsor; their change-of-frame matrices are exactly the displayed matrices in \(D\). The two classifying maps follow by forgetting the filtration or taking its quotient. The construction of \(m_n\) retains the original \(P_n\)-structured form \(\mathcal E_n\) on \(Y_C\) and only adds its quotient line. Its \(P_n\)-framing torsor is therefore the pullback of the associated framing torsor on \(Y_C\), proving the last assertion as a statement of actual torsors. ◻

Proof. Define \(\lambda_m\) by the powers \(h_m^j\), using adjunction and cup product. It is therefore a global map of derived pushforward objects, compatible with coefficient reduction and all descent data. For arithmetic descent, the linear action on \(\mathcal W\) induces descent on the actual line bundle \(\mathcal O(1)\). Naturality of the Kummer boundary makes \(h_m\) invariant in its twisted coefficient group.

Over \(\operatorname{Spa}(C,\mathcal O_C)\), the finite-constant projective-space calculation has basis \(1,h_1,\ldots,h_1^r\): apply (Li et al. 2024, Construction 3.2.1 and Proposition 3.2.2) with \(\mathbb F_p\) coefficients, since \(p\) is invertible on this base. The first Chern class there is the Kummer boundary (Li et al. 2024, Variant 3.1.8), so this basis uses our actual classes. To extend the calculation to arbitrary perfectoid test bases, apply the integral Riemann–Hilbert functor \(\operatorname{RH}_{\mathbb Z_p}\) of (Anschütz et al. 2026, Theorem 5.1.7). Smooth proper primitive comparison (Anschütz et al. 2026, Proposition 5.2.2 and Example 5.2.3) applies to \(\mathbb P_C^r\to\mathop{\mathrm{Spd}}C\) and the dualizable coefficient object \(\mathbb F_p\). Its mod-\(p\) target is \[\mathcal D_{\mathrm{FF}}(-,\mathbb F_p) =\mathcal D_{\widehat{\square}}(\mathcal O^\flat)^{\varphi},\] by (Anschütz et al. 2026, Remark 5.1.9 and the beginning of §5.2). The symmetric monoidal and pullback compatibility of the functor carries the Kummer basis to its coefficient images. It therefore gives the Frobenius-equivariant Chern-class equivalence with \(\mathcal O^\flat\) coefficients over \(C\).

After trivializing \(\mathcal W\) on a perfectoid test object, its projective bundle is a base change of \(\mathbb P_C^r\to\mathop{\mathrm{Spd}}C\). This projective morphism is \(p\)-bounded. Base change for solid pushforward, (Anschütz et al. 2026, Proposition 4.2.7(ii)) with \(\Lambda=\mathbb F_p\), transports the equivalence to that test object. The globally defined Chern-class map makes these local equivalences compatible on overlaps, so they descend.

We identify the resulting solid calculation with ordinary v-sheaf pushforward by testing on a perfectoid \(T\to X\). Apply tensor-unit Hom in the solid formalism on \(T\). Adjunction and the tensor-unit cohomology identification (Barthel et al. 2026, Theorem 3.2.1(iii)–(iv)) give \[\bigoplus_{j=0}^r R\Gamma_v(T,\mathcal O^\flat(-j))[-2j] \xrightarrow{\ \sim\ } R\Gamma_v(\mathbb P(\mathcal W)_T,\mathcal O^\flat).\] This is natural in \(T\) and induced by the same Chern classes. It thus identifies their coefficient-image map with the equivalence of ordinary derived v-sheaf pushforwards \[\bigoplus_{j=0}^r\mathcal O^\flat_X(-j)[-2j] \xrightarrow{\ \sim\ } Rg_*\mathcal O^\flat.\]

The displayed map is induced by the images of the actual mod-\(p\) étale classes \(h_1^j\). It therefore commutes with Frobenius on \(\mathcal O^\flat\), including its descent data. Take the fiber of \(F-1\) on both sides. The Artin–Schreier sequence on the v-site identifies this fiber with \(\mathbb F_p\); with twists it identifies the \(j\)th fiber with \(\mathbb F_p(-j)[-2j]\). Since derived pushforward preserves fibers, the resulting equivalence is exactly \(\lambda_1\). This step uses the Frobenius compatibility of the Chern-class map.

The Kummer classes \(h_m\) are compatible under reduction. The coefficient sequences \[0\longrightarrow\mathbb Z/p^{m-1}\xrightarrow{\,p\,}\mathbb Z/p^m \longrightarrow\mathbb F_p\longrightarrow0\] give compatible exact triangles on the source and target of \(\lambda_m\). Cup product with \(h_m^j\) commutes with both coefficient maps. Induction on \(m\) therefore proves [eq:projective-finite]. Apply global sections and then the derived inverse limit. The finite direct sum commutes with both operations and gives [eq:projective-integral]. Projection to its zeroth summand is a left inverse to pullback. Exact inversion of \(p\) preserves that left inverse. ◻

Proof. Use the common arithmetic action \(G_F\) from 14. The projective presentation is a projective bundle over \(B_CP_n\). Apply 16 first to this relative geometric base. Its two degree-three summands use the finite coefficients \(\Lambda_m\) and \(\Lambda_m(-1)\) on \(B_CP_n\), respectively, before their derived limits; higher projective summands would have negative cohomological degree and vanish. The compact geometric-point comparison (8), applied in the \(P_n\) bar degrees and with its Tate fiber, identifies these summands with the group cohomology groups in the \(G_F\)-equivariant direct sum \[ H^3_{\mathrm b}(\mathcal M_n) \cong H^3_{\mathrm b}(P_n) \oplus H^1_{\mathrm b}(P_n)(-1). \tag{21}\] The first summand is the image of the actual constant-cochain pullback, hence contains a nonzero \(e_{P_n}\). The comparison is natural for the Galois action on the \(\mathop{\mathrm{Spd}}C\) base. It identifies the untwisted group cochain factors with trivial Galois action because Galois fixes \(P_n\) and the untwisted constants. On the second summand it acts by \(\chi_{\mathrm{cyc}}^{-1}\). As \(\chi_{\mathrm{cyc}}(G_F)\) is open, this summand has no invariants. Thus \[ H^3_{\mathrm b}(\mathcal M_n)^{G_F} =\mathbb Q_p\,e_{P_n}|_{\mathcal M_n}. \tag{22}\]

It remains to prove that the pullback of \(e_{J_n}\) is nonzero. Choose a compact open uniform subgroup \(U\subset J_n\). Restriction sends \(e_{J_n}\) to a nonzero element of \(H^3_{\mathrm b}(U)\) by 4. We will detect it after the further pullback to \([\Omega_C^{n-1}/U]\).

Let \(K_n=\mathscr C_{\Omega_C^{n-1}}\) be the derived solid integral cohomology complex formed with profinite parameters. Descent, as in 3, gives the first-quadrant spectral sequence \[ E_2^{s,j}=H^s(U,H^j(K_n))(*) \ \Longrightarrow\ H_{\mathrm{int}}^{s+j}([\Omega_C^{n-1}/U]). \tag{23}\] The complex \(K_n\) is bounded below, and the group degree has the finite bound supplied by the completed resolution. The sequence therefore has a finite filtration in every degree and may be rationalized exactly. The additional geometric row bound uses only point values: 6 gives \(H^j(K_n)(*)=0\) for \(j>n-1\); the finite Hom calculation below then makes every ordinary point-valued \(E_2^{s,j}\) in that row zero. No vanishing assertion for the entire condensed module \(H^j(K_n)\) is needed.

Choose the finite projective resolution \(P_\bullet\to\mathbb Z_p\) over \(\Lambda=\mathbb Z_p[[U]]\) from 3. The unit \(\underline{\mathbb Z_p}\to K_n\) factors through \(H^0(K_n)[0]\), since \(K_n\) is concentrated in cohomological degrees at least zero. The degree-zero calculation and local constants argument following 6 show that this unit is an isomorphism on point values. Each \(\mathop{\mathrm{Hom}}_\Lambda(P_i,-)\) is a retract of a finite sum of point values, and the equivariant unit commutes with the idempotent defining that retract. It follows that the unit gives an isomorphism of the integral bottom-row computing complexes \[\mathop{\mathrm{Hom}}_\Lambda(P_\bullet,\underline{\mathbb Z_p}) \xrightarrow{\ \sim\ } \mathop{\mathrm{Hom}}_\Lambda(P_\bullet,H^0(K_n)).\] After \([1/p]\), (5) and (9) identify the restricted group class with the constants bottom-row edge class induced by \(u_{\Omega_C^{n-1},U,\mathrm b}\). This uses only the point-value isomorphism, not an identification of the entire condensed module.

For a fixed \(j\), the rationalized row is computed by \[\mathop{\mathrm{Hom}}_\Lambda(P_\bullet,H^j(K_n))[1/p].\] Every term is a direct summand of a finite sum of \(H^j(K_n)(*)[1/p]\). By the integral Drinfeld theorem, (Colmez et al. 2021, Theorem 1.1), a Galois element \(\gamma\) acts on the latter by the scalar \(\chi_{\mathrm{cyc}}(\gamma)^{-j}\). It commutes with \(\Lambda\) and with the idempotents defining the projective summands, so it has the same scalar action on the entire row and on its cohomology. No identification of a solid module by its point value is involved.

The constants row \(j=0\) has trivial Galois action. Choose \(\gamma\in G_F\) with \(\chi_{\mathrm{cyc}}(\gamma)\) of infinite order. The only possible incoming differentials to bidegree \((3,0)\) are \[d_2:E_2^{1,1}\longrightarrow E_2^{3,0}, \qquad d_3:E_3^{0,2}\longrightarrow E_3^{3,0}.\] The second source is absent when \(n=2\). After inversion of \(p\), the respective source scalars are \(\chi_{\mathrm{cyc}}(\gamma)^{-1}\) and \(\chi_{\mathrm{cyc}}(\gamma)^{-2}\), whereas the target scalar is \(1\). Every differential commutes with \(\gamma\), so both are zero. There are no outgoing differentials from the bottom row. Hence the nonzero bottom-row image of the restriction of \(e_{J_n}\) survives in \(H^3_{\mathrm b}([\Omega_C^{n-1}/U])\). Its abutment image is \(u_{\Omega_C^{n-1},U,\mathrm b}(\operatorname{res}^{J_n}_U e_{J_n})\) by the unit calculation above. Naturality (7) for \(U% \BeginAccSupp{method=hex,unicode,ActualText=21AA}% \hookrightarrow\EndAccSupp{}J_n\) identifies it with the pullback of \(u_{\Omega_C^{n-1},J_n,\mathrm b}(e_{J_n})\). Thus the class on \([\Omega_C^{n-1}/J_n]=\mathcal M_n\) is nonzero.

Finally, that class is \(G_F\)-invariant. Its classifying map to \(B_CJ_n\) is equivariant for the action on the \(\mathop{\mathrm{Spd}}C\) base, while \(G_F\) fixes \(J_n\) and the untwisted finite coefficients. Naturality of \(u_{B_C,J_n,m}\) for this base action passes through the derived coefficient limit and rationalization. It therefore sends the invariant group class to an invariant geometric class. Equation (22) therefore places both nonzero classes on the same one-dimensional \(\mathbb Q_p\)-line, proving [eq:primitive-comparison]. ◻

Proof. By 5, for the inclusion \(\rho_3:D\to J_3\) and projection \(\rho_2:D\to J_2\) there is a scalar \(b\in\mathbb Q_p^\times\) with \[\rho_3^*e_{J_3}=b\,\rho_2^*e_{J_2} \quad\text{in }H^3_{\mathrm b}(D).\] Apply \(u_{B_C,D,\mathrm b}\) to this group-cohomology equality and pull it back along \(d:\mathcal T\to B_CD\). The exact filtration in 15 identifies the two \(J_n\)-classifying maps with the composites induced by \(D\to J_n\), and (7) identifies their pullbacks with these same group classes. The \(P_n\)-framing torsors in that proposition give the first and last equalities below. Consequently \[\begin{aligned} \pi^*(e_{P_3}|_{Y_C}) &=m_3^*(e_{P_3}|_{\mathcal M_3})\\ &=\kappa_3\,d^*\rho_3^*e_{J_3}\\ &=\kappa_3b\,d^*\rho_2^*e_{J_2}\\ &=\frac{\kappa_3b}{\kappa_2}\, m_2^*(e_{P_2}|_{\mathcal M_2})\\ &=\frac{\kappa_3b}{\kappa_2}\,\pi^*(e_{P_2}|_{Y_C}). \end{aligned}\] All three scalars are nonzero. The projective morphism \(\pi:\mathcal T\to Y_C\) satisfies 16, so \(\pi^*\) is injective. Taking \(a=\kappa_3b/\kappa_2\) proves the theorem. ◻

Proof. For a characteristic-\(p\) ring \(R\), restriction of Witt components and Verschiebung give an exact sequence of additive groups \[ 0\longrightarrow R\xrightarrow{\ V^{m-1}\ }W_m(R) \longrightarrow W_{m-1}(R)\longrightarrow0 \qquad(m\geq2). \tag{27}\] The sequence is natural for ring maps and automorphisms. We use it as a sequence of additive coefficient objects; no assertion of linearity over \(R\) is needed. It also gives an exact sequence of the associated solid modules and of the corresponding sheaves on the v-site.

8 identifies the actual constant maps \[k\longrightarrow R\Gamma(P_3^1,\widehat L),\qquad C^\flat\longrightarrow R\Gamma_v(X_C,\mathcal O^\flat)\] as equivalences with their residual \(H\)-actions. As proved there, the characteristic-\(p\) vanishing uses only the central scalar restrictions of (Barthel et al. 2026, Propositions 3.5.6 and 3.6.9), identified with \(T_3\) by the determinant calculation in 7, and finite norm-unit invariants as in (Barthel et al. 2026, Lemma 5.2.5). Full \(H\)-equivariance comes from the actual maps of constants and quotient descent. Apply (27) to the source and target of each map. Induction on \(m\), using the resulting morphisms of exact triangles, gives equivalences \[\begin{align*} W_m(k)&\xrightarrow{\sim} R\Gamma(P_3^1,W_m(\widehat L)),\\ W_m(C^\flat)&\xrightarrow{\sim} R\Gamma_v(X_C,W_m(\mathcal O^\flat)). \end{align*}\] Every map used in this induction is the map induced by constants. In particular the equivalences are \(H\)-equivariant, including the \(\overline Z\)-action; \(H\) fixes the constant rings.

Derived global sections and derived invariants preserve limits. Passing to \(R\!\varprojlim_m\) therefore gives the same equivalences with full Witt coefficients. Here the ordinary Witt module is also the derived limit of its truncations: in Witt coordinates, restriction has the continuous set-theoretic section obtained by appending a zero. Consequently restriction is surjective on continuous functions from every profinite parameter space. The truncation towers have no higher derived-limit term in the solid coefficient category. The argument applies to \(k\), \(C^\flat\), and \(\widehat L\) with their respective topologies. Taking residual \(H\)-invariants gives the horizontal isomorphisms of (25). Naturality of the constant maps under \(k\to C^\flat\) gives its commutativity.

For (26), recall \(\widetilde Y=\mathop{\mathrm{Spd}}\widehat L\) from 3. This affinoid perfectoid space is acyclic for \(\mathcal O^\flat\), also with profinite parameters. Applying (27) gives, for every profinite set \(S\), the finite equivalence induced by sections \[\alpha_{S,m}:C_{\mathrm{cts}}(S,W_m(\widehat L))[0] \xrightarrow{\ \sim\ } R\Gamma_v(\widetilde Y\times\underline S,W_m(\mathcal O^\flat)).\] Here Witt coordinates identify \(W_m(C_{\mathrm{cts}}(S,\widehat L))\) with \(C_{\mathrm{cts}}(S,W_m(\widehat L))\). The equivalence is natural in \(S\) and in the \(G\)-action. Write \(u^a_{\widetilde Y,G,m}\) for the degree-\(a\) map of (6). The actual finite coefficient units give the commutative square \[ \begin{tikzcd}[column sep=large] C_{\mathrm{cts}}(G^a,\Lambda_m)[0] \arrow[r,"u^a_{\widetilde Y,G,m}"] \arrow[d,"\iota_m"'] &R\Gamma_v(\widetilde Y\times\underline{G^a},\Lambda_m) \arrow[d,"\omega_m"]\\ C_{\mathrm{cts}}(G^a,W_m(\widehat L))[0] \arrow[r,"\alpha_{G^a,m}","\sim"'] &R\Gamma_v(\widetilde Y\times\underline{G^a},W_m(\mathcal O^\flat)). \end{tikzcd} \tag{28}\] The left map applies \(\Lambda_m=W_m(\mathbb F_p)\to W_m(\widehat L)\) pointwise; the right map is induced by the corresponding sheaf unit. Both composites are the same locally constant Witt section. These squares commute with the bar faces, including the action face, and with restriction in \(m\). The same calculation with an extra profinite parameter retains the solid structure. The condensed standard resolution, quotient descent, and then \(R\!\varprojlim_m\) therefore give (26). The full Witt module is the derived limit of its truncations as shown above. Under point evaluation of this identification, the Witt image of \(u_{\widetilde Y,G,\mathrm{int}}\) is exactly the coefficient map induced by \(\mathbb Z_p\to W(\widehat L)\) and (5).

It remains to prove injectivity, since an equivalence of the horizontal groups alone would not imply it. The finite-resolution statement of 3 applies to \(H\). Indeed an element of order \(p\) in \(P_2\) would have minimal polynomial of degree \(p-1>2\) over \(\mathbb Q_p\), whereas the reduced characteristic polynomial in \(D_2\) has degree two. Thus \(P_2\), and hence \(H=P_2\times\mathbb Z_p\), has no \(p\)-torsion. Choose a bounded resolution \(P_\bullet\to\mathbb Z_p\) by finitely generated projective \(\mathbb Z_p[[H]]\)-modules. For a trivial solid coefficient module \(M\), the point-valued cochain complex is \[\mathop{\mathrm{Hom}}_{\mathbb Z_p[[H]]}(P_\bullet,M) =\mathop{\mathrm{Hom}}_{\mathbb Z_p}(Q_\bullet,M), \qquad Q_\bullet=\mathbb Z_p\otimes_{\mathbb Z_p[[H]]}P_\bullet.\] Every \(Q_j\) is finite free over \(\mathbb Z_p\). Thus this complex is obtained from a fixed bounded finite free \(\mathbb Z_p\)-complex by scalar extension to the underlying module of \(M\); its differentials are the same \(\mathbb Z_p\)-linear matrices. This description is natural in \(M\) and is valid for the valued Witt topology as well as for the \(p\)-adic one, by 3.

Take \(A=W(k)\) and \(B=W(C^\flat)\). The map \(A\to B\) is injective. Since \(A\) is a discrete valuation ring with uniformizer \(p\) and \(B\) is \(p\)-torsion-free, \(B\) is flat over \(A\). Applying the finite complex just described gives \[ H^i(H,A)[1/p]\otimes_{A[1/p]}B[1/p] \xrightarrow{\sim}H^i(H,B)[1/p]. \tag{29}\] The field extension \(A[1/p]% \BeginAccSupp{method=hex,unicode,ActualText=21AA}% \hookrightarrow\EndAccSupp{}B[1/p]\) is faithfully flat, so the natural map from the first cohomology group to its scalar extension is injective. The square now proves the assertion. All infinite limits preceded this finite-complex argument. ◻

Proof. Let \(\omega_{Z,m}\) be the map on cohomology complexes induced by \(\Lambda_m=W_m(\mathbb F_p)\to W_m(\mathcal O^\flat)\) for \(Z=Y\) or \(Y_C\). These maps are compatible with restriction in \(m\). Their derived inverse limit gives a map \[H^3_{\mathrm b}(Z)\longrightarrow H_W^3(Z)[1/p].\] Let \(j:Y_C\to Y\) denote base change, and put \[\beta_m=\omega_{Y,m}u_{\widetilde Y,G,m},\qquad \beta_{C,m}=\omega_{Y_C,m}u_{\widetilde Y_C,G,m}.\] The finite constant maps and the finite Witt units give \[ \begin{tikzcd}[column sep=large] C_{\mathrm{cts}}^{\bullet}(G,\Lambda_m) \arrow[r,"\beta_m"] \arrow[d,equal] &R\Gamma_v(Y,W_m(\mathcal O^\flat))\arrow[d,"j^*"]\\ C_{\mathrm{cts}}^{\bullet}(G,\Lambda_m) \arrow[r,"\beta_{C,m}"'] &R\Gamma_v(Y_C,W_m(\mathcal O^\flat)). \end{tikzcd} \tag{32}\] This square commutes in each cosimplicial degree before totalization, and is compatible with \(q_n:G\to P_n\) and reduction in \(m\). Take the derived coefficient limit, cohomology, and then invert \(p\). The classifying-pullback convention and (7) give \(j^*(e_{P_n}|_Y)=e_{P_n}|_{Y_C}\). Moreover, (28) identifies the Witt image of \(e_{P_n}|_Y\) with \(c_n^W\) under (26). Thus the geometric relation says that \(c_3^W-a c_2^W\) maps to zero in \(H_W^3(Y_C)[1/p]\). The right vertical map of (25) is injective after inverting \(p\), so this difference is already zero on \(Y\). This proves (30). Applying the coefficient map \(W(\widehat L)\to\pi_0\widehat B^{\square}\) then proves (31).

Finally, (14) is induced by the actual map \(E_2^{\square}\to\widehat B^{\square}\), by first taking \(P_3^1\)-invariants and then \(H\)-invariants. Its compatibility with constants is part of 9, whose finite Koszul argument proves the required coefficient instance of the descent construction in (Barthel et al. 2026, Proposition 4.6.12). The action of \(\overline Z\) on the source is trivial. Thus inflation from \(P_2\) followed by this map is exactly the coefficient map defining \(c_2\). This proves the last assertion while retaining the residual action. ◻

Proof. Set \(A_k=\pi_0E_2^{\square}(*)\). The class corresponding to \(c_2\) under (14) restricts along \(P_2% \BeginAccSupp{method=hex,unicode,ActualText=21AA}% \hookrightarrow\EndAccSupp{}H\) to the coefficient image of \(e_{P_2}\) in \(H^3(P_2,A_k)[1/p]\). It suffices to prove that this image is nonzero.

Choose an algebraic closure \(\overline k\) of \(k\) and let \(A_{\overline k}\) be the deformation ring of the same height-two formal group after this base extension. Deformation theory gives a natural continuous \(P_2\)-equivariant map \[A_k\longrightarrow A_{\overline k}.\] In compatible coordinates these rings are \(W(k)[[u'_1]]\) and \(W(\overline k)[[u'_1]]\), with their \((p,u'_1)\)-adic topologies; equivariance follows from deformation base change, without requiring the coordinate to be fixed. Since \(P_2\) is nonextended, it fixes \(W(\overline k)\) pointwise.

The splitting theorem of Barthel–Schlank–Stapleton–Weinstein (Barthel et al. 2025, Proposition 2.5.1 and Corollary 2.5.9) gives a continuous additive equivariant retraction \[r:A_{\overline k}\longrightarrow W(\overline k)\] of the inclusion of Witt constants. Restrict its equivariance to \(P_2\). Additivity and continuity make it \(\mathbb Z_p\)-linear. After condensation it is a morphism of solid \(P_2\)-modules, and hence induces a retraction on the continuous derived invariants used here.

The composite of the coefficient maps \[\mathbb Z_p\longrightarrow A_k\longrightarrow A_{\overline k} \xrightarrow{r}W(\overline k)\] is the usual inclusion of constants. Applying the finite free complex argument from 19 with \(P_2\) in place of \(H\) gives \[H^3(P_2,W(\overline k))[1/p] \cong H^3_{\mathrm b}(P_2) \otimes_{\mathbb Q_p}W(\overline k)[1/p].\] The image of \(e_{P_2}\) here is \(e_{P_2}\otimes1\), which is nonzero by 4. If its image in \(H^3(P_2,A_k)[1/p]\) vanished, the displayed composite would also send it to zero, a contradiction. Restriction to \(P_2\) therefore proves \(c_2\ne0\), and (31) with \(a\ne0\) proves \(c_3\ne0\). ◻

Proof. Let \(F(X)=(X^{hA})(*)\). This is an exact functor preserving limits and upper homotopy bounds. For the latter assertion, coconnective objects remain coconnective under evaluation and limits. Although \(X\) need not be a module spectrum over \(H\mathbb Z_p\), each Postnikov layer \(HM\) is one. The forgetful functor from module spectra preserves limits, so its continuous fixed points agree with the derived module invariants. On this Eilenberg–Mac Lane object, the finite resolution in 3 gives \[\pi_{-s}F(HM)=H^s(A,M),\qquad F(HM)\in\mathrm{Sp}_{[-d,0]}.\] The solid spectra are used with the hypersheaf convention of (Barthel et al. 2026, sec. 4.5, especially Proposition 4.5.5). Truncations remain solid, since they retain or kill each solid homotopy module. Evaluation on an extremally disconnected profinite set is \(t\)-exact: homotopy sheafification evaluated on this projective object agrees with pointwise homotopy. It therefore commutes with truncation. These evaluations detect equivalences, so Postnikov completeness of ordinary spectra gives \(X\simeq\varprojlim_n\tau_{\leq n}X\).

Put \(T_n=F(\tau_{\leq n}X)\). Exactness identifies the successive fiber as \[\operatorname{fib}(T_{n+1}\longrightarrow T_n) \simeq F\bigl(\Sigma^{n+1}H\pi_{n+1}X\bigr) \in\mathrm{Sp}_{[n+1-d,n+1]}.\] Consequently \(\pi_qT_{n+1}\to\pi_qT_n\) is an isomorphism for \(n\geq q+d\); the analogous tower in degree \(q+1\) is also eventually constant. The Milnor exact sequence therefore has zero derived-limit term, and gives \[ \pi_qF(X)\cong\pi_qF(\tau_{\leq q+d}X). \tag{37}\] The infinitely negative tail causes no further issue: since \(F(\tau_{\leq q-1}X)\) is concentrated in degrees at most \(q-1\), the fiber sequence for this lower cutoff gives \[\pi_qF(\tau_{[q,q+d]}X) \xrightarrow{\ \sim\ }\pi_qF(\tau_{\leq q+d}X).\] Hence each abutment degree is computed by a finite Postnikov interval.

The exact couple of this Postnikov construction has the \(E_2\)-term in (36). Its filtration in degree \(q\) is \[\operatorname{Fil}^{s}\pi_qF(X) =\ker\!\left(\pi_qF(X)\longrightarrow \pi_qF(\tau_{\leq q+s-1}X)\right), \quad 0\leq s\leq d+1.\] It starts with the whole group, ends in zero by (37), and has associated graded \(E_\infty^{s,q+s}\). This proves the asserted strong convergence and its naturality. Tensoring the exact couple and these finite filtrations with \(\mathbb Q\) is exact. It therefore gives the rational spectral sequence and its stated abutment; no exchange of rationalization with an infinite totalization is involved. The groups here are \(\mathbb Z_p\)-modules, so this tensor operation agrees with \([1/p]\). No uniform torsion exponent as \(t\) varies is required: each fixed homotopy degree has the finite filtration just constructed. ◻

Proof. Odd homotopy modules vanish. For \(t=2m\), scalar automorphisms \(a\in1+p\mathbb Z_p\) lift canonically to scalar automorphisms of the universal deformation. They fix its deformation ring, and act by \(a^m\) on its \(m\)th invariant-differential line \(\pi_{2m}E_n^{\square}\). This is the central character used in (Barthel et al. 2025, Lemma 2.6.1); replacing the periodicity convention by its dual replaces \(m\) by \(-m\) and has the same consequence.

Choose \(b=1+p\). It is central in \(G_3\), since the finite Galois action fixes these scalars. A central group element acting on a coefficient module induces the identity on its group cohomology. This follows already from the bar resolution: its coefficient action is chain homotopic to the action by conjugation, which is the identity for a central element. Consequently \((b^m-1)\) annihilates \(H^s(G_3,\pi_{2m}E_3^{\square})\). For \(m\ne0\) this is a nonzero \(p\)-adic scalar, and is invertible after \(p\) is inverted. This proves the first vanishing.

For the target, use the natural coefficient descent (14) to identify its cohomology with \(H^s(H,\pi_tE_2^{\square})\). The factor \(\overline Z\) acts trivially on these coefficients, and the central scalars in \(P_2\) act by the same character \(b^m\). The preceding argument applies to \(H\), and proves the second vanishing. ◻

Proof. First extend the source primitive to the extended stabilizer. By 4, \(e_{P_3}\) is represented under rational Lie comparison by a nonzero multiple of \[(X,Y,Z)% \BeginAccSupp{method=hex,unicode,ActualText=27FC}% \longmapsto\EndAccSupp{}\mathop{\mathrm{tr}}_{\mathrm{red}}\bigl(X[Y,Z]\bigr).\] In the standard presentation of the degree-three division algebra, take its maximal unramified subfield \(\mathbb Q_{p^3}\) and an element \(\Pi\) such that \[\Pi x=\sigma(x)\Pi\quad(x\in\mathbb Q_{p^3}),\qquad \Pi^3=p,\] where \(\sigma\) is arithmetic Frobenius of the unramified cubic extension \(\mathbb Q_{p^3}/\mathbb Q_p\), inducing \(z% \BeginAccSupp{method=hex,unicode,ActualText=21A6}% \mapsto\EndAccSupp{}z^p\) on the residue field. Its extension fixing \(\Pi\) equals conjugation \(y% \BeginAccSupp{method=hex,unicode,ActualText=21A6}% \mapsto\EndAccSupp{}\Pi y\Pi^{-1}\), as is checked on the generators \(\mathbb Q_{p^3}\) and \(\Pi\). It therefore preserves the reduced-trace form. Its action on \(P_3\) factors through the order-three quotient of \(\Delta\); the order-two kernel also fixes the \(\mathbb Z_p\)-coefficients. Naturality of Lie comparison therefore makes \(e_{P_3}\) invariant under \(\Delta\). Since \(6\) is a unit in \(\mathbb Z_p\), finite-group averaging and extension descent give \[H^3(G_3,\mathbb Z_p)[1/p] \xrightarrow{\ \sim\ } \bigl(H^3(P_3,\mathbb Z_p)[1/p]\bigr)^\Delta.\] Let \(\widetilde e_3\) be the class restricting to \(e_{P_3}\), and let \(z_3\) be its image under the unit-coefficient map \(\mathbb Z_p\to\pi_0E_3^{\square}\).

Under (5), restriction from \(G_3\) to \(P_3\) and inflation along \(G\to P_3\) on the constant classes are the derived limits of their finite continuous-cochain pullbacks. These are the group maps used in (7). All coefficient maps in the following square are the maps induced by (35): \[\begin{tikzcd}[column sep=large] H^3(G_3,\mathbb Z_p)[1/p] \arrow[r] \arrow[d,"{\mathrm{res},\,\mathrm{inf}}"'] & H^3(G_3,\pi_0E_3^{\square})[1/p]\arrow[d]\\ H^3(G,\mathbb Z_p)[1/p]\arrow[r] & H^3(G,\pi_0\widehat B^{\square})[1/p]. \end{tikzcd}\] On bar cochains, the right vertical map is restriction, pullback along \(G\to P_3\), and the coefficient map of \(E_3^{\square}\to\widehat B^{\square}\), in that order. Its composite with the source coefficient unit is the target unit, which is also \(\mathbb Z_p\to W(\widehat L)\to\pi_0\widehat B^{\square}\). Here the first map is the integral limit of the finite Witt units in (28); the map from full Witt vectors to deformation coefficients is applied only after that limit. The source descent identification used here is the canonical one in 9. In particular the image of \(z_3\) is exactly the class called \(c_3\) in 20: it is obtained from \(e_{P_3}\) by projection and the actual constant-coefficient inclusion. That proposition and 21 give \[c_3=a\,c_2\ne0,\qquad a\in\mathbb Q_p^{\times}.\] Thus the map between the rational descent \(E_2\)-terms is nonzero in bidegree \((s,t)=(3,0)\).

Apply 23 to the source and target of \(g\). By 24, only \(t=0\) survives after rationalization. The differential has bidegree \((r,r-1)\), so every rational differential vanishes. For total homotopy degree \(-3\), the finite filtration has just one possibly nonzero graded piece, namely \((s,t)=(3,0)\). The nonzero map on this piece is therefore a nonzero map on the abutments. These abutments are the ordinary rational homotopy groups asserted in the proposition. ◻

Proof of 1. Rationalize (34). If the canonical map \(L_0L_{K(3)}S\to L_0L_{K(2)}L_{K(3)}S\) were zero on \(\pi_{-3}\), its composite \(L_0g\) would also be zero there. This contradicts 25. Every step above applies to each prime \(p\geq5\), proving the theorem. ◻