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the actual profinite source. Continuous relation cohomology characterizes next-level lifts, and source inflation identifies when a finite-quotient obstruction disappears after that source resolution is refined. Compatible towers reconstruct continuously, and free pro-$p$ sources admit arbitrary coefficient lifts of their generator matrices.","For continuous representations of $\\mathbb Z_p^r$, removing a common scalar precision exposes normalized commutators. When one normalized generator is cyclic modulo $p$, their vanishing is necessary and sufficient for an integral lift, constructed by finite polynomial certificates. This condition covers every rank-two input. It yields a complete finite-matrix decision and a sharp uniform horizon $2n-1$ for lifting a representation modulo $p^n$. Explicit positive and negative examples distinguish genuine abelian relations from torsion relations introduced by fixing a finite quotient. A mixed-variance groupoid records coefficient reduction and source inflation without identifying their directions.","Z_p^r$, removing a common scalar precision exposes normalized commutators. When one normalized generator is cyclic modulo $p$, their vanishing is necessary and sufficient for an integral lift, constructed by finite polynomial certificates. This condition covers every rank-two input. It yields a complete finite-matrix decision and a sharp uniform horizon $2n-1$ for lifting a representation modulo $p^n$. Explicit positive and negative examples distinguish genuine abelian relations from torsion relations introduced by fixing a finite quotient. A mixed-variance groupoid records coefficient reduction and source inflation without identifying their directions.","The goal is to formalize the idea that a mathematical theory may contain two distinct ambient directions:\n\\[\n\\boxed{\\text{transformation: change of representation}}\n\\qquad\\text{and}\\qquad\n\\boxed{\\text{resolution: change of distinguishability}.}\n\\]\nThe distinction is operational rather than rhetorical. A Fourier transform, a duality, or a change of coordinates may alter representation while preserving the level of information being resolved. Coarsening a filtration, passing from modulus $p^{n+1}$ to modulus $p^n$, truncating a spectral expansion, or forgetting fine ramification data changes the level at which distinctions are visible.","t directions:\n\\[\n\\boxed{\\text{transformation: change of representation}}\n\\qquad\\text{and}\\qquad\n\\boxed{\\text{resolution: change of distinguishability}.}\n\\]\nThe distinction is operational rather than rhetorical. A Fourier transform, a duality, or a change of coordinates may alter representation while preserving the level of information being resolved. Coarsening a filtration, passing from modulus $p^{n+1}$ to modulus $p^n$, truncating a spectral expansion, or forgetting fine ramification data changes the level at which distinctions are visible.","ed{\\text{transformation: change of representation}}\n\\qquad\\text{and}\\qquad\n\\boxed{\\text{resolution: change of distinguishability}.}\n\\]\nThe distinction is operational rather than rhetorical. A Fourier transform, a duality, or a change of coordinates may alter representation while preserving the level of information being resolved. Coarsening a filtration, passing from modulus $p^{n+1}$ to modulus $p^n$, truncating a spectral expansion, or forgetting fine ramification data changes the level at which distinctions are visible.","\\paragraph{Categorical regimes and size.}\nThe initial indexed construction uses small ordinary categories and a strict contravariant functor. Every section involving functorial cofibers, higher coherence, stable envelopes, or stable Verdier quotients uses stable $\\infty$-categories, or a specified enhancement presenting them. These are small unless a section explicitly uses an unbounded ambient derived category in a fixed universe. All subcategories in closure constructions are full and replete. ``Stable'' means closed under finite limits and colimits, while ``thick'' additionally means closed under retracts. A stable seed zoo is not automatically idempotent-complete. Thick results are applied to a separately designated thick zoo or to its idempotent completion. Homotopy categories support the corresponding triangulated statements, not arbitrary canonical choices of cones. A fixed small presentation or larger ambient universe is understood whenever object","\\begin{definition}[Resolution category]\nA \\emph{resolution category} is a small poset $(R,\\le)$ regarded as a thin category $\\Rcat$. An arrow $r\\to s$ exists exactly when $r\\le s$. We interpret $s$ as at least as fine as $r$.\n\\end{definition}","The poset assumption is intentionally restrictive for the first formalization. A later version may replace $\\Rcat$ by a general category, a site, a filtered category, or a multi-parameter resolution geometry.","\\begin{definition}[Indexed transformation system]\nAn \\emph{indexed transformation system} over $\\Rcat$ is a strict functor\n\\[\nF:\\Rcat^{op}\\longrightarrow\\Cat.\n\\]\nFor $r\\in R$, write\n\\[\n\\Tcat_r:=F(r).\n\\]\nFor $r\\le s$, write\n\\[\nC_{s\\to r}:=F(r\\le s):\\Tcat_s\\longrightarrow\\Tcat_r.\n\\]\nWe call $C_{s\\to r}$ the \\emph{coarsening functor}.\n\\end{definition}","Strictness gives\n\\[\nC_{r\\to r}=\\id_{\\Tcat_r}\n\\]\nand, for $r\\le s\\le t$,\n\\[\nC_{t\\to r}=C_{s\\to r}\\circ C_{t\\to s}.\n\\]\nA pseudofunctor version is more natural in some applications. The strict version isolates the main structure without coherence bookkeeping.","\\begin{definition}[Resolved object]\nA \\emph{resolved object} is a pair $(r,X)$ with $X\\in\\Ob(\\Tcat_r)$.\n\\end{definition}","Let $X,Y\\in\\Tcat_t$ and let $r\\le t$.","\\begin{definition}[Indistinguishability at resolution $r$]\nWe write\n\\[\nX\\sim_r Y\n\\]\nwhen\n\\[\nC_{t\\to r}(X)\\cong C_{t\\to r}(Y)\n\\]\nin $\\Tcat_r$.\n\\end{definition}","For a fixed ambient fine level $t$, each $\\sim_r$ is an equivalence relation on isomorphism classes of objects of $\\Tcat_t$.","\\begin{theorem}[Resolution monotonicity]\\label{thm:resolution-monotonicity}\nLet $r\\le s\\le t$ and $X,Y\\in\\Tcat_t$. If\n\\[\nX\\sim_sY,\n\\]\nthen\n\\[\nX\\sim_rY.\n\\]\n\\end{theorem}","\\begin{proof}\nAn isomorphism\n\\[\nC_{t\\to s}(X)\\cong C_{t\\to s}(Y)\n\\]\nis carried by the functor $C_{s\\to r}$ to an isomorphism\n\\[\nC_{s\\to r}C_{t\\to s}(X)\n\\cong\nC_{s\\to r}C_{t\\to s}(Y).\n\\]\nBy functorial coherence this is\n\\[\nC_{t\\to r}(X)\\cong C_{t\\to r}(Y).\n\\]\nHence $X\\sim_rY$.\n\\end{proof}","\\begin{corollary}[Nested distinguishability]\\label{cor:nested}\nFor $r\\le s\\le t$, the equivalence relation $\\sim_s$ refines $\\sim_r$.\n\\end{corollary}","\\begin{definition}[Lift set]\nFor $r\\le s$ and $X\\in\\Tcat_r$, define the set of isomorphism classes of lifts\n\\[\n\\Lift_{r\\to s}(X)\n:=\n\\left\\{\nY\\in\\Ob(\\Tcat_s):C_{s\\to r}(Y)\\cong X\n\\right\\}/\\cong.\n\\]\n\\end{definition}","Three basic cases occur:\n\\[\n|\\Lift_{r\\to s}(X)|=0,\n\\qquad\n|\\Lift_{r\\to s}(X)|=1,\n\\qquad\n|\\Lift_{r\\to s}(X)|>1.\n\\]\nThey correspond respectively to obstruction to refinement, uniqueness of the fine object's isomorphism class, and hidden fine-scale multiplicity. The middle case does not imply a contractible space of refinements. It forgets the identification $C_{s\\to r}Y\\cong X$ and all automorphisms. When these matter, use the groupoid of pairs $(Y,\\theta)$ with $\\theta:C_{s\\to r}Y\\xrightarrow{\\sim}X$, and isomorphisms commuting with $\\theta$. In the enhanced setting this is the homotopy fiber of the functor on cores.","0,\n\\qquad\n|\\Lift_{r\\to s}(X)|=1,\n\\qquad\n|\\Lift_{r\\to s}(X)|>1.\n\\]\nThey correspond respectively to obstruction to refinement, uniqueness of the fine object's isomorphism class, and hidden fine-scale multiplicity. The middle case does not imply a contractible space of refinements. It forgets the identification $C_{s\\to r}Y\\cong X$ and all automorphisms. When these matter, use the groupoid of pairs $(Y,\\theta)$ with $\\theta:C_{s\\to r}Y\\xrightarrow{\\sim}X$, and isomorphisms commuting with $\\theta$. In the enhanced setting this is the homotopy fiber of the functor on cores.","r\\to s}(X)|=1,\n\\qquad\n|\\Lift_{r\\to s}(X)|>1.\n\\]\nThey correspond respectively to obstruction to refinement, uniqueness of the fine object's isomorphism class, and hidden fine-scale multiplicity. The middle case does not imply a contractible space of refinements. It forgets the identification $C_{s\\to r}Y\\cong X$ and all automorphisms. When these matter, use the groupoid of pairs $(Y,\\theta)$ with $\\theta:C_{s\\to r}Y\\xrightarrow{\\sim}X$, and isomorphisms commuting with $\\theta$. In the enhanced setting this is the homotopy fiber of the functor on cores.","ely to obstruction to refinement, uniqueness of the fine object's isomorphism class, and hidden fine-scale multiplicity. The middle case does not imply a contractible space of refinements. It forgets the identification $C_{s\\to r}Y\\cong X$ and all automorphisms. When these matter, use the groupoid of pairs $(Y,\\theta)$ with $\\theta:C_{s\\to r}Y\\xrightarrow{\\sim}X$, and isomorphisms commuting with $\\theta$. In the enhanced setting this is the homotopy fiber of the functor on cores.","\\begin{definition}[Resolution branching number]\nWhen $\\Lift_{r\\to s}(X)$ is finite, define\n\\[\nb_{r\\to s}(X):=|\\Lift_{r\\to s}(X)|.\n\\]\nWhen $b_{r\\to s}(X)>0$, its logarithm\n\\[\nH_{r\\to s}(X):=\\log b_{r\\to s}(X)\n\\]\nmay be regarded as a crude \\emph{resolution information} measuring how much ambiguity is hidden by coarsening.\n\\end{definition}","\\begin{construction}[Total Komorebi category]\\label{cons:groth}\nGiven $F:\\Rcat^{op}\\to\\Cat$, define the category\n\\[\n\\K:=\\int_{\\Rcat}F\n\\]\nas follows.","Its objects are pairs $(r,X)$ with $X\\in\\Tcat_r$.","For $r\\le s$, a morphism\n\\[\n(r,X)\\longrightarrow(s,Y)\n\\]\nis a morphism\n\\[\n\\alpha:X\\longrightarrow C_{s\\to r}(Y)\n\\]\nin $\\Tcat_r$. If $r\\nleq s$, there is no such morphism in this directed version.\n\\end{construction}","Given\n\\[\n(r,X)\\xrightarrow{\\alpha}(s,Y)\n\\quad\\text{and}\\quad\n(s,Y)\\xrightarrow{\\beta}(t,Z),\n\\]\ndefine\n\\[\n\\beta\\circ_{\\K}\\alpha\n:=\nC_{s\\to r}(\\beta)\\circ\\alpha:\nX\\to C_{t\\to r}(Z).\n\\]","\\begin{theorem}[Total category]\\label{thm:total-category}\nThe construction above defines a category $\\K$.\n\\end{theorem}","\\begin{proof}\nIdentity at $(r,X)$ is $\\id_X:X\\to C_{r\\to r}(X)=X$. For associativity, let\n\\[\n(r,X)\\xrightarrow{\\alpha}(s,Y)\n\\xrightarrow{\\beta}(t,Z)\n\\xrightarrow{\\gamma}(u,W).\n\\]\nOne composite gives\n\\[\nC_{t\\to r}(\\gamma)\\circ C_{s\\to r}(\\beta)\\circ\\alpha.\n\\]\nThe other gives the same expression by functoriality and\n\\[\nC_{t\\to r}=C_{s\\to r}C_{t\\to s}.\n\\]\nThus composition is associative.\n\\end{proof}","Define\n\\[\n\\pi:\\K\\to\\Rcat,\n\\qquad\n\\pi(r,X)=r.\n\\]\nThe fiber over $r$ is canonically equivalent to $\\Tcat_r$.","For $r\\le s$ and $Y\\in\\Tcat_s$, define the canonical morphism\n\\[\n\\kappa_{r,s,Y}:\n(r,C_{s\\to r}Y)\\longrightarrow(s,Y)\n\\]\nrepresented by the identity map\n\\[\n\\id_{C_{s\\to r}Y}.\n\\]","\\begin{theorem}[Two-axis factorization]\\label{thm:factorization}\nEvery morphism\n\\[\nf:(r,X)\\longrightarrow(s,Y)\n\\]\nin $\\K$ factors uniquely as\n\\[\n(r,X)\n\\xrightarrow{f_v}\n(r,C_{s\\to r}Y)\n\\xrightarrow{\\kappa_{r,s,Y}}\n(s,Y),\n\\]\nwhere $f_v$ lies entirely in the fiber $\\Tcat_r$.\n\\end{theorem}","\\begin{proof}\nBy definition, $f$ is exactly a morphism\n\\[\n\\alpha:X\\to C_{s\\to r}Y\n\\]\nin $\\Tcat_r$. Take $f_v=\\alpha$. Composition with $\\kappa_{r,s,Y}$ is represented by\n\\[\n\\id_{C_{s\\to r}Y}\\circ\\alpha=\\alpha,\n\\]\nso the composite is $f$. Any factorization through the fixed cartesian target $(r,C_{s\\to r}Y)$ must use the same fiber morphism $\\alpha$, hence is unique.\n\\end{proof}","\\begin{remark}\nThis is the formal version of the two-axis statement\n\\[\n\\boxed{\n\\text{mixed movement}\n=\n\\text{within-resolution transformation}\n+\n\\text{resolution movement}.\n}\n\\]\nCategorically, $\\pi:\\K\\to\\Rcat$ is the Grothendieck fibration associated with $F$. Indeed, a map $(u,Z)\\to(s,Y)$ above $u\\le r\\le s$ is exactly a map $Z\\to C_{s\\to u}Y=C_{r\\to u}C_{s\\to r}Y$. This uniquely factors through $\\kappa_{r,s,Y}$ as a morphism to $(r,C_{s\\to r}Y)$ above $u\\le r$, which is the cartesian lifting property. Notice the direction: the lift starts from an already specified fine object $Y$. It is not a procedure that selects a refinement of an arbitrary coarse object.\n\\end{remark}","begin{remark}\nThis is the formal version of the two-axis statement\n\\[\n\\boxed{\n\\text{mixed movement}\n=\n\\text{within-resolution transformation}\n+\n\\text{resolution movement}.\n}\n\\]\nCategorically, $\\pi:\\K\\to\\Rcat$ is the Grothendieck fibration associated with $F$. Indeed, a map $(u,Z)\\to(s,Y)$ above $u\\le r\\le s$ is exactly a map $Z\\to C_{s\\to u}Y=C_{r\\to u}C_{s\\to r}Y$. This uniquely factors through $\\kappa_{r,s,Y}$ as a morphism to $(r,C_{s\\to r}Y)$ above $u\\le r$, which is the cartesian lifting property. Notice the direction: the lift starts from an already specified fine object $Y$. It is not a procedure that selects a refinement of an arbitrary coarse object.\n\\end{remark}","ersion of the two-axis statement\n\\[\n\\boxed{\n\\text{mixed movement}\n=\n\\text{within-resolution transformation}\n+\n\\text{resolution movement}.\n}\n\\]\nCategorically, $\\pi:\\K\\to\\Rcat$ is the Grothendieck fibration associated with $F$. Indeed, a map $(u,Z)\\to(s,Y)$ above $u\\le r\\le s$ is exactly a map $Z\\to C_{s\\to u}Y=C_{r\\to u}C_{s\\to r}Y$. This uniquely factors through $\\kappa_{r,s,Y}$ as a morphism to $(r,C_{s\\to r}Y)$ above $u\\le r$, which is the cartesian lifting property. Notice the direction: the lift starts from an already specified fine object $Y$. It is not a procedure that selects a refinement of an arbitrary coarse object.\n\\end{remark}","tatement\n\\[\n\\boxed{\n\\text{mixed movement}\n=\n\\text{within-resolution transformation}\n+\n\\text{resolution movement}.\n}\n\\]\nCategorically, $\\pi:\\K\\to\\Rcat$ is the Grothendieck fibration associated with $F$. Indeed, a map $(u,Z)\\to(s,Y)$ above $u\\le r\\le s$ is exactly a map $Z\\to C_{s\\to u}Y=C_{r\\to u}C_{s\\to r}Y$. This uniquely factors through $\\kappa_{r,s,Y}$ as a morphism to $(r,C_{s\\to r}Y)$ above $u\\le r$, which is the cartesian lifting property. Notice the direction: the lift starts from an already specified fine object $Y$. It is not a procedure that selects a refinement of an arbitrary coarse object.\n\\end{remark}","cally, $\\pi:\\K\\to\\Rcat$ is the Grothendieck fibration associated with $F$. Indeed, a map $(u,Z)\\to(s,Y)$ above $u\\le r\\le s$ is exactly a map $Z\\to C_{s\\to u}Y=C_{r\\to u}C_{s\\to r}Y$. This uniquely factors through $\\kappa_{r,s,Y}$ as a morphism to $(r,C_{s\\to r}Y)$ above $u\\le r$, which is the cartesian lifting property. Notice the direction: the lift starts from an already specified fine object $Y$. It is not a procedure that selects a refinement of an arbitrary coarse object.\n\\end{remark}","\\begin{definition}[Separation depth]\nLet $S$ be a set of states, with equivalence relations $\\sim_n$ for every $n\\in\\mathbb N=\\{0,1,\\ldots\\}$. Assume\n\\[\n\\sim_{n+1}\\ \\subseteq\\ \\sim_n,\n\\qquad \\sim_0=S\\times S.\n\\]\nDefine\n\\[\n\\nu(X,Y):=\\sup\\{n\\in\\mathbb N:X\\sim_nY\\}\\in\\mathbb N\\cup\\{\\infty\\}.\n\\]\nThe universal zeroth level makes this set nonempty. Assume separation, meaning that $X\\sim_nY$ for every $n$ implies $X=Y$. The states can, for example, be coherent towers equipped with the chosen levelwise equivalence relations. No finiteness of the individual resolutions is required.\n\\end{definition}","nition}[Separation depth]\nLet $S$ be a set of states, with equivalence relations $\\sim_n$ for every $n\\in\\mathbb N=\\{0,1,\\ldots\\}$. Assume\n\\[\n\\sim_{n+1}\\ \\subseteq\\ \\sim_n,\n\\qquad \\sim_0=S\\times S.\n\\]\nDefine\n\\[\n\\nu(X,Y):=\\sup\\{n\\in\\mathbb N:X\\sim_nY\\}\\in\\mathbb N\\cup\\{\\infty\\}.\n\\]\nThe universal zeroth level makes this set nonempty. Assume separation, meaning that $X\\sim_nY$ for every $n$ implies $X=Y$. The states can, for example, be coherent towers equipped with the chosen levelwise equivalence relations. No finiteness of the individual resolutions is required.\n\\end{definition}","\\begin{theorem}[Resolution ultrametric]\\label{thm:ultrametric}\nDefine\n\\[\nd_R(X,Y):=\n\\begin{cases}\n0,&X=Y,\\\\\n2^{-\\nu(X,Y)},&X\\ne Y.\n\\end{cases}\n\\]\nThen $d_R$ is an ultrametric:\n\\[\nd_R(X,Z)\\le\\max\\{d_R(X,Y),d_R(Y,Z)\\}.\n\\]\n\\end{theorem}","2^{-\\nu(X,Y)},&X\\ne Y.\n\\end{cases}\n\\]\nThen $d_R$ is an ultrametric:\n\\[\nd_R(X,Z)\\le\\max\\{d_R(X,Y),d_R(Y,Z)\\}.\n\\]\n\\end{theorem}","\\begin{proof}\nLet\n\\[\nm=\\min\\{\\nu(X,Y),\\nu(Y,Z)\\}.\n\\]\nFor every $n\\le m$, we have $X\\sim_nY$ and $Y\\sim_nZ$. Since $\\sim_n$ is an equivalence relation, $X\\sim_nZ$. Hence\n\\[\n\\nu(X,Z)\\ge m.\n\\]\nTherefore\n\\[\n2^{-\\nu(X,Z)}\n\\le\n2^{-m}\n=\n\\max\\{2^{-\\nu(X,Y)},2^{-\\nu(Y,Z)}\\}.\n\\]\nThe identity and symmetry axioms are immediate, and separation gives positivity for distinct states. Without separation the same formula, with $2^{-\\infty}=0$, is an ultrapseudometric.\n\\end{proof}","Let\n\\[\nm=\\min\\{\\nu(X,Y),\\nu(Y,Z)\\}.\n\\]\nFor every $n\\le m$, we have $X\\sim_nY$ and $Y\\sim_nZ$. Since $\\sim_n$ is an equivalence relation, $X\\sim_nZ$. Hence\n\\[\n\\nu(X,Z)\\ge m.\n\\]\nTherefore\n\\[\n2^{-\\nu(X,Z)}\n\\le\n2^{-m}\n=\n\\max\\{2^{-\\nu(X,Y)},2^{-\\nu(Y,Z)}\\}.\n\\]\nThe identity and symmetry axioms are immediate, and separation gives positivity for distinct states. Without separation the same formula, with $2^{-\\infty}=0$, is an ultrapseudometric.\n\\end{proof}","\\begin{definition}[First distinguishing scale]\nWhen finite, define\n\\[\n\\delta(X,Y):=\\nu(X,Y)+1.\n\\]\nThis is the first resolution at which $X$ and $Y$ become distinguishable.\n\\end{definition}","This quantity may be useful in $p$-adic, dyadic, symbolic, filtration, and machine-representation settings.","Let $U$ be a set of isomorphism classes of resolved objects under consideration. Smallness is imposed only to use ordinary powerset fixed-point arguments.","\\begin{definition}[Seed signature]\nA \\emph{seed signature} on $U$ is data\n\\[\n\\mathfrak S=(\\Seeds_0,\\mathscr T,\\mathscr C,\\mathscr L,\\mathscr D),\n\\]\nwhere:\n\\begin{itemize}\n    \\item $\\Seeds_0\\subseteq U$ is the initial seed set;\n    \\item $\\mathscr T$ is a collection of partial unary transformation constructors on $U$;\n    \\item $\\mathscr C$ is the collection of allowed coarsening constructors;\n    \\item $\\mathscr L$ assigns to an object its admissible refinement classes;\n    \\item $\\mathscr D$ is a collection of partial residual constructors.\n\\end{itemize}\n\\end{definition}","efinition}[Seed signature]\nA \\emph{seed signature} on $U$ is data\n\\[\n\\mathfrak S=(\\Seeds_0,\\mathscr T,\\mathscr C,\\mathscr L,\\mathscr D),\n\\]\nwhere:\n\\begin{itemize}\n    \\item $\\Seeds_0\\subseteq U$ is the initial seed set;\n    \\item $\\mathscr T$ is a collection of partial unary transformation constructors on $U$;\n    \\item $\\mathscr C$ is the collection of allowed coarsening constructors;\n    \\item $\\mathscr L$ assigns to an object its admissible refinement classes;\n    \\item $\\mathscr D$ is a collection of partial residual constructors.\n\\end{itemize}\n\\end{definition}","S=(\\Seeds_0,\\mathscr T,\\mathscr C,\\mathscr L,\\mathscr D),\n\\]\nwhere:\n\\begin{itemize}\n    \\item $\\Seeds_0\\subseteq U$ is the initial seed set;\n    \\item $\\mathscr T$ is a collection of partial unary transformation constructors on $U$;\n    \\item $\\mathscr C$ is the collection of allowed coarsening constructors;\n    \\item $\\mathscr L$ assigns to an object its admissible refinement classes;\n    \\item $\\mathscr D$ is a collection of partial residual constructors.\n\\end{itemize}\n\\end{definition}","\\begin{definition}[One-step zoo operator]\nFor $A\\subseteq U$, define\n\\[\n\\Phi(A):=\nA\n\\cup \\mathscr T[A]\n\\cup \\mathscr C[A]\n\\cup \\mathscr L[A]\n\\cup \\mathscr D[A],\n\\]\nwhere each term denotes all defined outputs of the corresponding constructors applied to inputs from $A$.\n\\end{definition}","\\begin{lemma}[Monotonicity]\\label{lem:phi-monotone}\nIf $A\\subseteq B$, then\n\\[\n\\Phi(A)\\subseteq\\Phi(B).\n\\]\n\\end{lemma}","\\begin{proof}\nEvery constructor output available from inputs in $A$ is also available from the same inputs viewed as elements of $B$.\n\\end{proof}","\\begin{theorem}[Seed zoo existence]\\label{thm:zoo-existence}\nThere exists a least $\\Phi$-closed set containing $\\Seeds_0$. Equivalently, there is a least fixed point of the closure process above $\\Seeds_0$.\n\\end{theorem}","\\begin{proof}\nThe powerset $\\mathcal P(U)$ is a complete lattice. The map\n\\[\n\\Psi(A):=\\Seeds_0\\cup\\Phi(A)\n\\]\nis monotone by \\cref{lem:phi-monotone}. By the Knaster--Tarski fixed-point theorem, $\\Psi$ has a least fixed point. This is exactly the least constructor-closed set containing $\\Seeds_0$.\n\\end{proof}","\\begin{definition}[Generated zoo]\nThe least fixed point in \\cref{thm:zoo-existence} is denoted\n\\[\n\\Zoo(\\Seeds_0).\n\\]\n\\end{definition}","\\begin{proposition}[Finitary iteration]\\label{prop:finitary}\nAssume each constructor is given by a fixed input/output relation with finitely many inputs, independent of the stage $A$. Then $\\Phi$ preserves increasing unions of $\\omega$-chains. Then\n\\[\n\\Zoo(\\Seeds_0)\n=\n\\bigcup_{n<\\omega}\\Psi^n(\\varnothing).\n\\]\n\\end{proposition}","\\begin{proof}\nFor a constructor output from $\\bigcup_n A_n$, its finitely many inputs all belong to a common $A_N$. The output therefore already belongs to $\\Phi(A_N)$. This proves preservation of increasing unions. The union on the right is a fixed point because $\\Psi$ preserves that union. It contains $\\Seeds_0$ and is contained in every fixed point by induction on $n$, hence is least.\n\\end{proof}","This turns the seed calculus into an explicit generative process:\n\\[\n\\Seeds_0\n\\subseteq\n\\Seeds_1\n\\subseteq\n\\Seeds_2\n\\subseteq\\cdots,\n\\qquad\n\\Seeds_{n+1}=\\Psi(\\Seeds_n).\n\\]","\\begin{definition}[Growth function]\nWhen each finite stage contains finitely many isomorphism classes, define\n\\[\ng_{\\Seeds_0}(n):=|\\Psi^n(\\varnothing)|.\n\\]\n\\end{definition}","The asymptotic class of $g(n)$ may distinguish finite, polynomial, exponential, and faster-growing presentations. It depends on the chosen generator signature and cost convention, not merely on the generated category. This is a proposed invariant of a generated mathematical grammar rather than a theorem of the present note.","\\begin{definition}[Lax resolution-compatible transformation]\\label{def:laxtrans}\nA \\emph{lax resolution-compatible transformation} consists of endofunctors\n\\[\nA_r:\\Tcat_r\\to\\Tcat_r\n\\]\nfor every $r\\in R$, together with natural transformations\n\\[\n\\eta^A_{s,r}:\nA_rC_{s\\to r}\n\\Longrightarrow\nC_{s\\to r}A_s\n\\qquad(r\\le s),\n\\]\nsatisfying identity and composition coherence:\n\\[\n\\eta^A_{r,r}=\\id\n\\]\nand, for $r\\le s\\le t$,\n\\[\n\\eta^A_{t,r}\n=\n\\bigl(C_{s\\to r}\\eta^A_{t,s}\\bigr)\n\\circ\n\\bigl(\\eta^A_{s,r}C_{t\\to s}\\bigr).\n\\]\nIf every $\\eta^A_{s,r}$ is a natural isomorphism, $A$ is called \\emph{invertibly resolution-compatible}. This is distinct from exactness of a functor between stable categories.\n\\end{definition}","egin{definition}[Lax resolution-compatible transformation]\\label{def:laxtrans}\nA \\emph{lax resolution-compatible transformation} consists of endofunctors\n\\[\nA_r:\\Tcat_r\\to\\Tcat_r\n\\]\nfor every $r\\in R$, together with natural transformations\n\\[\n\\eta^A_{s,r}:\nA_rC_{s\\to r}\n\\Longrightarrow\nC_{s\\to r}A_s\n\\qquad(r\\le s),\n\\]\nsatisfying identity and composition coherence:\n\\[\n\\eta^A_{r,r}=\\id\n\\]\nand, for $r\\le s\\le t$,\n\\[\n\\eta^A_{t,r}\n=\n\\bigl(C_{s\\to r}\\eta^A_{t,s}\\bigr)\n\\circ\n\\bigl(\\eta^A_{s,r}C_{t\\to s}\\bigr).\n\\]\nIf every $\\eta^A_{s,r}$ is a natural isomorphism, $A$ is called \\emph{invertibly resolution-compatible}. This is distinct from exactness of a functor between stable categories.\n\\end{definition}","[Lax resolution-compatible transformation]\\label{def:laxtrans}\nA \\emph{lax resolution-compatible transformation} consists of endofunctors\n\\[\nA_r:\\Tcat_r\\to\\Tcat_r\n\\]\nfor every $r\\in R$, together with natural transformations\n\\[\n\\eta^A_{s,r}:\nA_rC_{s\\to r}\n\\Longrightarrow\nC_{s\\to r}A_s\n\\qquad(r\\le s),\n\\]\nsatisfying identity and composition coherence:\n\\[\n\\eta^A_{r,r}=\\id\n\\]\nand, for $r\\le s\\le t$,\n\\[\n\\eta^A_{t,r}\n=\n\\bigl(C_{s\\to r}\\eta^A_{t,s}\\bigr)\n\\circ\n\\bigl(\\eta^A_{s,r}C_{t\\to s}\\bigr).\n\\]\nIf every $\\eta^A_{s,r}$ is a natural isomorphism, $A$ is called \\emph{invertibly resolution-compatible}. This is distinct from exactness of a functor between stable categories.\n\\end{definition}","ation} consists of endofunctors\n\\[\nA_r:\\Tcat_r\\to\\Tcat_r\n\\]\nfor every $r\\in R$, together with natural transformations\n\\[\n\\eta^A_{s,r}:\nA_rC_{s\\to r}\n\\Longrightarrow\nC_{s\\to r}A_s\n\\qquad(r\\le s),\n\\]\nsatisfying identity and composition coherence:\n\\[\n\\eta^A_{r,r}=\\id\n\\]\nand, for $r\\le s\\le t$,\n\\[\n\\eta^A_{t,r}\n=\n\\bigl(C_{s\\to r}\\eta^A_{t,s}\\bigr)\n\\circ\n\\bigl(\\eta^A_{s,r}C_{t\\to s}\\bigr).\n\\]\nIf every $\\eta^A_{s,r}$ is a natural isomorphism, $A$ is called \\emph{invertibly resolution-compatible}. This is distinct from exactness of a functor between stable categories.\n\\end{definition}","cat_r\\to\\Tcat_r\n\\]\nfor every $r\\in R$, together with natural transformations\n\\[\n\\eta^A_{s,r}:\nA_rC_{s\\to r}\n\\Longrightarrow\nC_{s\\to r}A_s\n\\qquad(r\\le s),\n\\]\nsatisfying identity and composition coherence:\n\\[\n\\eta^A_{r,r}=\\id\n\\]\nand, for $r\\le s\\le t$,\n\\[\n\\eta^A_{t,r}\n=\n\\bigl(C_{s\\to r}\\eta^A_{t,s}\\bigr)\n\\circ\n\\bigl(\\eta^A_{s,r}C_{t\\to s}\\bigr).\n\\]\nIf every $\\eta^A_{s,r}$ is a natural isomorphism, $A$ is called \\emph{invertibly resolution-compatible}. This is distinct from exactness of a functor between stable categories.\n\\end{definition}","The square is\n\\[\n\\begin{tikzcd}[column sep=large,row sep=large]\nA_rC_{s\\to r}(X) \\arrow[r,\"\\eta^A_{s,r,X}\"] & C_{s\\to r}A_s(X).\n\\end{tikzcd}\n\\]\nIt compares transforming after coarsening with coarsening after transforming.","\\begin{theorem}[Lift to the total category]\\label{thm:lift-total}\nEvery lax resolution-compatible transformation $A$ induces an endofunctor\n\\[\n\\widehat A:\\K\\to\\K\n\\]\ndefined on objects by\n\\[\n\\widehat A(r,X)=(r,A_rX).\n\\]\n\\end{theorem}","\\begin{proof}\nA morphism $(r,X)\\to(s,Y)$ in $\\K$ is represented by\n\\[\n\\alpha:X\\to C_{s\\to r}Y.\n\\]\nDefine $\\widehat A(\\alpha)$ to be\n\\[\nA_rX\n\\xrightarrow{A_r(\\alpha)}\nA_rC_{s\\to r}Y\n\\xrightarrow{\\eta^A_{s,r,Y}}\nC_{s\\to r}A_sY.\n\\]\nIdentity preservation follows from $\\eta^A_{r,r}=\\id$. Composition preservation is exactly the coherence condition in \\cref{def:laxtrans}, together with naturality of $\\eta^A$.\n\\end{proof}","\\begin{definition}[Stable mixed residual]\nAssume each $\\Tcat_r$ is a small stable $\\infty$-category and the coarsening and transformation functors are exact, with homotopy-coherent exchange data. For $r\\le s$ and $X\\in\\Tcat_s$, define\n\\[\n\\Delta^A_{s,r}(X)\n:=\n\\Cofib\\bigl(\\eta^A_{s,r,X}\\bigr)\n\\in\\Tcat_r.\n\\]\n\\end{definition}","Then there is a cofiber sequence\n\\[\nA_rC_{s\\to r}X\n\\longrightarrow\nC_{s\\to r}A_sX\n\\longrightarrow\n\\Delta^A_{s,r}(X).\n\\]","\\begin{proposition}[Residual vanishing criterion]\\label{prop:residual-vanish}\nIn a stable fiber,\n\\[\n\\Delta^A_{s,r}(X)\\simeq0\n\\]\nif and only if\n\\[\n\\eta^A_{s,r,X}\n\\]\nis an equivalence.\n\\end{proposition}","In an abelian fiber one may instead record\n\\[\n\\Delta^-_{s,r}(X)=\\ker\\eta^A_{s,r,X},\n\\qquad\n\\Delta^+_{s,r}(X)=\\operatorname{coker}\\eta^A_{s,r,X}.\n\\]\nThese vanish simultaneously exactly when the comparison is an isomorphism.","\\begin{definition}[Residual promotion]\nIf $X$ is a seed-generated object and $\\Delta^A_{s,r}(X)$ is nonzero, the operation\n\\[\nX\\longmapsto\\Delta^A_{s,r}(X)\n\\]\nis called \\emph{residual promotion} when the residual is added to the active seed set.\n\\end{definition}","This creates the recursive pattern\n\\[\n\\text{seed}\n\\to\n\\text{mixed square}\n\\to\n\\text{residual}\n\\to\n\\text{new seed}\n\\to\n\\text{expanded zoo}.\n\\]","Consider a typed alphabet containing transformation symbols $T$ and resolution symbols $C$. A mixed word is a composable expression such as\n\\[\nW=T_1C_1T_2T_3C_2T_4.\n\\]\nAssume every well-typed adjacent transformation/coarsening inversion admits at least one length-preserving exchange rule of the form\n\\[\nTC\\;\\Rightarrow\\;C\\,T',\n\\]\nwhere $T'$ is the correctly reindexed transformation at the new resolution.","\\begin{definition}[Inversion number]\nFor a word $W$, let $\\iota(W)$ be the number of ordered pairs $(T,C)$ in which a transformation symbol occurs to the left of a resolution symbol when the chosen normal form requires all resolution symbols first.\n\\end{definition}","\\begin{lemma}[Exchange termination]\\label{lem:termination}\nEvery exchange rewrite\n\\[\nTC\\Rightarrow CT'\n\\]\nreduces $\\iota$ by one. Therefore every sequence of such rewrites terminates.\n\\end{lemma}","\\begin{definition}[Split residual exchange]\nA chosen exchange morphism $\\eta:TC\\Rightarrow CT'$ is \\emph{split residual} if it is naturally identified with the projection in a decomposition\n\\[\nTC(X)\\cong CT'(X)\\oplus E_{T,C}(X).\n\\]\nHere $E_{T,C}$ is the split \\emph{source-side remainder}. It is not denoted by $\\Delta$, because in the stable convention of this paper\n\\[\n\\Cofib(\\eta_X)\\simeq E_{T,C}(X)[1],\n\\qquad E_{T,C}(X)\\simeq\\Delta_{T,C}(X)[-1].\n\\]\n\\end{definition}","\\begin{theorem}[Finite split residual expansion]\\label{thm:split-expansion}\nFix a finite principal normalization path for a well-typed mixed word. Assume every step has a chosen split residual exchange and every surrounding operator preserves finite direct sums. Then\n\\[\nW(s)\\cong W_{\\mathrm{nf}}(s)\\oplus\\bigoplus_{i=1}^m E_i(s),\n\\]\nwhere $E_i(s)$ is the surrounding context applied to the source-side remainder of step $i$. In the stable case, $E_i(s)\\simeq D_i[-1]$ for the step cofibers $D_i$ defined below.\n\\end{theorem}","split residual expansion]\\label{thm:split-expansion}\nFix a finite principal normalization path for a well-typed mixed word. Assume every step has a chosen split residual exchange and every surrounding operator preserves finite direct sums. Then\n\\[\nW(s)\\cong W_{\\mathrm{nf}}(s)\\oplus\\bigoplus_{i=1}^m E_i(s),\n\\]\nwhere $E_i(s)$ is the surrounding context applied to the source-side remainder of step $i$. In the stable case, $E_i(s)\\simeq D_i[-1]$ for the step cofibers $D_i$ defined below.\n\\end{theorem}","]\\label{thm:split-expansion}\nFix a finite principal normalization path for a well-typed mixed word. Assume every step has a chosen split residual exchange and every surrounding operator preserves finite direct sums. Then\n\\[\nW(s)\\cong W_{\\mathrm{nf}}(s)\\oplus\\bigoplus_{i=1}^m E_i(s),\n\\]\nwhere $E_i(s)$ is the surrounding context applied to the source-side remainder of step $i$. In the stable case, $E_i(s)\\simeq D_i[-1]$ for the step cofibers $D_i$ defined below.\n\\end{theorem}","ncipal normalization path for a well-typed mixed word. Assume every step has a chosen split residual exchange and every surrounding operator preserves finite direct sums. Then\n\\[\nW(s)\\cong W_{\\mathrm{nf}}(s)\\oplus\\bigoplus_{i=1}^m E_i(s),\n\\]\nwhere $E_i(s)$ is the surrounding context applied to the source-side remainder of step $i$. In the stable case, $E_i(s)\\simeq D_i[-1]$ for the step cofibers $D_i$ defined below.\n\\end{theorem}","\\begin{proof}\nAt step $i$, the split exchange and additivity of the surrounding context give\n$X_{i-1}\\cong X_i\\oplus E_i(s)$. Substituting these decompositions successively yields\n$X_0\\cong X_m\\oplus E_m(s)\\oplus\\cdots\\oplus E_1(s)$.\nThe direct sum is finite because the chosen principal path is finite. The identification with shifted cofibers follows from the cofiber of a split projection.\n\\end{proof}","\\begin{definition}[Normalization path]\nLet $\\mathcal S$ be a stable $\\infty$-category, or a stable enhancement with functorial cofibers. A \\emph{normalization path} for a mixed word $W$ evaluated on $s$ is a finite chain\n\\[\nX_0=W(s)\\xrightarrow{f_1}X_1\\xrightarrow{f_2}\\cdots\\xrightarrow{f_m}X_m=W_{\\mathrm{nf}}(s),\n\\]\nwhere every $f_i$ is obtained by applying an exact surrounding context to one elementary exchange morphism. Write\n\\[\nD_i:=\\Cofib(f_i),\n\\qquad\nD_{\\mathrm{tot}}:=\\Cofib(X_0\\to X_m).\n\\]\n\\end{definition}","th]\nLet $\\mathcal S$ be a stable $\\infty$-category, or a stable enhancement with functorial cofibers. A \\emph{normalization path} for a mixed word $W$ evaluated on $s$ is a finite chain\n\\[\nX_0=W(s)\\xrightarrow{f_1}X_1\\xrightarrow{f_2}\\cdots\\xrightarrow{f_m}X_m=W_{\\mathrm{nf}}(s),\n\\]\nwhere every $f_i$ is obtained by applying an exact surrounding context to one elementary exchange morphism. Write\n\\[\nD_i:=\\Cofib(f_i),\n\\qquad\nD_{\\mathrm{tot}}:=\\Cofib(X_0\\to X_m).\n\\]\n\\end{definition}","\\begin{theorem}[Stable residual filtration along a normalization path]\\label{thm:stable-filtration}\nFor every finite normalization path in a stable enhancement there is a canonical finite filtration\n\\[\n0=F_0\\longrightarrow F_1\\longrightarrow\\cdots\\longrightarrow F_m=D_{\\mathrm{tot}}\n\\]\nwith cofiber sequences\n\\[\nF_{i-1}\\longrightarrow F_i\\longrightarrow D_i.\n\\]\nConsequently the associated graded pieces of the total normalization defect are precisely the transported elementary exchange residuals $D_i$.\n\\end{theorem}","For every finite normalization path in a stable enhancement there is a canonical finite filtration\n\\[\n0=F_0\\longrightarrow F_1\\longrightarrow\\cdots\\longrightarrow F_m=D_{\\mathrm{tot}}\n\\]\nwith cofiber sequences\n\\[\nF_{i-1}\\longrightarrow F_i\\longrightarrow D_i.\n\\]\nConsequently the associated graded pieces of the total normalization defect are precisely the transported elementary exchange residuals $D_i$.\n\\end{theorem}","\\begin{proof}\nDefine\n\\[\nF_i:=\\Cofib(X_0\\longrightarrow X_i).\n\\]\nThen $F_0\\simeq0$ and $F_m=D_{\\mathrm{tot}}$. For the composable maps\n\\[\nX_0\\longrightarrow X_{i-1}\\xrightarrow{f_i}X_i,\n\\]\nthe octahedral cofiber sequence gives\n\\[\n\\Cofib(X_0\\to X_{i-1})\n\\longrightarrow\n\\Cofib(X_0\\to X_i)\n\\longrightarrow\n\\Cofib(f_i).\n\\]\nThus\n\\[\nF_{i-1}\\longrightarrow F_i\\longrightarrow D_i\n\\]\nis a cofiber sequence. Since $f_i$ is obtained by placing an elementary exchange morphism inside an exact context, exactness identifies $D_i$ with the same context applied to the elementary exchange cofiber. Iterating gives the stated filtration.\n\\end{proof}","{proof}\nDefine\n\\[\nF_i:=\\Cofib(X_0\\longrightarrow X_i).\n\\]\nThen $F_0\\simeq0$ and $F_m=D_{\\mathrm{tot}}$. For the composable maps\n\\[\nX_0\\longrightarrow X_{i-1}\\xrightarrow{f_i}X_i,\n\\]\nthe octahedral cofiber sequence gives\n\\[\n\\Cofib(X_0\\to X_{i-1})\n\\longrightarrow\n\\Cofib(X_0\\to X_i)\n\\longrightarrow\n\\Cofib(f_i).\n\\]\nThus\n\\[\nF_{i-1}\\longrightarrow F_i\\longrightarrow D_i\n\\]\nis a cofiber sequence. Since $f_i$ is obtained by placing an elementary exchange morphism inside an exact context, exactness identifies $D_i$ with the same context applied to the elementary exchange cofiber. Iterating gives the stated filtration.\n\\end{proof}","the composable maps\n\\[\nX_0\\longrightarrow X_{i-1}\\xrightarrow{f_i}X_i,\n\\]\nthe octahedral cofiber sequence gives\n\\[\n\\Cofib(X_0\\to X_{i-1})\n\\longrightarrow\n\\Cofib(X_0\\to X_i)\n\\longrightarrow\n\\Cofib(f_i).\n\\]\nThus\n\\[\nF_{i-1}\\longrightarrow F_i\\longrightarrow D_i\n\\]\nis a cofiber sequence. Since $f_i$ is obtained by placing an elementary exchange morphism inside an exact context, exactness identifies $D_i$ with the same context applied to the elementary exchange cofiber. Iterating gives the stated filtration.\n\\end{proof}","\\begin{corollary}[Source-to-normal-form residual control]\\label{cor:source-normal-control}\nThere is a cofiber sequence\n\\[\nD_{\\mathrm{tot}}[-1]\\longrightarrow W(s)\\longrightarrow W_{\\mathrm{nf}}(s),\n\\]\nand $D_{\\mathrm{tot}}[-1]$ carries the shifted filtration with successive pieces $D_i[-1]$. Hence the failure of the original word to equal its normal form is an iterated extension of shifted elementary exchange residuals.\n\\end{corollary}","\\begin{proof}\nWork in the stable category of perfect complexes over a field $k$. Put $M=k[0]$ and $N=k^2[0]$, and consider the square\n\\[\n\\begin{tikzcd}[column sep=large,row sep=large]\n0 \\arrow[r] \\arrow[d] & M \\arrow[d,hook]","\\begin{proof}\nWork in the stable category of perfect complexes over a field $k$. Put $M=k[0]$ and $N=k^2[0]$, and consider the square\n\\[\n\\begin{tikzcd}[column sep=large,row sep=large]\n0 \\arrow[r] \\arrow[d] & M \\arrow[d,hook] \\\\\nN \\arrow[r,hook] & M\\oplus N.\n\\end{tikzcd}\n\\]\nThe maps into $M\\oplus N$ are the summand inclusions. This square is a pushout, hence also a pullback. Both composites are the unique map $0\\to M\\oplus N$.","N \\arrow[r,hook] & M\\oplus N.\n\\end{tikzcd}\n\\]\nThe maps into $M\\oplus N$ are the summand inclusions. This square is a pushout, hence also a pullback. Both composites are the unique map $0\\to M\\oplus N$.","The residual filtrations of its two boundary paths are\n\\[\n0\\longrightarrow M\\longrightarrow M\\oplus N,\n\\qquad\n0\\longrightarrow N\\longrightarrow M\\oplus N.\n\\]\nAn equivalence of these objects of $\\operatorname{Fun}([2],\\operatorname{Perf}(k))$ would induce an equivalence $M\\simeq N$ at the middle index. That is impossible because their degree-zero cohomology dimensions are $1$ and $2$. Nevertheless both total residuals are $M\\oplus N$.\n\\end{proof}","f its two boundary paths are\n\\[\n0\\longrightarrow M\\longrightarrow M\\oplus N,\n\\qquad\n0\\longrightarrow N\\longrightarrow M\\oplus N.\n\\]\nAn equivalence of these objects of $\\operatorname{Fun}([2],\\operatorname{Perf}(k))$ would induce an equivalence $M\\simeq N$ at the middle index. That is impossible because their degree-zero cohomology dimensions are $1$ and $2$. Nevertheless both total residuals are $M\\oplus N$.\n\\end{proof}","\\begin{example}[The counterexample occurs inside the two-axis grammar]\\label{ex:typed-diamond}\nThe preceding square is not merely an unrelated diagram. Let the resolution base be $0<1<2$, put\n\\[\n\\Tcat_0=\\Tcat_1=\\Tcat_2=\\operatorname{Perf}(k)\\times\\operatorname{Perf}(k),\n\\]\nand take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections\n\\[\nP_1(x,y)=(x,0),\\qquad P_2(x,y)=(0,y).\n\\]\nUse transformation families\n\\[\nA_0=P_2,\\quad A_1=A_2=\\id,\n\\qquad\nB_0=B_1=P_1,\\quad B_2=\\id.\n\\]\nFor $A$, the comparison across $1\\to0$ is the inclusion $P_2\\Rightarrow\\id$, and across $2\\to1$ it is the identity. For $B$, these choices are respectively the identity and $P_1\\Rightarrow\\id$. The long comparisons are their composites. Units and transitivity therefore hold.","mar]\\label{ex:typed-diamond}\nThe preceding square is not merely an unrelated diagram. Let the resolution base be $0<1<2$, put\n\\[\n\\Tcat_0=\\Tcat_1=\\Tcat_2=\\operatorname{Perf}(k)\\times\\operatorname{Perf}(k),\n\\]\nand take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections\n\\[\nP_1(x,y)=(x,0),\\qquad P_2(x,y)=(0,y).\n\\]\nUse transformation families\n\\[\nA_0=P_2,\\quad A_1=A_2=\\id,\n\\qquad\nB_0=B_1=P_1,\\quad B_2=\\id.\n\\]\nFor $A$, the comparison across $1\\to0$ is the inclusion $P_2\\Rightarrow\\id$, and across $2\\to1$ it is the identity. For $B$, these choices are respectively the identity and $P_1\\Rightarrow\\id$. The long comparisons are their composites. Units and transitivity therefore hold.","elated diagram. Let the resolution base be $0<1<2$, put\n\\[\n\\Tcat_0=\\Tcat_1=\\Tcat_2=\\operatorname{Perf}(k)\\times\\operatorname{Perf}(k),\n\\]\nand take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections\n\\[\nP_1(x,y)=(x,0),\\qquad P_2(x,y)=(0,y).\n\\]\nUse transformation families\n\\[\nA_0=P_2,\\quad A_1=A_2=\\id,\n\\qquad\nB_0=B_1=P_1,\\quad B_2=\\id.\n\\]\nFor $A$, the comparison across $1\\to0$ is the inclusion $P_2\\Rightarrow\\id$, and across $2\\to1$ it is the identity. For $B$, these choices are respectively the identity and $P_1\\Rightarrow\\id$. The long comparisons are their composites. Units and transitivity therefore hold.","olution base be $0<1<2$, put\n\\[\n\\Tcat_0=\\Tcat_1=\\Tcat_2=\\operatorname{Perf}(k)\\times\\operatorname{Perf}(k),\n\\]\nand take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections\n\\[\nP_1(x,y)=(x,0),\\qquad P_2(x,y)=(0,y).\n\\]\nUse transformation families\n\\[\nA_0=P_2,\\quad A_1=A_2=\\id,\n\\qquad\nB_0=B_1=P_1,\\quad B_2=\\id.\n\\]\nFor $A$, the comparison across $1\\to0$ is the inclusion $P_2\\Rightarrow\\id$, and across $2\\to1$ it is the identity. For $B$, these choices are respectively the identity and $P_1\\Rightarrow\\id$. The long comparisons are their composites. Units and transitivity therefore hold.","put\n\\[\n\\Tcat_0=\\Tcat_1=\\Tcat_2=\\operatorname{Perf}(k)\\times\\operatorname{Perf}(k),\n\\]\nand take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections\n\\[\nP_1(x,y)=(x,0),\\qquad P_2(x,y)=(0,y).\n\\]\nUse transformation families\n\\[\nA_0=P_2,\\quad A_1=A_2=\\id,\n\\qquad\nB_0=B_1=P_1,\\quad B_2=\\id.\n\\]\nFor $A$, the comparison across $1\\to0$ is the inclusion $P_2\\Rightarrow\\id$, and across $2\\to1$ it is the identity. For $B$, these choices are respectively the identity and $P_1\\Rightarrow\\id$. The long comparisons are their composites. Units and transitivity therefore hold.","operatorname{Perf}(k)\\times\\operatorname{Perf}(k),\n\\]\nand take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections\n\\[\nP_1(x,y)=(x,0),\\qquad P_2(x,y)=(0,y).\n\\]\nUse transformation families\n\\[\nA_0=P_2,\\quad A_1=A_2=\\id,\n\\qquad\nB_0=B_1=P_1,\\quad B_2=\\id.\n\\]\nFor $A$, the comparison across $1\\to0$ is the inclusion $P_2\\Rightarrow\\id$, and across $2\\to1$ it is the identity. For $B$, these choices are respectively the identity and $P_1\\Rightarrow\\id$. The long comparisons are their composites. Units and transitivity therefore hold.","atorname{Perf}(k),\n\\]\nand take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections\n\\[\nP_1(x,y)=(x,0),\\qquad P_2(x,y)=(0,y).\n\\]\nUse transformation families\n\\[\nA_0=P_2,\\quad A_1=A_2=\\id,\n\\qquad\nB_0=B_1=P_1,\\quad B_2=\\id.\n\\]\nFor $A$, the comparison across $1\\to0$ is the inclusion $P_2\\Rightarrow\\id$, and across $2\\to1$ it is the identity. For $B$, these choices are respectively the identity and $P_1\\Rightarrow\\id$. The long comparisons are their composites. Units and transitivity therefore hold.","or to be the identity between labelled copies. Define the exact coordinate projections\n\\[\nP_1(x,y)=(x,0),\\qquad P_2(x,y)=(0,y).\n\\]\nUse transformation families\n\\[\nA_0=P_2,\\quad A_1=A_2=\\id,\n\\qquad\nB_0=B_1=P_1,\\quad B_2=\\id.\n\\]\nFor $A$, the comparison across $1\\to0$ is the inclusion $P_2\\Rightarrow\\id$, and across $2\\to1$ it is the identity. For $B$, these choices are respectively the identity and $P_1\\Rightarrow\\id$. The long comparisons are their composites. Units and transitivity therefore hold.","On the seed $x=(k[0],k^2[0])$, the word\n\\[\nW=A_0C_{1\\to0}B_1C_{2\\to1}\n\\]\nevaluates to zero. Exchanging its left inversion first yields $P_1x$. Exchanging its right inversion first yields $P_2x$. The second exchange in either path yields $x$. Both paths then perform the same identity exchange to reach the normal word\n\\[\nC_{1\\to0}C_{2\\to1}A_2B_2.\n\\]\nThus the complete paths have residual filtrations\n\\[\n0\\to P_1x\\to x\\xrightarrow{\\id}x,\n\\qquad\n0\\to P_2x\\to x\\xrightarrow{\\id}x.\n\\]\nTheir first nonzero stages are not equivalent. The transformations, contexts, and coarsenings are exact, and all comparison data are coherent. Consequently extra coherence alone cannot repair the general claim of ordered filtration independence.\n\\","On the seed $x=(k[0],k^2[0])$, the word\n\\[\nW=A_0C_{1\\to0}B_1C_{2\\to1}\n\\]\nevaluates to zero. Exchanging its left inversion first yields $P_1x$. Exchanging its right inversion first yields $P_2x$. The second exchange in either path yields $x$. Both paths then perform the same identity exchange to reach the normal word\n\\[\nC_{1\\to0}C_{2\\to1}A_2B_2.\n\\]\nThus the complete paths have residual filtrations\n\\[\n0\\to P_1x\\to x\\xrightarrow{\\id}x,\n\\qquad\n0\\to P_2x\\to x\\xrightarrow{\\id}x.\n\\]\nTheir first nonzero stages are not equivalent. The transformations, contexts, and coarsenings are exact, and all comparison data are coherent. Consequently extra coherence alone cannot repair the general claim of ordered filtration independence.\n\\end{example}","the seed $x=(k[0],k^2[0])$, the word\n\\[\nW=A_0C_{1\\to0}B_1C_{2\\to1}\n\\]\nevaluates to zero. Exchanging its left inversion first yields $P_1x$. Exchanging its right inversion first yields $P_2x$. The second exchange in either path yields $x$. Both paths then perform the same identity exchange to reach the normal word\n\\[\nC_{1\\to0}C_{2\\to1}A_2B_2.\n\\]\nThus the complete paths have residual filtrations\n\\[\n0\\to P_1x\\to x\\xrightarrow{\\id}x,\n\\qquad\n0\\to P_2x\\to x\\xrightarrow{\\id}x.\n\\]\nTheir first nonzero stages are not equivalent. The transformations, contexts, and coarsenings are exact, and all comparison data are coherent. Consequently extra coherence alone cannot repair the general claim of ordered filtration independence.\n\\end{example}","The normal-form endpoint alone does not determine a residual. For example, the maps $\\id_k$ and $0:k\\to k$ have the same source and target, but cofibers $0$ and $k\\oplus k[1]$. Comparison maps therefore belong to the data.","\\begin{theorem}[Semantic diamond criterion]\\label{thm:semantic-diamond}\nLet a rewrite graph have a height $h$ in $\\mathbb N$ that decreases on every edge, and suppose every vertex $x$ has a unique reachable normal vertex $n(x)$. Assign an object $X_x$ in a stable $\\infty$-category to each vertex and a morphism to each edge. Assume every elementary peak\n\\[\nx\\xrightarrow{e}y,\\qquad x\\xrightarrow{f}z\n\\]\nhas joining paths $a:y\\leadsto u$ and $b:z\\leadsto u$ whose semantic composites satisfy\n\\[\n[a]\\,[e]=[b]\\,[f]\\quad\\text{in the homotopy category}.\n\\]\nThen every normalization path from $x$ to $n(x)$ induces the same homotopy class of comparison map. In particular, the total residuals of any two such paths are equivalent.\n\\end{theorem}","c diamond criterion]\\label{thm:semantic-diamond}\nLet a rewrite graph have a height $h$ in $\\mathbb N$ that decreases on every edge, and suppose every vertex $x$ has a unique reachable normal vertex $n(x)$. Assign an object $X_x$ in a stable $\\infty$-category to each vertex and a morphism to each edge. Assume every elementary peak\n\\[\nx\\xrightarrow{e}y,\\qquad x\\xrightarrow{f}z\n\\]\nhas joining paths $a:y\\leadsto u$ and $b:z\\leadsto u$ whose semantic composites satisfy\n\\[\n[a]\\,[e]=[b]\\,[f]\\quad\\text{in the homotopy category}.\n\\]\nThen every normalization path from $x$ to $n(x)$ induces the same homotopy class of comparison map. In particular, the total residuals of any two such paths are equivalent.\n\\end{theorem}","\\label{thm:semantic-diamond}\nLet a rewrite graph have a height $h$ in $\\mathbb N$ that decreases on every edge, and suppose every vertex $x$ has a unique reachable normal vertex $n(x)$. Assign an object $X_x$ in a stable $\\infty$-category to each vertex and a morphism to each edge. Assume every elementary peak\n\\[\nx\\xrightarrow{e}y,\\qquad x\\xrightarrow{f}z\n\\]\nhas joining paths $a:y\\leadsto u$ and $b:z\\leadsto u$ whose semantic composites satisfy\n\\[\n[a]\\,[e]=[b]\\,[f]\\quad\\text{in the homotopy category}.\n\\]\nThen every normalization path from $x$ to $n(x)$ induces the same homotopy class of comparison map. In particular, the total residuals of any two such paths are equivalent.\n\\end{theorem}","ve a height $h$ in $\\mathbb N$ that decreases on every edge, and suppose every vertex $x$ has a unique reachable normal vertex $n(x)$. Assign an object $X_x$ in a stable $\\infty$-category to each vertex and a morphism to each edge. Assume every elementary peak\n\\[\nx\\xrightarrow{e}y,\\qquad x\\xrightarrow{f}z\n\\]\nhas joining paths $a:y\\leadsto u$ and $b:z\\leadsto u$ whose semantic composites satisfy\n\\[\n[a]\\,[e]=[b]\\,[f]\\quad\\text{in the homotopy category}.\n\\]\nThen every normalization path from $x$ to $n(x)$ induces the same homotopy class of comparison map. In particular, the total residuals of any two such paths are equivalent.\n\\end{theorem}","unique reachable normal vertex $n(x)$. Assign an object $X_x$ in a stable $\\infty$-category to each vertex and a morphism to each edge. Assume every elementary peak\n\\[\nx\\xrightarrow{e}y,\\qquad x\\xrightarrow{f}z\n\\]\nhas joining paths $a:y\\leadsto u$ and $b:z\\leadsto u$ whose semantic composites satisfy\n\\[\n[a]\\,[e]=[b]\\,[f]\\quad\\text{in the homotopy category}.\n\\]\nThen every normalization path from $x$ to $n(x)$ induces the same homotopy class of comparison map. In particular, the total residuals of any two such paths are equivalent.\n\\end{theorem}","reachable normal vertex $n(x)$. Assign an object $X_x$ in a stable $\\infty$-category to each vertex and a morphism to each edge. Assume every elementary peak\n\\[\nx\\xrightarrow{e}y,\\qquad x\\xrightarrow{f}z\n\\]\nhas joining paths $a:y\\leadsto u$ and $b:z\\leadsto u$ whose semantic composites satisfy\n\\[\n[a]\\,[e]=[b]\\,[f]\\quad\\text{in the homotopy category}.\n\\]\nThen every normalization path from $x$ to $n(x)$ induces the same homotopy class of comparison map. In particular, the total residuals of any two such paths are equivalent.\n\\end{theorem}","\\begin{proof}\nInduct on $h(x)$. A normal vertex has only the empty normalization path. Otherwise take two normalization paths with initial edges $e$ and $f$. Choose the joining paths in the hypothesis and any normalization path $t:u\\leadsto n(x)$. Such a path exists, and its terminal vertex is $n(x)$ by uniqueness of normal forms.","Since $h(y),h(z)<h(x)$, the induction hypothesis replaces the two original suffixes by $ta$ and $tb$ respectively without changing their homotopy classes. The full comparison classes are therefore\n\\[\n[t][a][e]=[t][b][f].\n\\]\nThis proves equality for all normalization paths from $x$. Equality in the homotopy category gives a homotopy between the corresponding maps with fixed endpoints. A homotopy identifies the two arrows up to equivalence, and the cofiber functor preserves that equivalence.\n\\end{proof}","The pure exchange grammar has enough combinatorial structure to index all normalization histories without choosing their order. This subsection concerns only adjacent swaps $TC\\Rightarrow CT$ that retain the identity of each symbol. More elaborate rules that duplicate, delete, or replace generators are not included.","\\begin{definition}[Exchange event poset]\\label{def:event-poset}\nFix a finite word $W$ in two types of labelled symbols. Number its transformation symbols $T_1,\\ldots,T_a$ and its coarsening symbols $C_1,\\ldots,C_b$ in their original left-to-right order. An \\emph{exchange event} is a pair\n\\[\n(i,j)\\in E(W)\n\\quad\\Longleftrightarrow\\quad\nT_i\\text{ occurs to the left of }C_j\\text{ in }W.\n\\]\nThe event swaps the relative order of this pair exactly once. Define $P_W$ as the partial order generated by\n\\begin{align*}\n(i,k)&\\prec(i,j) &&\\text{when }k<j\\text{ and both events exist},","set}\nFix a finite word $W$ in two types of labelled symbols. Number its transformation symbols $T_1,\\ldots,T_a$ and its coarsening symbols $C_1,\\ldots,C_b$ in their original left-to-right order. An \\emph{exchange event} is a pair\n\\[\n(i,j)\\in E(W)\n\\quad\\Longleftrightarrow\\quad\nT_i\\text{ occurs to the left of }C_j\\text{ in }W.\n\\]\nThe event swaps the relative order of this pair exactly once. Define $P_W$ as the partial order generated by\n\\begin{align*}\n(i,k)&\\prec(i,j) &&\\text{when }k<j\\text{ and both events exist},","ymbols. Number its transformation symbols $T_1,\\ldots,T_a$ and its coarsening symbols $C_1,\\ldots,C_b$ in their original left-to-right order. An \\emph{exchange event} is a pair\n\\[\n(i,j)\\in E(W)\n\\quad\\Longleftrightarrow\\quad\nT_i\\text{ occurs to the left of }C_j\\text{ in }W.\n\\]\nThe event swaps the relative order of this pair exactly once. Define $P_W$ as the partial order generated by\n\\begin{align*}\n(i,k)&\\prec(i,j) &&\\text{when }k<j\\text{ and both events exist},\\\\\n(h,j)&\\prec(i,j) &&\\text{when }h>i\\text{ and both events exist}.\n\\end{align*}\nThese relations are acyclic: along a generating relation the transformation index decreases or the coarsening index increases, with at least one strict change. Let $J(P_W)$ be the poset of order ideals, ordered by inclusion.\n\\end{definition}","(h,j)&\\prec(i,j) &&\\text{when }h>i\\text{ and both events exist}.\n\\end{align*}\nThese relations are acyclic: along a generating relation the transformation index decreases or the coarsening index increases, with at least one strict change. Let $J(P_W)$ be the poset of order ideals, ordered by inclusion.\n\\end{definition}","\\begin{theorem}[Normalization states are order ideals]\\label{thm:event-ideals}\nFor the pure adjacent-swap grammar, the following correspondences are bijective:\n\\begin{enumerate}\n\\item words reachable from $W$ and ideals $I\\in J(P_W)$;\n\\item enabled elementary swaps at a reachable word and minimal elements of $P_W\\setminus I$;\n\\item complete normalization histories and linear extensions of $P_W$.\n\\end{enumerate}\nThe ideal attached to a reachable word is the set of events already performed. The original word corresponds to $\\varnothing$, and the normal word to $P_W$.\n\\end{theorem}","correspondences are bijective:\n\\begin{enumerate}\n\\item words reachable from $W$ and ideals $I\\in J(P_W)$;\n\\item enabled elementary swaps at a reachable word and minimal elements of $P_W\\setminus I$;\n\\item complete normalization histories and linear extensions of $P_W$.\n\\end{enumerate}\nThe ideal attached to a reachable word is the set of events already performed. The original word corresponds to $\\varnothing$, and the normal word to $P_W$.\n\\end{theorem}","{enumerate}\n\\item words reachable from $W$ and ideals $I\\in J(P_W)$;\n\\item enabled elementary swaps at a reachable word and minimal elements of $P_W\\setminus I$;\n\\item complete normalization histories and linear extensions of $P_W$.\n\\end{enumerate}\nThe ideal attached to a reachable word is the set of events already performed. The original word corresponds to $\\varnothing$, and the normal word to $P_W$.\n\\end{theorem}","Before event $(i,j)$ can occur, every initially intervening $C_k$ with $k<j$ must already have crossed $T_i$, and every initially intervening $T_h$ with $h>i$ must already have crossed $C_j$. Thus all predecessors of a performed event have already occurred. The performed events form an ideal.","That ideal determines the reachable word uniquely. It specifies whether each originally inverted pair has crossed, while pairs originally of the form $C_jT_i$ never change order. Together with the fixed orders inside each type, this determines every pairwise order of labelled symbols.","Conversely, suppose an ideal $I$ has been performed and $(i,j)$ is a minimal event outside it. In the current word $T_i$ still precedes $C_j$. If another symbol lies between them, it is either some $C_k$ with $k<j$ or some $T_h$ with $h>i$. In the first case $(i,k)$ is an unperformed predecessor. In the second case $(h,j)$ is an unperformed predecessor. The relevant pair was already inverted in the original word, since swaps never turn an original $CT$ pair into $TC$. Both cases contradict minimality. Hence $T_i$ and $C_j$ are adjacent and the event is enabled.","nd $(i,j)$ is a minimal event outside it. In the current word $T_i$ still precedes $C_j$. If another symbol lies between them, it is either some $C_k$ with $k<j$ or some $T_h$ with $h>i$. In the first case $(i,k)$ is an unperformed predecessor. In the second case $(h,j)$ is an unperformed predecessor. The relevant pair was already inverted in the original word, since swaps never turn an original $CT$ pair into $TC$. Both cases contradict minimality. Hence $T_i$ and $C_j$ are adjacent and the event is enabled.","minimal event outside it. In the current word $T_i$ still precedes $C_j$. If another symbol lies between them, it is either some $C_k$ with $k<j$ or some $T_h$ with $h>i$. In the first case $(i,k)$ is an unperformed predecessor. In the second case $(h,j)$ is an unperformed predecessor. The relevant pair was already inverted in the original word, since swaps never turn an original $CT$ pair into $TC$. Both cases contradict minimality. Hence $T_i$ and $C_j$ are adjacent and the event is enabled.","$T_i$ still precedes $C_j$. If another symbol lies between them, it is either some $C_k$ with $k<j$ or some $T_h$ with $h>i$. In the first case $(i,k)$ is an unperformed predecessor. In the second case $(h,j)$ is an unperformed predecessor. The relevant pair was already inverted in the original word, since swaps never turn an original $CT$ pair into $TC$. Both cases contradict minimality. Hence $T_i$ and $C_j$ are adjacent and the event is enabled.","ll precedes $C_j$. If another symbol lies between them, it is either some $C_k$ with $k<j$ or some $T_h$ with $h>i$. In the first case $(i,k)$ is an unperformed predecessor. In the second case $(h,j)$ is an unperformed predecessor. The relevant pair was already inverted in the original word, since swaps never turn an original $CT$ pair into $TC$. Both cases contradict minimality. Hence $T_i$ and $C_j$ are adjacent and the event is enabled.","Any ideal can now be constructed by adding its events in an order compatible with $P_W$. This proves surjectivity and the statement about edges. A complete path adds each event exactly once, so it is exactly a linear extension. Conversely each linear extension prescribes enabled swaps and therefore a complete path.\n\\end{proof}","\\begin{proof}\nLet $e$ be the first element of the second linear extension. It is minimal in the poset. In the first extension, every element appearing before $e$ is incomparable with it: it cannot precede $e$ in the poset by minimality, and it cannot follow $e$ in the poset while occurring before it in a linear extension. Move $e$ left across these incomparable elements. The two extensions now begin with the same element. Delete it and apply induction to the remaining finite poset.","It follows that the category of rewrite paths modulo these diamond relations is the thin category $J(P_W)$. The conclusion concerns the pure syntax. In a typed application, labels after crossing must be defined by the specified reindexing rules, and their semantic exchange cells must satisfy the corresponding comparisons.","\\begin{definition}[Coherent exchange-event diagram]\nFor a finite event poset $P$, a coherent exchange-event diagram is a functor\n\\[\nX:NJ(P)\\longrightarrow\\mathcal S\n\\]\ninto a stable $\\infty$-category. Here $N$ denotes the nerve. Its value $X_I$ is the semantic state after the ideal $I$ of events has been completed. Values on cover arrows realize the elementary exchanges, and its higher simplices supply the compatibility data.\n\\end{definition}","\\begin{theorem}[Ideal-indexed residual diagram]\\label{thm:ideal-residual}\nA coherent exchange-event diagram determines a functor\n\\[\n\\mathcal F_X:NJ(P)\\longrightarrow\\mathcal S,\n\\qquad\n\\mathcal F_X(I)=\\Cofib(X_{\\varnothing}\\longrightarrow X_I),\n\\]\nwith the following properties:\n\\begin{enumerate}\n\\item $\\mathcal F_X(\\varnothing)\\simeq0$;\n\\item along every maximal chain of ideals it restricts to the residual filtration of \\cref{thm:stable-filtration};\n\\item for every cover $I\\subset I\\cup\\{e\\}$ there is a cofiber sequence\n\\[\n\\mathcal F_X(I)\\longrightarrow\\mathcal F_X(I\\cup\\{e\\})\n\\longrightarrow\\Cofib(X_I\\to X_{I\\cup\\{e\\}});\n\\]\n\\item its terminal object is the total residual $\\Cofib(X_{\\varnothing}\\to X_P)$, independent of the linear extension used to reach it;\n\\item exact functors preserve the entire diagram, naturally in the specified coherent input.\n\\end{enumerate}\n\\end{theorem}","coherent exchange-event diagram determines a functor\n\\[\n\\mathcal F_X:NJ(P)\\longrightarrow\\mathcal S,\n\\qquad\n\\mathcal F_X(I)=\\Cofib(X_{\\varnothing}\\longrightarrow X_I),\n\\]\nwith the following properties:\n\\begin{enumerate}\n\\item $\\mathcal F_X(\\varnothing)\\simeq0$;\n\\item along every maximal chain of ideals it restricts to the residual filtration of \\cref{thm:stable-filtration};\n\\item for every cover $I\\subset I\\cup\\{e\\}$ there is a cofiber sequence\n\\[\n\\mathcal F_X(I)\\longrightarrow\\mathcal F_X(I\\cup\\{e\\})\n\\longrightarrow\\Cofib(X_I\\to X_{I\\cup\\{e\\}});\n\\]\n\\item its terminal object is the total residual $\\Cofib(X_{\\varnothing}\\to X_P)$, independent of the linear extension used to reach it;\n\\item exact functors preserve the entire diagram, naturally in the specified coherent input.\n\\end{enumerate}\n\\end{theorem}","r\n\\[\n\\mathcal F_X:NJ(P)\\longrightarrow\\mathcal S,\n\\qquad\n\\mathcal F_X(I)=\\Cofib(X_{\\varnothing}\\longrightarrow X_I),\n\\]\nwith the following properties:\n\\begin{enumerate}\n\\item $\\mathcal F_X(\\varnothing)\\simeq0$;\n\\item along every maximal chain of ideals it restricts to the residual filtration of \\cref{thm:stable-filtration};\n\\item for every cover $I\\subset I\\cup\\{e\\}$ there is a cofiber sequence\n\\[\n\\mathcal F_X(I)\\longrightarrow\\mathcal F_X(I\\cup\\{e\\})\n\\longrightarrow\\Cofib(X_I\\to X_{I\\cup\\{e\\}});\n\\]\n\\item its terminal object is the total residual $\\Cofib(X_{\\varnothing}\\to X_P)$, independent of the linear extension used to reach it;\n\\item exact functors preserve the entire diagram, naturally in the specified coherent input.\n\\end{enumerate}\n\\end{theorem}","gin{enumerate}\n\\item $\\mathcal F_X(\\varnothing)\\simeq0$;\n\\item along every maximal chain of ideals it restricts to the residual filtration of \\cref{thm:stable-filtration};\n\\item for every cover $I\\subset I\\cup\\{e\\}$ there is a cofiber sequence\n\\[\n\\mathcal F_X(I)\\longrightarrow\\mathcal F_X(I\\cup\\{e\\})\n\\longrightarrow\\Cofib(X_I\\to X_{I\\cup\\{e\\}});\n\\]\n\\item its terminal object is the total residual $\\Cofib(X_{\\varnothing}\\to X_P)$, independent of the linear extension used to reach it;\n\\item exact functors preserve the entire diagram, naturally in the specified coherent input.\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nThe arrows from the initial ideal $\\varnothing$ induce a natural transformation from the constant diagram at $X_{\\varnothing}$ to $X$. Take its cofiber pointwise in the stable functor category. This constructs $\\mathcal F_X$ together with its coherence, not just an assignment of objects.","At $\\varnothing$ the map is the identity, so its cofiber is zero. Restriction to a maximal chain gives exactly the pointwise formula defining the earlier path-relative filtration. For an inclusion of ideals, apply the cofiber sequence for the composable maps\n\\[\nX_{\\varnothing}\\longrightarrow X_I\\longrightarrow X_{I\\cup\\{e\\}}.\n\\]\nThis gives the displayed cover cofiber sequence. At the terminal ideal the same formula gives the total residual. Finally an exact functor preserves these finite colimits pointwise and hence preserves the diagram and all its structure maps.\n\\end{proof}","This is the appropriate replacement for a single path-independent ordered filtration. In \\cref{prop:filtration-no-go}, $P$ has two incomparable events, so $J(P)$ is the four-object diamond. Its two chains remember different intermediate residuals without disagreeing about the total residual. The richer diagram preserves information that the proposed identification of ordered filtrations would have erased.","\\begin{corollary}[History-independent total obstruction]\\label{cor:history-obstruction}\nLet $q:\\mathcal S\\to\\mathcal O$ be exact. For a fixed coherent exchange-event diagram, the object\n\\[\nq\\mathcal F_X(P)\n\\]\nis independent of the normalization history. Along every maximal chain it has the filtration of \\cref{thm:obstruction-localization}. Therefore its nonvanishing forces a nonzero elementary quotient residual on \\emph{each} normalization history. The particular event witnessing nonvanishing may depend on the history.\n\\end{corollary}","\\begin{definition}[Strict chain-natural realization]\\label{def:strict-chain-realization}\nFix a finite resolution chain $r_0\\le\\cdots\\le r_b$. At each level choose a category $\\mathcal E_j$ of complexes and a functor $L_j:\\mathcal E_j\\to\\mathcal S_j$ into a stable $\\infty$-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors\n\\[\n C_j:\\mathcal E_j\\to\\mathcal E_{j-1},\\qquad\n T_{i,j}:\\mathcal E_j\\to\\mathcal E_j,\n\\]\nwhich induce exact functors in the chosen stable realizations, and strict natural transformations\n\\[\n \\chi_{i,j}:T_{i,j-1}C_j\\Longrightarrow C_jT_{i,j}\n \\qquad(1\\le j\\le b).\n\\]\nThe components of $\\chi$ are actual chain maps and their naturality squares commute as chain maps. Functor composition and whiskering are taken in th","\\begin{definition}[Strict chain-natural realization]\\label{def:strict-chain-realization}\nFix a finite resolution chain $r_0\\le\\cdots\\le r_b$. At each level choose a category $\\mathcal E_j$ of complexes and a functor $L_j:\\mathcal E_j\\to\\mathcal S_j$ into a stable $\\infty$-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors\n\\[\n C_j:\\mathcal E_j\\to\\mathcal E_{j-1},\\qquad\n T_{i,j}:\\mathcal E_j\\to\\mathcal E_j,\n\\]\nwhich induce exact functors in the chosen stable realizations, and strict natural transformations\n\\[\n \\chi_{i,j}:T_{i,j-1}C_j\\Longrightarrow C_jT_{i,j}\n \\qquad(1\\le j\\le b).\n\\]\nThe components of $\\chi$ are actual chain maps and their naturality squares commute as chain maps. Functor composition and whiskering are taken in the ordinary strict 2-category of these models.\n\\e","\\begin{definition}[Strict chain-natural realization]\\label{def:strict-chain-realization}\nFix a finite resolution chain $r_0\\le\\cdots\\le r_b$. At each level choose a category $\\mathcal E_j$ of complexes and a functor $L_j:\\mathcal E_j\\to\\mathcal S_j$ into a stable $\\infty$-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors\n\\[\n C_j:\\mathcal E_j\\to\\mathcal E_{j-1},\\qquad\n T_{i,j}:\\mathcal E_j\\to\\mathcal E_j,\n\\]\nwhich induce exact functors in the chosen stable realizations, and strict natural transformations\n\\[\n \\chi_{i,j}:T_{i,j-1}C_j\\Longrightarrow C_jT_{i,j}\n \\qquad(1\\le j\\le b).\n\\]\nThe components of $\\chi$ are actual chain maps and their naturality squares commute as chain maps. Functor composition and whiskering are taken in the ordinary strict 2-category of these models.\n\\end{definition}","hain-natural realization]\\label{def:strict-chain-realization}\nFix a finite resolution chain $r_0\\le\\cdots\\le r_b$. At each level choose a category $\\mathcal E_j$ of complexes and a functor $L_j:\\mathcal E_j\\to\\mathcal S_j$ into a stable $\\infty$-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors\n\\[\n C_j:\\mathcal E_j\\to\\mathcal E_{j-1},\\qquad\n T_{i,j}:\\mathcal E_j\\to\\mathcal E_j,\n\\]\nwhich induce exact functors in the chosen stable realizations, and strict natural transformations\n\\[\n \\chi_{i,j}:T_{i,j-1}C_j\\Longrightarrow C_jT_{i,j}\n \\qquad(1\\le j\\le b).\n\\]\nThe components of $\\chi$ are actual chain maps and their naturality squares commute as chain maps. Functor composition and whiskering are taken in the ordinary strict 2-category of these models.\n\\end{definition}","or $L_j:\\mathcal E_j\\to\\mathcal S_j$ into a stable $\\infty$-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors\n\\[\n C_j:\\mathcal E_j\\to\\mathcal E_{j-1},\\qquad\n T_{i,j}:\\mathcal E_j\\to\\mathcal E_j,\n\\]\nwhich induce exact functors in the chosen stable realizations, and strict natural transformations\n\\[\n \\chi_{i,j}:T_{i,j-1}C_j\\Longrightarrow C_jT_{i,j}\n \\qquad(1\\le j\\le b).\n\\]\nThe components of $\\chi$ are actual chain maps and their naturality squares commute as chain maps. Functor composition and whiskering are taken in the ordinary strict 2-category of these models.\n\\end{definition}","complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors\n\\[\n C_j:\\mathcal E_j\\to\\mathcal E_{j-1},\\qquad\n T_{i,j}:\\mathcal E_j\\to\\mathcal E_j,\n\\]\nwhich induce exact functors in the chosen stable realizations, and strict natural transformations\n\\[\n \\chi_{i,j}:T_{i,j-1}C_j\\Longrightarrow C_jT_{i,j}\n \\qquad(1\\le j\\le b).\n\\]\nThe components of $\\chi$ are actual chain maps and their naturality squares commute as chain maps. Functor composition and whiskering are taken in the ordinary strict 2-category of these models.\n\\end{definition}","Coarsening symbols occur in their fixed order $C_1,\\ldots,C_b$. A transformation symbol with exactly $j$ coarsening symbols to its left is interpreted as $T_{i,j}$. Hence every reachable word represents a functor from $\\mathcal E_b$ to $\\mathcal E_0$. Moving $T_i$ across $C_j$ reindexes its level rather than creating a new untyped symbol. No rule in this subsection contracts coarsenings or introduces a new exchange relation.","\\begin{theorem}[Strict semantic realization of the event diagram]\\label{thm:strict-event-realization}\nFor a typed pure word $W$ in a strict chain-natural realization and a seed $s\\in\\mathcal E_b$, evaluation gives a functor\n\\[\n X_W:NJ(P_W)\\longrightarrow\\mathcal S_0.\n\\]\nIts vertices evaluate the reachable words and its cover arrows evaluate the whiskered exchange maps. Thus the residual diagram in \\cref{thm:ideal-residual} is constructed, rather than assumed, for this class of realizations.\n\\end{theorem}","\\begin{proof}\nFirst work in the ordinary category of functors $\\mathcal E_b\\to\\mathcal E_0$ and natural transformations. Assign to a rewrite edge its exchange transformation, whiskered by the surrounding word. Two simultaneously enabled swaps are disjoint. The two composites around their diamond are equal by the interchange law for natural transformations. This equality uses strict naturality, not invertibility of $\\chi$.","work in the ordinary category of functors $\\mathcal E_b\\to\\mathcal E_0$ and natural transformations. Assign to a rewrite edge its exchange transformation, whiskered by the surrounding word. Two simultaneously enabled swaps are disjoint. The two composites around their diamond are equal by the interchange law for natural transformations. This equality uses strict naturality, not invertibility of $\\chi$.","For ideals $I\\subseteq J$, two paths adding the events in $J\\setminus I$ are linear extensions of that interval's induced event order. The argument of \\cref{lem:linear-extension-swaps} connects them by adjacent interchanges of incomparable events. Each interchange is one of the commuting diamonds just checked. Hence all such paths induce the same natural transformation. Evaluation therefore factors through the thin category $J(P_W)$.","o paths adding the events in $J\\setminus I$ are linear extensions of that interval's induced event order. The argument of \\cref{lem:linear-extension-swaps} connects them by adjacent interchanges of incomparable events. Each interchange is one of the commuting diamonds just checked. Hence all such paths induce the same natural transformation. Evaluation therefore factors through the thin category $J(P_W)$.","Evaluate this ordinary functor on $s$, take its nerve, and compose with $L_0$. An ordinary functor gives all simplices of its nerve, so the output is a coherent diagram, not merely a diagram in a homotopy category. The prescribed vertices and edges are unchanged by this construction.\n\\end{proof}","Let a specified coherent square in a stable $\\infty$-category $\\mathcal S$ be\n\\[\n\\begin{tikzcd}\n A \\arrow[r,\"u\"] \\arrow[d,\"v\"'] & B \\arrow[d,\"v'\"]","Let a specified coherent square in a stable $\\infty$-category $\\mathcal S$ be\n\\[\n\\begin{tikzcd}\n A \\arrow[r,\"u\"] \\arrow[d,\"v\"'] & B \\arrow[d,\"v'\"]\\\\\n C \\arrow[r,\"u'\"'] & D.\n\\end{tikzcd}\n\\]\nWrite $E=\\Cofib(u)$ and $E'=\\Cofib(u')$. The square induces $E\\to E'$.","C \\arrow[r,\"u'\"'] & D.\n\\end{tikzcd}\n\\]\nWrite $E=\\Cofib(u)$ and $E'=\\Cofib(u')$. The square induces $E\\to E'$.","\\begin{definition}[Interaction residual]\\label{def:interaction-residual}\nThe interaction residual of this specified square is\n\\[\n \\Theta=\\Cofib\\bigl(\\Cofib(A\\to B)\\longrightarrow\n                         \\Cofib(C\\to D)\\bigr).\n\\]\nThe specified square, including its coherent homotopy, is part of the input.\n\\end{definition}","\\begin{theorem}[Interaction is the total cofiber]\\label{thm:interaction-cofiber}\nThere are natural equivalences\n\\[\n \\Theta\\simeq\\Cofib\\bigl(B\\amalg_A C\\longrightarrow D\\bigr)\n \\simeq\\Cofib\\bigl(\\Cofib(A\\to C)\\longrightarrow\n                         \\Cofib(B\\to D)\\bigr).\n\\]\nConsequently $\\Theta\\simeq0$ if and only if the square is a homotopy pushout, equivalently a homotopy pullback. An exact functor preserves these equivalences.\n\\end{theorem}","\\begin{proof}\nCofibers are pushouts along maps to zero. Interchanging the finite colimits in the two successive pushout constructions identifies the first expression with the cofiber of the map from the punctured-square colimit $B\\amalg_A C$ to $D$. Performing the same construction in the other coordinate gives the second expression. These identifications arise from the universal properties of the same finite diagrams and are natural in the given square.","A map in a stable category has zero cofiber exactly when it is an equivalence. Thus vanishing says precisely that the punctured-square colimit maps equivalently to $D$. Stability identifies pushout squares with pullback squares. Exact functors preserve zero objects and these finite colimits.\n\\end{proof}","Fix a finite poset $P$ and a coherent diagram $X:NJ(P)\\to\\mathcal S$. For $e\\in P$, put\n\\[\n D_e=\\{p\\in P:p<e\\},\\qquad\n \\downarrow e=D_e\\cup\\{e\\}.\n\\]\nThe eligible contexts for $e$ form the interval of ideals\n\\[\n \\mathcal A_e=\\{I\\in J(P):D_e\\subseteq I,\\ e\\notin I\\}.\n\\]\nIts initial object is $D_e$. Define the residual functor on this interval by\n\\[\n E_e(I)=\\Cofib(X_I\\to X_{I\\cup\\{e\\}}).\n\\]\nFor incomparable events $e,f$ enabled at $I$, write $\\Theta_{e,f}(I)$ for the interaction residual of their elementary square.","et $P$ and a coherent diagram $X:NJ(P)\\to\\mathcal S$. For $e\\in P$, put\n\\[\n D_e=\\{p\\in P:p<e\\},\\qquad\n \\downarrow e=D_e\\cup\\{e\\}.\n\\]\nThe eligible contexts for $e$ form the interval of ideals\n\\[\n \\mathcal A_e=\\{I\\in J(P):D_e\\subseteq I,\\ e\\notin I\\}.\n\\]\nIts initial object is $D_e$. Define the residual functor on this interval by\n\\[\n E_e(I)=\\Cofib(X_I\\to X_{I\\cup\\{e\\}}).\n\\]\nFor incomparable events $e,f$ enabled at $I$, write $\\Theta_{e,f}(I)$ for the interaction residual of their elementary square.","rent diagram $X:NJ(P)\\to\\mathcal S$. For $e\\in P$, put\n\\[\n D_e=\\{p\\in P:p<e\\},\\qquad\n \\downarrow e=D_e\\cup\\{e\\}.\n\\]\nThe eligible contexts for $e$ form the interval of ideals\n\\[\n \\mathcal A_e=\\{I\\in J(P):D_e\\subseteq I,\\ e\\notin I\\}.\n\\]\nIts initial object is $D_e$. Define the residual functor on this interval by\n\\[\n E_e(I)=\\Cofib(X_I\\to X_{I\\cup\\{e\\}}).\n\\]\nFor incomparable events $e,f$ enabled at $I$, write $\\Theta_{e,f}(I)$ for the interaction residual of their elementary square.","m $X:NJ(P)\\to\\mathcal S$. For $e\\in P$, put\n\\[\n D_e=\\{p\\in P:p<e\\},\\qquad\n \\downarrow e=D_e\\cup\\{e\\}.\n\\]\nThe eligible contexts for $e$ form the interval of ideals\n\\[\n \\mathcal A_e=\\{I\\in J(P):D_e\\subseteq I,\\ e\\notin I\\}.\n\\]\nIts initial object is $D_e$. Define the residual functor on this interval by\n\\[\n E_e(I)=\\Cofib(X_I\\to X_{I\\cup\\{e\\}}).\n\\]\nFor incomparable events $e,f$ enabled at $I$, write $\\Theta_{e,f}(I)$ for the interaction residual of their elementary square.","\\begin{theorem}[Criterion for context-independent event residuals]\\label{thm:event-context-invariance}\nThe following conditions are equivalent.\n\\begin{enumerate}\n\\item Every elementary independent-event square of $X$ is a homotopy pushout.\n\\item Every $\\Theta_{e,f}(I)$ vanishes.\n\\item For each event $e$, the natural comparison\n\\[\n \\operatorname{const}_{\\mathcal A_e}E_e(D_e)\\longrightarrow E_e\n\\]\nis an equivalence of diagrams.\n\\end{enumerate}\nIn this regime the successive cofibers of every normalization history are the event-labelled objects\n\\[\n d_e:=\\Cofib(X_{D_e}\\to X_{\\downarrow e}),\n\\]\nlisted in the order in which that history performs the events. Their labels and equivalence classes are independent of context, but their ordering and extension data need not disappear.\n\\end{theorem}","n{enumerate}\n\\item Every elementary independent-event square of $X$ is a homotopy pushout.\n\\item Every $\\Theta_{e,f}(I)$ vanishes.\n\\item For each event $e$, the natural comparison\n\\[\n \\operatorname{const}_{\\mathcal A_e}E_e(D_e)\\longrightarrow E_e\n\\]\nis an equivalence of diagrams.\n\\end{enumerate}\nIn this regime the successive cofibers of every normalization history are the event-labelled objects\n\\[\n d_e:=\\Cofib(X_{D_e}\\to X_{\\downarrow e}),\n\\]\nlisted in the order in which that history performs the events. Their labels and equivalence classes are independent of context, but their ordering and extension data need not disappear.\n\\end{theorem}","\\begin{proof}\nThe equivalence of (1) and (2) is \\cref{thm:interaction-cofiber}. A cover in $\\mathcal A_e$ adds an event $f$ incomparable with $e$. Indeed, predecessors of $e$ already lie in $I$, while a successor of $e$ cannot belong to an ideal omitting $e$. On this cover the map of event residuals has cofiber $\\Theta_{e,f}(I)$. Condition (2) therefore makes every cover map an equivalence. Any inclusion in the finite interval factors into covers. The comparisons from its initial object consequently give the objectwise, hence diagrammatic, equivalence in (3).","\\begin{lemma}[Elementary excision implies ideal excision]\\label{lem:ideal-excision}\nIf every elementary independent-event square of $X:NJ(P)\\to\\mathcal S$ is a homotopy pushout, then for all ideals $I,J$ the square\n\\[\n\\begin{tikzcd}\n X_{I\\cap J}\\arrow[r]\\arrow[d] &X_I\\arrow[d]","\\begin{lemma}[Elementary excision implies ideal excision]\\label{lem:ideal-excision}\nIf every elementary independent-event square of $X:NJ(P)\\to\\mathcal S$ is a homotopy pushout, then for all ideals $I,J$ the square\n\\[\n\\begin{tikzcd}\n X_{I\\cap J}\\arrow[r]\\arrow[d] &X_I\\arrow[d]\\\\\n X_J\\arrow[r]&X_{I\\cup J}\n\\end{tikzcd}\n\\]\nis a homotopy pushout.\n\\end{lemma}","\\begin{proof}\nElements of $I\\setminus J$ and $J\\setminus I$ are pairwise incomparable across the two sets. If $p\\in I\\setminus J$ satisfied $p\\le q\\in J\\setminus I$, downward closure of $J$ would imply $p\\in J$, a contradiction. The reverse comparison is excluded similarly.","Choose linear extensions for both exclusive sets, beginning from $I\\cap J$. Adjoining their successive initial segments gives a rectangular grid of ideals. Each cell adjoins one enabled event from each side and is an elementary independent-event square. It is a pushout by hypothesis. Repeated pasting of pushout squares gives the outer square displayed above. Empty exclusive sets give identity squares and cause no exception.\n\\end{proof}","\\begin{theorem}[Principal-ideal reconstruction]\\label{thm:principal-reconstruction}\nLet $P$ be finite and $\\mathcal S$ stable. Write $\\operatorname{Exc}_0(J(P),\\mathcal S)$ for the full subcategory of coherent diagrams $F:NJ(P)\\to\\mathcal S$ such that $F(\\varnothing)\\simeq0$ and every elementary independent-event square is a homotopy pushout. Restriction along\n\\[\n j:P\\longrightarrow J(P),\\qquad e\\longmapsto\\downarrow e\n\\]\ninduces an equivalence\n\\[\n \\operatorname{Exc}_0(J(P),\\mathcal S)\n \\simeq\\operatorname{Fun}(NP,\\mathcal S).\n\\]\nIts inverse sends $G:NP\\to\\mathcal S$ to\n\\[\n (j_!G)(I)=\\colim_{e\\in I}G(e),\n \\qquad (j_!G)(\\varnothing)=0.\n\\]\nAll colimits are finite. In particular, for an excisive event diagram $X$ its reduced residual diagram satisfies\n\\[\n \\boxed{\\mathcal F_X(I)\\sime","\\begin{theorem}[Principal-ideal reconstruction]\\label{thm:principal-reconstruction}\nLet $P$ be finite and $\\mathcal S$ stable. Write $\\operatorname{Exc}_0(J(P),\\mathcal S)$ for the full subcategory of coherent diagrams $F:NJ(P)\\to\\mathcal S$ such that $F(\\varnothing)\\simeq0$ and every elementary independent-event square is a homotopy pushout. Restriction along\n\\[\n j:P\\longrightarrow J(P),\\qquad e\\longmapsto\\downarrow e\n\\]\ninduces an equivalence\n\\[\n \\operatorname{Exc}_0(J(P),\\mathcal S)\n \\simeq\\operatorname{Fun}(NP,\\mathcal S).\n\\]\nIts inverse sends $G:NP\\to\\mathcal S$ to\n\\[\n (j_!G)(I)=\\colim_{e\\in I}G(e),\n \\qquad (j_!G)(\\varnothing)=0.\n\\]\nAll colimits are finite. In particular, for an excisive event diagram $X$ its reduced residual diagram satisfies\n\\[\n \\boxed{\\mathcal F_X(I)\\simeq\n \\colim_{e\\in I}\\mathcal","\\begin{theorem}[Principal-ideal reconstruction]\\label{thm:principal-reconstruction}\nLet $P$ be finite and $\\mathcal S$ stable. Write $\\operatorname{Exc}_0(J(P),\\mathcal S)$ for the full subcategory of coherent diagrams $F:NJ(P)\\to\\mathcal S$ such that $F(\\varnothing)\\simeq0$ and every elementary independent-event square is a homotopy pushout. Restriction along\n\\[\n j:P\\longrightarrow J(P),\\qquad e\\longmapsto\\downarrow e\n\\]\ninduces an equivalence\n\\[\n \\operatorname{Exc}_0(J(P),\\mathcal S)\n \\simeq\\operatorname{Fun}(NP,\\mathcal S).\n\\]\nIts inverse sends $G:NP\\to\\mathcal S$ to\n\\[\n (j_!G)(I)=\\colim_{e\\in I}G(e),\n \\qquad (j_!G)(\\varnothing)=0.\n\\]\nAll colimits are finite. In particular, for an excisive event diagram $X$ its reduced residual diagram satisfies\n\\[\n \\boxed{\\mathcal F_X(I)\\simeq\n \\colim_{e\\in I}\\mathcal F_X(\\downarrow e).}\n\\]\n\\end{theorem}","el{thm:principal-reconstruction}\nLet $P$ be finite and $\\mathcal S$ stable. Write $\\operatorname{Exc}_0(J(P),\\mathcal S)$ for the full subcategory of coherent diagrams $F:NJ(P)\\to\\mathcal S$ such that $F(\\varnothing)\\simeq0$ and every elementary independent-event square is a homotopy pushout. Restriction along\n\\[\n j:P\\longrightarrow J(P),\\qquad e\\longmapsto\\downarrow e\n\\]\ninduces an equivalence\n\\[\n \\operatorname{Exc}_0(J(P),\\mathcal S)\n \\simeq\\operatorname{Fun}(NP,\\mathcal S).\n\\]\nIts inverse sends $G:NP\\to\\mathcal S$ to\n\\[\n (j_!G)(I)=\\colim_{e\\in I}G(e),\n \\qquad (j_!G)(\\varnothing)=0.\n\\]\nAll colimits are finite. In particular, for an excisive event diagram $X$ its reduced residual diagram satisfies\n\\[\n \\boxed{\\mathcal F_X(I)\\simeq\n \\colim_{e\\in I}\\mathcal F_X(\\downarrow e).}\n\\]\n\\end{theorem}","J(P),\\mathcal S)$ for the full subcategory of coherent diagrams $F:NJ(P)\\to\\mathcal S$ such that $F(\\varnothing)\\simeq0$ and every elementary independent-event square is a homotopy pushout. Restriction along\n\\[\n j:P\\longrightarrow J(P),\\qquad e\\longmapsto\\downarrow e\n\\]\ninduces an equivalence\n\\[\n \\operatorname{Exc}_0(J(P),\\mathcal S)\n \\simeq\\operatorname{Fun}(NP,\\mathcal S).\n\\]\nIts inverse sends $G:NP\\to\\mathcal S$ to\n\\[\n (j_!G)(I)=\\colim_{e\\in I}G(e),\n \\qquad (j_!G)(\\varnothing)=0.\n\\]\nAll colimits are finite. In particular, for an excisive event diagram $X$ its reduced residual diagram satisfies\n\\[\n \\boxed{\\mathcal F_X(I)\\simeq\n \\colim_{e\\in I}\\mathcal F_X(\\downarrow e).}\n\\]\n\\end{theorem}","e full subcategory of coherent diagrams $F:NJ(P)\\to\\mathcal S$ such that $F(\\varnothing)\\simeq0$ and every elementary independent-event square is a homotopy pushout. Restriction along\n\\[\n j:P\\longrightarrow J(P),\\qquad e\\longmapsto\\downarrow e\n\\]\ninduces an equivalence\n\\[\n \\operatorname{Exc}_0(J(P),\\mathcal S)\n \\simeq\\operatorname{Fun}(NP,\\mathcal S).\n\\]\nIts inverse sends $G:NP\\to\\mathcal S$ to\n\\[\n (j_!G)(I)=\\colim_{e\\in I}G(e),\n \\qquad (j_!G)(\\varnothing)=0.\n\\]\nAll colimits are finite. In particular, for an excisive event diagram $X$ its reduced residual diagram satisfies\n\\[\n \\boxed{\\mathcal F_X(I)\\simeq\n \\colim_{e\\in I}\\mathcal F_X(\\downarrow e).}\n\\]\n\\end{theorem}","uare is a homotopy pushout. Restriction along\n\\[\n j:P\\longrightarrow J(P),\\qquad e\\longmapsto\\downarrow e\n\\]\ninduces an equivalence\n\\[\n \\operatorname{Exc}_0(J(P),\\mathcal S)\n \\simeq\\operatorname{Fun}(NP,\\mathcal S).\n\\]\nIts inverse sends $G:NP\\to\\mathcal S$ to\n\\[\n (j_!G)(I)=\\colim_{e\\in I}G(e),\n \\qquad (j_!G)(\\varnothing)=0.\n\\]\nAll colimits are finite. In particular, for an excisive event diagram $X$ its reduced residual diagram satisfies\n\\[\n \\boxed{\\mathcal F_X(I)\\simeq\n \\colim_{e\\in I}\\mathcal F_X(\\downarrow e).}\n\\]\n\\end{theorem}","g\n\\[\n j:P\\longrightarrow J(P),\\qquad e\\longmapsto\\downarrow e\n\\]\ninduces an equivalence\n\\[\n \\operatorname{Exc}_0(J(P),\\mathcal S)\n \\simeq\\operatorname{Fun}(NP,\\mathcal S).\n\\]\nIts inverse sends $G:NP\\to\\mathcal S$ to\n\\[\n (j_!G)(I)=\\colim_{e\\in I}G(e),\n \\qquad (j_!G)(\\varnothing)=0.\n\\]\nAll colimits are finite. In particular, for an excisive event diagram $X$ its reduced residual diagram satisfies\n\\[\n \\boxed{\\mathcal F_X(I)\\simeq\n \\colim_{e\\in I}\\mathcal F_X(\\downarrow e).}\n\\]\n\\end{theorem}","\\begin{proof}\nThe displayed formula is the pointwise left Kan extension along $j$. It follows directly from the colimit universal property: maps from $j_!G$ to $F$ are the same as compatible maps $G(e)\\to F(\\downarrow e)$. For a principal ideal $\\downarrow e$, its indexing poset has terminal object $e$, so $(j_!G)(\\downarrow e)\\simeq G(e)$.","For arbitrary ideals $I,J$, every chain in $I\\cup J$ lies wholly in $I$ or wholly in $J$. Its maximal element lies in one ideal and all its predecessors lie there too. Hence the nerve of their union is the union of their nerves along the intersection. Cocones on $G|_{I\\cup J}$ are precisely compatible cocones on $G|_I$ and $G|_J$. The resulting colimit is the pushout of the two colimits over the colimit on $I\\cap J$. Thus $j_!G$ is reduced and excisive.","ry ideals $I,J$, every chain in $I\\cup J$ lies wholly in $I$ or wholly in $J$. Its maximal element lies in one ideal and all its predecessors lie there too. Hence the nerve of their union is the union of their nerves along the intersection. Cocones on $G|_{I\\cup J}$ are precisely compatible cocones on $G|_I$ and $G|_J$. The resulting colimit is the pushout of the two colimits over the colimit on $I\\cap J$. Thus $j_!G$ is reduced and excisive.","every chain in $I\\cup J$ lies wholly in $I$ or wholly in $J$. Its maximal element lies in one ideal and all its predecessors lie there too. Hence the nerve of their union is the union of their nerves along the intersection. Cocones on $G|_{I\\cup J}$ are precisely compatible cocones on $G|_I$ and $G|_J$. The resulting colimit is the pushout of the two colimits over the colimit on $I\\cap J$. Thus $j_!G$ is reduced and excisive.","Now let $F$ be reduced and excisive. We prove the counit $j_!j^*F\\to F$ is an equivalence on an ideal $I$ by induction on $|I|$. The empty ideal is the zero object on both sides. Choose a maximal element $e\\in I$. Put $I'=I\\setminus\\{e\\}$. Then\n\\[\n I=I'\\cup\\downarrow e,\\qquad I'\\cap\\downarrow e=D_e.\n\\]\nThe ideal-excision lemma expresses $F(I)$ as the pushout of $F(I')$ and $F(\\downarrow e)$ over $F(D_e)$. The same holds for $j_!j^*F$. Induction handles $I'$ and $D_e$, and the terminal-object observation handles $\\downarrow e$. Therefore the counit is an equivalence at $I$. The unit and counit prove the asserted equivalence of functor categories, including natural transformations.","cisive. We prove the counit $j_!j^*F\\to F$ is an equivalence on an ideal $I$ by induction on $|I|$. The empty ideal is the zero object on both sides. Choose a maximal element $e\\in I$. Put $I'=I\\setminus\\{e\\}$. Then\n\\[\n I=I'\\cup\\downarrow e,\\qquad I'\\cap\\downarrow e=D_e.\n\\]\nThe ideal-excision lemma expresses $F(I)$ as the pushout of $F(I')$ and $F(\\downarrow e)$ over $F(D_e)$. The same holds for $j_!j^*F$. Induction handles $I'$ and $D_e$, and the terminal-object observation handles $\\downarrow e$. Therefore the counit is an equivalence at $I$. The unit and counit prove the asserted equivalence of functor categories, including natural transformations.","j_!j^*F\\to F$ is an equivalence on an ideal $I$ by induction on $|I|$. The empty ideal is the zero object on both sides. Choose a maximal element $e\\in I$. Put $I'=I\\setminus\\{e\\}$. Then\n\\[\n I=I'\\cup\\downarrow e,\\qquad I'\\cap\\downarrow e=D_e.\n\\]\nThe ideal-excision lemma expresses $F(I)$ as the pushout of $F(I')$ and $F(\\downarrow e)$ over $F(D_e)$. The same holds for $j_!j^*F$. Induction handles $I'$ and $D_e$, and the terminal-object observation handles $\\downarrow e$. Therefore the counit is an equivalence at $I$. The unit and counit prove the asserted equivalence of functor categories, including natural transformations.","o F$ is an equivalence on an ideal $I$ by induction on $|I|$. The empty ideal is the zero object on both sides. Choose a maximal element $e\\in I$. Put $I'=I\\setminus\\{e\\}$. Then\n\\[\n I=I'\\cup\\downarrow e,\\qquad I'\\cap\\downarrow e=D_e.\n\\]\nThe ideal-excision lemma expresses $F(I)$ as the pushout of $F(I')$ and $F(\\downarrow e)$ over $F(D_e)$. The same holds for $j_!j^*F$. Induction handles $I'$ and $D_e$, and the terminal-object observation handles $\\downarrow e$. Therefore the counit is an equivalence at $I$. The unit and counit prove the asserted equivalence of functor categories, including natural transformations.","on $|I|$. The empty ideal is the zero object on both sides. Choose a maximal element $e\\in I$. Put $I'=I\\setminus\\{e\\}$. Then\n\\[\n I=I'\\cup\\downarrow e,\\qquad I'\\cap\\downarrow e=D_e.\n\\]\nThe ideal-excision lemma expresses $F(I)$ as the pushout of $F(I')$ and $F(\\downarrow e)$ over $F(D_e)$. The same holds for $j_!j^*F$. Induction handles $I'$ and $D_e$, and the terminal-object observation handles $\\downarrow e$. Therefore the counit is an equivalence at $I$. The unit and counit prove the asserted equivalence of functor categories, including natural transformations.","s. Choose a maximal element $e\\in I$. Put $I'=I\\setminus\\{e\\}$. Then\n\\[\n I=I'\\cup\\downarrow e,\\qquad I'\\cap\\downarrow e=D_e.\n\\]\nThe ideal-excision lemma expresses $F(I)$ as the pushout of $F(I')$ and $F(\\downarrow e)$ over $F(D_e)$. The same holds for $j_!j^*F$. Induction handles $I'$ and $D_e$, and the terminal-object observation handles $\\downarrow e$. Therefore the counit is an equivalence at $I$. The unit and counit prove the asserted equivalence of functor categories, including natural transformations.","Finally, the total-cofiber construction is exact on square diagrams. Applied to the cofiber sequence of diagrams $\\operatorname{const}X_{\\varnothing}\\to X\\to\\mathcal F_X$, it shows that the elementary total cofibers of $X$ and $\\mathcal F_X$ agree, since a constant square has zero total cofiber. Thus $\\mathcal F_X$ is reduced and excisive, and the first part applies.\n\\end{proof}","\\begin{corollary}[Independent events split, dependencies retain extensions]\\label{cor:independent-splitting}\nUnder \\cref{thm:principal-reconstruction}, if $P$ is an antichain then\n\\[\n \\mathcal F_X(P)\\simeq\\bigoplus_{e\\in P}d_e.\n\\]\nFor a general event poset, a direct-sum formula does not follow.\n\\end{corollary}","\\begin{proof}\nFor an antichain, each principal ideal is a singleton, $D_e=\\varnothing$, and a finite colimit over the discrete poset is a finite direct sum. For the negative statement, take $P$ to be a two-element chain and a reduced diagram in $\\operatorname{Perf}(\\mathbb Z)$\n\\[\n 0\\longrightarrow\\mathbb Z/p\\xrightarrow{\\,p\\,}\\mathbb Z/p^2\n\\]\nfor a prime $p$, where the second map sends the class of $x$ to the class of $px$. Both successive cofibers are $\\mathbb Z/p$ in degree zero, but the terminal object is not $\\mathbb Z/p\\oplus\\mathbb Z/p$: the latter is annihilated by $p$ and the former is not. There are no independent-event squares in a chain, so excision holds without imposing a split extension. This diagram is also the residual diagram of $\\mathbb Z\\xrightarrow{p}\\mat","\\begin{proof}\nFor an antichain, each principal ideal is a singleton, $D_e=\\varnothing$, and a finite colimit over the discrete poset is a finite direct sum. For the negative statement, take $P$ to be a two-element chain and a reduced diagram in $\\operatorname{Perf}(\\mathbb Z)$\n\\[\n 0\\longrightarrow\\mathbb Z/p\\xrightarrow{\\,p\\,}\\mathbb Z/p^2\n\\]\nfor a prime $p$, where the second map sends the class of $x$ to the class of $px$. Both successive cofibers are $\\mathbb Z/p$ in degree zero, but the terminal object is not $\\mathbb Z/p\\oplus\\mathbb Z/p$: the latter is annihilated by $p$ and the former is not. There are no independent-event squares in a chain, so excision holds without imposing a split extension. This diagram is also the residual diagram of $\\mathbb Z\\xrightarrow{p}\\mathbb Z\\xrightarrow{p}\\mathbb Z$.\n\\end{proof}","gin{proof}\nFor an antichain, each principal ideal is a singleton, $D_e=\\varnothing$, and a finite colimit over the discrete poset is a finite direct sum. For the negative statement, take $P$ to be a two-element chain and a reduced diagram in $\\operatorname{Perf}(\\mathbb Z)$\n\\[\n 0\\longrightarrow\\mathbb Z/p\\xrightarrow{\\,p\\,}\\mathbb Z/p^2\n\\]\nfor a prime $p$, where the second map sends the class of $x$ to the class of $px$. Both successive cofibers are $\\mathbb Z/p$ in degree zero, but the terminal object is not $\\mathbb Z/p\\oplus\\mathbb Z/p$: the latter is annihilated by $p$ and the former is not. There are no independent-event squares in a chain, so excision holds without imposing a split extension. This diagram is also the residual diagram of $\\mathbb Z\\xrightarrow{p}\\mathbb Z\\xrightarrow{p}\\mathbb Z$.\n\\end{proof}","tichain, each principal ideal is a singleton, $D_e=\\varnothing$, and a finite colimit over the discrete poset is a finite direct sum. For the negative statement, take $P$ to be a two-element chain and a reduced diagram in $\\operatorname{Perf}(\\mathbb Z)$\n\\[\n 0\\longrightarrow\\mathbb Z/p\\xrightarrow{\\,p\\,}\\mathbb Z/p^2\n\\]\nfor a prime $p$, where the second map sends the class of $x$ to the class of $px$. Both successive cofibers are $\\mathbb Z/p$ in degree zero, but the terminal object is not $\\mathbb Z/p\\oplus\\mathbb Z/p$: the latter is annihilated by $p$ and the former is not. There are no independent-event squares in a chain, so excision holds without imposing a split extension. This diagram is also the residual diagram of $\\mathbb Z\\xrightarrow{p}\\mathbb Z\\xrightarrow{p}\\mathbb Z$.\n\\end{proof}","singleton, $D_e=\\varnothing$, and a finite colimit over the discrete poset is a finite direct sum. For the negative statement, take $P$ to be a two-element chain and a reduced diagram in $\\operatorname{Perf}(\\mathbb Z)$\n\\[\n 0\\longrightarrow\\mathbb Z/p\\xrightarrow{\\,p\\,}\\mathbb Z/p^2\n\\]\nfor a prime $p$, where the second map sends the class of $x$ to the class of $px$. Both successive cofibers are $\\mathbb Z/p$ in degree zero, but the terminal object is not $\\mathbb Z/p\\oplus\\mathbb Z/p$: the latter is annihilated by $p$ and the former is not. There are no independent-event squares in a chain, so excision holds without imposing a split extension. This diagram is also the residual diagram of $\\mathbb Z\\xrightarrow{p}\\mathbb Z\\xrightarrow{p}\\mathbb Z$.\n\\end{proof}","iscrete poset is a finite direct sum. For the negative statement, take $P$ to be a two-element chain and a reduced diagram in $\\operatorname{Perf}(\\mathbb Z)$\n\\[\n 0\\longrightarrow\\mathbb Z/p\\xrightarrow{\\,p\\,}\\mathbb Z/p^2\n\\]\nfor a prime $p$, where the second map sends the class of $x$ to the class of $px$. Both successive cofibers are $\\mathbb Z/p$ in degree zero, but the terminal object is not $\\mathbb Z/p\\oplus\\mathbb Z/p$: the latter is annihilated by $p$ and the former is not. There are no independent-event squares in a chain, so excision holds without imposing a split extension. This diagram is also the residual diagram of $\\mathbb Z\\xrightarrow{p}\\mathbb Z\\xrightarrow{p}\\mathbb Z$.\n\\end{proof}","ent, take $P$ to be a two-element chain and a reduced diagram in $\\operatorname{Perf}(\\mathbb Z)$\n\\[\n 0\\longrightarrow\\mathbb Z/p\\xrightarrow{\\,p\\,}\\mathbb Z/p^2\n\\]\nfor a prime $p$, where the second map sends the class of $x$ to the class of $px$. Both successive cofibers are $\\mathbb Z/p$ in degree zero, but the terminal object is not $\\mathbb Z/p\\oplus\\mathbb Z/p$: the latter is annihilated by $p$ and the former is not. There are no independent-event squares in a chain, so excision holds without imposing a split extension. This diagram is also the residual diagram of $\\mathbb Z\\xrightarrow{p}\\mathbb Z\\xrightarrow{p}\\mathbb Z$.\n\\end{proof}","$p$, where the second map sends the class of $x$ to the class of $px$. Both successive cofibers are $\\mathbb Z/p$ in degree zero, but the terminal object is not $\\mathbb Z/p\\oplus\\mathbb Z/p$: the latter is annihilated by $p$ and the former is not. There are no independent-event squares in a chain, so excision holds without imposing a split extension. This diagram is also the residual diagram of $\\mathbb Z\\xrightarrow{p}\\mathbb Z\\xrightarrow{p}\\mathbb Z$.\n\\end{proof}","\\begin{definition}[Killable residual]\nLet $\\delta\\in\\Zoo(\\Seeds_0)$. Call $\\delta$ \\emph{killable} if there exists an admissible finite constructor word $W$ such that\n\\[\nW(\\delta)\\simeq0.\n\\]\n\\end{definition}","\\begin{definition}[Persistent obstruction]\nA residual seed $\\delta$ is a \\emph{persistent obstruction} relative to a constructor zoo if it is not killable by any admissible finite word.\n\\end{definition}","A first taxonomy is:\n\\[\n\\boxed{\n\\Omega=(\\Omega_R,\\Omega_T,\\Omega_{TR},\\Omega_G)\n}\n\\]\nwith:\n\\begin{itemize}\n    \\item $\\Omega_R$: no admissible refinement exists;\n    \\item $\\Omega_T$: a required within-resolution transformation does not exist;\n    \\item $\\Omega_{TR}$: exchange cannot be made coherent or its residual persists;\n    \\item $\\Omega_G$: locally compatible data fail to reconstruct globally.\n\\end{itemize}","Let $\\mathcal Z$ be a small idempotent-complete stable $\\infty$-category representing a thick seed zoo, and let $\\mathscr M$ be a collection of exact constructor endofunctors of $\\mathcal Z$ containing the identity and closed under composition.","\\begin{definition}[Operationally killable subcategory]\nDefine\n\\[\n\\mathcal K_{\\mathscr M}\n:=\n\\{X\\in\\mathcal Z:\\exists W\\in\\mathscr M,\\;W(X)\\simeq0\\}.\n\\]\nLet\n\\[\n\\mathcal N_{\\mathscr M}\n:=\n\\operatorname{Thick}(\\mathcal K_{\\mathscr M})\n\\]\nbe the smallest thick stable subcategory containing all operationally killable objects.\n\\end{definition}","\\begin{definition}[Stable obstruction category and obstruction class]\\label{def:obstruction-quotient}\nThe \\emph{stable obstruction category} is the Verdier quotient\n\\[\n\\mathcal O_{\\mathscr M}\n:=\n\\mathcal Z/\\mathcal N_{\\mathscr M},\n\\]\nwith quotient functor\n\\[\nq:\\mathcal Z\\longrightarrow\\mathcal O_{\\mathscr M}.\n\\]\nFor $X\\in\\mathcal Z$, define its \\emph{obstruction class}\n\\[\n\\Obs_{\\mathscr M}(X):=q(X).\n\\]\n\\end{definition}","\\begin{theorem}[Universal stable obstruction quotient]\\label{thm:obstruction-quotient}\nThe quotient $q$ is exact and satisfies:\n\\begin{enumerate}\n    \\item every operationally killable object maps to zero;\n    \\item $q(X)\\simeq0$ if and only if $X\\in\\mathcal N_{\\mathscr M}$;\n    \\item for every exact functor $E:\\mathcal Z\\to\\mathcal D$ into a stable $\\infty$-category such that $E(K)\\simeq0$ for every $K\\in\\mathcal K_{\\mathscr M}$, there is an exact factorization, unique up to contractible choice together with its comparison to $E$,\n    \\[\n    \\mathcal Z\\xrightarrow{q}\\mathcal O_{\\mathscr M}\\xrightarrow{\\bar E}\\mathcal D.\n    \\]\n\\end{enumerate}\nThus $\\mathcal O_{\\mathscr M}$ is universal among exact targets in which all operationally killable residuals vanish.\n\\end{theorem}","n{theorem}[Universal stable obstruction quotient]\\label{thm:obstruction-quotient}\nThe quotient $q$ is exact and satisfies:\n\\begin{enumerate}\n    \\item every operationally killable object maps to zero;\n    \\item $q(X)\\simeq0$ if and only if $X\\in\\mathcal N_{\\mathscr M}$;\n    \\item for every exact functor $E:\\mathcal Z\\to\\mathcal D$ into a stable $\\infty$-category such that $E(K)\\simeq0$ for every $K\\in\\mathcal K_{\\mathscr M}$, there is an exact factorization, unique up to contractible choice together with its comparison to $E$,\n    \\[\n    \\mathcal Z\\xrightarrow{q}\\mathcal O_{\\mathscr M}\\xrightarrow{\\bar E}\\mathcal D.\n    \\]\n\\end{enumerate}\nThus $\\mathcal O_{\\mathscr M}$ is universal among exact targets in which all operationally killable residuals vanish.\n\\end{theorem}","{enumerate}\n    \\item every operationally killable object maps to zero;\n    \\item $q(X)\\simeq0$ if and only if $X\\in\\mathcal N_{\\mathscr M}$;\n    \\item for every exact functor $E:\\mathcal Z\\to\\mathcal D$ into a stable $\\infty$-category such that $E(K)\\simeq0$ for every $K\\in\\mathcal K_{\\mathscr M}$, there is an exact factorization, unique up to contractible choice together with its comparison to $E$,\n    \\[\n    \\mathcal Z\\xrightarrow{q}\\mathcal O_{\\mathscr M}\\xrightarrow{\\bar E}\\mathcal D.\n    \\]\n\\end{enumerate}\nThus $\\mathcal O_{\\mathscr M}$ is universal among exact targets in which all operationally killable residuals vanish.\n\\end{theorem}","}\n    \\item every operationally killable object maps to zero;\n    \\item $q(X)\\simeq0$ if and only if $X\\in\\mathcal N_{\\mathscr M}$;\n    \\item for every exact functor $E:\\mathcal Z\\to\\mathcal D$ into a stable $\\infty$-category such that $E(K)\\simeq0$ for every $K\\in\\mathcal K_{\\mathscr M}$, there is an exact factorization, unique up to contractible choice together with its comparison to $E$,\n    \\[\n    \\mathcal Z\\xrightarrow{q}\\mathcal O_{\\mathscr M}\\xrightarrow{\\bar E}\\mathcal D.\n    \\]\n\\end{enumerate}\nThus $\\mathcal O_{\\mathscr M}$ is universal among exact targets in which all operationally killable residuals vanish.\n\\end{theorem}","N_{\\mathscr M}$;\n    \\item for every exact functor $E:\\mathcal Z\\to\\mathcal D$ into a stable $\\infty$-category such that $E(K)\\simeq0$ for every $K\\in\\mathcal K_{\\mathscr M}$, there is an exact factorization, unique up to contractible choice together with its comparison to $E$,\n    \\[\n    \\mathcal Z\\xrightarrow{q}\\mathcal O_{\\mathscr M}\\xrightarrow{\\bar E}\\mathcal D.\n    \\]\n\\end{enumerate}\nThus $\\mathcal O_{\\mathscr M}$ is universal among exact targets in which all operationally killable residuals vanish.\n\\end{theorem}","\\begin{proof}\nThe Verdier quotient by the thick subcategory $\\mathcal N_{\\mathscr M}$ exists and the quotient functor is exact. Its kernel is exactly $\\mathcal N_{\\mathscr M}$. This gives (1)--(2). For (3), the kernel of an exact functor is thick. Since $E$ kills $\\mathcal K_{\\mathscr M}$, it kills the thick subcategory generated by that class, namely $\\mathcal N_{\\mathscr M}$. The universal property of the stable Verdier quotient gives the factorization. More precisely, precomposition with $q$ identifies the category of exact functors out of $\\mathcal O_{\\mathscr M}$ with the full subcategory of exact functors out of $\\mathcal Z$ annihilating $\\mathcal N_{\\mathscr M}$. This also specifies the sense of uniqueness. See Drew~\\cite{Drew2015} for the stable localization and \\ci","\\begin{proof}\nThe Verdier quotient by the thick subcategory $\\mathcal N_{\\mathscr M}$ exists and the quotient functor is exact. Its kernel is exactly $\\mathcal N_{\\mathscr M}$. This gives (1)--(2). For (3), the kernel of an exact functor is thick. Since $E$ kills $\\mathcal K_{\\mathscr M}$, it kills the thick subcategory generated by that class, namely $\\mathcal N_{\\mathscr M}$. The universal property of the stable Verdier quotient gives the factorization. More precisely, precomposition with $q$ identifies the category of exact functors out of $\\mathcal O_{\\mathscr M}$ with the full subcategory of exact functors out of $\\mathcal Z$ annihilating $\\mathcal N_{\\mathscr M}$. This also specifies the sense of uniqueness. See Drew~\\cite{Drew2015} for the stable localization and \\cite{StacksVerdier} for the kernel calculation in the triangulated homotopy category.\n\\end{p","\\begin{proof}\nThe Verdier quotient by the thick subcategory $\\mathcal N_{\\mathscr M}$ exists and the quotient functor is exact. Its kernel is exactly $\\mathcal N_{\\mathscr M}$. This gives (1)--(2). For (3), the kernel of an exact functor is thick. Since $E$ kills $\\mathcal K_{\\mathscr M}$, it kills the thick subcategory generated by that class, namely $\\mathcal N_{\\mathscr M}$. The universal property of the stable Verdier quotient gives the factorization. More precisely, precomposition with $q$ identifies the category of exact functors out of $\\mathcal O_{\\mathscr M}$ with the full subcategory of exact functors out of $\\mathcal Z$ annihilating $\\mathcal N_{\\mathscr M}$. This also specifies the sense of uniqueness. See Drew~\\cite{Drew2015} for the stable localization and \\cite{StacksVerdier} for the kernel calculation in the triangulated homotopy category.\n\\end{proof}","the quotient functor is exact. Its kernel is exactly $\\mathcal N_{\\mathscr M}$. This gives (1)--(2). For (3), the kernel of an exact functor is thick. Since $E$ kills $\\mathcal K_{\\mathscr M}$, it kills the thick subcategory generated by that class, namely $\\mathcal N_{\\mathscr M}$. The universal property of the stable Verdier quotient gives the factorization. More precisely, precomposition with $q$ identifies the category of exact functors out of $\\mathcal O_{\\mathscr M}$ with the full subcategory of exact functors out of $\\mathcal Z$ annihilating $\\mathcal N_{\\mathscr M}$. This also specifies the sense of uniqueness. See Drew~\\cite{Drew2015} for the stable localization and \\cite{StacksVerdier} for the kernel calculation in the triangulated homotopy category.\n\\end{proof}","$\\mathcal N_{\\mathscr M}$. This gives (1)--(2). For (3), the kernel of an exact functor is thick. Since $E$ kills $\\mathcal K_{\\mathscr M}$, it kills the thick subcategory generated by that class, namely $\\mathcal N_{\\mathscr M}$. The universal property of the stable Verdier quotient gives the factorization. More precisely, precomposition with $q$ identifies the category of exact functors out of $\\mathcal O_{\\mathscr M}$ with the full subcategory of exact functors out of $\\mathcal Z$ annihilating $\\mathcal N_{\\mathscr M}$. This also specifies the sense of uniqueness. See Drew~\\cite{Drew2015} for the stable localization and \\cite{StacksVerdier} for the kernel calculation in the triangulated homotopy category.\n\\end{proof}","ct functor is thick. Since $E$ kills $\\mathcal K_{\\mathscr M}$, it kills the thick subcategory generated by that class, namely $\\mathcal N_{\\mathscr M}$. The universal property of the stable Verdier quotient gives the factorization. More precisely, precomposition with $q$ identifies the category of exact functors out of $\\mathcal O_{\\mathscr M}$ with the full subcategory of exact functors out of $\\mathcal Z$ annihilating $\\mathcal N_{\\mathscr M}$. This also specifies the sense of uniqueness. See Drew~\\cite{Drew2015} for the stable localization and \\cite{StacksVerdier} for the kernel calculation in the triangulated homotopy category.\n\\end{proof}","$E$ kills $\\mathcal K_{\\mathscr M}$, it kills the thick subcategory generated by that class, namely $\\mathcal N_{\\mathscr M}$. The universal property of the stable Verdier quotient gives the factorization. More precisely, precomposition with $q$ identifies the category of exact functors out of $\\mathcal O_{\\mathscr M}$ with the full subcategory of exact functors out of $\\mathcal Z$ annihilating $\\mathcal N_{\\mathscr M}$. This also specifies the sense of uniqueness. See Drew~\\cite{Drew2015} for the stable localization and \\cite{StacksVerdier} for the kernel calculation in the triangulated homotopy category.\n\\end{proof}","A zero obstruction here means membership in the chosen thick kernel. It need not mean that one constructor word kills the object. Nor is $q(X)$ a chosen irreducible representative back inside $\\mathcal Z$. A section or reflective localization would be extra data. The quotient also need not inherit the constructors automatically, as \\cref{ex:quotient-collapse} will show.","\\begin{example}[Why thick closure is genuinely necessary]\\label{ex:killable-not-thick}\nLet $\\mathcal C$ be any nonzero stable category and set\n\\[\n\\mathcal Z=\\mathcal C^3.\n\\]\nDefine exact idempotent endofunctors\n\\[\nU(X_1,X_2,X_3)=(0,0,X_2\\oplus X_3),\n\\qquad\nV(X_1,X_2,X_3)=(0,0,X_1\\oplus X_3).\n\\]\nThey satisfy\n\\[\nU^2\\simeq U,\n\\qquad\nV^2\\simeq V,\n\\qquad\nU\\circ V\\simeq V,\n\\qquad\nV\\circ U\\simeq U,\n\\]\nso $\\mathscr M=\\{\\id,U,V\\}$ is closed under composition. Choose $0\\not\\simeq A\\in\\mathcal C$ and set\n\\[\nX=(A,0,0),\n\\qquad\nY=(0,A,0).\n\\]\nThen $U(X)\\simeq0$ and $V(Y)\\simeq0$, so both $X$ and $Y$ are operationally killable. However\n\\[\nX\\oplus Y=(A,A,0),\n\\]\nand both $U$ and $V$ send this object to $(0,0,A)$, which is fixed by $U$ and $V$. Hence no constructor word kills $X\\oplus Y$. Thus $\\mathcal K_{\\ma","\\begin{example}[Why thick closure is genuinely necessary]\\label{ex:killable-not-thick}\nLet $\\mathcal C$ be any nonzero stable category and set\n\\[\n\\mathcal Z=\\mathcal C^3.\n\\]\nDefine exact idempotent endofunctors\n\\[\nU(X_1,X_2,X_3)=(0,0,X_2\\oplus X_3),\n\\qquad\nV(X_1,X_2,X_3)=(0,0,X_1\\oplus X_3).\n\\]\nThey satisfy\n\\[\nU^2\\simeq U,\n\\qquad\nV^2\\simeq V,\n\\qquad\nU\\circ V\\simeq V,\n\\qquad\nV\\circ U\\simeq U,\n\\]\nso $\\mathscr M=\\{\\id,U,V\\}$ is closed under composition. Choose $0\\not\\simeq A\\in\\mathcal C$ and set\n\\[\nX=(A,0,0),\n\\qquad\nY=(0,A,0).\n\\]\nThen $U(X)\\simeq0$ and $V(Y)\\simeq0$, so both $X$ and $Y$ are operationally killable. However\n\\[\nX\\oplus Y=(A,A,0),\n\\]\nand both $U$ and $V$ send this object to $(0,0,A)$, which is fixed by $U$ and $V$. Hence no constructor word kills $X\\oplus Y$. Thus $\\mathcal K_{\\mathscr M}$ need not be additive, much less thick.\n\\end{example}","egin{example}[Why thick closure is genuinely necessary]\\label{ex:killable-not-thick}\nLet $\\mathcal C$ be any nonzero stable category and set\n\\[\n\\mathcal Z=\\mathcal C^3.\n\\]\nDefine exact idempotent endofunctors\n\\[\nU(X_1,X_2,X_3)=(0,0,X_2\\oplus X_3),\n\\qquad\nV(X_1,X_2,X_3)=(0,0,X_1\\oplus X_3).\n\\]\nThey satisfy\n\\[\nU^2\\simeq U,\n\\qquad\nV^2\\simeq V,\n\\qquad\nU\\circ V\\simeq V,\n\\qquad\nV\\circ U\\simeq U,\n\\]\nso $\\mathscr M=\\{\\id,U,V\\}$ is closed under composition. Choose $0\\not\\simeq A\\in\\mathcal C$ and set\n\\[\nX=(A,0,0),\n\\qquad\nY=(0,A,0).\n\\]\nThen $U(X)\\simeq0$ and $V(Y)\\simeq0$, so both $X$ and $Y$ are operationally killable. However\n\\[\nX\\oplus Y=(A,A,0),\n\\]\nand both $U$ and $V$ send this object to $(0,0,A)$, which is fixed by $U$ and $V$. Hence no constructor word kills $X\\oplus Y$. Thus $\\mathcal K_{\\mathscr M}$ need not be additive, much less thick.\n\\end{example}","{ex:killable-not-thick}\nLet $\\mathcal C$ be any nonzero stable category and set\n\\[\n\\mathcal Z=\\mathcal C^3.\n\\]\nDefine exact idempotent endofunctors\n\\[\nU(X_1,X_2,X_3)=(0,0,X_2\\oplus X_3),\n\\qquad\nV(X_1,X_2,X_3)=(0,0,X_1\\oplus X_3).\n\\]\nThey satisfy\n\\[\nU^2\\simeq U,\n\\qquad\nV^2\\simeq V,\n\\qquad\nU\\circ V\\simeq V,\n\\qquad\nV\\circ U\\simeq U,\n\\]\nso $\\mathscr M=\\{\\id,U,V\\}$ is closed under composition. Choose $0\\not\\simeq A\\in\\mathcal C$ and set\n\\[\nX=(A,0,0),\n\\qquad\nY=(0,A,0).\n\\]\nThen $U(X)\\simeq0$ and $V(Y)\\simeq0$, so both $X$ and $Y$ are operationally killable. However\n\\[\nX\\oplus Y=(A,A,0),\n\\]\nand both $U$ and $V$ send this object to $(0,0,A)$, which is fixed by $U$ and $V$. Hence no constructor word kills $X\\oplus Y$. Thus $\\mathcal K_{\\mathscr M}$ need not be additive, much less thick.\n\\end{example}","C$ be any nonzero stable category and set\n\\[\n\\mathcal Z=\\mathcal C^3.\n\\]\nDefine exact idempotent endofunctors\n\\[\nU(X_1,X_2,X_3)=(0,0,X_2\\oplus X_3),\n\\qquad\nV(X_1,X_2,X_3)=(0,0,X_1\\oplus X_3).\n\\]\nThey satisfy\n\\[\nU^2\\simeq U,\n\\qquad\nV^2\\simeq V,\n\\qquad\nU\\circ V\\simeq V,\n\\qquad\nV\\circ U\\simeq U,\n\\]\nso $\\mathscr M=\\{\\id,U,V\\}$ is closed under composition. Choose $0\\not\\simeq A\\in\\mathcal C$ and set\n\\[\nX=(A,0,0),\n\\qquad\nY=(0,A,0).\n\\]\nThen $U(X)\\simeq0$ and $V(Y)\\simeq0$, so both $X$ and $Y$ are operationally killable. However\n\\[\nX\\oplus Y=(A,A,0),\n\\]\nand both $U$ and $V$ send this object to $(0,0,A)$, which is fixed by $U$ and $V$. Hence no constructor word kills $X\\oplus Y$. Thus $\\mathcal K_{\\mathscr M}$ need not be additive, much less thick.\n\\end{example}","et\n\\[\n\\mathcal Z=\\mathcal C^3.\n\\]\nDefine exact idempotent endofunctors\n\\[\nU(X_1,X_2,X_3)=(0,0,X_2\\oplus X_3),\n\\qquad\nV(X_1,X_2,X_3)=(0,0,X_1\\oplus X_3).\n\\]\nThey satisfy\n\\[\nU^2\\simeq U,\n\\qquad\nV^2\\simeq V,\n\\qquad\nU\\circ V\\simeq V,\n\\qquad\nV\\circ U\\simeq U,\n\\]\nso $\\mathscr M=\\{\\id,U,V\\}$ is closed under composition. Choose $0\\not\\simeq A\\in\\mathcal C$ and set\n\\[\nX=(A,0,0),\n\\qquad\nY=(0,A,0).\n\\]\nThen $U(X)\\simeq0$ and $V(Y)\\simeq0$, so both $X$ and $Y$ are operationally killable. However\n\\[\nX\\oplus Y=(A,A,0),\n\\]\nand both $U$ and $V$ send this object to $(0,0,A)$, which is fixed by $U$ and $V$. Hence no constructor word kills $X\\oplus Y$. Thus $\\mathcal K_{\\mathscr M}$ need not be additive, much less thick.\n\\end{example}","mathcal C^3.\n\\]\nDefine exact idempotent endofunctors\n\\[\nU(X_1,X_2,X_3)=(0,0,X_2\\oplus X_3),\n\\qquad\nV(X_1,X_2,X_3)=(0,0,X_1\\oplus X_3).\n\\]\nThey satisfy\n\\[\nU^2\\simeq U,\n\\qquad\nV^2\\simeq V,\n\\qquad\nU\\circ V\\simeq V,\n\\qquad\nV\\circ U\\simeq U,\n\\]\nso $\\mathscr M=\\{\\id,U,V\\}$ is closed under composition. Choose $0\\not\\simeq A\\in\\mathcal C$ and set\n\\[\nX=(A,0,0),\n\\qquad\nY=(0,A,0).\n\\]\nThen $U(X)\\simeq0$ and $V(Y)\\simeq0$, so both $X$ and $Y$ are operationally killable. However\n\\[\nX\\oplus Y=(A,A,0),\n\\]\nand both $U$ and $V$ send this object to $(0,0,A)$, which is fixed by $U$ and $V$. Hence no constructor word kills $X\\oplus Y$. Thus $\\mathcal K_{\\mathscr M}$ need not be additive, much less thick.\n\\end{example}","exact idempotent endofunctors\n\\[\nU(X_1,X_2,X_3)=(0,0,X_2\\oplus X_3),\n\\qquad\nV(X_1,X_2,X_3)=(0,0,X_1\\oplus X_3).\n\\]\nThey satisfy\n\\[\nU^2\\simeq U,\n\\qquad\nV^2\\simeq V,\n\\qquad\nU\\circ V\\simeq V,\n\\qquad\nV\\circ U\\simeq U,\n\\]\nso $\\mathscr M=\\{\\id,U,V\\}$ is closed under composition. Choose $0\\not\\simeq A\\in\\mathcal C$ and set\n\\[\nX=(A,0,0),\n\\qquad\nY=(0,A,0).\n\\]\nThen $U(X)\\simeq0$ and $V(Y)\\simeq0$, so both $X$ and $Y$ are operationally killable. However\n\\[\nX\\oplus Y=(A,A,0),\n\\]\nand both $U$ and $V$ send this object to $(0,0,A)$, which is fixed by $U$ and $V$. Hence no constructor word kills $X\\oplus Y$. Thus $\\mathcal K_{\\mathscr M}$ need not be additive, much less thick.\n\\end{example}","empotent endofunctors\n\\[\nU(X_1,X_2,X_3)=(0,0,X_2\\oplus X_3),\n\\qquad\nV(X_1,X_2,X_3)=(0,0,X_1\\oplus X_3).\n\\]\nThey satisfy\n\\[\nU^2\\simeq U,\n\\qquad\nV^2\\simeq V,\n\\qquad\nU\\circ V\\simeq V,\n\\qquad\nV\\circ U\\simeq U,\n\\]\nso $\\mathscr M=\\{\\id,U,V\\}$ is closed under composition. Choose $0\\not\\simeq A\\in\\mathcal C$ and set\n\\[\nX=(A,0,0),\n\\qquad\nY=(0,A,0).\n\\]\nThen $U(X)\\simeq0$ and $V(Y)\\simeq0$, so both $X$ and $Y$ are operationally killable. However\n\\[\nX\\oplus Y=(A,A,0),\n\\]\nand both $U$ and $V$ send this object to $(0,0,A)$, which is fixed by $U$ and $V$. Hence no constructor word kills $X\\oplus Y$. Thus $\\mathcal K_{\\mathscr M}$ need not be additive, much less thick.\n\\end{example}","0,0,X_2\\oplus X_3),\n\\qquad\nV(X_1,X_2,X_3)=(0,0,X_1\\oplus X_3).\n\\]\nThey satisfy\n\\[\nU^2\\simeq U,\n\\qquad\nV^2\\simeq V,\n\\qquad\nU\\circ V\\simeq V,\n\\qquad\nV\\circ U\\simeq U,\n\\]\nso $\\mathscr M=\\{\\id,U,V\\}$ is closed under composition. Choose $0\\not\\simeq A\\in\\mathcal C$ and set\n\\[\nX=(A,0,0),\n\\qquad\nY=(0,A,0).\n\\]\nThen $U(X)\\simeq0$ and $V(Y)\\simeq0$, so both $X$ and $Y$ are operationally killable. However\n\\[\nX\\oplus Y=(A,A,0),\n\\]\nand both $U$ and $V$ send this object to $(0,0,A)$, which is fixed by $U$ and $V$. Hence no constructor word kills $X\\oplus Y$. Thus $\\mathcal K_{\\mathscr M}$ need not be additive, much less thick.\n\\end{example}","X_2,X_3)=(0,0,X_1\\oplus X_3).\n\\]\nThey satisfy\n\\[\nU^2\\simeq U,\n\\qquad\nV^2\\simeq V,\n\\qquad\nU\\circ V\\simeq V,\n\\qquad\nV\\circ U\\simeq U,\n\\]\nso $\\mathscr M=\\{\\id,U,V\\}$ is closed under composition. Choose $0\\not\\simeq A\\in\\mathcal C$ and set\n\\[\nX=(A,0,0),\n\\qquad\nY=(0,A,0).\n\\]\nThen $U(X)\\simeq0$ and $V(Y)\\simeq0$, so both $X$ and $Y$ are operationally killable. However\n\\[\nX\\oplus Y=(A,A,0),\n\\]\nand both $U$ and $V$ send this object to $(0,0,A)$, which is fixed by $U$ and $V$. Hence no constructor word kills $X\\oplus Y$. Thus $\\mathcal K_{\\mathscr M}$ need not be additive, much less thick.\n\\end{example}","=(0,0,X_1\\oplus X_3).\n\\]\nThey satisfy\n\\[\nU^2\\simeq U,\n\\qquad\nV^2\\simeq V,\n\\qquad\nU\\circ V\\simeq V,\n\\qquad\nV\\circ U\\simeq U,\n\\]\nso $\\mathscr M=\\{\\id,U,V\\}$ is closed under composition. Choose $0\\not\\simeq A\\in\\mathcal C$ and set\n\\[\nX=(A,0,0),\n\\qquad\nY=(0,A,0).\n\\]\nThen $U(X)\\simeq0$ and $V(Y)\\simeq0$, so both $X$ and $Y$ are operationally killable. However\n\\[\nX\\oplus Y=(A,A,0),\n\\]\nand both $U$ and $V$ send this object to $(0,0,A)$, which is fixed by $U$ and $V$. Hence no constructor word kills $X\\oplus Y$. Thus $\\mathcal K_{\\mathscr M}$ need not be additive, much less thick.\n\\end{example}","They satisfy\n\\[\nU^2\\simeq U,\n\\qquad\nV^2\\simeq V,\n\\qquad\nU\\circ V\\simeq V,\n\\qquad\nV\\circ U\\simeq U,\n\\]\nso $\\mathscr M=\\{\\id,U,V\\}$ is closed under composition. Choose $0\\not\\simeq A\\in\\mathcal C$ and set\n\\[\nX=(A,0,0),\n\\qquad\nY=(0,A,0).\n\\]\nThen $U(X)\\simeq0$ and $V(Y)\\simeq0$, so both $X$ and $Y$ are operationally killable. However\n\\[\nX\\oplus Y=(A,A,0),\n\\]\nand both $U$ and $V$ send this object to $(0,0,A)$, which is fixed by $U$ and $V$. Hence no constructor word kills $X\\oplus Y$. Thus $\\mathcal K_{\\mathscr M}$ need not be additive, much less thick.\n\\end{example}","\\qquad\nV^2\\simeq V,\n\\qquad\nU\\circ V\\simeq V,\n\\qquad\nV\\circ U\\simeq U,\n\\]\nso $\\mathscr M=\\{\\id,U,V\\}$ is closed under composition. Choose $0\\not\\simeq A\\in\\mathcal C$ and set\n\\[\nX=(A,0,0),\n\\qquad\nY=(0,A,0).\n\\]\nThen $U(X)\\simeq0$ and $V(Y)\\simeq0$, so both $X$ and $Y$ are operationally killable. However\n\\[\nX\\oplus Y=(A,A,0),\n\\]\nand both $U$ and $V$ send this object to $(0,0,A)$, which is fixed by $U$ and $V$. Hence no constructor word kills $X\\oplus Y$. Thus $\\mathcal K_{\\mathscr M}$ need not be additive, much less thick.\n\\end{example}","V^2\\simeq V,\n\\qquad\nU\\circ V\\simeq V,\n\\qquad\nV\\circ U\\simeq U,\n\\]\nso $\\mathscr M=\\{\\id,U,V\\}$ is closed under composition. Choose $0\\not\\simeq A\\in\\mathcal C$ and set\n\\[\nX=(A,0,0),\n\\qquad\nY=(0,A,0).\n\\]\nThen $U(X)\\simeq0$ and $V(Y)\\simeq0$, so both $X$ and $Y$ are operationally killable. However\n\\[\nX\\oplus Y=(A,A,0),\n\\]\nand both $U$ and $V$ send this object to $(0,0,A)$, which is fixed by $U$ and $V$. Hence no constructor word kills $X\\oplus Y$. Thus $\\mathcal K_{\\mathscr M}$ need not be additive, much less thick.\n\\end{example}","\\qquad\nV\\circ U\\simeq U,\n\\]\nso $\\mathscr M=\\{\\id,U,V\\}$ is closed under composition. Choose $0\\not\\simeq A\\in\\mathcal C$ and set\n\\[\nX=(A,0,0),\n\\qquad\nY=(0,A,0).\n\\]\nThen $U(X)\\simeq0$ and $V(Y)\\simeq0$, so both $X$ and $Y$ are operationally killable. However\n\\[\nX\\oplus Y=(A,A,0),\n\\]\nand both $U$ and $V$ send this object to $(0,0,A)$, which is fixed by $U$ and $V$. Hence no constructor word kills $X\\oplus Y$. Thus $\\mathcal K_{\\mathscr M}$ need not be additive, much less thick.\n\\end{example}","imeq U,\n\\]\nso $\\mathscr M=\\{\\id,U,V\\}$ is closed under composition. Choose $0\\not\\simeq A\\in\\mathcal C$ and set\n\\[\nX=(A,0,0),\n\\qquad\nY=(0,A,0).\n\\]\nThen $U(X)\\simeq0$ and $V(Y)\\simeq0$, so both $X$ and $Y$ are operationally killable. However\n\\[\nX\\oplus Y=(A,A,0),\n\\]\nand both $U$ and $V$ send this object to $(0,0,A)$, which is fixed by $U$ and $V$. Hence no constructor word kills $X\\oplus Y$. Thus $\\mathcal K_{\\mathscr M}$ need not be additive, much less thick.\n\\end{example}","This example shows that replacing $\\mathcal K_{\\mathscr M}$ by its thick closure in \\cref{def:obstruction-quotient} is forced if the quotient is required to be universal for exact functors.","\\begin{definition}[Directed annihilator property]\nThe constructor system $\\mathscr M$ has the \\emph{directed annihilator property} if for every $U,V\\in\\mathscr M$ there exist $P,Q,W\\in\\mathscr M$ and natural equivalences\n\\[\nW\\simeq P\\circ U\\simeq Q\\circ V.\n\\]\nThus any two constructor words admit a common postcomposition. With multiplication written as composition, $P\\circ U=Q\\circ V$ is a common left multiple. The displayed equation, rather than a left/right name, fixes the convention.\n\\end{definition}","\\begin{theorem}[When operational killability is already thick]\\label{thm:killable-thick}\nIf $\\mathscr M$ consists of exact endofunctors and has the directed annihilator property, then $\\mathcal K_{\\mathscr M}$ is a thick stable subcategory. Hence\n\\[\n\\mathcal N_{\\mathscr M}=\\mathcal K_{\\mathscr M}\n\\]\nand\n\\[\n\\boxed{\n\\Obs_{\\mathscr M}(X)\\simeq0\n\\iff\nX\\text{ is operationally killable}.}\n\\]\n\\end{theorem}","\\begin{proof}\nThe zero object is killed by every exact constructor. If $U(X)\\simeq0$, exactness gives $U(X[n])\\simeq U(X)[n]\\simeq0$ for every integer $n$, so the class is closed under both suspensions and desuspensions. It is closed under retracts because exact functors preserve retract diagrams.","For cofibers, suppose $X$ is killed by $U$ and $Y$ is killed by $V$. Choose a common postcomposition\n\\[\nW\\simeq P\\circ U\\simeq Q\\circ V.\n\\]\nThen $W(X)\\simeq0\\simeq W(Y)$. For any morphism $f:X\\to Y$, exactness gives\n\\[\nW(\\Cofib(f))\\simeq\\Cofib(Wf)\\simeq\\Cofib(0\\to0)\\simeq0.\n\\]\nThus the killable objects form a stable subcategory closed under retracts, hence a thick subcategory.\n\\end{proof}","\\begin{example}[Idempotent projection model]\nLet $\\mathcal Z=\\mathcal A\\times\\mathcal B$ for stable categories $\\mathcal A,\\mathcal B$, and let\n\\[\nL(a,b)=(a,0).\n\\]\nWith constructor monoid generated by the idempotent exact functor $L$, an object is operationally killable exactly when it lies in $0\\times\\mathcal B$. The obstruction quotient is equivalent to $\\mathcal A$, and\n\\[\n\\Obs(a,b)\\simeq a.\n\\]\nThus the quotient literally retains the component that cannot be annihilated by the admissible constructor.\n\\end{example}","\\begin{theorem}[Functorial transport of obstruction]\\label{thm:obstruction-transport}\nLet $(\\mathcal Z_A,\\mathscr M_A)$ and $(\\mathcal Z_B,\\mathscr M_B)$ be stable obstruction systems, and let\n\\[\n\\Lambda:\\mathcal Z_A\\longrightarrow\\mathcal Z_B\n\\]\nbe exact. If\n\\[\n\\Lambda(\\mathcal K_{\\mathscr M_A})\\subseteq\\mathcal N_{\\mathscr M_B},\n\\]\nthen $\\Lambda$ induces an exact functor\n\\[\n\\bar\\Lambda:\\mathcal O_{\\mathscr M_A}\\longrightarrow\\mathcal O_{\\mathscr M_B}.\n\\]\nIf $\\Lambda$ is an equivalence and carries $\\mathcal N_{\\mathscr M_A}$ onto $\\mathcal N_{\\mathscr M_B}$, then $\\bar\\Lambda$ is an equivalence. In that case obstruction vanishing is preserved and reflected.\n\\end{theorem}","struction-transport}\nLet $(\\mathcal Z_A,\\mathscr M_A)$ and $(\\mathcal Z_B,\\mathscr M_B)$ be stable obstruction systems, and let\n\\[\n\\Lambda:\\mathcal Z_A\\longrightarrow\\mathcal Z_B\n\\]\nbe exact. If\n\\[\n\\Lambda(\\mathcal K_{\\mathscr M_A})\\subseteq\\mathcal N_{\\mathscr M_B},\n\\]\nthen $\\Lambda$ induces an exact functor\n\\[\n\\bar\\Lambda:\\mathcal O_{\\mathscr M_A}\\longrightarrow\\mathcal O_{\\mathscr M_B}.\n\\]\nIf $\\Lambda$ is an equivalence and carries $\\mathcal N_{\\mathscr M_A}$ onto $\\mathcal N_{\\mathscr M_B}$, then $\\bar\\Lambda$ is an equivalence. In that case obstruction vanishing is preserved and reflected.\n\\end{theorem}","\\mathscr M_A)$ and $(\\mathcal Z_B,\\mathscr M_B)$ be stable obstruction systems, and let\n\\[\n\\Lambda:\\mathcal Z_A\\longrightarrow\\mathcal Z_B\n\\]\nbe exact. If\n\\[\n\\Lambda(\\mathcal K_{\\mathscr M_A})\\subseteq\\mathcal N_{\\mathscr M_B},\n\\]\nthen $\\Lambda$ induces an exact functor\n\\[\n\\bar\\Lambda:\\mathcal O_{\\mathscr M_A}\\longrightarrow\\mathcal O_{\\mathscr M_B}.\n\\]\nIf $\\Lambda$ is an equivalence and carries $\\mathcal N_{\\mathscr M_A}$ onto $\\mathcal N_{\\mathscr M_B}$, then $\\bar\\Lambda$ is an equivalence. In that case obstruction vanishing is preserved and reflected.\n\\end{theorem}",",\\mathscr M_B)$ be stable obstruction systems, and let\n\\[\n\\Lambda:\\mathcal Z_A\\longrightarrow\\mathcal Z_B\n\\]\nbe exact. If\n\\[\n\\Lambda(\\mathcal K_{\\mathscr M_A})\\subseteq\\mathcal N_{\\mathscr M_B},\n\\]\nthen $\\Lambda$ induces an exact functor\n\\[\n\\bar\\Lambda:\\mathcal O_{\\mathscr M_A}\\longrightarrow\\mathcal O_{\\mathscr M_B}.\n\\]\nIf $\\Lambda$ is an equivalence and carries $\\mathcal N_{\\mathscr M_A}$ onto $\\mathcal N_{\\mathscr M_B}$, then $\\bar\\Lambda$ is an equivalence. In that case obstruction vanishing is preserved and reflected.\n\\end{theorem}","ion systems, and let\n\\[\n\\Lambda:\\mathcal Z_A\\longrightarrow\\mathcal Z_B\n\\]\nbe exact. If\n\\[\n\\Lambda(\\mathcal K_{\\mathscr M_A})\\subseteq\\mathcal N_{\\mathscr M_B},\n\\]\nthen $\\Lambda$ induces an exact functor\n\\[\n\\bar\\Lambda:\\mathcal O_{\\mathscr M_A}\\longrightarrow\\mathcal O_{\\mathscr M_B}.\n\\]\nIf $\\Lambda$ is an equivalence and carries $\\mathcal N_{\\mathscr M_A}$ onto $\\mathcal N_{\\mathscr M_B}$, then $\\bar\\Lambda$ is an equivalence. In that case obstruction vanishing is preserved and reflected.\n\\end{theorem}","\\begin{proof}\nThe hypothesis and exactness imply\n\\[\n\\Lambda(\\mathcal N_{\\mathscr M_A})\\subseteq\\mathcal N_{\\mathscr M_B},\n\\]\nbecause exact functors preserve the operations used to generate a thick subcategory. Therefore $q_B\\Lambda$ kills $\\mathcal N_{\\mathscr M_A}$, so the universal property in \\cref{thm:obstruction-quotient} gives $\\bar\\Lambda$. If $\\Lambda$ is an equivalence identifying the two thick kernels, a quasi-inverse descends in the same way and yields a quasi-inverse to $\\bar\\Lambda$.\n\\end{proof}","ubseteq\\mathcal N_{\\mathscr M_B},\n\\]\nbecause exact functors preserve the operations used to generate a thick subcategory. Therefore $q_B\\Lambda$ kills $\\mathcal N_{\\mathscr M_A}$, so the universal property in \\cref{thm:obstruction-quotient} gives $\\bar\\Lambda$. If $\\Lambda$ is an equivalence identifying the two thick kernels, a quasi-inverse descends in the same way and yields a quasi-inverse to $\\bar\\Lambda$.\n\\end{proof}","\\begin{theorem}[Obstruction localization along normalization]\\label{thm:obstruction-localization}\nLet\n\\[\nX_0\\xrightarrow{f_1}X_1\\xrightarrow{}\\cdots\\xrightarrow{f_m}X_m\n\\]\nbe a normalization path as in \\cref{thm:stable-filtration}, with elementary transported residuals $D_i=\\Cofib(f_i)$ and total residual\n\\[\nD_{\\mathrm{tot}}=\\Cofib(X_0\\to X_m).\n\\]\nLet\n\\[\nq:\\mathcal Z\\to\\mathcal O_{\\mathscr M}\n\\]\nbe the stable obstruction quotient. Then:\n\\begin{enumerate}\n    \\item $q(D_{\\mathrm{tot}})$ inherits a finite filtration whose successive cofibers are $q(D_i)$;\n    \\item if $q(D_i)\\simeq0$ for every $i$, then\n    \\[\n    q(D_{\\mathrm{tot}})\\simeq0;\n    \\]\n    \\item equivalently,\n    \\[\n    q(D_{\\mathrm{tot}})\\not\\simeq0\n    \\Longrightarrow\n    \\exists i,\\;q(D_i)\\not\\simeq0.\n    \\]\n\\end{enumerate}\nIf the directed annihilator property of \\cref{thm:killable-thick} hold","\\begin{theorem}[Obstruction localization along normalization]\\label{thm:obstruction-localization}\nLet\n\\[\nX_0\\xrightarrow{f_1}X_1\\xrightarrow{}\\cdots\\xrightarrow{f_m}X_m\n\\]\nbe a normalization path as in \\cref{thm:stable-filtration}, with elementary transported residuals $D_i=\\Cofib(f_i)$ and total residual\n\\[\nD_{\\mathrm{tot}}=\\Cofib(X_0\\to X_m).\n\\]\nLet\n\\[\nq:\\mathcal Z\\to\\mathcal O_{\\mathscr M}\n\\]\nbe the stable obstruction quotient. Then:\n\\begin{enumerate}\n    \\item $q(D_{\\mathrm{tot}})$ inherits a finite filtration whose successive cofibers are $q(D_i)$;\n    \\item if $q(D_i)\\simeq0$ for every $i$, then\n    \\[\n    q(D_{\\mathrm{tot}})\\simeq0;\n    \\]\n    \\item equivalently,\n    \\[\n    q(D_{\\mathrm{tot}})\\not\\simeq0\n    \\Longrightarrow\n    \\exists i,\\;q(D_i)\\not\\simeq0.\n    \\]\n\\end{enumerate}\nIf the directed annihilator property of \\cref{thm:killable-thick} holds, the phrase ``$q(D)\\simeq0$'' can be replaced throughout by ``$D$ is operationally killable.''\n\\end{theorem}","m:obstruction-localization}\nLet\n\\[\nX_0\\xrightarrow{f_1}X_1\\xrightarrow{}\\cdots\\xrightarrow{f_m}X_m\n\\]\nbe a normalization path as in \\cref{thm:stable-filtration}, with elementary transported residuals $D_i=\\Cofib(f_i)$ and total residual\n\\[\nD_{\\mathrm{tot}}=\\Cofib(X_0\\to X_m).\n\\]\nLet\n\\[\nq:\\mathcal Z\\to\\mathcal O_{\\mathscr M}\n\\]\nbe the stable obstruction quotient. Then:\n\\begin{enumerate}\n    \\item $q(D_{\\mathrm{tot}})$ inherits a finite filtration whose successive cofibers are $q(D_i)$;\n    \\item if $q(D_i)\\simeq0$ for every $i$, then\n    \\[\n    q(D_{\\mathrm{tot}})\\simeq0;\n    \\]\n    \\item equivalently,\n    \\[\n    q(D_{\\mathrm{tot}})\\not\\simeq0\n    \\Longrightarrow\n    \\exists i,\\;q(D_i)\\not\\simeq0.\n    \\]\n\\end{enumerate}\nIf the directed annihilator property of \\cref{thm:killable-thick} holds, the phrase ``$q(D)\\simeq0$'' can be replaced throughout by ``$D$ is operationally killable.''\n\\end{theorem}","ightarrow{f_m}X_m\n\\]\nbe a normalization path as in \\cref{thm:stable-filtration}, with elementary transported residuals $D_i=\\Cofib(f_i)$ and total residual\n\\[\nD_{\\mathrm{tot}}=\\Cofib(X_0\\to X_m).\n\\]\nLet\n\\[\nq:\\mathcal Z\\to\\mathcal O_{\\mathscr M}\n\\]\nbe the stable obstruction quotient. Then:\n\\begin{enumerate}\n    \\item $q(D_{\\mathrm{tot}})$ inherits a finite filtration whose successive cofibers are $q(D_i)$;\n    \\item if $q(D_i)\\simeq0$ for every $i$, then\n    \\[\n    q(D_{\\mathrm{tot}})\\simeq0;\n    \\]\n    \\item equivalently,\n    \\[\n    q(D_{\\mathrm{tot}})\\not\\simeq0\n    \\Longrightarrow\n    \\exists i,\\;q(D_i)\\not\\simeq0.\n    \\]\n\\end{enumerate}\nIf the directed annihilator property of \\cref{thm:killable-thick} holds, the phrase ``$q(D)\\simeq0$'' can be replaced throughout by ``$D$ is operationally killable.''\n\\end{theorem}","a normalization path as in \\cref{thm:stable-filtration}, with elementary transported residuals $D_i=\\Cofib(f_i)$ and total residual\n\\[\nD_{\\mathrm{tot}}=\\Cofib(X_0\\to X_m).\n\\]\nLet\n\\[\nq:\\mathcal Z\\to\\mathcal O_{\\mathscr M}\n\\]\nbe the stable obstruction quotient. Then:\n\\begin{enumerate}\n    \\item $q(D_{\\mathrm{tot}})$ inherits a finite filtration whose successive cofibers are $q(D_i)$;\n    \\item if $q(D_i)\\simeq0$ for every $i$, then\n    \\[\n    q(D_{\\mathrm{tot}})\\simeq0;\n    \\]\n    \\item equivalently,\n    \\[\n    q(D_{\\mathrm{tot}})\\not\\simeq0\n    \\Longrightarrow\n    \\exists i,\\;q(D_i)\\not\\simeq0.\n    \\]\n\\end{enumerate}\nIf the directed annihilator property of \\cref{thm:killable-thick} holds, the phrase ``$q(D)\\simeq0$'' can be replaced throughout by ``$D$ is operationally killable.''\n\\end{theorem}","n \\cref{thm:stable-filtration}, with elementary transported residuals $D_i=\\Cofib(f_i)$ and total residual\n\\[\nD_{\\mathrm{tot}}=\\Cofib(X_0\\to X_m).\n\\]\nLet\n\\[\nq:\\mathcal Z\\to\\mathcal O_{\\mathscr M}\n\\]\nbe the stable obstruction quotient. Then:\n\\begin{enumerate}\n    \\item $q(D_{\\mathrm{tot}})$ inherits a finite filtration whose successive cofibers are $q(D_i)$;\n    \\item if $q(D_i)\\simeq0$ for every $i$, then\n    \\[\n    q(D_{\\mathrm{tot}})\\simeq0;\n    \\]\n    \\item equivalently,\n    \\[\n    q(D_{\\mathrm{tot}})\\not\\simeq0\n    \\Longrightarrow\n    \\exists i,\\;q(D_i)\\not\\simeq0.\n    \\]\n\\end{enumerate}\nIf the directed annihilator property of \\cref{thm:killable-thick} holds, the phrase ``$q(D)\\simeq0$'' can be replaced throughout by ``$D$ is operationally killable.''\n\\end{theorem}","ble-filtration}, with elementary transported residuals $D_i=\\Cofib(f_i)$ and total residual\n\\[\nD_{\\mathrm{tot}}=\\Cofib(X_0\\to X_m).\n\\]\nLet\n\\[\nq:\\mathcal Z\\to\\mathcal O_{\\mathscr M}\n\\]\nbe the stable obstruction quotient. Then:\n\\begin{enumerate}\n    \\item $q(D_{\\mathrm{tot}})$ inherits a finite filtration whose successive cofibers are $q(D_i)$;\n    \\item if $q(D_i)\\simeq0$ for every $i$, then\n    \\[\n    q(D_{\\mathrm{tot}})\\simeq0;\n    \\]\n    \\item equivalently,\n    \\[\n    q(D_{\\mathrm{tot}})\\not\\simeq0\n    \\Longrightarrow\n    \\exists i,\\;q(D_i)\\not\\simeq0.\n    \\]\n\\end{enumerate}\nIf the directed annihilator property of \\cref{thm:killable-thick} holds, the phrase ``$q(D)\\simeq0$'' can be replaced throughout by ``$D$ is operationally killable.''\n\\end{theorem}",")$ and total residual\n\\[\nD_{\\mathrm{tot}}=\\Cofib(X_0\\to X_m).\n\\]\nLet\n\\[\nq:\\mathcal Z\\to\\mathcal O_{\\mathscr M}\n\\]\nbe the stable obstruction quotient. Then:\n\\begin{enumerate}\n    \\item $q(D_{\\mathrm{tot}})$ inherits a finite filtration whose successive cofibers are $q(D_i)$;\n    \\item if $q(D_i)\\simeq0$ for every $i$, then\n    \\[\n    q(D_{\\mathrm{tot}})\\simeq0;\n    \\]\n    \\item equivalently,\n    \\[\n    q(D_{\\mathrm{tot}})\\not\\simeq0\n    \\Longrightarrow\n    \\exists i,\\;q(D_i)\\not\\simeq0.\n    \\]\n\\end{enumerate}\nIf the directed annihilator property of \\cref{thm:killable-thick} holds, the phrase ``$q(D)\\simeq0$'' can be replaced throughout by ``$D$ is operationally killable.''\n\\end{theorem}","}})$ inherits a finite filtration whose successive cofibers are $q(D_i)$;\n    \\item if $q(D_i)\\simeq0$ for every $i$, then\n    \\[\n    q(D_{\\mathrm{tot}})\\simeq0;\n    \\]\n    \\item equivalently,\n    \\[\n    q(D_{\\mathrm{tot}})\\not\\simeq0\n    \\Longrightarrow\n    \\exists i,\\;q(D_i)\\not\\simeq0.\n    \\]\n\\end{enumerate}\nIf the directed annihilator property of \\cref{thm:killable-thick} holds, the phrase ``$q(D)\\simeq0$'' can be replaced throughout by ``$D$ is operationally killable.''\n\\end{theorem}","ve cofibers are $q(D_i)$;\n    \\item if $q(D_i)\\simeq0$ for every $i$, then\n    \\[\n    q(D_{\\mathrm{tot}})\\simeq0;\n    \\]\n    \\item equivalently,\n    \\[\n    q(D_{\\mathrm{tot}})\\not\\simeq0\n    \\Longrightarrow\n    \\exists i,\\;q(D_i)\\not\\simeq0.\n    \\]\n\\end{enumerate}\nIf the directed annihilator property of \\cref{thm:killable-thick} holds, the phrase ``$q(D)\\simeq0$'' can be replaced throughout by ``$D$ is operationally killable.''\n\\end{theorem}","\\begin{proof}\nThe residual filtration in \\cref{thm:stable-filtration} consists of cofiber sequences\n\\[\nF_{i-1}\\longrightarrow F_i\\longrightarrow D_i,\n\\qquad\nF_0\\simeq0,\n\\qquad\nF_m\\simeq D_{\\mathrm{tot}}.\n\\]\nBecause the Verdier quotient functor $q$ is exact, applying $q$ gives cofiber sequences\n\\[\nq(F_{i-1})\\longrightarrow q(F_i)\\longrightarrow q(D_i).\n\\]\nThis proves (1). If every $q(D_i)$ vanishes, then $q(F_{i-1})\\to q(F_i)$ is an equivalence for every $i$. Starting from $q(F_0)\\simeq0$ yields $q(F_m)\\simeq0$, proving (2). Statement (3) is its contrapositive. Under the directed annihilator hypothesis, \\cref{thm:killable-thick} identifies quotient-vanishing with operational killability.\n\\end{proof}","roof}\nThe residual filtration in \\cref{thm:stable-filtration} consists of cofiber sequences\n\\[\nF_{i-1}\\longrightarrow F_i\\longrightarrow D_i,\n\\qquad\nF_0\\simeq0,\n\\qquad\nF_m\\simeq D_{\\mathrm{tot}}.\n\\]\nBecause the Verdier quotient functor $q$ is exact, applying $q$ gives cofiber sequences\n\\[\nq(F_{i-1})\\longrightarrow q(F_i)\\longrightarrow q(D_i).\n\\]\nThis proves (1). If every $q(D_i)$ vanishes, then $q(F_{i-1})\\to q(F_i)$ is an equivalence for every $i$. Starting from $q(F_0)\\simeq0$ yields $q(F_m)\\simeq0$, proving (2). Statement (3) is its contrapositive. Under the directed annihilator hypothesis, \\cref{thm:killable-thick} identifies quotient-vanishing with operational killability.\n\\end{proof}","iltration} consists of cofiber sequences\n\\[\nF_{i-1}\\longrightarrow F_i\\longrightarrow D_i,\n\\qquad\nF_0\\simeq0,\n\\qquad\nF_m\\simeq D_{\\mathrm{tot}}.\n\\]\nBecause the Verdier quotient functor $q$ is exact, applying $q$ gives cofiber sequences\n\\[\nq(F_{i-1})\\longrightarrow q(F_i)\\longrightarrow q(D_i).\n\\]\nThis proves (1). If every $q(D_i)$ vanishes, then $q(F_{i-1})\\to q(F_i)$ is an equivalence for every $i$. Starting from $q(F_0)\\simeq0$ yields $q(F_m)\\simeq0$, proving (2). Statement (3) is its contrapositive. Under the directed annihilator hypothesis, \\cref{thm:killable-thick} identifies quotient-vanishing with operational killability.\n\\end{proof}","of cofiber sequences\n\\[\nF_{i-1}\\longrightarrow F_i\\longrightarrow D_i,\n\\qquad\nF_0\\simeq0,\n\\qquad\nF_m\\simeq D_{\\mathrm{tot}}.\n\\]\nBecause the Verdier quotient functor $q$ is exact, applying $q$ gives cofiber sequences\n\\[\nq(F_{i-1})\\longrightarrow q(F_i)\\longrightarrow q(D_i).\n\\]\nThis proves (1). If every $q(D_i)$ vanishes, then $q(F_{i-1})\\to q(F_i)$ is an equivalence for every $i$. Starting from $q(F_0)\\simeq0$ yields $q(F_m)\\simeq0$, proving (2). Statement (3) is its contrapositive. Under the directed annihilator hypothesis, \\cref{thm:killable-thick} identifies quotient-vanishing with operational killability.\n\\end{proof}","[\nF_{i-1}\\longrightarrow F_i\\longrightarrow D_i,\n\\qquad\nF_0\\simeq0,\n\\qquad\nF_m\\simeq D_{\\mathrm{tot}}.\n\\]\nBecause the Verdier quotient functor $q$ is exact, applying $q$ gives cofiber sequences\n\\[\nq(F_{i-1})\\longrightarrow q(F_i)\\longrightarrow q(D_i).\n\\]\nThis proves (1). If every $q(D_i)$ vanishes, then $q(F_{i-1})\\to q(F_i)$ is an equivalence for every $i$. Starting from $q(F_0)\\simeq0$ yields $q(F_m)\\simeq0$, proving (2). Statement (3) is its contrapositive. Under the directed annihilator hypothesis, \\cref{thm:killable-thick} identifies quotient-vanishing with operational killability.\n\\end{proof}","The universal property of $\\mathcal Z/\\mathcal N_{\\mathscr M}$ concerns exact functors that kill a prescribed class. It does not automatically preserve the full transformation grammar. To make the quotient a new environment for that grammar, the kernel must be invariant under the constructors.","\\begin{example}[A constructor can fail to descend, and invariant closure can collapse]\\label{ex:quotient-collapse}\nReturn to \\cref{ex:killable-not-thick}, with $\\mathcal Z=\\mathcal C^3$ and $\\mathscr M=\\{\\id,U,V\\}$. An object is killed by $U$ precisely when its second and third components vanish, and killed by $V$ precisely when its first and third components vanish. Indeed, a summand of a zero direct sum is zero. Consequently\n\\[\n\\mathcal N_{\\mathscr M}=\\mathcal C\\times\\mathcal C\\times0,\n\\qquad\n\\mathcal Z/\\mathcal N_{\\mathscr M}\\simeq\\mathcal C.\n\\]\nNevertheless\n\\[\nU(0,A,0)=(0,0,A)\n\\]\nfor every $A\\in\\mathcal C$. The input lies in $\\mathcal N_{\\mathscr M}$ and the output generally does not. If $U$ descended to an exact $\\bar U$ on the quotient, then\n$qU(0,A,0)\\simeq\\bar Uq(0,A,0)\\simeq0$, a contradiction for $A\\not\\simeq0$.","fail to descend, and invariant closure can collapse]\\label{ex:quotient-collapse}\nReturn to \\cref{ex:killable-not-thick}, with $\\mathcal Z=\\mathcal C^3$ and $\\mathscr M=\\{\\id,U,V\\}$. An object is killed by $U$ precisely when its second and third components vanish, and killed by $V$ precisely when its first and third components vanish. Indeed, a summand of a zero direct sum is zero. Consequently\n\\[\n\\mathcal N_{\\mathscr M}=\\mathcal C\\times\\mathcal C\\times0,\n\\qquad\n\\mathcal Z/\\mathcal N_{\\mathscr M}\\simeq\\mathcal C.\n\\]\nNevertheless\n\\[\nU(0,A,0)=(0,0,A)\n\\]\nfor every $A\\in\\mathcal C$. The input lies in $\\mathcal N_{\\mathscr M}$ and the output generally does not. If $U$ descended to an exact $\\bar U$ on the quotient, then\n$qU(0,A,0)\\simeq\\bar Uq(0,A,0)\\simeq0$, a contradiction for $A\\not\\simeq0$.","Z=\\mathcal C^3$ and $\\mathscr M=\\{\\id,U,V\\}$. An object is killed by $U$ precisely when its second and third components vanish, and killed by $V$ precisely when its first and third components vanish. Indeed, a summand of a zero direct sum is zero. Consequently\n\\[\n\\mathcal N_{\\mathscr M}=\\mathcal C\\times\\mathcal C\\times0,\n\\qquad\n\\mathcal Z/\\mathcal N_{\\mathscr M}\\simeq\\mathcal C.\n\\]\nNevertheless\n\\[\nU(0,A,0)=(0,0,A)\n\\]\nfor every $A\\in\\mathcal C$. The input lies in $\\mathcal N_{\\mathscr M}$ and the output generally does not. If $U$ descended to an exact $\\bar U$ on the quotient, then\n$qU(0,A,0)\\simeq\\bar Uq(0,A,0)\\simeq0$, a contradiction for $A\\not\\simeq0$.","\\{\\id,U,V\\}$. An object is killed by $U$ precisely when its second and third components vanish, and killed by $V$ precisely when its first and third components vanish. Indeed, a summand of a zero direct sum is zero. Consequently\n\\[\n\\mathcal N_{\\mathscr M}=\\mathcal C\\times\\mathcal C\\times0,\n\\qquad\n\\mathcal Z/\\mathcal N_{\\mathscr M}\\simeq\\mathcal C.\n\\]\nNevertheless\n\\[\nU(0,A,0)=(0,0,A)\n\\]\nfor every $A\\in\\mathcal C$. The input lies in $\\mathcal N_{\\mathscr M}$ and the output generally does not. If $U$ descended to an exact $\\bar U$ on the quotient, then\n$qU(0,A,0)\\simeq\\bar Uq(0,A,0)\\simeq0$, a contradiction for $A\\not\\simeq0$.","$U$ precisely when its second and third components vanish, and killed by $V$ precisely when its first and third components vanish. Indeed, a summand of a zero direct sum is zero. Consequently\n\\[\n\\mathcal N_{\\mathscr M}=\\mathcal C\\times\\mathcal C\\times0,\n\\qquad\n\\mathcal Z/\\mathcal N_{\\mathscr M}\\simeq\\mathcal C.\n\\]\nNevertheless\n\\[\nU(0,A,0)=(0,0,A)\n\\]\nfor every $A\\in\\mathcal C$. The input lies in $\\mathcal N_{\\mathscr M}$ and the output generally does not. If $U$ descended to an exact $\\bar U$ on the quotient, then\n$qU(0,A,0)\\simeq\\bar Uq(0,A,0)\\simeq0$, a contradiction for $A\\not\\simeq0$.","killed by $V$ precisely when its first and third components vanish. Indeed, a summand of a zero direct sum is zero. Consequently\n\\[\n\\mathcal N_{\\mathscr M}=\\mathcal C\\times\\mathcal C\\times0,\n\\qquad\n\\mathcal Z/\\mathcal N_{\\mathscr M}\\simeq\\mathcal C.\n\\]\nNevertheless\n\\[\nU(0,A,0)=(0,0,A)\n\\]\nfor every $A\\in\\mathcal C$. The input lies in $\\mathcal N_{\\mathscr M}$ and the output generally does not. If $U$ descended to an exact $\\bar U$ on the quotient, then\n$qU(0,A,0)\\simeq\\bar Uq(0,A,0)\\simeq0$, a contradiction for $A\\not\\simeq0$.","en its first and third components vanish. Indeed, a summand of a zero direct sum is zero. Consequently\n\\[\n\\mathcal N_{\\mathscr M}=\\mathcal C\\times\\mathcal C\\times0,\n\\qquad\n\\mathcal Z/\\mathcal N_{\\mathscr M}\\simeq\\mathcal C.\n\\]\nNevertheless\n\\[\nU(0,A,0)=(0,0,A)\n\\]\nfor every $A\\in\\mathcal C$. The input lies in $\\mathcal N_{\\mathscr M}$ and the output generally does not. If $U$ descended to an exact $\\bar U$ on the quotient, then\n$qU(0,A,0)\\simeq\\bar Uq(0,A,0)\\simeq0$, a contradiction for $A\\not\\simeq0$.","ts vanish. Indeed, a summand of a zero direct sum is zero. Consequently\n\\[\n\\mathcal N_{\\mathscr M}=\\mathcal C\\times\\mathcal C\\times0,\n\\qquad\n\\mathcal Z/\\mathcal N_{\\mathscr M}\\simeq\\mathcal C.\n\\]\nNevertheless\n\\[\nU(0,A,0)=(0,0,A)\n\\]\nfor every $A\\in\\mathcal C$. The input lies in $\\mathcal N_{\\mathscr M}$ and the output generally does not. If $U$ descended to an exact $\\bar U$ on the quotient, then\n$qU(0,A,0)\\simeq\\bar Uq(0,A,0)\\simeq0$, a contradiction for $A\\not\\simeq0$.","onsequently\n\\[\n\\mathcal N_{\\mathscr M}=\\mathcal C\\times\\mathcal C\\times0,\n\\qquad\n\\mathcal Z/\\mathcal N_{\\mathscr M}\\simeq\\mathcal C.\n\\]\nNevertheless\n\\[\nU(0,A,0)=(0,0,A)\n\\]\nfor every $A\\in\\mathcal C$. The input lies in $\\mathcal N_{\\mathscr M}$ and the output generally does not. If $U$ descended to an exact $\\bar U$ on the quotient, then\n$qU(0,A,0)\\simeq\\bar Uq(0,A,0)\\simeq0$, a contradiction for $A\\not\\simeq0$.","Moreover, any constructor-invariant thick subcategory containing all killable objects contains both the first and second coordinate subcategories. Applying $U$ to the second coordinate forces it to contain the third. Closure under direct sums then gives all of $\\mathcal Z$. Thus the constructor-compatible quotient in this example is the zero category.\n\\end{example}","\\begin{definition}[Constructor-invariant thick saturation]\\label{def:invariant-saturation}\nFor a small stable constructor system $(\\mathcal Z,\\mathscr M)$ as above, put\n\\[\n\\mathcal J_{\\mathscr M}\n:=\\operatorname{Thick}\\{W(K):W\\in\\mathscr M,\n\\ K\\in\\mathcal K_{\\mathscr M}\\}.\n\\]\nIts quotient is denoted\n\\[\nq^{\\mathrm{inv}}:\\mathcal Z\\longrightarrow\n\\mathcal O^{\\mathrm{inv}}_{\\mathscr M}\n:=\\mathcal Z/\\mathcal J_{\\mathscr M}.\n\\]\nThe identity constructor ensures $\\mathcal N_{\\mathscr M}\\subseteq\\mathcal J_{\\mathscr M}$.\n\\end{definition}","bel{def:invariant-saturation}\nFor a small stable constructor system $(\\mathcal Z,\\mathscr M)$ as above, put\n\\[\n\\mathcal J_{\\mathscr M}\n:=\\operatorname{Thick}\\{W(K):W\\in\\mathscr M,\n\\ K\\in\\mathcal K_{\\mathscr M}\\}.\n\\]\nIts quotient is denoted\n\\[\nq^{\\mathrm{inv}}:\\mathcal Z\\longrightarrow\n\\mathcal O^{\\mathrm{inv}}_{\\mathscr M}\n:=\\mathcal Z/\\mathcal J_{\\mathscr M}.\n\\]\nThe identity constructor ensures $\\mathcal N_{\\mathscr M}\\subseteq\\mathcal J_{\\mathscr M}$.\n\\end{definition}","\\begin{theorem}[Universal constructor-compatible obstruction quotient]\\label{thm:invariant-quotient}\nThe subcategory $\\mathcal J_{\\mathscr M}$ is the least thick subcategory that contains $\\mathcal K_{\\mathscr M}$ and is preserved by every constructor. Each $T\\in\\mathscr M$ descends to an exact endofunctor of $\\mathcal O^{\\mathrm{inv}}_{\\mathscr M}$. If the constructor action is supplied with coherent unit and composition data, those data descend as well.","The quotient is universal among exact functors that annihilate every object $W(K)$ in the displayed generating family. In particular, every exact realization that annihilates killable objects and intertwines the given constructors with exact constructors of its target factors through $q^{\\mathrm{inv}}$.\n\\end{theorem}","\\begin{proof}\nFix $T\\in\\mathscr M$. It takes a generator $W(K)$ to $(T\\circ W)(K)$, again a generator because $\\mathscr M$ is closed under composition. Exactness and preservation of retracts imply\n\\[\nT(\\mathcal J_{\\mathscr M})\\subseteq\\mathcal J_{\\mathscr M}.\n\\]\nConversely any invariant thick subcategory containing $\\mathcal K_{\\mathscr M}$ contains every $W(K)$ and therefore their thick closure. This proves minimality.","Now $q^{\\mathrm{inv}}T$ annihilates $\\mathcal J_{\\mathscr M}$, so the quotient universal property gives $\\bar T$ with\n\\[\n\\bar T q^{\\mathrm{inv}}\\simeq q^{\\mathrm{inv}}T.\n\\]\nThe full faithfulness of precomposition in the functor-category universal property descends any supplied unit, composition, and higher coherence data. It does not create action data absent from the original system.","The last universal property follows directly from thick closure and Verdier localization. If $E$ is equivariant and kills $K$, then\n$E(WK)\\simeq W'E(K)\\simeq W'(0)\\simeq0$. Thus it kills all the generators and factors through the quotient.\n\\end{proof}","\\begin{theorem}[Directed annihilators also give constructor invariance]\\label{thm:directed-invariant}\nUnder the directed annihilator hypothesis of \\cref{thm:killable-thick},\n\\[\n\\mathcal J_{\\mathscr M}=\\mathcal N_{\\mathscr M}\n=\\mathcal K_{\\mathscr M}.\n\\]\nIn particular the same quotient both supports the admitted constructors and detects operational killability exactly.\n\\end{theorem}","\\begin{proof}\nThe earlier theorem shows that $\\mathcal K_{\\mathscr M}$ is thick. It remains to prove invariance. Suppose $U(X)\\simeq0$ and fix a constructor $T$. Directedness supplies constructors $P,Q,W$ with\n\\[\nW\\simeq P\\circ U\\simeq Q\\circ T.\n\\]\nTherefore\n\\[\nQ(TX)\\simeq P(UX)\\simeq P(0)\\simeq0.\n\\]\nThus $TX$ is killable. The killable subcategory is invariant and thick, so minimality of $\\mathcal J_{\\mathscr M}$ gives the asserted equalities.\n\\end{proof}","\\begin{proposition}[A noncollapse certificate]\\label{prop:noncollapse}\nSuppose $E:\\mathcal Z\\to\\mathcal D$ is an exact equivariant realization into a stable constructor system, $E$ kills every operationally killable object, and $E(X)\\not\\simeq0$ for some $X$. Then\n\\[\nq^{\\mathrm{inv}}(X)\\not\\simeq0,\n\\qquad\n\\mathcal O^{\\mathrm{inv}}_{\\mathscr M}\\not\\simeq0.\n\\]\n\\end{proposition}","\\begin{proof}\nBy \\cref{thm:invariant-quotient}, write $E\\simeq\\bar E q^{\\mathrm{inv}}$. Exactness of $\\bar E$ would imply $E(X)\\simeq0$ if $q^{\\mathrm{inv}}(X)\\simeq0$, a contradiction.\n\\end{proof}","For the projection example $\\mathcal Z=\\mathcal A\\times\\mathcal B$ and $L(a,b)=(a,0)$, projection to $\\mathcal A$ is such a detector whenever $\\mathcal A$ is nonzero. This gives a nontrivial model, in contrast with \\cref{ex:quotient-collapse}. Admissibility matters: allowing the zero functor as a repair makes every object killable and necessarily collapses the quotient.","The refined distinction is therefore\n\\[\n\\begin{aligned}\n\\mathcal K_{\\mathscr M}&:\\ \\text{objects killed by a single admitted word},\\\\\n\\mathcal N_{\\mathscr M}&:\\ \\text{their thick closure},\\\\\n\\mathcal J_{\\mathscr M}&:\\ \\text{their constructor-invariant thick closure}.\n\\end{aligned}\n\\]\nOnly under additional hypotheses can these three classes be identified.","Let $\\mathcal Z$ be a small stable category, let $\\mathscr M$ be a coherently acting monoid of exact endofunctors, and let $\\mathscr X$ be a small family of finite coherent event diagrams in $\\mathcal Z$. Denote by $\\mathscr H$ the family of interaction residuals of all their elementary independent-event squares. Put\n\\[\n \\mathcal J_{\\square}\n =\\operatorname{Thick}\\{T(H):T\\in\\mathscr M,\\ H\\in\\mathscr H\\}.\n\\]\nThis notation does not assert that $H$ was operationally killable before localization.","t $\\mathscr M$ be a coherently acting monoid of exact endofunctors, and let $\\mathscr X$ be a small family of finite coherent event diagrams in $\\mathcal Z$. Denote by $\\mathscr H$ the family of interaction residuals of all their elementary independent-event squares. Put\n\\[\n \\mathcal J_{\\square}\n =\\operatorname{Thick}\\{T(H):T\\in\\mathscr M,\\ H\\in\\mathscr H\\}.\n\\]\nThis notation does not assert that $H$ was operationally killable before localization.","\\begin{theorem}[Universal constructor-compatible excisive quotient]\\label{thm:excisive-quotient}\nThe stable Verdier quotient\n\\[\n q_{\\square}:\\mathcal Z\\longrightarrow\\mathcal Q_{\\square}\n :=\\mathcal Z/\\mathcal J_{\\square}\n\\]\nhas the following properties.\n\\begin{enumerate}\n\\item The constructors descend, with their supplied coherent action data.\n\\item Every diagram $q_{\\square}X$, for $X\\in\\mathscr X$, has context-independent event residuals and satisfies principal-ideal reconstruction.\n\\item Every exact equivariant functor $E:\\mathcal Z\\to\\mathcal D$ for which all the diagrams $EX$ are excisive factors through $q_{\\square}$, with the enhanced universal uniqueness of Verdier localization.\n\\item The kernel is the least constructor-invariant thick subcategory forcing those squares to be excisive.\n\\end{enumerate}\nThe quotient is not automatically nonzero. If such an $E$ has $E(Y)\\not\\simeq0$, then $q_{\\square}(Y)","\\begin{theorem}[Universal constructor-compatible excisive quotient]\\label{thm:excisive-quotient}\nThe stable Verdier quotient\n\\[\n q_{\\square}:\\mathcal Z\\longrightarrow\\mathcal Q_{\\square}\n :=\\mathcal Z/\\mathcal J_{\\square}\n\\]\nhas the following properties.\n\\begin{enumerate}\n\\item The constructors descend, with their supplied coherent action data.\n\\item Every diagram $q_{\\square}X$, for $X\\in\\mathscr X$, has context-independent event residuals and satisfies principal-ideal reconstruction.\n\\item Every exact equivariant functor $E:\\mathcal Z\\to\\mathcal D$ for which all the diagrams $EX$ are excisive factors through $q_{\\square}$, with the enhanced universal uniqueness of Verdier localization.\n\\item The kernel is the least constructor-invariant thick subcategory forcing those squares to be excisive.\n\\end{enumerate}\nThe quotient is not automatically nonzero. If such an $E$ has $E(Y)\\not\\simeq0$, then $q_{\\square}(Y)\\not\\simeq0$.\n\\end{theorem}","_{\\square}:\\mathcal Z\\longrightarrow\\mathcal Q_{\\square}\n :=\\mathcal Z/\\mathcal J_{\\square}\n\\]\nhas the following properties.\n\\begin{enumerate}\n\\item The constructors descend, with their supplied coherent action data.\n\\item Every diagram $q_{\\square}X$, for $X\\in\\mathscr X$, has context-independent event residuals and satisfies principal-ideal reconstruction.\n\\item Every exact equivariant functor $E:\\mathcal Z\\to\\mathcal D$ for which all the diagrams $EX$ are excisive factors through $q_{\\square}$, with the enhanced universal uniqueness of Verdier localization.\n\\item The kernel is the least constructor-invariant thick subcategory forcing those squares to be excisive.\n\\end{enumerate}\nThe quotient is not automatically nonzero. If such an $E$ has $E(Y)\\not\\simeq0$, then $q_{\\square}(Y)\\not\\simeq0$.\n\\end{theorem}","}\n :=\\mathcal Z/\\mathcal J_{\\square}\n\\]\nhas the following properties.\n\\begin{enumerate}\n\\item The constructors descend, with their supplied coherent action data.\n\\item Every diagram $q_{\\square}X$, for $X\\in\\mathscr X$, has context-independent event residuals and satisfies principal-ideal reconstruction.\n\\item Every exact equivariant functor $E:\\mathcal Z\\to\\mathcal D$ for which all the diagrams $EX$ are excisive factors through $q_{\\square}$, with the enhanced universal uniqueness of Verdier localization.\n\\item The kernel is the least constructor-invariant thick subcategory forcing those squares to be excisive.\n\\end{enumerate}\nThe quotient is not automatically nonzero. If such an $E$ has $E(Y)\\not\\simeq0$, then $q_{\\square}(Y)\\not\\simeq0$.\n\\end{theorem}","e}\n\\]\nhas the following properties.\n\\begin{enumerate}\n\\item The constructors descend, with their supplied coherent action data.\n\\item Every diagram $q_{\\square}X$, for $X\\in\\mathscr X$, has context-independent event residuals and satisfies principal-ideal reconstruction.\n\\item Every exact equivariant functor $E:\\mathcal Z\\to\\mathcal D$ for which all the diagrams $EX$ are excisive factors through $q_{\\square}$, with the enhanced universal uniqueness of Verdier localization.\n\\item The kernel is the least constructor-invariant thick subcategory forcing those squares to be excisive.\n\\end{enumerate}\nThe quotient is not automatically nonzero. If such an $E$ has $E(Y)\\not\\simeq0$, then $q_{\\square}(Y)\\not\\simeq0$.\n\\end{theorem}","econstruction.\n\\item Every exact equivariant functor $E:\\mathcal Z\\to\\mathcal D$ for which all the diagrams $EX$ are excisive factors through $q_{\\square}$, with the enhanced universal uniqueness of Verdier localization.\n\\item The kernel is the least constructor-invariant thick subcategory forcing those squares to be excisive.\n\\end{enumerate}\nThe quotient is not automatically nonzero. If such an $E$ has $E(Y)\\not\\simeq0$, then $q_{\\square}(Y)\\not\\simeq0$.\n\\end{theorem}","ction.\n\\item Every exact equivariant functor $E:\\mathcal Z\\to\\mathcal D$ for which all the diagrams $EX$ are excisive factors through $q_{\\square}$, with the enhanced universal uniqueness of Verdier localization.\n\\item The kernel is the least constructor-invariant thick subcategory forcing those squares to be excisive.\n\\end{enumerate}\nThe quotient is not automatically nonzero. If such an $E$ has $E(Y)\\not\\simeq0$, then $q_{\\square}(Y)\\not\\simeq0$.\n\\end{theorem}","equivariant functor $E:\\mathcal Z\\to\\mathcal D$ for which all the diagrams $EX$ are excisive factors through $q_{\\square}$, with the enhanced universal uniqueness of Verdier localization.\n\\item The kernel is the least constructor-invariant thick subcategory forcing those squares to be excisive.\n\\end{enumerate}\nThe quotient is not automatically nonzero. If such an $E$ has $E(Y)\\not\\simeq0$, then $q_{\\square}(Y)\\not\\simeq0$.\n\\end{theorem}","\\begin{proof}\nThe proof of \\cref{thm:invariant-quotient} applies to any designated family of objects, not only killable objects. A constructor sends a generator $T(H)$ to another generator. Exactness then preserves its thick closure. The Verdier universal property therefore descends constructors and their already specified coherence.","Since $q_{\\square}(H)\\simeq0$ and exact functors preserve interaction residuals, every prescribed elementary square in $q_{\\square}X$ is a pushout. Apply \\cref{thm:event-context-invariance,thm:principal-reconstruction} to obtain (2).","If $EX$ is excisive, then $E(H)\\simeq0$. Equivariance implies $E(T(H))\\simeq T'(E(H))\\simeq0$. Exactness annihilates the thick closure, so $E$ factors through the quotient. Conversely, the kernel of any exact equivariant localization making the diagrams excisive must contain $\\mathscr H$ and its invariant thick closure. This proves the universal and minimality assertions. Finally a factorization $E\\simeq\\bar E q_{\\square}$ would send $Y$ to zero if $q_{\\square}(Y)$ were zero.\n\\end{proof}",". Equivariance implies $E(T(H))\\simeq T'(E(H))\\simeq0$. Exactness annihilates the thick closure, so $E$ factors through the quotient. Conversely, the kernel of any exact equivariant localization making the diagrams excisive must contain $\\mathscr H$ and its invariant thick closure. This proves the universal and minimality assertions. Finally a factorization $E\\simeq\\bar E q_{\\square}$ would send $Y$ to zero if $q_{\\square}(Y)$ were zero.\n\\end{proof}","implies $E(T(H))\\simeq T'(E(H))\\simeq0$. Exactness annihilates the thick closure, so $E$ factors through the quotient. Conversely, the kernel of any exact equivariant localization making the diagrams excisive must contain $\\mathscr H$ and its invariant thick closure. This proves the universal and minimality assertions. Finally a factorization $E\\simeq\\bar E q_{\\square}$ would send $Y$ to zero if $q_{\\square}(Y)$ were zero.\n\\end{proof}","We now give a completely specified arithmetic instance. It uses perfect complexes over $\\mathbb Z$, not Galois representations or automorphic representations. Its purpose is to make the coherence, residual, excision, noncollapse, and finite-generator obligations explicit in one model.","\\begin{theorem}[Tensor-cube residual formula]\\label{thm:tensor-cube}\nLet $\\mathcal S$ be a symmetric monoidal stable category whose tensor product is exact separately in each variable. For $m\\ge1$ and arrows $f_i:A_i\\to B_i$, form the coherent $m$-cube\n\\[\n Y(S)=\\bigotimes_{i=1}^m Z_i(S),\\qquad\n Z_i(S)=\\begin{cases}B_i&i\\in S,","\\begin{theorem}[Tensor-cube residual formula]\\label{thm:tensor-cube}\nLet $\\mathcal S$ be a symmetric monoidal stable category whose tensor product is exact separately in each variable. For $m\\ge1$ and arrows $f_i:A_i\\to B_i$, form the coherent $m$-cube\n\\[\n Y(S)=\\bigotimes_{i=1}^m Z_i(S),\\qquad\n Z_i(S)=\\begin{cases}B_i&i\\in S,\\\\A_i&i\\notin S.\\end{cases}\n\\]\nIts total cofiber, obtained by taking the cofiber in every coordinate, satisfies\n\\[\n \\operatorname{tcofib}(Y)\\simeq\n \\bigotimes_{i=1}^m\\Cofib(f_i).\n\\]\nEquivalently it is the cofiber of the canonical map from the colimit of the punctured cube to its terminal vertex. The equivalences are natural in the arrows, with the coherent symmetries of the monoidal structure, rather than literal equality of chain matrices for different coordinate orders.\n\\end{theorem}","A_i&i\\notin S.\\end{cases}\n\\]\nIts total cofiber, obtained by taking the cofiber in every coordinate, satisfies\n\\[\n \\operatorname{tcofib}(Y)\\simeq\n \\bigotimes_{i=1}^m\\Cofib(f_i).\n\\]\nEquivalently it is the cofiber of the canonical map from the colimit of the punctured cube to its terminal vertex. The equivalences are natural in the arrows, with the coherent symmetries of the monoidal structure, rather than literal equality of chain matrices for different coordinate orders.\n\\end{theorem}","\\begin{proof}\nFor $m=1$ this is the definition. Taking cofibers in the last coordinate replaces its factor $A_m\\to B_m$ by $\\Cofib(f_m)$ because tensoring by the other factors is exact. Repeat for the remaining coordinates. Interchanging finite colimits shows that this iterated construction is the cofiber of the punctured-cube colimit map. The universal properties give naturality and coherent interchange. In a complex model, these interchange maps incorporate the tensor differential's graded signs.\n\\end{proof}","For a commutative ring $R$ and $a\\in R$, set\n\\[\n C_R(a)=\\bigl[R\\xrightarrow{a}R\\bigr],\n\\]\nwith the source in cohomological degree $-1$ and the target in degree $0$. The tensor product $\\bigotimes_i C_R(a_i)$ is the Koszul complex in this cohomological convention. Its tensor, cone, and change-of-basis properties are classical; see \\cite{StacksKoszul}. We use them with an explicit sign convention and direct proofs below.","Fix a finite poset $P$ and assign a positive integer $n_e$ to every event $e\\in P$. For $M\\in\\operatorname{Perf}(\\mathbb Z)$ define\n\\[\n X_M(I)=M,\\qquad\n X_M(I\\subseteq J)=\\left(\\prod_{e\\in J\\setminus I}n_e\\right)\\id_M.\n\\]\nEmpty products are $1$. For nested ideals these products multiply exactly, so $X_M$ is a strict chain-level diagram followed by the stable realization. When $P=P_W$, take identity transformation and coarsening functors and let the exchange cell for event $e$ be multiplication by $n_e$. This realizes the diagram inside the typed two-axis grammar of \\cref{thm:strict-event-realization}. The object at every vertex is the same, but its comparison maps retain nontrivial arithmetic.","n P$. For $M\\in\\operatorname{Perf}(\\mathbb Z)$ define\n\\[\n X_M(I)=M,\\qquad\n X_M(I\\subseteq J)=\\left(\\prod_{e\\in J\\setminus I}n_e\\right)\\id_M.\n\\]\nEmpty products are $1$. For nested ideals these products multiply exactly, so $X_M$ is a strict chain-level diagram followed by the stable realization. When $P=P_W$, take identity transformation and coarsening functors and let the exchange cell for event $e$ be multiplication by $n_e$. This realizes the diagram inside the typed two-axis grammar of \\cref{thm:strict-event-realization}. The object at every vertex is the same, but its comparison maps retain nontrivial arithmetic.","ame{Perf}(\\mathbb Z)$ define\n\\[\n X_M(I)=M,\\qquad\n X_M(I\\subseteq J)=\\left(\\prod_{e\\in J\\setminus I}n_e\\right)\\id_M.\n\\]\nEmpty products are $1$. For nested ideals these products multiply exactly, so $X_M$ is a strict chain-level diagram followed by the stable realization. When $P=P_W$, take identity transformation and coarsening functors and let the exchange cell for event $e$ be multiplication by $n_e$. This realizes the diagram inside the typed two-axis grammar of \\cref{thm:strict-event-realization}. The object at every vertex is the same, but its comparison maps retain nontrivial arithmetic.","For an enabled event $e$,\n\\[\n E_e(I)\\simeq M\\otimes^{\\mathbf L}_{\\mathbb Z}C_{\\mathbb Z}(n_e).\n\\]\nThe comparison after an unrelated event $f$ is multiplication by $n_f$ on that complex. Consequently\n\\[\n \\boxed{\\Theta_{e,f}(I)\\simeq\n M\\otimes^{\\mathbf L}_{\\mathbb Z}\n C_{\\mathbb Z}(n_e)\\otimes^{\\mathbf L}_{\\mathbb Z}C_{\\mathbb Z}(n_f).}\n\\]\nThe complexes $C_{\\mathbb Z}(n)$ are bounded and free, so ordinary tensor complexes compute these derived tensors. The total normalization residual is instead\n\\[\n \\mathcal F_{X_M}(P)\\simeq\n M\\otimes^{\\mathbf L}_{\\mathbb Z}C_{\\mathbb Z}(N),\n \\qquad N=\\prod_{e\\in P}n_e.\n\\]\nThis is not the total cofiber of the entire event cube. The normalization residual compares the initial and fin","For an enabled event $e$,\n\\[\n E_e(I)\\simeq M\\otimes^{\\mathbf L}_{\\mathbb Z}C_{\\mathbb Z}(n_e).\n\\]\nThe comparison after an unrelated event $f$ is multiplication by $n_f$ on that complex. Consequently\n\\[\n \\boxed{\\Theta_{e,f}(I)\\simeq\n M\\otimes^{\\mathbf L}_{\\mathbb Z}\n C_{\\mathbb Z}(n_e)\\otimes^{\\mathbf L}_{\\mathbb Z}C_{\\mathbb Z}(n_f).}\n\\]\nThe complexes $C_{\\mathbb Z}(n)$ are bounded and free, so ordinary tensor complexes compute these derived tensors. The total normalization residual is instead\n\\[\n \\mathcal F_{X_M}(P)\\simeq\n M\\otimes^{\\mathbf L}_{\\mathbb Z}C_{\\mathbb Z}(N),\n \\qquad N=\\prod_{e\\in P}n_e.\n\\]\nThis is not the total cofiber of the entire event cube. The normalization residual compares the initial and final vertices, whereas a cube's total cofiber measures their higher intera","For an enabled event $e$,\n\\[\n E_e(I)\\simeq M\\otimes^{\\mathbf L}_{\\mathbb Z}C_{\\mathbb Z}(n_e).\n\\]\nThe comparison after an unrelated event $f$ is multiplication by $n_f$ on that complex. Consequently\n\\[\n \\boxed{\\Theta_{e,f}(I)\\simeq\n M\\otimes^{\\mathbf L}_{\\mathbb Z}\n C_{\\mathbb Z}(n_e)\\otimes^{\\mathbf L}_{\\mathbb Z}C_{\\mathbb Z}(n_f).}\n\\]\nThe complexes $C_{\\mathbb Z}(n)$ are bounded and free, so ordinary tensor complexes compute these derived tensors. The total normalization residual is instead\n\\[\n \\mathcal F_{X_M}(P)\\simeq\n M\\otimes^{\\mathbf L}_{\\mathbb Z}C_{\\mathbb Z}(N),\n \\qquad N=\\prod_{e\\in P}n_e.\n\\]\nThis is not the total cofiber of the entire event cube. The normalization residual compares the initial and final vertices, whereas a cube's total cofiber measures their higher interaction. These are different constructions.","comparison after an unrelated event $f$ is multiplication by $n_f$ on that complex. Consequently\n\\[\n \\boxed{\\Theta_{e,f}(I)\\simeq\n M\\otimes^{\\mathbf L}_{\\mathbb Z}\n C_{\\mathbb Z}(n_e)\\otimes^{\\mathbf L}_{\\mathbb Z}C_{\\mathbb Z}(n_f).}\n\\]\nThe complexes $C_{\\mathbb Z}(n)$ are bounded and free, so ordinary tensor complexes compute these derived tensors. The total normalization residual is instead\n\\[\n \\mathcal F_{X_M}(P)\\simeq\n M\\otimes^{\\mathbf L}_{\\mathbb Z}C_{\\mathbb Z}(N),\n \\qquad N=\\prod_{e\\in P}n_e.\n\\]\nThis is not the total cofiber of the entire event cube. The normalization residual compares the initial and final vertices, whereas a cube's total cofiber measures their higher interaction. These are different constructions.","\\begin{theorem}[Integral interaction is supported at the gcd]\\label{thm:integral-gcd}\nFor positive integers $a_1,\\ldots,a_m$, $m\\ge1$, put $g=\\gcd(a_1,\\ldots,a_m)$. Then\n\\[\n K(a_1,\\ldots,a_m):=\n \\bigotimes_{i=1}^m C_{\\mathbb Z}(a_i)\n \\simeq\\bigoplus_{j=0}^{m-1}\n C_{\\mathbb Z}(g)[j]^{\\oplus\\binom{m-1}{j}}.\n\\]\nThe displayed splitting depends on a unimodular change of basis and is not claimed canonical. Therefore\n\\[\n H^{-j}K(a_1,\\ldots,a_m)\\cong\n (\\mathbb Z/g)^{\\oplus\\binom{m-1}{j}}\n \\quad(0\\le j\\le m-1),\n\\]\nand every other cohomology group vanishes. In particular the interaction vanishes exactly when $g=1$.\n\\end{theorem}","Integral interaction is supported at the gcd]\\label{thm:integral-gcd}\nFor positive integers $a_1,\\ldots,a_m$, $m\\ge1$, put $g=\\gcd(a_1,\\ldots,a_m)$. Then\n\\[\n K(a_1,\\ldots,a_m):=\n \\bigotimes_{i=1}^m C_{\\mathbb Z}(a_i)\n \\simeq\\bigoplus_{j=0}^{m-1}\n C_{\\mathbb Z}(g)[j]^{\\oplus\\binom{m-1}{j}}.\n\\]\nThe displayed splitting depends on a unimodular change of basis and is not claimed canonical. Therefore\n\\[\n H^{-j}K(a_1,\\ldots,a_m)\\cong\n (\\mathbb Z/g)^{\\oplus\\binom{m-1}{j}}\n \\quad(0\\le j\\le m-1),\n\\]\nand every other cohomology group vanishes. In particular the interaction vanishes exactly when $g=1$.\n\\end{theorem}","i=1}^m C_{\\mathbb Z}(a_i)\n \\simeq\\bigoplus_{j=0}^{m-1}\n C_{\\mathbb Z}(g)[j]^{\\oplus\\binom{m-1}{j}}.\n\\]\nThe displayed splitting depends on a unimodular change of basis and is not claimed canonical. Therefore\n\\[\n H^{-j}K(a_1,\\ldots,a_m)\\cong\n (\\mathbb Z/g)^{\\oplus\\binom{m-1}{j}}\n \\quad(0\\le j\\le m-1),\n\\]\nand every other cohomology group vanishes. In particular the interaction vanishes exactly when $g=1$.\n\\end{theorem}","\\begin{proof}\nThe Euclidean algorithm produces a unimodular matrix $U\\in\\operatorname{GL}_m(\\mathbb Z)$ with\n\\[\n (a_1,\\ldots,a_m)U=(g,0,\\ldots,0).\n\\]\nThe Koszul complex is the exterior algebra on a free module in degree $-1$, with differential the derivation determined by this row vector. Hence the maps $\\bigwedge^jU$ give an isomorphism of complexes from $K(g,0,\\ldots,0)$ to $K(a_1,\\ldots,a_m)$.","Each zero entry contributes $C_{\\mathbb Z}(0)\\cong\\mathbb Z\\oplus\\mathbb Z[1]$ with zero differential. Distributing these $m-1$ factors gives the stated shifted copies, with the standard shift isomorphisms. Since $g>0$, the map $g:\\mathbb Z\\to\\mathbb Z$ is injective. Thus $C_{\\mathbb Z}(g)$ has only $H^0=\\mathbb Z/g$, proving the cohomology formula and the vanishing criterion.\n\\end{proof}","For two events the entire calculation is visible:\n\\[\n K(a,b)=\\left[\n \\mathbb Z\\xrightarrow{\\binom{-b}{a}}\n \\mathbb Z^2\\xrightarrow{(a\\ \\ b)}\\mathbb Z\n \\right],\n \\qquad\\text{degrees }-2,-1,0.\n\\]\nThe product of the two differentials is zero. The kernel of $(a\\ b)$ is generated by $(-b/g,a/g)$, and the incoming image is $g$ times that generator. Both $H^{-1}$ and $H^0$ are $\\mathbb Z/g$. Equivalently, if $au+bv=g$, then\n\\[\n U=\\begin{pmatrix}u&-b/g","o events the entire calculation is visible:\n\\[\n K(a,b)=\\left[\n \\mathbb Z\\xrightarrow{\\binom{-b}{a}}\n \\mathbb Z^2\\xrightarrow{(a\\ \\ b)}\\mathbb Z\n \\right],\n \\qquad\\text{degrees }-2,-1,0.\n\\]\nThe product of the two differentials is zero. The kernel of $(a\\ b)$ is generated by $(-b/g,a/g)$, and the incoming image is $g$ times that generator. Both $H^{-1}$ and $H^0$ are $\\mathbb Z/g$. Equivalently, if $au+bv=g$, then\n\\[\n U=\\begin{pmatrix}u&-b/g","re calculation is visible:\n\\[\n K(a,b)=\\left[\n \\mathbb Z\\xrightarrow{\\binom{-b}{a}}\n \\mathbb Z^2\\xrightarrow{(a\\ \\ b)}\\mathbb Z\n \\right],\n \\qquad\\text{degrees }-2,-1,0.\n\\]\nThe product of the two differentials is zero. The kernel of $(a\\ b)$ is generated by $(-b/g,a/g)$, and the incoming image is $g$ times that generator. Both $H^{-1}$ and $H^0$ are $\\mathbb Z/g$. Equivalently, if $au+bv=g$, then\n\\[\n U=\\begin{pmatrix}u&-b/g\\\\v&a/g\\end{pmatrix},\n \\qquad\\det U=1,\n \\qquad(a\\ b)U=(g\\ 0).\n\\]","Thus a strictly commuting scalar exchange square is a homotopy pushout on the seed $\\mathbb Z$ exactly when the two exchange integers are coprime. Its commutativity alone says nothing about this condition.","\\begin{example}[Higher vanishing does not remove lower interactions]\\label{ex:higher-not-lower}\nFor $(6,10,15)$, all pair interactions are nonzero: their gcds are $2,3,5$. But the threefold interaction is zero because the joint gcd is $1$. Hence vanishing of a higher total cofiber does not imply vanishing of the lower square interactions. It cannot replace the square-by-square excision condition in \\cref{thm:event-context-invariance}.\n\\end{example}","Let $\\mathscr M$ consist of tensoring by perfect complexes, closed under composition with its coherent tensor action. This is a constructor system, not the repair monoid defining operational killability in the earlier sections. We now kill only the designated interaction residuals.","For the fixed labelled poset define the finite prime set\n\\[\n S_P=\\bigcup_{e\\parallel f}\n \\{p\\text{ prime}:p\\mid\\gcd(n_e,n_f)\\},\n\\]\nwhere $e\\parallel f$ means incomparable. Let $\\mathcal J_P$ be the invariant thick closure of the square interaction residuals for the unit seed $\\mathbb Z$.","\\begin{theorem}[Prime-overlap classification of the excisive kernel]\\label{thm:prime-overlap-quotient}\nIn $\\operatorname{Perf}(\\mathbb Z)$,\n\\[\n \\boxed{\\mathcal J_P=\n \\operatorname{Thick}\\{\\mathbb Z/p:p\\in S_P\\}.}\n\\]\nEquivalently, $\\mathcal J_P$ consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in $S_P$. For the stable Verdier quotient $q_P$, and every perfect complex $M$,\n\\[\n q_P(M)\\simeq0\n \\quad\\Longleftrightarrow\\quad\n M\\otimes^{\\mathbf L}_{\\mathbb Z}\\mathbb Z[S_P^{-1}]\\simeq0.\n\\]\nThe quotient is nonzero. Its constructor action is defined, every $q_PX_M$ is excisive, and its reduced residual diagram is reconstructed from principal ideals.\n\\end{theorem}","classification of the excisive kernel]\\label{thm:prime-overlap-quotient}\nIn $\\operatorname{Perf}(\\mathbb Z)$,\n\\[\n \\boxed{\\mathcal J_P=\n \\operatorname{Thick}\\{\\mathbb Z/p:p\\in S_P\\}.}\n\\]\nEquivalently, $\\mathcal J_P$ consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in $S_P$. For the stable Verdier quotient $q_P$, and every perfect complex $M$,\n\\[\n q_P(M)\\simeq0\n \\quad\\Longleftrightarrow\\quad\n M\\otimes^{\\mathbf L}_{\\mathbb Z}\\mathbb Z[S_P^{-1}]\\simeq0.\n\\]\nThe quotient is nonzero. Its constructor action is defined, every $q_PX_M$ is excisive, and its reduced residual diagram is reconstructed from principal ideals.\n\\end{theorem}","ification of the excisive kernel]\\label{thm:prime-overlap-quotient}\nIn $\\operatorname{Perf}(\\mathbb Z)$,\n\\[\n \\boxed{\\mathcal J_P=\n \\operatorname{Thick}\\{\\mathbb Z/p:p\\in S_P\\}.}\n\\]\nEquivalently, $\\mathcal J_P$ consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in $S_P$. For the stable Verdier quotient $q_P$, and every perfect complex $M$,\n\\[\n q_P(M)\\simeq0\n \\quad\\Longleftrightarrow\\quad\n M\\otimes^{\\mathbf L}_{\\mathbb Z}\\mathbb Z[S_P^{-1}]\\simeq0.\n\\]\nThe quotient is nonzero. Its constructor action is defined, every $q_PX_M$ is excisive, and its reduced residual diagram is reconstructed from principal ideals.\n\\end{theorem}","\\]\nEquivalently, $\\mathcal J_P$ consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in $S_P$. For the stable Verdier quotient $q_P$, and every perfect complex $M$,\n\\[\n q_P(M)\\simeq0\n \\quad\\Longleftrightarrow\\quad\n M\\otimes^{\\mathbf L}_{\\mathbb Z}\\mathbb Z[S_P^{-1}]\\simeq0.\n\\]\nThe quotient is nonzero. Its constructor action is defined, every $q_PX_M$ is excisive, and its reduced residual diagram is reconstructed from principal ideals.\n\\end{theorem}","\\begin{proof}\nFirst, every pair of incomparable events occurs in an elementary square. Take the ideal $D_e\\cup D_f$. It contains neither $e$ nor $f$, contains their predecessors, and enables both. The square residual at the unit is $K(n_e,n_f)$.","By \\cref{thm:integral-gcd}, this complex has the same thick closure as $C_{\\mathbb Z}(g)$ for $g=\\gcd(n_e,n_f)$: it is a direct sum of that complex and its shift, and contains it as a retract. The thick closure of any family in $\\operatorname{Perf}(\\mathbb Z)$ is already a tensor ideal. Indeed $\\operatorname{Perf}(\\mathbb Z)=\\operatorname{Thick}(\\mathbb Z)$, so tensoring a fixed generator by a perfect complex builds its image using finite cones, shifts, and retracts from that same generator.","For $p\\mid g$, tensor $C_{\\mathbb Z}(g)$ by $C_{\\mathbb Z}(p)$. The gcd theorem identifies the result with $C_{\\mathbb Z}(p)\\oplus C_{\\mathbb Z}(p)[1]$. Hence $\\mathbb Z/p\\simeq C_{\\mathbb Z}(p)$ belongs to the thick closure. Conversely, the Chinese remainder decomposition of $\\mathbb Z/g$ and the short exact sequences building $\\mathbb Z/p^j$ from copies of $\\mathbb Z/p$ put $C_{\\mathbb Z}(g)\\simeq\\mathbb Z/g$ in the thick subcategory generated by its prime divisors. This proves the boxed equality.","Having finite $S_P$-primary cohomology is preserved by finite triangles and retracts, by the long exact sequence in cohomology. Conversely, each such finite abelian cohomology group is built by finite extensions from the indicated $\\mathbb Z/p$. A perfect complex is bounded, and its finite truncation triangles then build it from these groups. These truncations are perfect over $\\mathbb Z$: every finitely generated abelian group has a finite free resolution of length at most one. This proves the cohomological characterization.","Localization of abelian groups is exact. Thus tensoring with $\\mathbb Z[S_P^{-1}]$ has cohomology obtained by localizing each cohomology group. A finitely generated abelian group localizes to zero exactly when it is finite with all prime divisors in $S_P$. The localization kernel is therefore $\\mathcal J_P$, as is the kernel of the Verdier quotient by this thick subcategory. This proves the equivalence of vanishing tests on every source object.","Finally $\\mathbb Z[S_P^{-1}]$ is a nonzero ring, so localization detects that $q_P(\\mathbb Z)\\not\\simeq0$. It is exact and intertwines tensor constructors with their localized counterparts. The universal excisive-quotient theorem supplies the remaining assertions. For general $M$, each square residual is its unit-seed counterpart tensored with $M$, so it lies in the same invariant kernel.\n\\end{proof}","The positivity hypothesis excludes a real collapse case. If two independent labels are both zero, then $K(0,0)\\simeq\\mathbb Z\\oplus\\mathbb Z[1]^{\\oplus2}\\oplus\\mathbb Z[2]$ contains the unit as a retract. Killing that interaction forces the whole perfect category to vanish. Thus noncollapse is proved for the specified positive-label model, not for arbitrary scalar labels.","\\begin{example}[Remove shared torsion without deleting the primary residuals]\\label{ex:six-ten}\nTake two independent events labelled $6$ and $10$. Their interaction is\n\\[\n K(6,10)\\simeq\\mathbb Z/2\\oplus(\\mathbb Z/2)[1].\n\\]\nHere $S_P=\\{2\\}$. Inverting $2$ kills this interaction while retaining\n\\[\n C_{\\mathbb Z}(6)\\otimes\\mathbb Z[1/2]\\simeq\\mathbb Z/3,\n \\qquad\n C_{\\mathbb Z}(10)\\otimes\\mathbb Z[1/2]\\simeq\\mathbb Z/5.\n\\]\nThe total normalization residual is $C_{\\mathbb Z}(60)$. After localization it is $\\mathbb Z/15$, and principal-ideal reconstruction gives\n\\[\n \\mathbb Z/15\\simeq\\mathbb Z/3\\oplus\\mathbb Z/5.\n\\]\nThis is a classical Chinese-remainder instance of the calculus. Before localization the two paths have first residuals $\\mathbb Z/6$ and $\\mathbb Z/10$. After localization those first stages remain different, so the or","\\begin{example}[Remove shared torsion without deleting the primary residuals]\\label{ex:six-ten}\nTake two independent events labelled $6$ and $10$. Their interaction is\n\\[\n K(6,10)\\simeq\\mathbb Z/2\\oplus(\\mathbb Z/2)[1].\n\\]\nHere $S_P=\\{2\\}$. Inverting $2$ kills this interaction while retaining\n\\[\n C_{\\mathbb Z}(6)\\otimes\\mathbb Z[1/2]\\simeq\\mathbb Z/3,\n \\qquad\n C_{\\mathbb Z}(10)\\otimes\\mathbb Z[1/2]\\simeq\\mathbb Z/5.\n\\]\nThe total normalization residual is $C_{\\mathbb Z}(60)$. After localization it is $\\mathbb Z/15$, and principal-ideal reconstruction gives\n\\[\n \\mathbb Z/15\\simeq\\mathbb Z/3\\oplus\\mathbb Z/5.\n\\]\nThis is a classical Chinese-remainder instance of the calculus. Before localization the two paths have first residuals $\\mathbb Z/6$ and $\\mathbb Z/10$. After localization those first stages remain different, so the original or","\\begin{example}[Remove shared torsion without deleting the primary residuals]\\label{ex:six-ten}\nTake two independent events labelled $6$ and $10$. Their interaction is\n\\[\n K(6,10)\\simeq\\mathbb Z/2\\oplus(\\mathbb Z/2)[1].\n\\]\nHere $S_P=\\{2\\}$. Inverting $2$ kills this interaction while retaining\n\\[\n C_{\\mathbb Z}(6)\\otimes\\mathbb Z[1/2]\\simeq\\mathbb Z/3,\n \\qquad\n C_{\\mathbb Z}(10)\\otimes\\mathbb Z[1/2]\\simeq\\mathbb Z/5.\n\\]\nThe total normalization residual is $C_{\\mathbb Z}(60)$. After localization it is $\\mathbb Z/15$, and principal-ideal reconstruction gives\n\\[\n \\mathbb Z/15\\simeq\\mathbb Z/3\\oplus\\mathbb Z/5.\n\\]\nThis is a classical Chinese-remainder instance of the calculus. Before localization the two paths have first residuals $\\mathbb Z/6$ and $\\mathbb Z/10$. After localization those first stages remain different, so the original ordered-filtration independence claim is still not being restored.\n\\end{exampl","\\begin{example}[Remove shared torsion without deleting the primary residuals]\\label{ex:six-ten}\nTake two independent events labelled $6$ and $10$. Their interaction is\n\\[\n K(6,10)\\simeq\\mathbb Z/2\\oplus(\\mathbb Z/2)[1].\n\\]\nHere $S_P=\\{2\\}$. Inverting $2$ kills this interaction while retaining\n\\[\n C_{\\mathbb Z}(6)\\otimes\\mathbb Z[1/2]\\simeq\\mathbb Z/3,\n \\qquad\n C_{\\mathbb Z}(10)\\otimes\\mathbb Z[1/2]\\simeq\\mathbb Z/5.\n\\]\nThe total normalization residual is $C_{\\mathbb Z}(60)$. After localization it is $\\mathbb Z/15$, and principal-ideal reconstruction gives\n\\[\n \\mathbb Z/15\\simeq\\mathbb Z/3\\oplus\\mathbb Z/5.\n\\]\nThis is a classical Chinese-remainder instance of the calculus. Before localization the two paths have first residuals $\\mathbb Z/6$ and $\\mathbb Z/10$. After localization those first stages remain different, so the original ordered-filtration independence claim is still not being restored.\n\\end{example}","rimary residuals]\\label{ex:six-ten}\nTake two independent events labelled $6$ and $10$. Their interaction is\n\\[\n K(6,10)\\simeq\\mathbb Z/2\\oplus(\\mathbb Z/2)[1].\n\\]\nHere $S_P=\\{2\\}$. Inverting $2$ kills this interaction while retaining\n\\[\n C_{\\mathbb Z}(6)\\otimes\\mathbb Z[1/2]\\simeq\\mathbb Z/3,\n \\qquad\n C_{\\mathbb Z}(10)\\otimes\\mathbb Z[1/2]\\simeq\\mathbb Z/5.\n\\]\nThe total normalization residual is $C_{\\mathbb Z}(60)$. After localization it is $\\mathbb Z/15$, and principal-ideal reconstruction gives\n\\[\n \\mathbb Z/15\\simeq\\mathbb Z/3\\oplus\\mathbb Z/5.\n\\]\nThis is a classical Chinese-remainder instance of the calculus. Before localization the two paths have first residuals $\\mathbb Z/6$ and $\\mathbb Z/10$. After localization those first stages remain different, so the original ordered-filtration independence claim is still not being restored.\n\\end{example}","ndependent events labelled $6$ and $10$. Their interaction is\n\\[\n K(6,10)\\simeq\\mathbb Z/2\\oplus(\\mathbb Z/2)[1].\n\\]\nHere $S_P=\\{2\\}$. Inverting $2$ kills this interaction while retaining\n\\[\n C_{\\mathbb Z}(6)\\otimes\\mathbb Z[1/2]\\simeq\\mathbb Z/3,\n \\qquad\n C_{\\mathbb Z}(10)\\otimes\\mathbb Z[1/2]\\simeq\\mathbb Z/5.\n\\]\nThe total normalization residual is $C_{\\mathbb Z}(60)$. After localization it is $\\mathbb Z/15$, and principal-ideal reconstruction gives\n\\[\n \\mathbb Z/15\\simeq\\mathbb Z/3\\oplus\\mathbb Z/5.\n\\]\nThis is a classical Chinese-remainder instance of the calculus. Before localization the two paths have first residuals $\\mathbb Z/6$ and $\\mathbb Z/10$. After localization those first stages remain different, so the original ordered-filtration independence claim is still not being restored.\n\\end{example}","is\n\\[\n K(6,10)\\simeq\\mathbb Z/2\\oplus(\\mathbb Z/2)[1].\n\\]\nHere $S_P=\\{2\\}$. Inverting $2$ kills this interaction while retaining\n\\[\n C_{\\mathbb Z}(6)\\otimes\\mathbb Z[1/2]\\simeq\\mathbb Z/3,\n \\qquad\n C_{\\mathbb Z}(10)\\otimes\\mathbb Z[1/2]\\simeq\\mathbb Z/5.\n\\]\nThe total normalization residual is $C_{\\mathbb Z}(60)$. After localization it is $\\mathbb Z/15$, and principal-ideal reconstruction gives\n\\[\n \\mathbb Z/15\\simeq\\mathbb Z/3\\oplus\\mathbb Z/5.\n\\]\nThis is a classical Chinese-remainder instance of the calculus. Before localization the two paths have first residuals $\\mathbb Z/6$ and $\\mathbb Z/10$. After localization those first stages remain different, so the original ordered-filtration independence claim is still not being restored.\n\\end{example}","imeq\\mathbb Z/3,\n \\qquad\n C_{\\mathbb Z}(10)\\otimes\\mathbb Z[1/2]\\simeq\\mathbb Z/5.\n\\]\nThe total normalization residual is $C_{\\mathbb Z}(60)$. After localization it is $\\mathbb Z/15$, and principal-ideal reconstruction gives\n\\[\n \\mathbb Z/15\\simeq\\mathbb Z/3\\oplus\\mathbb Z/5.\n\\]\nThis is a classical Chinese-remainder instance of the calculus. Before localization the two paths have first residuals $\\mathbb Z/6$ and $\\mathbb Z/10$. After localization those first stages remain different, so the original ordered-filtration independence claim is still not being restored.\n\\end{example}","\\qquad\n C_{\\mathbb Z}(10)\\otimes\\mathbb Z[1/2]\\simeq\\mathbb Z/5.\n\\]\nThe total normalization residual is $C_{\\mathbb Z}(60)$. After localization it is $\\mathbb Z/15$, and principal-ideal reconstruction gives\n\\[\n \\mathbb Z/15\\simeq\\mathbb Z/3\\oplus\\mathbb Z/5.\n\\]\nThis is a classical Chinese-remainder instance of the calculus. Before localization the two paths have first residuals $\\mathbb Z/6$ and $\\mathbb Z/10$. After localization those first stages remain different, so the original ordered-filtration independence claim is still not being restored.\n\\end{example}","\\begin{corollary}[Finite-signature arithmetic certification]\\label{cor:finite-signature}\nFix finitely many positive integer exchange labels $n_1,\\ldots,n_t$ and a specified symmetric relation $\\mathcal I$ of label pairs allowed to occur on incomparable events. The relation may contain a diagonal pair. Put\n\\[\n S_{\\mathcal I}=\\bigcup_{(i,j)\\in\\mathcal I}\n \\{p:p\\mid\\gcd(n_i,n_j)\\}.\n\\]\nKilling the finitely many unit-seed interactions $K(n_i,n_j)$ for $(i,j)\\in\\mathcal I$, with invariant thick closure, makes every scalar event diagram on every finite labelled poset satisfying this independence restriction excisive, for every perfect input $M$. The quotient is nonzero and its kernel is the finite-prime torsion class supported on $S_{\\mathcal I}$.\n\\end{corollary}","arithmetic certification]\\label{cor:finite-signature}\nFix finitely many positive integer exchange labels $n_1,\\ldots,n_t$ and a specified symmetric relation $\\mathcal I$ of label pairs allowed to occur on incomparable events. The relation may contain a diagonal pair. Put\n\\[\n S_{\\mathcal I}=\\bigcup_{(i,j)\\in\\mathcal I}\n \\{p:p\\mid\\gcd(n_i,n_j)\\}.\n\\]\nKilling the finitely many unit-seed interactions $K(n_i,n_j)$ for $(i,j)\\in\\mathcal I$, with invariant thick closure, makes every scalar event diagram on every finite labelled poset satisfying this independence restriction excisive, for every perfect input $M$. The quotient is nonzero and its kernel is the finite-prime torsion class supported on $S_{\\mathcal I}$.\n\\end{corollary}","ication]\\label{cor:finite-signature}\nFix finitely many positive integer exchange labels $n_1,\\ldots,n_t$ and a specified symmetric relation $\\mathcal I$ of label pairs allowed to occur on incomparable events. The relation may contain a diagonal pair. Put\n\\[\n S_{\\mathcal I}=\\bigcup_{(i,j)\\in\\mathcal I}\n \\{p:p\\mid\\gcd(n_i,n_j)\\}.\n\\]\nKilling the finitely many unit-seed interactions $K(n_i,n_j)$ for $(i,j)\\in\\mathcal I$, with invariant thick closure, makes every scalar event diagram on every finite labelled poset satisfying this independence restriction excisive, for every perfect input $M$. The quotient is nonzero and its kernel is the finite-prime torsion class supported on $S_{\\mathcal I}$.\n\\end{corollary}","omparable events. The relation may contain a diagonal pair. Put\n\\[\n S_{\\mathcal I}=\\bigcup_{(i,j)\\in\\mathcal I}\n \\{p:p\\mid\\gcd(n_i,n_j)\\}.\n\\]\nKilling the finitely many unit-seed interactions $K(n_i,n_j)$ for $(i,j)\\in\\mathcal I$, with invariant thick closure, makes every scalar event diagram on every finite labelled poset satisfying this independence restriction excisive, for every perfect input $M$. The quotient is nonzero and its kernel is the finite-prime torsion class supported on $S_{\\mathcal I}$.\n\\end{corollary}","{(i,j)\\in\\mathcal I}\n \\{p:p\\mid\\gcd(n_i,n_j)\\}.\n\\]\nKilling the finitely many unit-seed interactions $K(n_i,n_j)$ for $(i,j)\\in\\mathcal I$, with invariant thick closure, makes every scalar event diagram on every finite labelled poset satisfying this independence restriction excisive, for every perfect input $M$. The quotient is nonzero and its kernel is the finite-prime torsion class supported on $S_{\\mathcal I}$.\n\\end{corollary}","\\begin{proof}\nThere are only finitely many prescribed pair types. Every elementary square in every admitted diagram has one of these types, and its residual is the corresponding unit-seed Koszul complex tensored with $M$. All such residuals lie in the same invariant thick kernel. Apply \\cref{thm:excisive-quotient,thm:prime-overlap-quotient}. The latter proof depends only on the finite family of gcds, so it applies unchanged to the signature-wide family.\n\\end{proof}","This is a genuine finite reduction of an unbounded word-length obligation, but only in the scalar integral regime with the stated independence relation. If a label can occur independently of itself, its primes belong to $S_{\\mathcal I}$ and its primary torsion is erased. Declaring independence therefore changes the quotient's information content. Dependent events can share a prime without forcing its removal, as the $p$-then-$p$ chain in \\cref{cor:independent-splitting} demonstrates.","ite reduction of an unbounded word-length obligation, but only in the scalar integral regime with the stated independence relation. If a label can occur independently of itself, its primes belong to $S_{\\mathcal I}$ and its primary torsion is erased. Declaring independence therefore changes the quotient's information content. Dependent events can share a prime without forcing its removal, as the $p$-then-$p$ chain in \\cref{cor:independent-splitting} demonstrates.","tion of an unbounded word-length obligation, but only in the scalar integral regime with the stated independence relation. If a label can occur independently of itself, its primes belong to $S_{\\mathcal I}$ and its primary torsion is erased. Declaring independence therefore changes the quotient's information content. Dependent events can share a prime without forcing its removal, as the $p$-then-$p$ chain in \\cref{cor:independent-splitting} demonstrates.","\\begin{theorem}[Finite integral matrix collapse criterion]\\label{thm:matrix-collapse}\nLet $f_i:\\mathbb Z^{a_i}\\to\\mathbb Z^{b_i}$ be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank $r_i$ and positive nonzero diagonal entries $d_{i,1},\\ldots,d_{i,r_i}$. Put\n\\[\n \\epsilon_i=\\begin{cases}\n 1&a_i+b_i-2r_i>0,","\\begin{theorem}[Finite integral matrix collapse criterion]\\label{thm:matrix-collapse}\nLet $f_i:\\mathbb Z^{a_i}\\to\\mathbb Z^{b_i}$ be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank $r_i$ and positive nonzero diagonal entries $d_{i,1},\\ldots,d_{i,r_i}$. Put\n\\[\n \\epsilon_i=\\begin{cases}\n 1&a_i+b_i-2r_i>0,\\\\\n 0&a_i+b_i-2r_i=0,\n \\end{cases}\n \\qquad\n P_i=\\{p:p\\mid d_{i,k}\\text{ for some }k\\}.\n\\]\nLet $\\mathcal J$ be the tensor-invariant thick subcategory generated by $\\Cofib(f_i)\\otimes^{\\mathbf L}\\Cofib(f_j)$ over the allowed independent pairs.\n\\begin{enumerate}\n\\item If an allowed pair has $\\epsilon_i=\\epsilon_j=1$, then $\\mathcal J=\\operatorname{Perf}(\\mathbb Z)$ and its quotient is zero.\n\\item Otherwise the quotient is nonzero and\n\\[\n \\mathcal J=\\operatorname{Thick}\\{\\mathbb Z/p:p\\in S\\},\n\\]\nwhere $S$ is the union over allowed pairs of\n\\[\n S_{ij}=\\begin{cases}\n P_i\\cap P_j&\\epsilon_i=\\epsilon_j=0,","0&a_i+b_i-2r_i=0,\n \\end{cases}\n \\qquad\n P_i=\\{p:p\\mid d_{i,k}\\text{ for some }k\\}.\n\\]\nLet $\\mathcal J$ be the tensor-invariant thick subcategory generated by $\\Cofib(f_i)\\otimes^{\\mathbf L}\\Cofib(f_j)$ over the allowed independent pairs.\n\\begin{enumerate}\n\\item If an allowed pair has $\\epsilon_i=\\epsilon_j=1$, then $\\mathcal J=\\operatorname{Perf}(\\mathbb Z)$ and its quotient is zero.\n\\item Otherwise the quotient is nonzero and\n\\[\n \\mathcal J=\\operatorname{Thick}\\{\\mathbb Z/p:p\\in S\\},\n\\]\nwhere $S$ is the union over allowed pairs of\n\\[\n S_{ij}=\\begin{cases}\n P_i\\cap P_j&\\epsilon_i=\\epsilon_j=0,","qquad\n P_i=\\{p:p\\mid d_{i,k}\\text{ for some }k\\}.\n\\]\nLet $\\mathcal J$ be the tensor-invariant thick subcategory generated by $\\Cofib(f_i)\\otimes^{\\mathbf L}\\Cofib(f_j)$ over the allowed independent pairs.\n\\begin{enumerate}\n\\item If an allowed pair has $\\epsilon_i=\\epsilon_j=1$, then $\\mathcal J=\\operatorname{Perf}(\\mathbb Z)$ and its quotient is zero.\n\\item Otherwise the quotient is nonzero and\n\\[\n \\mathcal J=\\operatorname{Thick}\\{\\mathbb Z/p:p\\in S\\},\n\\]\nwhere $S$ is the union over allowed pairs of\n\\[\n S_{ij}=\\begin{cases}\n P_i\\cap P_j&\\epsilon_i=\\epsilon_j=0,","J$ be the tensor-invariant thick subcategory generated by $\\Cofib(f_i)\\otimes^{\\mathbf L}\\Cofib(f_j)$ over the allowed independent pairs.\n\\begin{enumerate}\n\\item If an allowed pair has $\\epsilon_i=\\epsilon_j=1$, then $\\mathcal J=\\operatorname{Perf}(\\mathbb Z)$ and its quotient is zero.\n\\item Otherwise the quotient is nonzero and\n\\[\n \\mathcal J=\\operatorname{Thick}\\{\\mathbb Z/p:p\\in S\\},\n\\]\nwhere $S$ is the union over allowed pairs of\n\\[\n S_{ij}=\\begin{cases}\n P_i\\cap P_j&\\epsilon_i=\\epsilon_j=0,","gory generated by $\\Cofib(f_i)\\otimes^{\\mathbf L}\\Cofib(f_j)$ over the allowed independent pairs.\n\\begin{enumerate}\n\\item If an allowed pair has $\\epsilon_i=\\epsilon_j=1$, then $\\mathcal J=\\operatorname{Perf}(\\mathbb Z)$ and its quotient is zero.\n\\item Otherwise the quotient is nonzero and\n\\[\n \\mathcal J=\\operatorname{Thick}\\{\\mathbb Z/p:p\\in S\\},\n\\]\nwhere $S$ is the union over allowed pairs of\n\\[\n S_{ij}=\\begin{cases}\n P_i\\cap P_j&\\epsilon_i=\\epsilon_j=0,\\\\\n P_j&\\epsilon_i=1,\\ \\epsilon_j=0,\\\\\n P_i&\\epsilon_i=0,\\ \\epsilon_j=1.\n \\end{cases}\n\\]\nLocalization at $S$ is an exact nonzero detector with that kernel. It makes the prescribed tensor-cube squares excisive.\n\\end{enumerate}\nA supplied Smith form is a finite certificate: its row and column changes must be unimodular and their matrix product must give the claimed diagonal form. The existence of Smith forms over $\\mathbb Z$ follows from the Euclidean algorithm; the argument below needs only such a verified form.\n\\end{theorem}","P_i&\\epsilon_i=0,\\ \\epsilon_j=1.\n \\end{cases}\n\\]\nLocalization at $S$ is an exact nonzero detector with that kernel. It makes the prescribed tensor-cube squares excisive.\n\\end{enumerate}\nA supplied Smith form is a finite certificate: its row and column changes must be unimodular and their matrix product must give the claimed diagonal form. The existence of Smith forms over $\\mathbb Z$ follows from the Euclidean algorithm; the argument below needs only such a verified form.\n\\end{theorem}","\\begin{proof}\nChanging bases in the source and target yields a chain isomorphism\n\\[\n \\Cofib(f_i)\\simeq\n \\mathbb Z^{b_i-r_i}\\oplus\\mathbb Z^{a_i-r_i}[1]\n \\oplus\\bigoplus_{k=1}^{r_i}C_{\\mathbb Z}(d_{i,k}).\n\\]\nUnit diagonal entries contribute contractible complexes. Thus $\\epsilon_i=1$ means that a shift of the unit occurs as a direct summand. If two independent cones each have such a summand, their tensor product contains a shift of the unit. A thick subcategory containing it is all of $\\operatorname{Perf}(\\mathbb Z)$.","b(f_i)\\simeq\n \\mathbb Z^{b_i-r_i}\\oplus\\mathbb Z^{a_i-r_i}[1]\n \\oplus\\bigoplus_{k=1}^{r_i}C_{\\mathbb Z}(d_{i,k}).\n\\]\nUnit diagonal entries contribute contractible complexes. Thus $\\epsilon_i=1$ means that a shift of the unit occurs as a direct summand. If two independent cones each have such a summand, their tensor product contains a shift of the unit. A thick subcategory containing it is all of $\\operatorname{Perf}(\\mathbb Z)$.","If no allowed pair has two free summands, every generating tensor is torsion. A free summand tensored with a torsion cone preserves that torsion's prime support. Two torsion summands have tensor product $K(d_{i,k},d_{j,l})$, whose thick support is the prime intersection by \\cref{thm:integral-gcd}. Taking finite direct sums gives exactly the listed sets $S_{ij}$. The argument of \\cref{thm:prime-overlap-quotient} identifies the generated thick subcategory with finite $S$-primary cohomology and identifies its kernel with localization at $S$. The localized unit is nonzero. The tensor-cube formula and the excisive-quotient theorem finish the proof.\n\\end{proof}","rsion. A free summand tensored with a torsion cone preserves that torsion's prime support. Two torsion summands have tensor product $K(d_{i,k},d_{j,l})$, whose thick support is the prime intersection by \\cref{thm:integral-gcd}. Taking finite direct sums gives exactly the listed sets $S_{ij}$. The argument of \\cref{thm:prime-overlap-quotient} identifies the generated thick subcategory with finite $S$-primary cohomology and identifies its kernel with localization at $S$. The localized unit is nonzero. The tensor-cube formula and the excisive-quotient theorem finish the proof.\n\\end{proof}","ion's prime support. Two torsion summands have tensor product $K(d_{i,k},d_{j,l})$, whose thick support is the prime intersection by \\cref{thm:integral-gcd}. Taking finite direct sums gives exactly the listed sets $S_{ij}$. The argument of \\cref{thm:prime-overlap-quotient} identifies the generated thick subcategory with finite $S$-primary cohomology and identifies its kernel with localization at $S$. The localized unit is nonzero. The tensor-cube formula and the excisive-quotient theorem finish the proof.\n\\end{proof}","Fix a finite nonempty set $\\mathcal P$ of primes and put\n\\[\n D=\\prod_{p\\in\\mathcal P}p,\\qquad B_n=\\mathbb Z/D^n\\mathbb Z\\quad(n\\geq1).\n\\]\nWrite $B_{p,n}=\\mathbb Z/p^n\\mathbb Z$. The Chinese remainder isomorphisms\n\\[\n B_n\\cong\\prod_{p\\in\\mathcal P}B_{p,n}\n\\]\nare compatible with reduction. Use the enhanced category $\\mathcal S_n=\\operatorname{Perf}(B_n)$ and the coarsening\n\\[\n C_{m,n}(M)=M\\otimes^{\\mathbf L}_{B_m}B_n,\\qquad m\\geq n.\n\\]\nDerived tensor associativity supplies coherent equivalences $C_{l,n}\\simeq C_{m,n}C_{l,m}$. These are the data of an enhanced indexed system. Literal equalities are not asserted between arbitrarily chosen derived-functor models. On specified bounded projective complexes the operation is termwise re","Fix a finite nonempty set $\\mathcal P$ of primes and put\n\\[\n D=\\prod_{p\\in\\mathcal P}p,\\qquad B_n=\\mathbb Z/D^n\\mathbb Z\\quad(n\\geq1).\n\\]\nWrite $B_{p,n}=\\mathbb Z/p^n\\mathbb Z$. The Chinese remainder isomorphisms\n\\[\n B_n\\cong\\prod_{p\\in\\mathcal P}B_{p,n}\n\\]\nare compatible with reduction. Use the enhanced category $\\mathcal S_n=\\operatorname{Perf}(B_n)$ and the coarsening\n\\[\n C_{m,n}(M)=M\\otimes^{\\mathbf L}_{B_m}B_n,\\qquad m\\geq n.\n\\]\nDerived tensor associativity supplies coherent equivalences $C_{l,n}\\simeq C_{m,n}C_{l,m}$. These are the data of an enhanced indexed system. Literal equalities are not asserted between arbitrarily chosen derived-functor models. On specified bounded projective complexes the operation is termwise reduction. Preservation of perfect complexes under derived base change is standard \\cite{StacksPerfe","Fix a finite nonempty set $\\mathcal P$ of primes and put\n\\[\n D=\\prod_{p\\in\\mathcal P}p,\\qquad B_n=\\mathbb Z/D^n\\mathbb Z\\quad(n\\geq1).\n\\]\nWrite $B_{p,n}=\\mathbb Z/p^n\\mathbb Z$. The Chinese remainder isomorphisms\n\\[\n B_n\\cong\\prod_{p\\in\\mathcal P}B_{p,n}\n\\]\nare compatible with reduction. Use the enhanced category $\\mathcal S_n=\\operatorname{Perf}(B_n)$ and the coarsening\n\\[\n C_{m,n}(M)=M\\otimes^{\\mathbf L}_{B_m}B_n,\\qquad m\\geq n.\n\\]\nDerived tensor associativity supplies coherent equivalences $C_{l,n}\\simeq C_{m,n}C_{l,m}$. These are the data of an enhanced indexed system. Literal equalities are not asserted between arbitrarily chosen derived-functor models. On specified bounded projective complexes the operation is termwise reduction. Preservation of perfect complexes under derived base change is standard \\cite{StacksPerfect}; the pointwise cone model also shows","Fix a finite nonempty set $\\mathcal P$ of primes and put\n\\[\n D=\\prod_{p\\in\\mathcal P}p,\\qquad B_n=\\mathbb Z/D^n\\mathbb Z\\quad(n\\geq1).\n\\]\nWrite $B_{p,n}=\\mathbb Z/p^n\\mathbb Z$. The Chinese remainder isomorphisms\n\\[\n B_n\\cong\\prod_{p\\in\\mathcal P}B_{p,n}\n\\]\nare compatible with reduction. Use the enhanced category $\\mathcal S_n=\\operatorname{Perf}(B_n)$ and the coarsening\n\\[\n C_{m,n}(M)=M\\otimes^{\\mathbf L}_{B_m}B_n,\\qquad m\\geq n.\n\\]\nDerived tensor associativity supplies coherent equivalences $C_{l,n}\\simeq C_{m,n}C_{l,m}$. These are the data of an enhanced indexed system. Literal equalities are not asserted between arbitrarily chosen derived-functor models. On specified bounded projective complexes the operation is termwise reduction. Preservation of perfect complexes under derived base change is standard \\cite{StacksPerfect}; the pointwise cone model also shows exactness directly.","mathbb Z/D^n\\mathbb Z\\quad(n\\geq1).\n\\]\nWrite $B_{p,n}=\\mathbb Z/p^n\\mathbb Z$. The Chinese remainder isomorphisms\n\\[\n B_n\\cong\\prod_{p\\in\\mathcal P}B_{p,n}\n\\]\nare compatible with reduction. Use the enhanced category $\\mathcal S_n=\\operatorname{Perf}(B_n)$ and the coarsening\n\\[\n C_{m,n}(M)=M\\otimes^{\\mathbf L}_{B_m}B_n,\\qquad m\\geq n.\n\\]\nDerived tensor associativity supplies coherent equivalences $C_{l,n}\\simeq C_{m,n}C_{l,m}$. These are the data of an enhanced indexed system. Literal equalities are not asserted between arbitrarily chosen derived-functor models. On specified bounded projective complexes the operation is termwise reduction. Preservation of perfect complexes under derived base change is standard \\cite{StacksPerfect}; the pointwise cone model also shows exactness directly.","\\begin{theorem}[Nonidentity conservative resolution tower]\\label{thm:adic-tower}\nThe functors $C_{m,n}$ are exact and conservative on perfect complexes. They need not be faithful. In particular, for $m>n$, the nonzero endomorphism $D^n:B_m\\to B_m$ becomes zero under $C_{m,n}$.\n\\end{theorem}\n\\begin{proof}\nExactness follows by reducing bounded projective cone diagrams. For conservativity it suffices, by the Chinese remainder decomposition and composition with reduction to level one, to prove that a perfect complex over $R=\\mathbb Z/p^m$ is zero if its derived reduction to $\\mathbb F_p$ is zero.","vative on perfect complexes. They need not be faithful. In particular, for $m>n$, the nonzero endomorphism $D^n:B_m\\to B_m$ becomes zero under $C_{m,n}$.\n\\end{theorem}\n\\begin{proof}\nExactness follows by reducing bounded projective cone diagrams. For conservativity it suffices, by the Chinese remainder decomposition and composition with reduction to level one, to prove that a perfect complex over $R=\\mathbb Z/p^m$ is zero if its derived reduction to $\\mathbb F_p$ is zero.",". In particular, for $m>n$, the nonzero endomorphism $D^n:B_m\\to B_m$ becomes zero under $C_{m,n}$.\n\\end{theorem}\n\\begin{proof}\nExactness follows by reducing bounded projective cone diagrams. For conservativity it suffices, by the Chinese remainder decomposition and composition with reduction to level one, to prove that a perfect complex over $R=\\mathbb Z/p^m$ is zero if its derived reduction to $\\mathbb F_p$ is zero.","A finite projective module over $R$ is free. Indeed, lift a basis modulo $(p)$ to obtain a surjection $R^r\\to P$. It is surjective because its cokernel $Q$ satisfies $Q=pQ$, whence $Q=p^mQ=0$. The surjection splits, and its kernel has zero reduction modulo $p$, so the same argument kills the kernel. A perfect complex therefore has a bounded finite free representative.","If a differential has a unit entry, elementary changes of bases split off the contractible two-term complex $R\\xrightarrow{1}R$. The adjacent differentials have zero components into and out of that summand because their composites are zero. Repeating strictly decreases the total rank and terminates in a complex all of whose differential entries lie in $(p)$. Its reduction has zero differentials. If that reduction is acyclic, every remaining rank is zero. The original complex is contractible. Applied to a cone, this also proves that reduction reflects equivalences.","Finally $D^n$ is not zero modulo $D^m$, but is zero modulo $D^n$. Since $\\operatorname{Hom}_{D(B_m)}(B_m,B_m)=B_m$, this is genuine loss of morphism information, not just a change of representatives.\n\\end{proof}","\\begin{example}[Why ordinary tensor and cohomology truncation are not substitutes]\\label{ex:derived-not-naive}\nOver $R=\\mathbb Z/p^2$, the exact sequence $0\\to(p)\\to R\\to\\mathbb F_p\\to0$ ceases to be left exact after ordinary tensor with $\\mathbb F_p$: the first map becomes zero. The module $\\mathbb F_p$ has the infinite periodic free resolution with every differential multiplication by $p$. Tensoring that resolution with $\\mathbb F_p$ gives zero differentials, hence $\\operatorname{Tor}_i^R(\\mathbb F_p,\\mathbb F_p)=\\mathbb F_p$ for every $i\\geq0$.","y tensor and cohomology truncation are not substitutes]\\label{ex:derived-not-naive}\nOver $R=\\mathbb Z/p^2$, the exact sequence $0\\to(p)\\to R\\to\\mathbb F_p\\to0$ ceases to be left exact after ordinary tensor with $\\mathbb F_p$: the first map becomes zero. The module $\\mathbb F_p$ has the infinite periodic free resolution with every differential multiplication by $p$. Tensoring that resolution with $\\mathbb F_p$ gives zero differentials, hence $\\operatorname{Tor}_i^R(\\mathbb F_p,\\mathbb F_p)=\\mathbb F_p$ for every $i\\geq0$.","stitutes]\\label{ex:derived-not-naive}\nOver $R=\\mathbb Z/p^2$, the exact sequence $0\\to(p)\\to R\\to\\mathbb F_p\\to0$ ceases to be left exact after ordinary tensor with $\\mathbb F_p$: the first map becomes zero. The module $\\mathbb F_p$ has the infinite periodic free resolution with every differential multiplication by $p$. Tensoring that resolution with $\\mathbb F_p$ gives zero differentials, hence $\\operatorname{Tor}_i^R(\\mathbb F_p,\\mathbb F_p)=\\mathbb F_p$ for every $i\\geq0$.","sequence $0\\to(p)\\to R\\to\\mathbb F_p\\to0$ ceases to be left exact after ordinary tensor with $\\mathbb F_p$: the first map becomes zero. The module $\\mathbb F_p$ has the infinite periodic free resolution with every differential multiplication by $p$. Tensoring that resolution with $\\mathbb F_p$ gives zero differentials, hence $\\operatorname{Tor}_i^R(\\mathbb F_p,\\mathbb F_p)=\\mathbb F_p$ for every $i\\geq0$.","In contrast, $C_R(p)=[R\\xrightarrow p R]$ is perfect, but has both $H^{-1}=\\mathbb F_p$ and $H^0=\\mathbb F_p$. It is not the module $\\mathbb F_p$ placed in degree zero. We retain the actual two-term complex throughout the tower. Derived tensor is computed on these projective representatives, not on a truncated cohomology surrogate \\cite{StacksDerivedTensor}.\n\\end{example}","For a fixed perfect integral complex $L$, put\n\\[\n A_{L,n}(M)=(L\\otimes^{\\mathbf L}_{\\mathbb Z}B_n)\\otimes^{\\mathbf L}_{B_n}M.\n\\]\nBase change gives a natural equivalence\n\\[\n \\theta_{L;m,n}:A_{L,n}C_{m,n}\\simeq C_{m,n}A_{L,m}.\n\\]\nThus the canonical tensor/base-change exchange already has zero residual. The tower alone does not force a defect. To test a noninvertible exchange, we explicitly add a positive integer label $b$ and use the cell\n\\[\n \\eta_{L,b;m,n}=b^{m-n}\\theta_{L;m,n}.\n\\]\nFor scalar unit templates take $L=\\mathbb Z$, so that the transformation functors are identities but the coarsening functors are not.","ntegral complex $L$, put\n\\[\n A_{L,n}(M)=(L\\otimes^{\\mathbf L}_{\\mathbb Z}B_n)\\otimes^{\\mathbf L}_{B_n}M.\n\\]\nBase change gives a natural equivalence\n\\[\n \\theta_{L;m,n}:A_{L,n}C_{m,n}\\simeq C_{m,n}A_{L,m}.\n\\]\nThus the canonical tensor/base-change exchange already has zero residual. The tower alone does not force a defect. To test a noninvertible exchange, we explicitly add a positive integer label $b$ and use the cell\n\\[\n \\eta_{L,b;m,n}=b^{m-n}\\theta_{L;m,n}.\n\\]\nFor scalar unit templates take $L=\\mathbb Z$, so that the transformation functors are identities but the coarsening functors are not.","ut\n\\[\n A_{L,n}(M)=(L\\otimes^{\\mathbf L}_{\\mathbb Z}B_n)\\otimes^{\\mathbf L}_{B_n}M.\n\\]\nBase change gives a natural equivalence\n\\[\n \\theta_{L;m,n}:A_{L,n}C_{m,n}\\simeq C_{m,n}A_{L,m}.\n\\]\nThus the canonical tensor/base-change exchange already has zero residual. The tower alone does not force a defect. To test a noninvertible exchange, we explicitly add a positive integer label $b$ and use the cell\n\\[\n \\eta_{L,b;m,n}=b^{m-n}\\theta_{L;m,n}.\n\\]\nFor scalar unit templates take $L=\\mathbb Z$, so that the transformation functors are identities but the coarsening functors are not.","{B_n}M.\n\\]\nBase change gives a natural equivalence\n\\[\n \\theta_{L;m,n}:A_{L,n}C_{m,n}\\simeq C_{m,n}A_{L,m}.\n\\]\nThus the canonical tensor/base-change exchange already has zero residual. The tower alone does not force a defect. To test a noninvertible exchange, we explicitly add a positive integer label $b$ and use the cell\n\\[\n \\eta_{L,b;m,n}=b^{m-n}\\theta_{L;m,n}.\n\\]\nFor scalar unit templates take $L=\\mathbb Z$, so that the transformation functors are identities but the coarsening functors are not.","\\begin{theorem}[Coherent weighted exchange and residual transport]\\label{thm:weighted-adic-exchange}\nThe cells $\\eta_{L,b;m,n}$ are natural and coherent with composition of resolution changes. Their residual at $M\\in\\mathcal S_m$ is\n\\[\n \\Cofib(\\eta_{L,b;m,n,M})\\simeq\n A_{L,n}C_{m,n}(M)\\otimes^{\\mathbf L}_{B_n}C_{B_n}(b^{m-n}).\n\\]\nFor a fixed scalar $a$, subsequent base change transports its cone exactly:\n\\[\n C_{n,k}\\bigl(N\\otimes^{\\mathbf L}_{B_n}C_{B_n}(a)\\bigr)\n \\simeq C_{n,k}(N)\\otimes^{\\mathbf L}_{B_k}C_{B_k}(a).\n\\]\nAt a common target level $n$, two independent scalar events of positive lengths $d,e$ and labels $a,b$ have interaction\n\\[\n N\\otimes^{\\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\\qquad\n K_R(u,v)=C_R(u)\\otimes^{\\mathbf L}_R C_R(v).\n\\]\nThese formulas also hold with compatible tensor templates absorbed into $N$.","\\begin{theorem}[Coherent weighted exchange and residual transport]\\label{thm:weighted-adic-exchange}\nThe cells $\\eta_{L,b;m,n}$ are natural and coherent with composition of resolution changes. Their residual at $M\\in\\mathcal S_m$ is\n\\[\n \\Cofib(\\eta_{L,b;m,n,M})\\simeq\n A_{L,n}C_{m,n}(M)\\otimes^{\\mathbf L}_{B_n}C_{B_n}(b^{m-n}).\n\\]\nFor a fixed scalar $a$, subsequent base change transports its cone exactly:\n\\[\n C_{n,k}\\bigl(N\\otimes^{\\mathbf L}_{B_n}C_{B_n}(a)\\bigr)\n \\simeq C_{n,k}(N)\\otimes^{\\mathbf L}_{B_k}C_{B_k}(a).\n\\]\nAt a common target level $n$, two independent scalar events of positive lengths $d,e$ and labels $a,b$ have interaction\n\\[\n N\\otimes^{\\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\\qquad\n K_R(u,v)=C_R(u)\\otimes^{\\mathbf L}_R C_R(v).\n\\]\nThese formulas also hold with compatible tensor templates absorbed into $N$.\n\\end{theorem}\n\\begin{proof}\nNaturality follows from central scalar multiplication. For $l\\geq m\\geq n$","\\begin{theorem}[Coherent weighted exchange and residual transport]\\label{thm:weighted-adic-exchange}\nThe cells $\\eta_{L,b;m,n}$ are natural and coherent with composition of resolution changes. Their residual at $M\\in\\mathcal S_m$ is\n\\[\n \\Cofib(\\eta_{L,b;m,n,M})\\simeq\n A_{L,n}C_{m,n}(M)\\otimes^{\\mathbf L}_{B_n}C_{B_n}(b^{m-n}).\n\\]\nFor a fixed scalar $a$, subsequent base change transports its cone exactly:\n\\[\n C_{n,k}\\bigl(N\\otimes^{\\mathbf L}_{B_n}C_{B_n}(a)\\bigr)\n \\simeq C_{n,k}(N)\\otimes^{\\mathbf L}_{B_k}C_{B_k}(a).\n\\]\nAt a common target level $n$, two independent scalar events of positive lengths $d,e$ and labels $a,b$ have interaction\n\\[\n N\\otimes^{\\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\\qquad\n K_R(u,v)=C_R(u)\\otimes^{\\mathbf L}_R C_R(v).\n\\]\nThese formulas also hold with compatible tensor templates absorbed into $N$.\n\\end{theorem}\n\\begin{proof}\nNaturality follows from central scalar multiplication. For $l\\geq m\\geq n$, the two-step cell has scalar $b^{m-n}b^{l-m}=b^{l-n}$. The associativity coherence of derived tenso","\\begin{theorem}[Coherent weighted exchange and residual transport]\\label{thm:weighted-adic-exchange}\nThe cells $\\eta_{L,b;m,n}$ are natural and coherent with composition of resolution changes. Their residual at $M\\in\\mathcal S_m$ is\n\\[\n \\Cofib(\\eta_{L,b;m,n,M})\\simeq\n A_{L,n}C_{m,n}(M)\\otimes^{\\mathbf L}_{B_n}C_{B_n}(b^{m-n}).\n\\]\nFor a fixed scalar $a$, subsequent base change transports its cone exactly:\n\\[\n C_{n,k}\\bigl(N\\otimes^{\\mathbf L}_{B_n}C_{B_n}(a)\\bigr)\n \\simeq C_{n,k}(N)\\otimes^{\\mathbf L}_{B_k}C_{B_k}(a).\n\\]\nAt a common target level $n$, two independent scalar events of positive lengths $d,e$ and labels $a,b$ have interaction\n\\[\n N\\otimes^{\\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\\qquad\n K_R(u,v)=C_R(u)\\otimes^{\\mathbf L}_R C_R(v).\n\\]\nThese formulas also hold with compatible tensor templates absorbed into $N$.\n\\end{theorem}\n\\begin{proof}\nNaturality follows from central scalar multiplication. For $l\\geq m\\geq n$, the two-step cell has scalar $b^{m-n}b^{l-m}=b^{l-n}$. The associativity coherence of derived tensor identifies its underly","orem}[Coherent weighted exchange and residual transport]\\label{thm:weighted-adic-exchange}\nThe cells $\\eta_{L,b;m,n}$ are natural and coherent with composition of resolution changes. Their residual at $M\\in\\mathcal S_m$ is\n\\[\n \\Cofib(\\eta_{L,b;m,n,M})\\simeq\n A_{L,n}C_{m,n}(M)\\otimes^{\\mathbf L}_{B_n}C_{B_n}(b^{m-n}).\n\\]\nFor a fixed scalar $a$, subsequent base change transports its cone exactly:\n\\[\n C_{n,k}\\bigl(N\\otimes^{\\mathbf L}_{B_n}C_{B_n}(a)\\bigr)\n \\simeq C_{n,k}(N)\\otimes^{\\mathbf L}_{B_k}C_{B_k}(a).\n\\]\nAt a common target level $n$, two independent scalar events of positive lengths $d,e$ and labels $a,b$ have interaction\n\\[\n N\\otimes^{\\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\\qquad\n K_R(u,v)=C_R(u)\\otimes^{\\mathbf L}_R C_R(v).\n\\]\nThese formulas also hold with compatible tensor templates absorbed into $N$.\n\\end{theorem}\n\\begin{proof}\nNaturality follows from central scalar multiplication. For $l\\geq m\\geq n$, the two-step cell has scalar $b^{m-n}b^{l-m}=b^{l-n}$. The associativity coherence of derived tensor identifies its underlying $\\theta$ with $\\theta_{L;l,n}$. Identity intervals ha…","with composition of resolution changes. Their residual at $M\\in\\mathcal S_m$ is\n\\[\n \\Cofib(\\eta_{L,b;m,n,M})\\simeq\n A_{L,n}C_{m,n}(M)\\otimes^{\\mathbf L}_{B_n}C_{B_n}(b^{m-n}).\n\\]\nFor a fixed scalar $a$, subsequent base change transports its cone exactly:\n\\[\n C_{n,k}\\bigl(N\\otimes^{\\mathbf L}_{B_n}C_{B_n}(a)\\bigr)\n \\simeq C_{n,k}(N)\\otimes^{\\mathbf L}_{B_k}C_{B_k}(a).\n\\]\nAt a common target level $n$, two independent scalar events of positive lengths $d,e$ and labels $a,b$ have interaction\n\\[\n N\\otimes^{\\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\\qquad\n K_R(u,v)=C_R(u)\\otimes^{\\mathbf L}_R C_R(v).\n\\]\nThese formulas also hold with compatible tensor templates absorbed into $N$.\n\\end{theorem}\n\\begin{proof}\nNaturality follows from central scalar multiplication. For $l\\geq m\\geq n$, the two-step cell has scalar $b^{m-n}b^{l-m}=b^{l-n}$. The associativity coherence of derived tensor identifies its underlying $\\theta$ with $\\theta_{L;l,n}$. Identity intervals have scalar one. Tensoring transformation templates gives their usual associative action, with scalar weights multiplied; no additional cells are","at $M\\in\\mathcal S_m$ is\n\\[\n \\Cofib(\\eta_{L,b;m,n,M})\\simeq\n A_{L,n}C_{m,n}(M)\\otimes^{\\mathbf L}_{B_n}C_{B_n}(b^{m-n}).\n\\]\nFor a fixed scalar $a$, subsequent base change transports its cone exactly:\n\\[\n C_{n,k}\\bigl(N\\otimes^{\\mathbf L}_{B_n}C_{B_n}(a)\\bigr)\n \\simeq C_{n,k}(N)\\otimes^{\\mathbf L}_{B_k}C_{B_k}(a).\n\\]\nAt a common target level $n$, two independent scalar events of positive lengths $d,e$ and labels $a,b$ have interaction\n\\[\n N\\otimes^{\\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\\qquad\n K_R(u,v)=C_R(u)\\otimes^{\\mathbf L}_R C_R(v).\n\\]\nThese formulas also hold with compatible tensor templates absorbed into $N$.\n\\end{theorem}\n\\begin{proof}\nNaturality follows from central scalar multiplication. For $l\\geq m\\geq n$, the two-step cell has scalar $b^{m-n}b^{l-m}=b^{l-n}$. The associativity coherence of derived tensor identifies its underlying $\\theta$ with $\\theta_{L;l,n}$. Identity intervals have scalar one. Tensoring transformation templates gives their usual associative action, with scalar weights multiplied; no additional cells are inferred without these specified data.","S_m$ is\n\\[\n \\Cofib(\\eta_{L,b;m,n,M})\\simeq\n A_{L,n}C_{m,n}(M)\\otimes^{\\mathbf L}_{B_n}C_{B_n}(b^{m-n}).\n\\]\nFor a fixed scalar $a$, subsequent base change transports its cone exactly:\n\\[\n C_{n,k}\\bigl(N\\otimes^{\\mathbf L}_{B_n}C_{B_n}(a)\\bigr)\n \\simeq C_{n,k}(N)\\otimes^{\\mathbf L}_{B_k}C_{B_k}(a).\n\\]\nAt a common target level $n$, two independent scalar events of positive lengths $d,e$ and labels $a,b$ have interaction\n\\[\n N\\otimes^{\\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\\qquad\n K_R(u,v)=C_R(u)\\otimes^{\\mathbf L}_R C_R(v).\n\\]\nThese formulas also hold with compatible tensor templates absorbed into $N$.\n\\end{theorem}\n\\begin{proof}\nNaturality follows from central scalar multiplication. For $l\\geq m\\geq n$, the two-step cell has scalar $b^{m-n}b^{l-m}=b^{l-n}$. The associativity coherence of derived tensor identifies its underlying $\\theta$ with $\\theta_{L;l,n}$. Identity intervals have scalar one. Tensoring transformation templates gives their usual associative action, with scalar weights multiplied; no additional cells are inferred without these specified data.","_{L,b;m,n,M})\\simeq\n A_{L,n}C_{m,n}(M)\\otimes^{\\mathbf L}_{B_n}C_{B_n}(b^{m-n}).\n\\]\nFor a fixed scalar $a$, subsequent base change transports its cone exactly:\n\\[\n C_{n,k}\\bigl(N\\otimes^{\\mathbf L}_{B_n}C_{B_n}(a)\\bigr)\n \\simeq C_{n,k}(N)\\otimes^{\\mathbf L}_{B_k}C_{B_k}(a).\n\\]\nAt a common target level $n$, two independent scalar events of positive lengths $d,e$ and labels $a,b$ have interaction\n\\[\n N\\otimes^{\\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\\qquad\n K_R(u,v)=C_R(u)\\otimes^{\\mathbf L}_R C_R(v).\n\\]\nThese formulas also hold with compatible tensor templates absorbed into $N$.\n\\end{theorem}\n\\begin{proof}\nNaturality follows from central scalar multiplication. For $l\\geq m\\geq n$, the two-step cell has scalar $b^{m-n}b^{l-m}=b^{l-n}$. The associativity coherence of derived tensor identifies its underlying $\\theta$ with $\\theta_{L;l,n}$. Identity intervals have scalar one. Tensoring transformation templates gives their usual associative action, with scalar weights multiplied; no additional cells are inferred without these specified data.","gl(N\\otimes^{\\mathbf L}_{B_n}C_{B_n}(a)\\bigr)\n \\simeq C_{n,k}(N)\\otimes^{\\mathbf L}_{B_k}C_{B_k}(a).\n\\]\nAt a common target level $n$, two independent scalar events of positive lengths $d,e$ and labels $a,b$ have interaction\n\\[\n N\\otimes^{\\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\\qquad\n K_R(u,v)=C_R(u)\\otimes^{\\mathbf L}_R C_R(v).\n\\]\nThese formulas also hold with compatible tensor templates absorbed into $N$.\n\\end{theorem}\n\\begin{proof}\nNaturality follows from central scalar multiplication. For $l\\geq m\\geq n$, the two-step cell has scalar $b^{m-n}b^{l-m}=b^{l-n}$. The associativity coherence of derived tensor identifies its underlying $\\theta$ with $\\theta_{L;l,n}$. Identity intervals have scalar one. Tensoring transformation templates gives their usual associative action, with scalar weights multiplied; no additional cells are inferred without these specified data.","B_k}(a).\n\\]\nAt a common target level $n$, two independent scalar events of positive lengths $d,e$ and labels $a,b$ have interaction\n\\[\n N\\otimes^{\\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\\qquad\n K_R(u,v)=C_R(u)\\otimes^{\\mathbf L}_R C_R(v).\n\\]\nThese formulas also hold with compatible tensor templates absorbed into $N$.\n\\end{theorem}\n\\begin{proof}\nNaturality follows from central scalar multiplication. For $l\\geq m\\geq n$, the two-step cell has scalar $b^{m-n}b^{l-m}=b^{l-n}$. The associativity coherence of derived tensor identifies its underlying $\\theta$ with $\\theta_{L;l,n}$. Identity intervals have scalar one. Tensoring transformation templates gives their usual associative action, with scalar weights multiplied; no additional cells are inferred without these specified data.","independent scalar events of positive lengths $d,e$ and labels $a,b$ have interaction\n\\[\n N\\otimes^{\\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\\qquad\n K_R(u,v)=C_R(u)\\otimes^{\\mathbf L}_R C_R(v).\n\\]\nThese formulas also hold with compatible tensor templates absorbed into $N$.\n\\end{theorem}\n\\begin{proof}\nNaturality follows from central scalar multiplication. For $l\\geq m\\geq n$, the two-step cell has scalar $b^{m-n}b^{l-m}=b^{l-n}$. The associativity coherence of derived tensor identifies its underlying $\\theta$ with $\\theta_{L;l,n}$. Identity intervals have scalar one. Tensoring transformation templates gives their usual associative action, with scalar weights multiplied; no additional cells are inferred without these specified data.","^{\\mathbf L}_{B_n}K_{B_n}(a^d,b^e),\\qquad\n K_R(u,v)=C_R(u)\\otimes^{\\mathbf L}_R C_R(v).\n\\]\nThese formulas also hold with compatible tensor templates absorbed into $N$.\n\\end{theorem}\n\\begin{proof}\nNaturality follows from central scalar multiplication. For $l\\geq m\\geq n$, the two-step cell has scalar $b^{m-n}b^{l-m}=b^{l-n}$. The associativity coherence of derived tensor identifies its underlying $\\theta$ with $\\theta_{L;l,n}$. Identity intervals have scalar one. Tensoring transformation templates gives their usual associative action, with scalar weights multiplied; no additional cells are inferred without these specified data.","B_n}K_{B_n}(a^d,b^e),\\qquad\n K_R(u,v)=C_R(u)\\otimes^{\\mathbf L}_R C_R(v).\n\\]\nThese formulas also hold with compatible tensor templates absorbed into $N$.\n\\end{theorem}\n\\begin{proof}\nNaturality follows from central scalar multiplication. For $l\\geq m\\geq n$, the two-step cell has scalar $b^{m-n}b^{l-m}=b^{l-n}$. The associativity coherence of derived tensor identifies its underlying $\\theta$ with $\\theta_{L;l,n}$. Identity intervals have scalar one. Tensoring transformation templates gives their usual associative action, with scalar weights multiplied; no additional cells are inferred without these specified data.","After using $\\theta$ to identify the target, the exchange is multiplication by $b^{m-n}$ on $A_{L,n}C_{m,n}M$. Exact tensor identifies its cone with the first formula. Associativity of derived tensor proves the second. The two-variable interaction formula follows by taking the two cofibers, as in \\cref{thm:tensor-cube}. For a pure adjacent-swap word, tensor templates keep their relative order and all coarsenings combine into the same final reduction. Identifying those endpoints by the supplied tensor associativities leaves precisely the products of the scalar cells. This constructs the corresponding coherent scalar event diagram at the target level.\n\\end{proof}","\\begin{theorem}[Prime-power interaction cohomology]\\label{thm:prime-power-koszul}\nLet $R=\\mathbb Z/p^n$, let $a_1,\\ldots,a_t$ be positive integers with $t\\geq1$, and put\n\\[\n h=\\min\\{n,v_p(a_1),\\ldots,v_p(a_t)\\},\\qquad Q_h=R/(p^h).\n\\]\nHere $Q_0=0$ and $Q_n=R$. There is a generally noncanonical chain isomorphism\n\\[\n K_R(a_1,\\ldots,a_t)\\cong\n \\bigoplus_{j=0}^{t-1}C_R(p^h)[j]^{\\oplus\\binom{t-1}{j}}.\n\\]\nConsequently\n\\[\n \\boxed{H^{-j}K_R(a_1,\\ldots,a_t)\\cong Q_h^{\\oplus\\binom tj}\n \\quad(0\\leq j\\leq t),}\n\\]\nand all other cohomology groups vanish. The complex is zero in $D(R)$ exactly when at least one label is a unit modulo $p$.\n\\end{theorem}\n\\begin{proof}\nIf $h=n$, all entries are zero in $R$. Otherwise choose an entry of minimal valuation, scale its unit part away, and subtract suitable multiples","\\begin{theorem}[Prime-power interaction cohomology]\\label{thm:prime-power-koszul}\nLet $R=\\mathbb Z/p^n$, let $a_1,\\ldots,a_t$ be positive integers with $t\\geq1$, and put\n\\[\n h=\\min\\{n,v_p(a_1),\\ldots,v_p(a_t)\\},\\qquad Q_h=R/(p^h).\n\\]\nHere $Q_0=0$ and $Q_n=R$. There is a generally noncanonical chain isomorphism\n\\[\n K_R(a_1,\\ldots,a_t)\\cong\n \\bigoplus_{j=0}^{t-1}C_R(p^h)[j]^{\\oplus\\binom{t-1}{j}}.\n\\]\nConsequently\n\\[\n \\boxed{H^{-j}K_R(a_1,\\ldots,a_t)\\cong Q_h^{\\oplus\\binom tj}\n \\quad(0\\leq j\\leq t),}\n\\]\nand all other cohomology groups vanish. The complex is zero in $D(R)$ exactly when at least one label is a unit modulo $p$.\n\\end{theorem}\n\\begin{proof}\nIf $h=n$, all entries are zero in $R$. Otherwise choose an entry of minimal valuation, scale its unit part away, and subtract suitable multiples of it from the other","\\begin{theorem}[Prime-power interaction cohomology]\\label{thm:prime-power-koszul}\nLet $R=\\mathbb Z/p^n$, let $a_1,\\ldots,a_t$ be positive integers with $t\\geq1$, and put\n\\[\n h=\\min\\{n,v_p(a_1),\\ldots,v_p(a_t)\\},\\qquad Q_h=R/(p^h).\n\\]\nHere $Q_0=0$ and $Q_n=R$. There is a generally noncanonical chain isomorphism\n\\[\n K_R(a_1,\\ldots,a_t)\\cong\n \\bigoplus_{j=0}^{t-1}C_R(p^h)[j]^{\\oplus\\binom{t-1}{j}}.\n\\]\nConsequently\n\\[\n \\boxed{H^{-j}K_R(a_1,\\ldots,a_t)\\cong Q_h^{\\oplus\\binom tj}\n \\quad(0\\leq j\\leq t),}\n\\]\nand all other cohomology groups vanish. The complex is zero in $D(R)$ exactly when at least one label is a unit modulo $p$.\n\\end{theorem}\n\\begin{proof}\nIf $h=n$, all entries are zero in $R$. Otherwise choose an entry of minimal valuation, scale its unit part away, and subtract suitable multiples of it from the other entries. These operations form an i","\\begin{theorem}[Prime-power interaction cohomology]\\label{thm:prime-power-koszul}\nLet $R=\\mathbb Z/p^n$, let $a_1,\\ldots,a_t$ be positive integers with $t\\geq1$, and put\n\\[\n h=\\min\\{n,v_p(a_1),\\ldots,v_p(a_t)\\},\\qquad Q_h=R/(p^h).\n\\]\nHere $Q_0=0$ and $Q_n=R$. There is a generally noncanonical chain isomorphism\n\\[\n K_R(a_1,\\ldots,a_t)\\cong\n \\bigoplus_{j=0}^{t-1}C_R(p^h)[j]^{\\oplus\\binom{t-1}{j}}.\n\\]\nConsequently\n\\[\n \\boxed{H^{-j}K_R(a_1,\\ldots,a_t)\\cong Q_h^{\\oplus\\binom tj}\n \\quad(0\\leq j\\leq t),}\n\\]\nand all other cohomology groups vanish. The complex is zero in $D(R)$ exactly when at least one label is a unit modulo $p$.\n\\end{theorem}\n\\begin{proof}\nIf $h=n$, all entries are zero in $R$. Otherwise choose an entry of minimal valuation, scale its unit part away, and subtract suitable multiples of it from the other entries. These operations form an invertible change of the coefficient row and produce $(p^h,0,\\ldots,0)$.","\\begin{theorem}[Prime-power interaction cohomology]\\label{thm:prime-power-koszul}\nLet $R=\\mathbb Z/p^n$, let $a_1,\\ldots,a_t$ be positive integers with $t\\geq1$, and put\n\\[\n h=\\min\\{n,v_p(a_1),\\ldots,v_p(a_t)\\},\\qquad Q_h=R/(p^h).\n\\]\nHere $Q_0=0$ and $Q_n=R$. There is a generally noncanonical chain isomorphism\n\\[\n K_R(a_1,\\ldots,a_t)\\cong\n \\bigoplus_{j=0}^{t-1}C_R(p^h)[j]^{\\oplus\\binom{t-1}{j}}.\n\\]\nConsequently\n\\[\n \\boxed{H^{-j}K_R(a_1,\\ldots,a_t)\\cong Q_h^{\\oplus\\binom tj}\n \\quad(0\\leq j\\leq t),}\n\\]\nand all other cohomology groups vanish. The complex is zero in $D(R)$ exactly when at least one label is a unit modulo $p$.\n\\end{theorem}\n\\begin{proof}\nIf $h=n$, all entries are zero in $R$. Otherwise choose an entry of minimal valuation, scale its unit part away, and subtract suitable multiples of it from the other entries. These operations form an invertible change of the coefficient row and produce $(p^h,0,\\ldots,0)$. The exterior p","\\begin{theorem}[Prime-power interaction cohomology]\\label{thm:prime-power-koszul}\nLet $R=\\mathbb Z/p^n$, let $a_1,\\ldots,a_t$ be positive integers with $t\\geq1$, and put\n\\[\n h=\\min\\{n,v_p(a_1),\\ldots,v_p(a_t)\\},\\qquad Q_h=R/(p^h).\n\\]\nHere $Q_0=0$ and $Q_n=R$. There is a generally noncanonical chain isomorphism\n\\[\n K_R(a_1,\\ldots,a_t)\\cong\n \\bigoplus_{j=0}^{t-1}C_R(p^h)[j]^{\\oplus\\binom{t-1}{j}}.\n\\]\nConsequently\n\\[\n \\boxed{H^{-j}K_R(a_1,\\ldots,a_t)\\cong Q_h^{\\oplus\\binom tj}\n \\quad(0\\leq j\\leq t),}\n\\]\nand all other cohomology groups vanish. The complex is zero in $D(R)$ exactly when at least one label is a unit modulo $p$.\n\\end{theorem}\n\\begin{proof}\nIf $h=n$, all entries are zero in $R$. Otherwise choose an entry of minimal valuation, scale its unit part away, and subtract suitable multiples of it from the other entries. These operations form an invertible change of the coefficient row and produce $(p^h,0,\\ldots,0)$. The exterior powers of the","\\begin{theorem}[Prime-power interaction cohomology]\\label{thm:prime-power-koszul}\nLet $R=\\mathbb Z/p^n$, let $a_1,\\ldots,a_t$ be positive integers with $t\\geq1$, and put\n\\[\n h=\\min\\{n,v_p(a_1),\\ldots,v_p(a_t)\\},\\qquad Q_h=R/(p^h).\n\\]\nHere $Q_0=0$ and $Q_n=R$. There is a generally noncanonical chain isomorphism\n\\[\n K_R(a_1,\\ldots,a_t)\\cong\n \\bigoplus_{j=0}^{t-1}C_R(p^h)[j]^{\\oplus\\binom{t-1}{j}}.\n\\]\nConsequently\n\\[\n \\boxed{H^{-j}K_R(a_1,\\ldots,a_t)\\cong Q_h^{\\oplus\\binom tj}\n \\quad(0\\leq j\\leq t),}\n\\]\nand all other cohomology groups vanish. The complex is zero in $D(R)$ exactly when at least one label is a unit modulo $p$.\n\\end{theorem}\n\\begin{proof}\nIf $h=n$, all entries are zero in $R$. Otherwise choose an entry of minimal valuation, scale its unit part away, and subtract suitable multiples of it from the other entries. These operations form an invertible change of the coefficient row and produce $(p^h,0,\\ldots,0)$. The exterior powers of the basis change give a Koszul chain isomorphism, with the convention of \\cref{thm:integral-gcd} and \\cite{StacksKoszul}.","rime-power interaction cohomology]\\label{thm:prime-power-koszul}\nLet $R=\\mathbb Z/p^n$, let $a_1,\\ldots,a_t$ be positive integers with $t\\geq1$, and put\n\\[\n h=\\min\\{n,v_p(a_1),\\ldots,v_p(a_t)\\},\\qquad Q_h=R/(p^h).\n\\]\nHere $Q_0=0$ and $Q_n=R$. There is a generally noncanonical chain isomorphism\n\\[\n K_R(a_1,\\ldots,a_t)\\cong\n \\bigoplus_{j=0}^{t-1}C_R(p^h)[j]^{\\oplus\\binom{t-1}{j}}.\n\\]\nConsequently\n\\[\n \\boxed{H^{-j}K_R(a_1,\\ldots,a_t)\\cong Q_h^{\\oplus\\binom tj}\n \\quad(0\\leq j\\leq t),}\n\\]\nand all other cohomology groups vanish. The complex is zero in $D(R)$ exactly when at least one label is a unit modulo $p$.\n\\end{theorem}\n\\begin{proof}\nIf $h=n$, all entries are zero in $R$. Otherwise choose an entry of minimal valuation, scale its unit part away, and subtract suitable multiples of it from the other entries. These operations form an invertible change of the coefficient row and produce $(p^h,0,\\ldots,0)$. The exterior powers of the basis change give a Koszul chain isomorphism, with the convention of \\cref{thm:integral-gcd} and \\cite{StacksKoszul}.","\\[\n h=\\min\\{n,v_p(a_1),\\ldots,v_p(a_t)\\},\\qquad Q_h=R/(p^h).\n\\]\nHere $Q_0=0$ and $Q_n=R$. There is a generally noncanonical chain isomorphism\n\\[\n K_R(a_1,\\ldots,a_t)\\cong\n \\bigoplus_{j=0}^{t-1}C_R(p^h)[j]^{\\oplus\\binom{t-1}{j}}.\n\\]\nConsequently\n\\[\n \\boxed{H^{-j}K_R(a_1,\\ldots,a_t)\\cong Q_h^{\\oplus\\binom tj}\n \\quad(0\\leq j\\leq t),}\n\\]\nand all other cohomology groups vanish. The complex is zero in $D(R)$ exactly when at least one label is a unit modulo $p$.\n\\end{theorem}\n\\begin{proof}\nIf $h=n$, all entries are zero in $R$. Otherwise choose an entry of minimal valuation, scale its unit part away, and subtract suitable multiples of it from the other entries. These operations form an invertible change of the coefficient row and produce $(p^h,0,\\ldots,0)$. The exterior powers of the basis change give a Koszul chain isomorphism, with the convention of \\cref{thm:integral-gcd} and \\cite{StacksKoszul}.","^h).\n\\]\nHere $Q_0=0$ and $Q_n=R$. There is a generally noncanonical chain isomorphism\n\\[\n K_R(a_1,\\ldots,a_t)\\cong\n \\bigoplus_{j=0}^{t-1}C_R(p^h)[j]^{\\oplus\\binom{t-1}{j}}.\n\\]\nConsequently\n\\[\n \\boxed{H^{-j}K_R(a_1,\\ldots,a_t)\\cong Q_h^{\\oplus\\binom tj}\n \\quad(0\\leq j\\leq t),}\n\\]\nand all other cohomology groups vanish. The complex is zero in $D(R)$ exactly when at least one label is a unit modulo $p$.\n\\end{theorem}\n\\begin{proof}\nIf $h=n$, all entries are zero in $R$. Otherwise choose an entry of minimal valuation, scale its unit part away, and subtract suitable multiples of it from the other entries. These operations form an invertible change of the coefficient row and produce $(p^h,0,\\ldots,0)$. The exterior powers of the basis change give a Koszul chain isomorphism, with the convention of \\cref{thm:integral-gcd} and \\cite{StacksKoszul}.","ere is a generally noncanonical chain isomorphism\n\\[\n K_R(a_1,\\ldots,a_t)\\cong\n \\bigoplus_{j=0}^{t-1}C_R(p^h)[j]^{\\oplus\\binom{t-1}{j}}.\n\\]\nConsequently\n\\[\n \\boxed{H^{-j}K_R(a_1,\\ldots,a_t)\\cong Q_h^{\\oplus\\binom tj}\n \\quad(0\\leq j\\leq t),}\n\\]\nand all other cohomology groups vanish. The complex is zero in $D(R)$ exactly when at least one label is a unit modulo $p$.\n\\end{theorem}\n\\begin{proof}\nIf $h=n$, all entries are zero in $R$. Otherwise choose an entry of minimal valuation, scale its unit part away, and subtract suitable multiples of it from the other entries. These operations form an invertible change of the coefficient row and produce $(p^h,0,\\ldots,0)$. The exterior powers of the basis change give a Koszul chain isomorphism, with the convention of \\cref{thm:integral-gcd} and \\cite{StacksKoszul}.","chain isomorphism\n\\[\n K_R(a_1,\\ldots,a_t)\\cong\n \\bigoplus_{j=0}^{t-1}C_R(p^h)[j]^{\\oplus\\binom{t-1}{j}}.\n\\]\nConsequently\n\\[\n \\boxed{H^{-j}K_R(a_1,\\ldots,a_t)\\cong Q_h^{\\oplus\\binom tj}\n \\quad(0\\leq j\\leq t),}\n\\]\nand all other cohomology groups vanish. The complex is zero in $D(R)$ exactly when at least one label is a unit modulo $p$.\n\\end{theorem}\n\\begin{proof}\nIf $h=n$, all entries are zero in $R$. Otherwise choose an entry of minimal valuation, scale its unit part away, and subtract suitable multiples of it from the other entries. These operations form an invertible change of the coefficient row and produce $(p^h,0,\\ldots,0)$. The exterior powers of the basis change give a Koszul chain isomorphism, with the convention of \\cref{thm:integral-gcd} and \\cite{StacksKoszul}.","l other cohomology groups vanish. The complex is zero in $D(R)$ exactly when at least one label is a unit modulo $p$.\n\\end{theorem}\n\\begin{proof}\nIf $h=n$, all entries are zero in $R$. Otherwise choose an entry of minimal valuation, scale its unit part away, and subtract suitable multiples of it from the other entries. These operations form an invertible change of the coefficient row and produce $(p^h,0,\\ldots,0)$. The exterior powers of the basis change give a Koszul chain isomorphism, with the convention of \\cref{thm:integral-gcd} and \\cite{StacksKoszul}.","Each zero entry contributes $R\\oplus R[1]$. This proves the complex decomposition. Multiplication by $p^h$ on $R$ has cokernel $Q_h$ and kernel $p^{n-h}R$, the latter isomorphic to $Q_h$ by multiplication by $p^{n-h}$. Hence $C_R(p^h)$ has $Q_h$ in degrees $-1$ and $0$. Counting the shifted copies and using Pascal's identity gives the formula. The unit case is contractible, and otherwise $H^0=Q_h\\ne0$.\n\\end{proof}","For two events the cohomology ranks over $Q_h$ are $1,2,1$ in degrees $-2,-1,0$, rather than the $0,1,1$ pattern over the integral or complete domain. These extra classes are real finite-level cohomology. They may not be discarded while retaining the same complex. In particular, a cohomology group appearing here is not automatically a perfect $R$-module.","\\begin{theorem}[Nilpotent interaction forces excisive collapse]\\label{thm:nilpotent-collapse}\nFor any commutative ring $R$ and nilpotent $x\\in R$, the thick subcategory generated by $C_R(x)$ is all of $\\operatorname{Perf}(R)$. Therefore, over $R=\\mathbb Z/p^n$, forcing the interaction of two nonunit scalar labels to vanish in an exact quotient forces that quotient to be zero.\n\\end{theorem}\n\\begin{proof}\nChoose $N$ with $x^N=0$. For composable scalar multiplications, the octahedral cofiber relation builds $C_R(x^k)$ from $C_R(x^{k-1})$ and $C_R(x)$. Induction puts $C_R(x^N)$ in $\\operatorname{Thick}(C_R(x))$. But\n\\[\n C_R(x^N)=C_R(0)\\cong R\\oplus R[1].\n\\]\nRetract closure therefore includes $R$, and $R$ thickly generates $\\operatorname{Perf}(R)$. This cone-of-a-product mechanism is also part of the standard Koszu","\\begin{theorem}[Nilpotent interaction forces excisive collapse]\\label{thm:nilpotent-collapse}\nFor any commutative ring $R$ and nilpotent $x\\in R$, the thick subcategory generated by $C_R(x)$ is all of $\\operatorname{Perf}(R)$. Therefore, over $R=\\mathbb Z/p^n$, forcing the interaction of two nonunit scalar labels to vanish in an exact quotient forces that quotient to be zero.\n\\end{theorem}\n\\begin{proof}\nChoose $N$ with $x^N=0$. For composable scalar multiplications, the octahedral cofiber relation builds $C_R(x^k)$ from $C_R(x^{k-1})$ and $C_R(x)$. Induction puts $C_R(x^N)$ in $\\operatorname{Thick}(C_R(x))$. But\n\\[\n C_R(x^N)=C_R(0)\\cong R\\oplus R[1].\n\\]\nRetract closure therefore includes $R$, and $R$ thickly generates $\\operatorname{Perf}(R)$. This cone-of-a-product mechanism is also part of the standard Koszul calculus \\cite{Stacks","\\begin{theorem}[Nilpotent interaction forces excisive collapse]\\label{thm:nilpotent-collapse}\nFor any commutative ring $R$ and nilpotent $x\\in R$, the thick subcategory generated by $C_R(x)$ is all of $\\operatorname{Perf}(R)$. Therefore, over $R=\\mathbb Z/p^n$, forcing the interaction of two nonunit scalar labels to vanish in an exact quotient forces that quotient to be zero.\n\\end{theorem}\n\\begin{proof}\nChoose $N$ with $x^N=0$. For composable scalar multiplications, the octahedral cofiber relation builds $C_R(x^k)$ from $C_R(x^{k-1})$ and $C_R(x)$. Induction puts $C_R(x^N)$ in $\\operatorname{Thick}(C_R(x))$. But\n\\[\n C_R(x^N)=C_R(0)\\cong R\\oplus R[1].\n\\]\nRetract closure therefore includes $R$, and $R$ thickly generates $\\operatorname{Perf}(R)$. This cone-of-a-product mechanism is also part of the standard Koszul calculus \\cite{StacksKoszul}.","[Nilpotent interaction forces excisive collapse]\\label{thm:nilpotent-collapse}\nFor any commutative ring $R$ and nilpotent $x\\in R$, the thick subcategory generated by $C_R(x)$ is all of $\\operatorname{Perf}(R)$. Therefore, over $R=\\mathbb Z/p^n$, forcing the interaction of two nonunit scalar labels to vanish in an exact quotient forces that quotient to be zero.\n\\end{theorem}\n\\begin{proof}\nChoose $N$ with $x^N=0$. For composable scalar multiplications, the octahedral cofiber relation builds $C_R(x^k)$ from $C_R(x^{k-1})$ and $C_R(x)$. Induction puts $C_R(x^N)$ in $\\operatorname{Thick}(C_R(x))$. But\n\\[\n C_R(x^N)=C_R(0)\\cong R\\oplus R[1].\n\\]\nRetract closure therefore includes $R$, and $R$ thickly generates $\\operatorname{Perf}(R)$. This cone-of-a-product mechanism is also part of the standard Koszul calculus \\cite{StacksKoszul}.","t interaction forces excisive collapse]\\label{thm:nilpotent-collapse}\nFor any commutative ring $R$ and nilpotent $x\\in R$, the thick subcategory generated by $C_R(x)$ is all of $\\operatorname{Perf}(R)$. Therefore, over $R=\\mathbb Z/p^n$, forcing the interaction of two nonunit scalar labels to vanish in an exact quotient forces that quotient to be zero.\n\\end{theorem}\n\\begin{proof}\nChoose $N$ with $x^N=0$. For composable scalar multiplications, the octahedral cofiber relation builds $C_R(x^k)$ from $C_R(x^{k-1})$ and $C_R(x)$. Induction puts $C_R(x^N)$ in $\\operatorname{Thick}(C_R(x))$. But\n\\[\n C_R(x^N)=C_R(0)\\cong R\\oplus R[1].\n\\]\nRetract closure therefore includes $R$, and $R$ thickly generates $\\operatorname{Perf}(R)$. This cone-of-a-product mechanism is also part of the standard Koszul calculus \\cite{StacksKoszul}.","ve ring $R$ and nilpotent $x\\in R$, the thick subcategory generated by $C_R(x)$ is all of $\\operatorname{Perf}(R)$. Therefore, over $R=\\mathbb Z/p^n$, forcing the interaction of two nonunit scalar labels to vanish in an exact quotient forces that quotient to be zero.\n\\end{theorem}\n\\begin{proof}\nChoose $N$ with $x^N=0$. For composable scalar multiplications, the octahedral cofiber relation builds $C_R(x^k)$ from $C_R(x^{k-1})$ and $C_R(x)$. Induction puts $C_R(x^N)$ in $\\operatorname{Thick}(C_R(x))$. But\n\\[\n C_R(x^N)=C_R(0)\\cong R\\oplus R[1].\n\\]\nRetract closure therefore includes $R$, and $R$ thickly generates $\\operatorname{Perf}(R)$. This cone-of-a-product mechanism is also part of the standard Koszul calculus \\cite{StacksKoszul}.","nilpotent $x\\in R$, the thick subcategory generated by $C_R(x)$ is all of $\\operatorname{Perf}(R)$. Therefore, over $R=\\mathbb Z/p^n$, forcing the interaction of two nonunit scalar labels to vanish in an exact quotient forces that quotient to be zero.\n\\end{theorem}\n\\begin{proof}\nChoose $N$ with $x^N=0$. For composable scalar multiplications, the octahedral cofiber relation builds $C_R(x^k)$ from $C_R(x^{k-1})$ and $C_R(x)$. Induction puts $C_R(x^N)$ in $\\operatorname{Thick}(C_R(x))$. But\n\\[\n C_R(x^N)=C_R(0)\\cong R\\oplus R[1].\n\\]\nRetract closure therefore includes $R$, and $R$ thickly generates $\\operatorname{Perf}(R)$. This cone-of-a-product mechanism is also part of the standard Koszul calculus \\cite{StacksKoszul}.","the thick subcategory generated by $C_R(x)$ is all of $\\operatorname{Perf}(R)$. Therefore, over $R=\\mathbb Z/p^n$, forcing the interaction of two nonunit scalar labels to vanish in an exact quotient forces that quotient to be zero.\n\\end{theorem}\n\\begin{proof}\nChoose $N$ with $x^N=0$. For composable scalar multiplications, the octahedral cofiber relation builds $C_R(x^k)$ from $C_R(x^{k-1})$ and $C_R(x)$. Induction puts $C_R(x^N)$ in $\\operatorname{Thick}(C_R(x))$. But\n\\[\n C_R(x^N)=C_R(0)\\cong R\\oplus R[1].\n\\]\nRetract closure therefore includes $R$, and $R$ thickly generates $\\operatorname{Perf}(R)$. This cone-of-a-product mechanism is also part of the standard Koszul calculus \\cite{StacksKoszul}.","nerated by $C_R(x)$ is all of $\\operatorname{Perf}(R)$. Therefore, over $R=\\mathbb Z/p^n$, forcing the interaction of two nonunit scalar labels to vanish in an exact quotient forces that quotient to be zero.\n\\end{theorem}\n\\begin{proof}\nChoose $N$ with $x^N=0$. For composable scalar multiplications, the octahedral cofiber relation builds $C_R(x^k)$ from $C_R(x^{k-1})$ and $C_R(x)$. Induction puts $C_R(x^N)$ in $\\operatorname{Thick}(C_R(x))$. But\n\\[\n C_R(x^N)=C_R(0)\\cong R\\oplus R[1].\n\\]\nRetract closure therefore includes $R$, and $R$ thickly generates $\\operatorname{Perf}(R)$. This cone-of-a-product mechanism is also part of the standard Koszul calculus \\cite{StacksKoszul}.","R(x)$ is all of $\\operatorname{Perf}(R)$. Therefore, over $R=\\mathbb Z/p^n$, forcing the interaction of two nonunit scalar labels to vanish in an exact quotient forces that quotient to be zero.\n\\end{theorem}\n\\begin{proof}\nChoose $N$ with $x^N=0$. For composable scalar multiplications, the octahedral cofiber relation builds $C_R(x^k)$ from $C_R(x^{k-1})$ and $C_R(x)$. Induction puts $C_R(x^N)$ in $\\operatorname{Thick}(C_R(x))$. But\n\\[\n C_R(x^N)=C_R(0)\\cong R\\oplus R[1].\n\\]\nRetract closure therefore includes $R$, and $R$ thickly generates $\\operatorname{Perf}(R)$. This cone-of-a-product mechanism is also part of the standard Koszul calculus \\cite{StacksKoszul}.","R)$. Therefore, over $R=\\mathbb Z/p^n$, forcing the interaction of two nonunit scalar labels to vanish in an exact quotient forces that quotient to be zero.\n\\end{theorem}\n\\begin{proof}\nChoose $N$ with $x^N=0$. For composable scalar multiplications, the octahedral cofiber relation builds $C_R(x^k)$ from $C_R(x^{k-1})$ and $C_R(x)$. Induction puts $C_R(x^N)$ in $\\operatorname{Thick}(C_R(x))$. But\n\\[\n C_R(x^N)=C_R(0)\\cong R\\oplus R[1].\n\\]\nRetract closure therefore includes $R$, and $R$ thickly generates $\\operatorname{Perf}(R)$. This cone-of-a-product mechanism is also part of the standard Koszul calculus \\cite{StacksKoszul}.","it scalar labels to vanish in an exact quotient forces that quotient to be zero.\n\\end{theorem}\n\\begin{proof}\nChoose $N$ with $x^N=0$. For composable scalar multiplications, the octahedral cofiber relation builds $C_R(x^k)$ from $C_R(x^{k-1})$ and $C_R(x)$. Induction puts $C_R(x^N)$ in $\\operatorname{Thick}(C_R(x))$. But\n\\[\n C_R(x^N)=C_R(0)\\cong R\\oplus R[1].\n\\]\nRetract closure therefore includes $R$, and $R$ thickly generates $\\operatorname{Perf}(R)$. This cone-of-a-product mechanism is also part of the standard Koszul calculus \\cite{StacksKoszul}.","labels to vanish in an exact quotient forces that quotient to be zero.\n\\end{theorem}\n\\begin{proof}\nChoose $N$ with $x^N=0$. For composable scalar multiplications, the octahedral cofiber relation builds $C_R(x^k)$ from $C_R(x^{k-1})$ and $C_R(x)$. Induction puts $C_R(x^N)$ in $\\operatorname{Thick}(C_R(x))$. But\n\\[\n C_R(x^N)=C_R(0)\\cong R\\oplus R[1].\n\\]\nRetract closure therefore includes $R$, and $R$ thickly generates $\\operatorname{Perf}(R)$. This cone-of-a-product mechanism is also part of the standard Koszul calculus \\cite{StacksKoszul}.","n exact quotient forces that quotient to be zero.\n\\end{theorem}\n\\begin{proof}\nChoose $N$ with $x^N=0$. For composable scalar multiplications, the octahedral cofiber relation builds $C_R(x^k)$ from $C_R(x^{k-1})$ and $C_R(x)$. Induction puts $C_R(x^N)$ in $\\operatorname{Thick}(C_R(x))$. But\n\\[\n C_R(x^N)=C_R(0)\\cong R\\oplus R[1].\n\\]\nRetract closure therefore includes $R$, and $R$ thickly generates $\\operatorname{Perf}(R)$. This cone-of-a-product mechanism is also part of the standard Koszul calculus \\cite{StacksKoszul}.","For two nonunit labels, \\cref{thm:prime-power-koszul} makes their interaction a sum of $C_R(p^h)$ and its shift, with $1\\leq h\\leq n$. Thus it contains $C_R(p^h)$ as a retract. This element is nilpotent. Any exact functor annihilating the interaction annihilates the thick subcategory it generates, hence every perfect object.\n\\end{proof}","This is stronger than the zero-label counterexample over $\\mathbb Z$. At finite $p$-power resolution, even nonzero labels such as $p$ and $p^2$ can force total collapse. Localizing $\\mathbb Z/p^n$ at $p$ gives the zero ring. Noncollapse over the integral coefficients does not survive this base change automatically.","Specify finitely many positive scalar labels and a collection $\\mathcal I$ of allowed independent pairs. Include their unit-seed interaction diagrams as the designated tests. This hypothesis ensures that the minimal kernel below is not being inferred from templates that accidentally annihilate an interaction already. Put\n\\[\n S=\\{p\\in\\mathcal P:\\text{some }(a,b)\\in\\mathcal I\n                    \\text{ satisfies }p\\mid a\\text{ and }p\\mid b\\},\n \\qquad D_0=\\prod_{p\\in\\mathcal P\\setminus S}p.\n\\]\nLet $\\mathcal J_n$ be the tensor-invariant thick closure of those interactions in $\\operatorname{Perf}(B_n)$. Positive lengths in \\cref{thm:weighted-adic-exchange} do not change $S$.","rs. Include their unit-seed interaction diagrams as the designated tests. This hypothesis ensures that the minimal kernel below is not being inferred from templates that accidentally annihilate an interaction already. Put\n\\[\n S=\\{p\\in\\mathcal P:\\text{some }(a,b)\\in\\mathcal I\n                    \\text{ satisfies }p\\mid a\\text{ and }p\\mid b\\},\n \\qquad D_0=\\prod_{p\\in\\mathcal P\\setminus S}p.\n\\]\nLet $\\mathcal J_n$ be the tensor-invariant thick closure of those interactions in $\\operatorname{Perf}(B_n)$. Positive lengths in \\cref{thm:weighted-adic-exchange} do not change $S$.","hypothesis ensures that the minimal kernel below is not being inferred from templates that accidentally annihilate an interaction already. Put\n\\[\n S=\\{p\\in\\mathcal P:\\text{some }(a,b)\\in\\mathcal I\n                    \\text{ satisfies }p\\mid a\\text{ and }p\\mid b\\},\n \\qquad D_0=\\prod_{p\\in\\mathcal P\\setminus S}p.\n\\]\nLet $\\mathcal J_n$ be the tensor-invariant thick closure of those interactions in $\\operatorname{Perf}(B_n)$. Positive lengths in \\cref{thm:weighted-adic-exchange} do not change $S$.","dentally annihilate an interaction already. Put\n\\[\n S=\\{p\\in\\mathcal P:\\text{some }(a,b)\\in\\mathcal I\n                    \\text{ satisfies }p\\mid a\\text{ and }p\\mid b\\},\n \\qquad D_0=\\prod_{p\\in\\mathcal P\\setminus S}p.\n\\]\nLet $\\mathcal J_n$ be the tensor-invariant thick closure of those interactions in $\\operatorname{Perf}(B_n)$. Positive lengths in \\cref{thm:weighted-adic-exchange} do not change $S$.","\\begin{theorem}[Prime-component quotient and tower compatibility]\\label{thm:finite-prime-quotient}\nUnder the Chinese remainder equivalence,\n\\[\n \\boxed{\\mathcal J_n=\n \\prod_{p\\in S}\\operatorname{Perf}(\\mathbb Z/p^n)\n \\times\\prod_{p\\in\\mathcal P\\setminus S}0.}\n\\]\nIn particular there is an equivalence of stable categories\n\\[\n \\operatorname{Perf}(B_n)/\\mathcal J_n\n \\simeq \\prod_{p\\in\\mathcal P\\setminus S}\\operatorname{Perf}(\\mathbb Z/p^n)\n \\simeq\\operatorname{Perf}(\\mathbb Z/D_0^n).\n\\]\nThe equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If $q_n$ denotes projection, then\n\\[\n q_nC_{m,n}\\simeq C^0_{m,n}q_m\n\\]\nwith the supplied base-change coherence. The quotient is nonzero exactly when $S\\ne\\mathcal P$. When $S=\\mathcal P$, use the zero category and the zero ring $\\mathbb Z/1$.\n\\end{theorem}\n\\begin{proof}\nAt a prime outside $S$, each designated interaction ha","\\begin{theorem}[Prime-component quotient and tower compatibility]\\label{thm:finite-prime-quotient}\nUnder the Chinese remainder equivalence,\n\\[\n \\boxed{\\mathcal J_n=\n \\prod_{p\\in S}\\operatorname{Perf}(\\mathbb Z/p^n)\n \\times\\prod_{p\\in\\mathcal P\\setminus S}0.}\n\\]\nIn particular there is an equivalence of stable categories\n\\[\n \\operatorname{Perf}(B_n)/\\mathcal J_n\n \\simeq \\prod_{p\\in\\mathcal P\\setminus S}\\operatorname{Perf}(\\mathbb Z/p^n)\n \\simeq\\operatorname{Perf}(\\mathbb Z/D_0^n).\n\\]\nThe equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If $q_n$ denotes projection, then\n\\[\n q_nC_{m,n}\\simeq C^0_{m,n}q_m\n\\]\nwith the supplied base-change coherence. The quotient is nonzero exactly when $S\\ne\\mathcal P$. When $S=\\mathcal P$, use the zero category and the zero ring $\\mathbb Z/1$.\n\\end{theorem}\n\\begin{proof}\nAt a prime outside $S$, each designated interaction has a unit label and is acyclic. At a prime in $S$, at least one pair has both labels nonunit. Tensor by the corresponding Chinese remainder id…","cal P\\setminus S}0.}\n\\]\nIn particular there is an equivalence of stable categories\n\\[\n \\operatorname{Perf}(B_n)/\\mathcal J_n\n \\simeq \\prod_{p\\in\\mathcal P\\setminus S}\\operatorname{Perf}(\\mathbb Z/p^n)\n \\simeq\\operatorname{Perf}(\\mathbb Z/D_0^n).\n\\]\nThe equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If $q_n$ denotes projection, then\n\\[\n q_nC_{m,n}\\simeq C^0_{m,n}q_m\n\\]\nwith the supplied base-change coherence. The quotient is nonzero exactly when $S\\ne\\mathcal P$. When $S=\\mathcal P$, use the zero category and the zero ring $\\mathbb Z/1$.\n\\end{theorem}\n\\begin{proof}\nAt a prime outside $S$, each designated interaction has a unit label and is acyclic. At a prime in $S$, at least one pair has both labels nonunit. Tensor by the corresponding Chinese remainder idempotent to isolate that component. The nilpotent-collapse theorem then puts its unit in $\\mathcal J_n$. Its whole perfect category follows. These two observations prove equality of the kernel with the displayed product subcategory.","cular there is an equivalence of stable categories\n\\[\n \\operatorname{Perf}(B_n)/\\mathcal J_n\n \\simeq \\prod_{p\\in\\mathcal P\\setminus S}\\operatorname{Perf}(\\mathbb Z/p^n)\n \\simeq\\operatorname{Perf}(\\mathbb Z/D_0^n).\n\\]\nThe equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If $q_n$ denotes projection, then\n\\[\n q_nC_{m,n}\\simeq C^0_{m,n}q_m\n\\]\nwith the supplied base-change coherence. The quotient is nonzero exactly when $S\\ne\\mathcal P$. When $S=\\mathcal P$, use the zero category and the zero ring $\\mathbb Z/1$.\n\\end{theorem}\n\\begin{proof}\nAt a prime outside $S$, each designated interaction has a unit label and is acyclic. At a prime in $S$, at least one pair has both labels nonunit. Tensor by the corresponding Chinese remainder idempotent to isolate that component. The nilpotent-collapse theorem then puts its unit in $\\mathcal J_n$. Its whole perfect category follows. These two observations prove equality of the kernel with the displayed product subcategory.","athcal P\\setminus S}\\operatorname{Perf}(\\mathbb Z/p^n)\n \\simeq\\operatorname{Perf}(\\mathbb Z/D_0^n).\n\\]\nThe equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If $q_n$ denotes projection, then\n\\[\n q_nC_{m,n}\\simeq C^0_{m,n}q_m\n\\]\nwith the supplied base-change coherence. The quotient is nonzero exactly when $S\\ne\\mathcal P$. When $S=\\mathcal P$, use the zero category and the zero ring $\\mathbb Z/1$.\n\\end{theorem}\n\\begin{proof}\nAt a prime outside $S$, each designated interaction has a unit label and is acyclic. At a prime in $S$, at least one pair has both labels nonunit. Tensor by the corresponding Chinese remainder idempotent to isolate that component. The nilpotent-collapse theorem then puts its unit in $\\mathcal J_n$. Its whole perfect category follows. These two observations prove equality of the kernel with the displayed product subcategory.","eratorname{Perf}(\\mathbb Z/p^n)\n \\simeq\\operatorname{Perf}(\\mathbb Z/D_0^n).\n\\]\nThe equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If $q_n$ denotes projection, then\n\\[\n q_nC_{m,n}\\simeq C^0_{m,n}q_m\n\\]\nwith the supplied base-change coherence. The quotient is nonzero exactly when $S\\ne\\mathcal P$. When $S=\\mathcal P$, use the zero category and the zero ring $\\mathbb Z/1$.\n\\end{theorem}\n\\begin{proof}\nAt a prime outside $S$, each designated interaction has a unit label and is acyclic. At a prime in $S$, at least one pair has both labels nonunit. Tensor by the corresponding Chinese remainder idempotent to isolate that component. The nilpotent-collapse theorem then puts its unit in $\\mathcal J_n$. Its whole perfect category follows. These two observations prove equality of the kernel with the displayed product subcategory.","f}(\\mathbb Z/D_0^n).\n\\]\nThe equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If $q_n$ denotes projection, then\n\\[\n q_nC_{m,n}\\simeq C^0_{m,n}q_m\n\\]\nwith the supplied base-change coherence. The quotient is nonzero exactly when $S\\ne\\mathcal P$. When $S=\\mathcal P$, use the zero category and the zero ring $\\mathbb Z/1$.\n\\end{theorem}\n\\begin{proof}\nAt a prime outside $S$, each designated interaction has a unit label and is acyclic. At a prime in $S$, at least one pair has both labels nonunit. Tensor by the corresponding Chinese remainder idempotent to isolate that component. The nilpotent-collapse theorem then puts its unit in $\\mathcal J_n$. Its whole perfect category follows. These two observations prove equality of the kernel with the displayed product subcategory.","ion to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If $q_n$ denotes projection, then\n\\[\n q_nC_{m,n}\\simeq C^0_{m,n}q_m\n\\]\nwith the supplied base-change coherence. The quotient is nonzero exactly when $S\\ne\\mathcal P$. When $S=\\mathcal P$, use the zero category and the zero ring $\\mathbb Z/1$.\n\\end{theorem}\n\\begin{proof}\nAt a prime outside $S$, each designated interaction has a unit label and is acyclic. At a prime in $S$, at least one pair has both labels nonunit. Tensor by the corresponding Chinese remainder idempotent to isolate that component. The nilpotent-collapse theorem then puts its unit in $\\mathcal J_n$. Its whole perfect category follows. These two observations prove equality of the kernel with the displayed product subcategory.","eserved by every coarsening. If $q_n$ denotes projection, then\n\\[\n q_nC_{m,n}\\simeq C^0_{m,n}q_m\n\\]\nwith the supplied base-change coherence. The quotient is nonzero exactly when $S\\ne\\mathcal P$. When $S=\\mathcal P$, use the zero category and the zero ring $\\mathbb Z/1$.\n\\end{theorem}\n\\begin{proof}\nAt a prime outside $S$, each designated interaction has a unit label and is acyclic. At a prime in $S$, at least one pair has both labels nonunit. Tensor by the corresponding Chinese remainder idempotent to isolate that component. The nilpotent-collapse theorem then puts its unit in $\\mathcal J_n$. Its whole perfect category follows. These two observations prove equality of the kernel with the displayed product subcategory.","athcal P$. When $S=\\mathcal P$, use the zero category and the zero ring $\\mathbb Z/1$.\n\\end{theorem}\n\\begin{proof}\nAt a prime outside $S$, each designated interaction has a unit label and is acyclic. At a prime in $S$, at least one pair has both labels nonunit. Tensor by the corresponding Chinese remainder idempotent to isolate that component. The nilpotent-collapse theorem then puts its unit in $\\mathcal J_n$. Its whole perfect category follows. These two observations prove equality of the kernel with the displayed product subcategory.","Write\n\\[\n \\widehat B=\\varprojlim_n B_n\\cong\\prod_{p\\in\\mathcal P}\\mathbb Z_p.\n\\]\nThe isomorphism follows by taking inverse limits of the compatible Chinese remainder decompositions. Here $\\mathbb Z_p$ is the ring of compatible residue sequences modulo $p^n$, and $\\mathbb Q_p=\\mathbb Z_p[1/p]$.","\\begin{theorem}[Uniform based-complex tower reconstruction]\\label{thm:based-tower-reconstruction}\nSuppose $M_n$ are based finite free cochain complexes over $B_n$, concentrated in a common finite interval with fixed ranks in each degree. Suppose their differential matrices reduce exactly from level $m$ to level $n$ for $m\\geq n$. Then there is a unique based complex $\\widehat M$ with those ranks over $\\widehat B$ whose matrices reduce to the supplied ones.","n{theorem}[Uniform based-complex tower reconstruction]\\label{thm:based-tower-reconstruction}\nSuppose $M_n$ are based finite free cochain complexes over $B_n$, concentrated in a common finite interval with fixed ranks in each degree. Suppose their differential matrices reduce exactly from level $m$ to level $n$ for $m\\geq n$. Then there is a unique based complex $\\widehat M$ with those ranks over $\\widehat B$ whose matrices reduce to the supplied ones.","Compatible chain maps and specified compatible chain homotopies reconstruct coefficientwise. Moreover, as complexes of abelian groups,\n\\[\n \\widehat M\\simeq R\\varprojlim_n M_n.\n\\]\nFinite tensor products, finite direct sums, shifts, and cones of compatible based maps commute with this reconstruction. In particular this holds for the scalar residual and interaction complexes with fixed integral labels.\n\\end{theorem}\n\\begin{proof}\nEvery matrix entry is a compatible residue sequence and therefore determines a unique element of $\\widehat B$. The equation $d^{i+1}d^i=0$ holds because it holds modulo every $D^n$ and the inverse-limit ring is separated. The same argument proves the equations for a reconstructed chain map or homotopy.","oups,\n\\[\n \\widehat M\\simeq R\\varprojlim_n M_n.\n\\]\nFinite tensor products, finite direct sums, shifts, and cones of compatible based maps commute with this reconstruction. In particular this holds for the scalar residual and interaction complexes with fixed integral labels.\n\\end{theorem}\n\\begin{proof}\nEvery matrix entry is a compatible residue sequence and therefore determines a unique element of $\\widehat B$. The equation $d^{i+1}d^i=0$ holds because it holds modulo every $D^n$ and the inverse-limit ring is separated. The same argument proves the equations for a reconstructed chain map or homotopy.","R\\varprojlim_n M_n.\n\\]\nFinite tensor products, finite direct sums, shifts, and cones of compatible based maps commute with this reconstruction. In particular this holds for the scalar residual and interaction complexes with fixed integral labels.\n\\end{theorem}\n\\begin{proof}\nEvery matrix entry is a compatible residue sequence and therefore determines a unique element of $\\widehat B$. The equation $d^{i+1}d^i=0$ holds because it holds modulo every $D^n$ and the inverse-limit ring is separated. The same argument proves the equations for a reconstructed chain map or homotopy.","te direct sums, shifts, and cones of compatible based maps commute with this reconstruction. In particular this holds for the scalar residual and interaction complexes with fixed integral labels.\n\\end{theorem}\n\\begin{proof}\nEvery matrix entry is a compatible residue sequence and therefore determines a unique element of $\\widehat B$. The equation $d^{i+1}d^i=0$ holds because it holds modulo every $D^n$ and the inverse-limit ring is separated. The same argument proves the equations for a reconstructed chain map or homotopy.","Degreewise transition maps of the finite free modules are surjective. In the product model of the homotopy inverse limit, the map\n\\[\n 1-\\mathrm{shift}:\\prod_n M_n^i\\longrightarrow\\prod_n M_n^i\n\\]\nis surjective in each degree: choose an initial entry and successively lift the required difference across the surjective transition map. Its kernel is the ordinary inverse limit. Thus the homotopy fiber is represented by that kernel complex. This is the degreewise Mittag--Leffler case of derived inverse limits \\cite{StacksRlim}.","\\begin{proposition}[Transient negative cohomology in the resolution direction]\\label{prop:transient-koszul}\nFix $p$ and an integer $a\\geq1$. For $n\\geq a$ put $R_n=\\mathbb Z/p^n$. Then\n\\[\n H^0C_{R_n}(p^a)\\cong\\mathbb Z/p^a,\n \\qquad H^{-1}C_{R_n}(p^a)\\cong\\mathbb Z/p^a.\n\\]\nUnder reduction from $m$ to $n$ with $m\\geq n\\geq a$, the map on $H^0$ is the identity in these coordinates. On $H^{-1}$ it is multiplication by $p^{m-n}$. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$.","\\begin{proposition}[Transient negative cohomology in the resolution direction]\\label{prop:transient-koszul}\nFix $p$ and an integer $a\\geq1$. For $n\\geq a$ put $R_n=\\mathbb Z/p^n$. Then\n\\[\n H^0C_{R_n}(p^a)\\cong\\mathbb Z/p^a,\n \\qquad H^{-1}C_{R_n}(p^a)\\cong\\mathbb Z/p^a.\n\\]\nUnder reduction from $m$ to $n$ with $m\\geq n\\geq a$, the map on $H^0$ is the identity in these coordinates. On $H^{-1}$ it is multiplication by $p^{m-n}$. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target le","\\begin{proposition}[Transient negative cohomology in the resolution direction]\\label{prop:transient-koszul}\nFix $p$ and an integer $a\\geq1$. For $n\\geq a$ put $R_n=\\mathbb Z/p^n$. Then\n\\[\n H^0C_{R_n}(p^a)\\cong\\mathbb Z/p^a,\n \\qquad H^{-1}C_{R_n}(p^a)\\cong\\mathbb Z/p^a.\n\\]\nUnder reduction from $m$ to $n$ with $m\\geq n\\geq a$, the map on $H^0$ is the identity in these coordinates. On $H^{-1}$ it is multiplication by $p^{m-n}$. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every","\\begin{proposition}[Transient negative cohomology in the resolution direction]\\label{prop:transient-koszul}\nFix $p$ and an integer $a\\geq1$. For $n\\geq a$ put $R_n=\\mathbb Z/p^n$. Then\n\\[\n H^0C_{R_n}(p^a)\\cong\\mathbb Z/p^a,\n \\qquad H^{-1}C_{R_n}(p^a)\\cong\\mathbb Z/p^a.\n\\]\nUnder reduction from $m$ to $n$ with $m\\geq n\\geq a$, the map on $H^0$ is the identity in these coordinates. On $H^{-1}$ it is multiplication by $p^{m-n}$. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$","\\begin{proposition}[Transient negative cohomology in the resolution direction]\\label{prop:transient-koszul}\nFix $p$ and an integer $a\\geq1$. For $n\\geq a$ put $R_n=\\mathbb Z/p^n$. Then\n\\[\n H^0C_{R_n}(p^a)\\cong\\mathbb Z/p^a,\n \\qquad H^{-1}C_{R_n}(p^a)\\cong\\mathbb Z/p^a.\n\\]\nUnder reduction from $m$ to $n$ with $m\\geq n\\geq a$, the map on $H^0$ is the identity in these coordinates. On $H^{-1}$ it is multiplication by $p^{m-n}$. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identi","\\begin{proposition}[Transient negative cohomology in the resolution direction]\\label{prop:transient-koszul}\nFix $p$ and an integer $a\\geq1$. For $n\\geq a$ put $R_n=\\mathbb Z/p^n$. Then\n\\[\n H^0C_{R_n}(p^a)\\cong\\mathbb Z/p^a,\n \\qquad H^{-1}C_{R_n}(p^a)\\cong\\mathbb Z/p^a.\n\\]\nUnder reduction from $m$ to $n$ with $m\\geq n\\geq a$, the map on $H^0$ is the identity in these coordinates. On $H^{-1}$ it is multiplication by $p^{m-n}$. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theor","\\begin{proposition}[Transient negative cohomology in the resolution direction]\\label{prop:transient-koszul}\nFix $p$ and an integer $a\\geq1$. For $n\\geq a$ put $R_n=\\mathbb Z/p^n$. Then\n\\[\n H^0C_{R_n}(p^a)\\cong\\mathbb Z/p^a,\n \\qquad H^{-1}C_{R_n}(p^a)\\cong\\mathbb Z/p^a.\n\\]\nUnder reduction from $m$ to $n$ with $m\\geq n\\geq a$, the map on $H^0$ is the identity in these coordinates. On $H^{-1}$ it is multiplication by $p^{m-n}$. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theorem reco","\\begin{proposition}[Transient negative cohomology in the resolution direction]\\label{prop:transient-koszul}\nFix $p$ and an integer $a\\geq1$. For $n\\geq a$ put $R_n=\\mathbb Z/p^n$. Then\n\\[\n H^0C_{R_n}(p^a)\\cong\\mathbb Z/p^a,\n \\qquad H^{-1}C_{R_n}(p^a)\\cong\\mathbb Z/p^a.\n\\]\nUnder reduction from $m$ to $n$ with $m\\geq n\\geq a$, the map on $H^0$ is the identity in these coordinates. On $H^{-1}$ it is multiplication by $p^{m-n}$. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theorem reconstructs the complex","\\begin{proposition}[Transient negative cohomology in the resolution direction]\\label{prop:transient-koszul}\nFix $p$ and an integer $a\\geq1$. For $n\\geq a$ put $R_n=\\mathbb Z/p^n$. Then\n\\[\n H^0C_{R_n}(p^a)\\cong\\mathbb Z/p^a,\n \\qquad H^{-1}C_{R_n}(p^a)\\cong\\mathbb Z/p^a.\n\\]\nUnder reduction from $m$ to $n$ with $m\\geq n\\geq a$, the map on $H^0$ is the identity in these coordinates. On $H^{-1}$ it is multiplication by $p^{m-n}$. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theorem reconstructs the complex coefficientwise. M","\\begin{proposition}[Transient negative cohomology in the resolution direction]\\label{prop:transient-koszul}\nFix $p$ and an integer $a\\geq1$. For $n\\geq a$ put $R_n=\\mathbb Z/p^n$. Then\n\\[\n H^0C_{R_n}(p^a)\\cong\\mathbb Z/p^a,\n \\qquad H^{-1}C_{R_n}(p^a)\\cong\\mathbb Z/p^a.\n\\]\nUnder reduction from $m$ to $n$ with $m\\geq n\\geq a$, the map on $H^0$ is the identity in these coordinates. On $H^{-1}$ it is multiplication by $p^{m-n}$. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theorem reconstructs the complex coefficientwise. Multiplication by $p^a$ on $\\mathbb Z_p$ is injecti","[Transient negative cohomology in the resolution direction]\\label{prop:transient-koszul}\nFix $p$ and an integer $a\\geq1$. For $n\\geq a$ put $R_n=\\mathbb Z/p^n$. Then\n\\[\n H^0C_{R_n}(p^a)\\cong\\mathbb Z/p^a,\n \\qquad H^{-1}C_{R_n}(p^a)\\cong\\mathbb Z/p^a.\n\\]\nUnder reduction from $m$ to $n$ with $m\\geq n\\geq a$, the map on $H^0$ is the identity in these coordinates. On $H^{-1}$ it is multiplication by $p^{m-n}$. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theorem reconstructs the complex coefficientwise. Multiplication by $p^a$ on $\\mathbb Z_p$ is injective, so its completed kernel…","put $R_n=\\mathbb Z/p^n$. Then\n\\[\n H^0C_{R_n}(p^a)\\cong\\mathbb Z/p^a,\n \\qquad H^{-1}C_{R_n}(p^a)\\cong\\mathbb Z/p^a.\n\\]\nUnder reduction from $m$ to $n$ with $m\\geq n\\geq a$, the map on $H^0$ is the identity in these coordinates. On $H^{-1}$ it is multiplication by $p^{m-n}$. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theorem reconstructs the complex coefficientwise. Multiplication by $p^a$ on $\\mathbb Z_p$ is injective, so its completed kernel vanishes.\n\\end{proof}","p^a.\n\\]\nUnder reduction from $m$ to $n$ with $m\\geq n\\geq a$, the map on $H^0$ is the identity in these coordinates. On $H^{-1}$ it is multiplication by $p^{m-n}$. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theorem reconstructs the complex coefficientwise. Multiplication by $p^a$ on $\\mathbb Z_p$ is injective, so its completed kernel vanishes.\n\\end{proof}","$m$ to $n$ with $m\\geq n\\geq a$, the map on $H^0$ is the identity in these coordinates. On $H^{-1}$ it is multiplication by $p^{m-n}$. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theorem reconstructs the complex coefficientwise. Multiplication by $p^a$ on $\\mathbb Z_p$ is injective, so its completed kernel vanishes.\n\\end{proof}","geq a$, the map on $H^0$ is the identity in these coordinates. On $H^{-1}$ it is multiplication by $p^{m-n}$. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theorem reconstructs the complex coefficientwise. Multiplication by $p^a$ on $\\mathbb Z_p$ is injective, so its completed kernel vanishes.\n\\end{proof}","the identity in these coordinates. On $H^{-1}$ it is multiplication by $p^{m-n}$. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theorem reconstructs the complex coefficientwise. Multiplication by $p^a$ on $\\mathbb Z_p$ is injective, so its completed kernel vanishes.\n\\end{proof}",". On $H^{-1}$ it is multiplication by $p^{m-n}$. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theorem reconstructs the complex coefficientwise. Multiplication by $p^a$ on $\\mathbb Z_p$ is injective, so its completed kernel vanishes.\n\\end{proof}","is multiplication by $p^{m-n}$. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theorem reconstructs the complex coefficientwise. Multiplication by $p^a$ on $\\mathbb Z_p$ is injective, so its completed kernel vanishes.\n\\end{proof}","ohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theorem reconstructs the complex coefficientwise. Multiplication by $p^a$ on $\\mathbb Z_p$ is injective, so its completed kernel vanishes.\n\\end{proof}","efore pro-zero, although every one of its terms is nonzero. The completed complex is\n\\[\n C_{\\mathbb Z_p}(p^a),\\qquad\n H^{-1}=0,\\quad H^0=\\mathbb Z/p^a.\n\\]\n\\end{proposition}\n\\begin{proof}\nRepresent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theorem reconstructs the complex coefficientwise. Multiplication by $p^a$ on $\\mathbb Z_p$ is injective, so its completed kernel vanishes.\n\\end{proof}","ent $H^{-1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theorem reconstructs the complex coefficientwise. Multiplication by $p^a$ on $\\mathbb Z_p$ is injective, so its completed kernel vanishes.\n\\end{proof}","1}$ by the annihilator $p^{n-a}R_n$ and identify $\\mathbb Z/p^a$ with it by $x\\mapsto p^{n-a}x$. Reduction of $p^{m-a}x$ gives $p^{n-a}p^{m-n}x$. Thus for each target level $n$, every source level $m\\geq n+a$ maps to zero. This is exactly the pro-zero assertion. The cokernel coordinates reduce identically. The preceding theorem reconstructs the complex coefficientwise. Multiplication by $p^a$ on $\\mathbb Z_p$ is injective, so its completed kernel vanishes.\n\\end{proof}","This calculation concerns the actual transition maps on homology, not merely the group appearing at each level. A list of isomorphism types $(\\mathbb Z/p^a,\\mathbb Z/p^a,\\ldots)$ cannot distinguish the identity system from the pro-zero system. The two-axis data must retain those maps.","Let $S\\subseteq\\mathcal P$ be the interaction-prime set in \\cref{thm:finite-prime-quotient}, and set $T=\\prod_{p\\in S}p$, with $T=1$ if $S$ is empty. Localizing the finite rings and then completing gives\n\\[\n \\varprojlim_n B_n[1/T]\\cong\\prod_{p\\in\\mathcal P\\setminus S}\\mathbb Z_p.\n\\]\nCompleting and then localizing instead gives\n\\[\n \\widehat B[1/T]\\cong\n \\prod_{p\\in S}\\mathbb Q_p\\times\n \\prod_{p\\in\\mathcal P\\setminus S}\\mathbb Z_p.\n\\]\nThe latter is also an exact realization that annihilates the specified scalar interactions, because each selected prime becomes invertible. There is a canonical comparison map between these rings induced by the reductions. Localization is the usual algebraic operation \\cite{StacksLocaliz","Let $S\\subseteq\\mathcal P$ be the interaction-prime set in \\cref{thm:finite-prime-quotient}, and set $T=\\prod_{p\\in S}p$, with $T=1$ if $S$ is empty. Localizing the finite rings and then completing gives\n\\[\n \\varprojlim_n B_n[1/T]\\cong\\prod_{p\\in\\mathcal P\\setminus S}\\mathbb Z_p.\n\\]\nCompleting and then localizing instead gives\n\\[\n \\widehat B[1/T]\\cong\n \\prod_{p\\in S}\\mathbb Q_p\\times\n \\prod_{p\\in\\mathcal P\\setminus S}\\mathbb Z_p.\n\\]\nThe latter is also an exact realization that annihilates the specified scalar interactions, because each selected prime becomes invertible. There is a canonical comparison map between these rings induced by the reductions. Localization is the usual algebraic operation \\cite{StacksLocalization}, not an additional choice of quotient semantics.","\\begin{theorem}[Explicit completion--repair defect]\\label{thm:completion-repair-defect}\nThe canonical map\n\\[\n \\chi:\\widehat B[1/T]\\longrightarrow\\varprojlim_n B_n[1/T]\n\\]\nis the projection that forgets the factors $\\mathbb Q_p$ for $p\\in S$. As $\\widehat B$-modules it fits into the split exact sequence\n\\[\n 0\\longrightarrow\\prod_{p\\in S}\\mathbb Q_p\n \\longrightarrow\\widehat B[1/T]\n \\xrightarrow{\\chi}\\prod_{p\\in\\mathcal P\\setminus S}\\mathbb Z_p\n \\longrightarrow0.\n\\]\nThe splitting is a module splitting, not a claimed unital ring section.","For every perfect $\\widehat B$-complex $M$, the natural comparison\n\\[\n \\chi_M:M\\otimes^{\\mathbf L}_{\\widehat B}\\widehat B[1/T]\n \\longrightarrow R\\varprojlim_n\n \\bigl(M\\otimes^{\\mathbf L}_{\\widehat B}B_n[1/T]\\bigr)\n\\]\nhas fiber\n\\[\n \\boxed{\\operatorname{Fib}(\\chi_M)\\simeq\n \\bigoplus_{p\\in S}M\\otimes^{\\mathbf L}_{\\widehat B}\\mathbb Q_p.}\n\\]\nThus $\\chi_M$ is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by $[1]$ of that fiber. These statements take place in $D(\\widehat B)$; the field summands need not be perfect over $\\widehat B$.\n\\end{theorem}\n\\begin{proof}\nIn the $p$-primary factor of $B_n$, inverting $T$ gives zero if $p\\in S$ and does nothing otherwise. In the $p$-primary fa","For every perfect $\\widehat B$-complex $M$, the natural comparison\n\\[\n \\chi_M:M\\otimes^{\\mathbf L}_{\\widehat B}\\widehat B[1/T]\n \\longrightarrow R\\varprojlim_n\n \\bigl(M\\otimes^{\\mathbf L}_{\\widehat B}B_n[1/T]\\bigr)\n\\]\nhas fiber\n\\[\n \\boxed{\\operatorname{Fib}(\\chi_M)\\simeq\n \\bigoplus_{p\\in S}M\\otimes^{\\mathbf L}_{\\widehat B}\\mathbb Q_p.}\n\\]\nThus $\\chi_M$ is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by $[1]$ of that fiber. These statements take place in $D(\\widehat B)$; the field summands need not be perfect over $\\widehat B$.\n\\end{theorem}\n\\begin{proof}\nIn the $p$-primary factor of $B_n$, inverting $T$ gives zero if $p\\in S$ and does nothing otherwise. In the $p$-primary factor of $\\wi","For every perfect $\\widehat B$-complex $M$, the natural comparison\n\\[\n \\chi_M:M\\otimes^{\\mathbf L}_{\\widehat B}\\widehat B[1/T]\n \\longrightarrow R\\varprojlim_n\n \\bigl(M\\otimes^{\\mathbf L}_{\\widehat B}B_n[1/T]\\bigr)\n\\]\nhas fiber\n\\[\n \\boxed{\\operatorname{Fib}(\\chi_M)\\simeq\n \\bigoplus_{p\\in S}M\\otimes^{\\mathbf L}_{\\widehat B}\\mathbb Q_p.}\n\\]\nThus $\\chi_M$ is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by $[1]$ of that fiber. These statements take place in $D(\\widehat B)$; the field summands need not be perfect over $\\widehat B$.\n\\end{theorem}\n\\begin{proof}\nIn the $p$-primary factor of $B_n$, inverting $T$ gives zero if $p\\in S$ and does nothing otherwise. In the $p$-primary factor of $\\widehat B$, it gives $\\mathbb Q_p$ if $p\\in S$ and leaves $\\mathbb Z_p$ otherwise. This proves both ring formulas. The induced map is zero on the first set of factors and the","For every perfect $\\widehat B$-complex $M$, the natural comparison\n\\[\n \\chi_M:M\\otimes^{\\mathbf L}_{\\widehat B}\\widehat B[1/T]\n \\longrightarrow R\\varprojlim_n\n \\bigl(M\\otimes^{\\mathbf L}_{\\widehat B}B_n[1/T]\\bigr)\n\\]\nhas fiber\n\\[\n \\boxed{\\operatorname{Fib}(\\chi_M)\\simeq\n \\bigoplus_{p\\in S}M\\otimes^{\\mathbf L}_{\\widehat B}\\mathbb Q_p.}\n\\]\nThus $\\chi_M$ is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by $[1]$ of that fiber. These statements take place in $D(\\widehat B)$; the field summands need not be perfect over $\\widehat B$.\n\\end{theorem}\n\\begin{proof}\nIn the $p$-primary factor of $B_n$, inverting $T$ gives zero if $p\\in S$ and does nothing otherwise. In the $p$-primary factor of $\\widehat B$, it gives $\\mathbb Q_p$ if $p\\in S$ and leaves $\\mathbb Z_p$ otherwise. This proves both ring formulas. The induced map is zero on the first set of factors and the identity on the others, proving the split sequence.","al comparison\n\\[\n \\chi_M:M\\otimes^{\\mathbf L}_{\\widehat B}\\widehat B[1/T]\n \\longrightarrow R\\varprojlim_n\n \\bigl(M\\otimes^{\\mathbf L}_{\\widehat B}B_n[1/T]\\bigr)\n\\]\nhas fiber\n\\[\n \\boxed{\\operatorname{Fib}(\\chi_M)\\simeq\n \\bigoplus_{p\\in S}M\\otimes^{\\mathbf L}_{\\widehat B}\\mathbb Q_p.}\n\\]\nThus $\\chi_M$ is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by $[1]$ of that fiber. These statements take place in $D(\\widehat B)$; the field summands need not be perfect over $\\widehat B$.\n\\end{theorem}\n\\begin{proof}\nIn the $p$-primary factor of $B_n$, inverting $T$ gives zero if $p\\in S$ and does nothing otherwise. In the $p$-primary factor of $\\widehat B$, it gives $\\mathbb Q_p$ if $p\\in S$ and leaves $\\mathbb Z_p$ otherwise. This proves both ring formulas. The induced map is zero on the first set of factors and the identity on the others, proving the split sequence.","hat B}\\widehat B[1/T]\n \\longrightarrow R\\varprojlim_n\n \\bigl(M\\otimes^{\\mathbf L}_{\\widehat B}B_n[1/T]\\bigr)\n\\]\nhas fiber\n\\[\n \\boxed{\\operatorname{Fib}(\\chi_M)\\simeq\n \\bigoplus_{p\\in S}M\\otimes^{\\mathbf L}_{\\widehat B}\\mathbb Q_p.}\n\\]\nThus $\\chi_M$ is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by $[1]$ of that fiber. These statements take place in $D(\\widehat B)$; the field summands need not be perfect over $\\widehat B$.\n\\end{theorem}\n\\begin{proof}\nIn the $p$-primary factor of $B_n$, inverting $T$ gives zero if $p\\in S$ and does nothing otherwise. In the $p$-primary factor of $\\widehat B$, it gives $\\mathbb Q_p$ if $p\\in S$ and leaves $\\mathbb Z_p$ otherwise. This proves both ring formulas. The induced map is zero on the first set of factors and the identity on the others, proving the split sequence.","M\\otimes^{\\mathbf L}_{\\widehat B}B_n[1/T]\\bigr)\n\\]\nhas fiber\n\\[\n \\boxed{\\operatorname{Fib}(\\chi_M)\\simeq\n \\bigoplus_{p\\in S}M\\otimes^{\\mathbf L}_{\\widehat B}\\mathbb Q_p.}\n\\]\nThus $\\chi_M$ is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by $[1]$ of that fiber. These statements take place in $D(\\widehat B)$; the field summands need not be perfect over $\\widehat B$.\n\\end{theorem}\n\\begin{proof}\nIn the $p$-primary factor of $B_n$, inverting $T$ gives zero if $p\\in S$ and does nothing otherwise. In the $p$-primary factor of $\\widehat B$, it gives $\\mathbb Q_p$ if $p\\in S$ and leaves $\\mathbb Z_p$ otherwise. This proves both ring formulas. The induced map is zero on the first set of factors and the identity on the others, proving the split sequence.","]\nhas fiber\n\\[\n \\boxed{\\operatorname{Fib}(\\chi_M)\\simeq\n \\bigoplus_{p\\in S}M\\otimes^{\\mathbf L}_{\\widehat B}\\mathbb Q_p.}\n\\]\nThus $\\chi_M$ is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by $[1]$ of that fiber. These statements take place in $D(\\widehat B)$; the field summands need not be perfect over $\\widehat B$.\n\\end{theorem}\n\\begin{proof}\nIn the $p$-primary factor of $B_n$, inverting $T$ gives zero if $p\\in S$ and does nothing otherwise. In the $p$-primary factor of $\\widehat B$, it gives $\\mathbb Q_p$ if $p\\in S$ and leaves $\\mathbb Z_p$ otherwise. This proves both ring formulas. The induced map is zero on the first set of factors and the identity on the others, proving the split sequence.","{\\operatorname{Fib}(\\chi_M)\\simeq\n \\bigoplus_{p\\in S}M\\otimes^{\\mathbf L}_{\\widehat B}\\mathbb Q_p.}\n\\]\nThus $\\chi_M$ is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by $[1]$ of that fiber. These statements take place in $D(\\widehat B)$; the field summands need not be perfect over $\\widehat B$.\n\\end{theorem}\n\\begin{proof}\nIn the $p$-primary factor of $B_n$, inverting $T$ gives zero if $p\\in S$ and does nothing otherwise. In the $p$-primary factor of $\\widehat B$, it gives $\\mathbb Q_p$ if $p\\in S$ and leaves $\\mathbb Z_p$ otherwise. This proves both ring formulas. The induced map is zero on the first set of factors and the identity on the others, proving the split sequence.","b}(\\chi_M)\\simeq\n \\bigoplus_{p\\in S}M\\otimes^{\\mathbf L}_{\\widehat B}\\mathbb Q_p.}\n\\]\nThus $\\chi_M$ is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by $[1]$ of that fiber. These statements take place in $D(\\widehat B)$; the field summands need not be perfect over $\\widehat B$.\n\\end{theorem}\n\\begin{proof}\nIn the $p$-primary factor of $B_n$, inverting $T$ gives zero if $p\\in S$ and does nothing otherwise. In the $p$-primary factor of $\\widehat B$, it gives $\\mathbb Q_p$ if $p\\in S$ and leaves $\\mathbb Z_p$ otherwise. This proves both ring formulas. The induced map is zero on the first set of factors and the identity on the others, proving the split sequence.","\\bigoplus_{p\\in S}M\\otimes^{\\mathbf L}_{\\widehat B}\\mathbb Q_p.}\n\\]\nThus $\\chi_M$ is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by $[1]$ of that fiber. These statements take place in $D(\\widehat B)$; the field summands need not be perfect over $\\widehat B$.\n\\end{theorem}\n\\begin{proof}\nIn the $p$-primary factor of $B_n$, inverting $T$ gives zero if $p\\in S$ and does nothing otherwise. In the $p$-primary factor of $\\widehat B$, it gives $\\mathbb Q_p$ if $p\\in S$ and leaves $\\mathbb Z_p$ otherwise. This proves both ring formulas. The induced map is zero on the first set of factors and the identity on the others, proving the split sequence.","ehat B}\\mathbb Q_p.}\n\\]\nThus $\\chi_M$ is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by $[1]$ of that fiber. These statements take place in $D(\\widehat B)$; the field summands need not be perfect over $\\widehat B$.\n\\end{theorem}\n\\begin{proof}\nIn the $p$-primary factor of $B_n$, inverting $T$ gives zero if $p\\in S$ and does nothing otherwise. In the $p$-primary factor of $\\widehat B$, it gives $\\mathbb Q_p$ if $p\\in S$ and leaves $\\mathbb Z_p$ otherwise. This proves both ring formulas. The induced map is zero on the first set of factors and the identity on the others, proving the split sequence.","]\nThus $\\chi_M$ is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by $[1]$ of that fiber. These statements take place in $D(\\widehat B)$; the field summands need not be perfect over $\\widehat B$.\n\\end{theorem}\n\\begin{proof}\nIn the $p$-primary factor of $B_n$, inverting $T$ gives zero if $p\\in S$ and does nothing otherwise. In the $p$-primary factor of $\\widehat B$, it gives $\\mathbb Q_p$ if $p\\in S$ and leaves $\\mathbb Z_p$ otherwise. This proves both ring formulas. The induced map is zero on the first set of factors and the identity on the others, proving the split sequence.","quivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by $[1]$ of that fiber. These statements take place in $D(\\widehat B)$; the field summands need not be perfect over $\\widehat B$.\n\\end{theorem}\n\\begin{proof}\nIn the $p$-primary factor of $B_n$, inverting $T$ gives zero if $p\\in S$ and does nothing otherwise. In the $p$-primary factor of $\\widehat B$, it gives $\\mathbb Q_p$ if $p\\in S$ and leaves $\\mathbb Z_p$ otherwise. This proves both ring formulas. The induced map is zero on the first set of factors and the identity on the others, proving the split sequence.","y when every displayed generic fiber is zero. Its cofiber is the shift by $[1]$ of that fiber. These statements take place in $D(\\widehat B)$; the field summands need not be perfect over $\\widehat B$.\n\\end{theorem}\n\\begin{proof}\nIn the $p$-primary factor of $B_n$, inverting $T$ gives zero if $p\\in S$ and does nothing otherwise. In the $p$-primary factor of $\\widehat B$, it gives $\\mathbb Q_p$ if $p\\in S$ and leaves $\\mathbb Z_p$ otherwise. This proves both ring formulas. The induced map is zero on the first set of factors and the identity on the others, proving the split sequence.","ed generic fiber is zero. Its cofiber is the shift by $[1]$ of that fiber. These statements take place in $D(\\widehat B)$; the field summands need not be perfect over $\\widehat B$.\n\\end{theorem}\n\\begin{proof}\nIn the $p$-primary factor of $B_n$, inverting $T$ gives zero if $p\\in S$ and does nothing otherwise. In the $p$-primary factor of $\\widehat B$, it gives $\\mathbb Q_p$ if $p\\in S$ and leaves $\\mathbb Z_p$ otherwise. This proves both ring formulas. The induced map is zero on the first set of factors and the identity on the others, proving the split sequence.","For $M=\\widehat B$, the derived inverse limit on the right is the ordinary one, since the retained residue-module tower is degreewise surjective. The asserted comparison and fiber are therefore exactly the split sequence. For a general perfect $M$, all functors in the claimed identity are exact and preserve finite sums and retracts. Homotopy inverse limits are exact in a stable derived category. The natural comparison identifies its right side with the retained summand of $M$: this holds on the unit by the just-computed inverse limit and hence on its thick closure, which is $\\operatorname{Perf}(\\widehat B)$. Tensoring the split sequence now identifies the fiber with the indicated finite sum. A morphism in a","For $M=\\widehat B$, the derived inverse limit on the right is the ordinary one, since the retained residue-module tower is degreewise surjective. The asserted comparison and fiber are therefore exactly the split sequence. For a general perfect $M$, all functors in the claimed identity are exact and preserve finite sums and retracts. Homotopy inverse limits are exact in a stable derived category. The natural comparison identifies its right side with the retained summand of $M$: this holds on the unit by the just-computed inverse limit and hence on its thick closure, which is $\\operatorname{Perf}(\\widehat B)$. Tensoring the split sequence now identifies the fiber with the indicated finite sum. A morphism in a stable category is an equivalence precisely when its fiber is zero.\n\\end{proof}",", since the retained residue-module tower is degreewise surjective. The asserted comparison and fiber are therefore exactly the split sequence. For a general perfect $M$, all functors in the claimed identity are exact and preserve finite sums and retracts. Homotopy inverse limits are exact in a stable derived category. The natural comparison identifies its right side with the retained summand of $M$: this holds on the unit by the just-computed inverse limit and hence on its thick closure, which is $\\operatorname{Perf}(\\widehat B)$. Tensoring the split sequence now identifies the fiber with the indicated finite sum. A morphism in a stable category is an equivalence precisely when its fiber is zero.\n\\end{proof}","e therefore exactly the split sequence. For a general perfect $M$, all functors in the claimed identity are exact and preserve finite sums and retracts. Homotopy inverse limits are exact in a stable derived category. The natural comparison identifies its right side with the retained summand of $M$: this holds on the unit by the just-computed inverse limit and hence on its thick closure, which is $\\operatorname{Perf}(\\widehat B)$. Tensoring the split sequence now identifies the fiber with the indicated finite sum. A morphism in a stable category is an equivalence precisely when its fiber is zero.\n\\end{proof}","\\begin{example}[A nonzero generic fiber is lost at every finite repair]\\label{ex:repair-before-completion}\nFor $\\mathcal P=\\{p\\}$ and two independent labels divisible by $p$, every finite excisive quotient is zero, whereas\n\\[\n \\bigl(\\varprojlim_n\\mathbb Z/p^n\\bigr)[1/p]=\\mathbb Q_p\\ne0.\n\\]\nFor a nonzero finite-level quotient, take $D=30$ and labels $6,10$. Only the prime $2$ is shared, so the finite repair is\n\\[\n B_n[1/2]\\cong\\mathbb Z/15^n.\n\\]\nThe two completed outputs are $\\mathbb Q_2\\times\\mathbb Z_3\\times\\mathbb Z_5$ and $\\mathbb Z_3\\times\\mathbb Z_5$, with fiber $\\mathbb Q_2$. This is an explicit failure of an unconditional bi-completion principle, even though the finite quotient commutes with every finite coarsening.\n\\end{example}","[A nonzero generic fiber is lost at every finite repair]\\label{ex:repair-before-completion}\nFor $\\mathcal P=\\{p\\}$ and two independent labels divisible by $p$, every finite excisive quotient is zero, whereas\n\\[\n \\bigl(\\varprojlim_n\\mathbb Z/p^n\\bigr)[1/p]=\\mathbb Q_p\\ne0.\n\\]\nFor a nonzero finite-level quotient, take $D=30$ and labels $6,10$. Only the prime $2$ is shared, so the finite repair is\n\\[\n B_n[1/2]\\cong\\mathbb Z/15^n.\n\\]\nThe two completed outputs are $\\mathbb Q_2\\times\\mathbb Z_3\\times\\mathbb Z_5$ and $\\mathbb Z_3\\times\\mathbb Z_5$, with fiber $\\mathbb Q_2$. This is an explicit failure of an unconditional bi-completion principle, even though the finite quotient commutes with every finite coarsening.\n\\end{example}",":repair-before-completion}\nFor $\\mathcal P=\\{p\\}$ and two independent labels divisible by $p$, every finite excisive quotient is zero, whereas\n\\[\n \\bigl(\\varprojlim_n\\mathbb Z/p^n\\bigr)[1/p]=\\mathbb Q_p\\ne0.\n\\]\nFor a nonzero finite-level quotient, take $D=30$ and labels $6,10$. Only the prime $2$ is shared, so the finite repair is\n\\[\n B_n[1/2]\\cong\\mathbb Z/15^n.\n\\]\nThe two completed outputs are $\\mathbb Q_2\\times\\mathbb Z_3\\times\\mathbb Z_5$ and $\\mathbb Z_3\\times\\mathbb Z_5$, with fiber $\\mathbb Q_2$. This is an explicit failure of an unconditional bi-completion principle, even though the finite quotient commutes with every finite coarsening.\n\\end{example}","d two independent labels divisible by $p$, every finite excisive quotient is zero, whereas\n\\[\n \\bigl(\\varprojlim_n\\mathbb Z/p^n\\bigr)[1/p]=\\mathbb Q_p\\ne0.\n\\]\nFor a nonzero finite-level quotient, take $D=30$ and labels $6,10$. Only the prime $2$ is shared, so the finite repair is\n\\[\n B_n[1/2]\\cong\\mathbb Z/15^n.\n\\]\nThe two completed outputs are $\\mathbb Q_2\\times\\mathbb Z_3\\times\\mathbb Z_5$ and $\\mathbb Z_3\\times\\mathbb Z_5$, with fiber $\\mathbb Q_2$. This is an explicit failure of an unconditional bi-completion principle, even though the finite quotient commutes with every finite coarsening.\n\\end{example}","every finite excisive quotient is zero, whereas\n\\[\n \\bigl(\\varprojlim_n\\mathbb Z/p^n\\bigr)[1/p]=\\mathbb Q_p\\ne0.\n\\]\nFor a nonzero finite-level quotient, take $D=30$ and labels $6,10$. Only the prime $2$ is shared, so the finite repair is\n\\[\n B_n[1/2]\\cong\\mathbb Z/15^n.\n\\]\nThe two completed outputs are $\\mathbb Q_2\\times\\mathbb Z_3\\times\\mathbb Z_5$ and $\\mathbb Z_3\\times\\mathbb Z_5$, with fiber $\\mathbb Q_2$. This is an explicit failure of an unconditional bi-completion principle, even though the finite quotient commutes with every finite coarsening.\n\\end{example}","The generic-fiber criterion also gives a positive case. For $M=C_{\\widehat B}(u)$ with a nonzero integer $u$, its selected generic fibers vanish, because $u$ becomes invertible in each $\\mathbb Q_p$. Thus this comparison is an equivalence on that torsion residual even though it is not an equivalence on the unit. Commutation depends on the object, not only on the existence of coherent finite-level squares. The distinction is consistent with the standard derived-completion fact that a complex on which a defining element is invertible has zero completion \\cite{StacksDerivedCompletion}.","Put\n\\[\n A=\\mathbb Z[1/D],\\qquad B=\\prod_{p\\in\\mathcal P}\\mathbb Z_p,\n \\qquad G=\\prod_{p\\in\\mathcal P}\\mathbb Q_p.\n\\]","\\begin{theorem}[Arithmetic fracture reconstruction]\\label{thm:arithmetic-fracture}\nThe diagonal maps and their difference give an exact sequence of abelian groups\n\\[\n 0\\longrightarrow\\mathbb Z\n \\longrightarrow A\\oplus B\n \\xrightarrow{(a,b)\\mapsto a-b}G\n \\longrightarrow0.\n\\]\nConsequently, for every perfect integral complex $M$ there is a natural homotopy pullback square in $D(\\mathbb Z)$,\n\\[\n \\begin{tikzcd}\n M\\arrow[r]\\arrow[d]&M\\otimes^{\\mathbf L}_{\\mathbb Z}A\\arrow[d]","\\begin{theorem}[Arithmetic fracture reconstruction]\\label{thm:arithmetic-fracture}\nThe diagonal maps and their difference give an exact sequence of abelian groups\n\\[\n 0\\longrightarrow\\mathbb Z\n \\longrightarrow A\\oplus B\n \\xrightarrow{(a,b)\\mapsto a-b}G\n \\longrightarrow0.\n\\]\nConsequently, for every perfect integral complex $M$ there is a natural homotopy pullback square in $D(\\mathbb Z)$,\n\\[\n \\begin{tikzcd}\n M\\arrow[r]\\arrow[d]&M\\otimes^{\\mathbf L}_{\\mathbb Z}A\\arrow[d]\\\\\n M\\otimes^{\\mathbf L}_{\\mathbb Z}B\\arrow[r]&M\\otimes^{\\mathbf L}_{\\mathbb Z}G.\n \\end{tikzcd}\n\\]\nIts bottom-left term is $R\\varprojlim_n(M\\otimes^{\\mathbf L}_{\\mathbb Z}B_n)$. In particular $M$ is zero if and only if its localized and completed terms are both zero.\n\\end{theorem}\n\\begin{proof}\nAn element of $A$ integral at every $p\\in\\mathcal P$ is an integer: its possible denominator primes all lie in $\\mathcal P$, and integrality removes each of them. This identifies the kernel with the diagonal copy of $\\mathbb Z$.","M\\otimes^{\\mathbf L}_{\\mathbb Z}B\\arrow[r]&M\\otimes^{\\mathbf L}_{\\mathbb Z}G.\n \\end{tikzcd}\n\\]\nIts bottom-left term is $R\\varprojlim_n(M\\otimes^{\\mathbf L}_{\\mathbb Z}B_n)$. In particular $M$ is zero if and only if its localized and completed terms are both zero.\n\\end{theorem}\n\\begin{proof}\nAn element of $A$ integral at every $p\\in\\mathcal P$ is an integer: its possible denominator primes all lie in $\\mathcal P$, and integrality removes each of them. This identifies the kernel with the diagonal copy of $\\mathbb Z$.","imes^{\\mathbf L}_{\\mathbb Z}B\\arrow[r]&M\\otimes^{\\mathbf L}_{\\mathbb Z}G.\n \\end{tikzcd}\n\\]\nIts bottom-left term is $R\\varprojlim_n(M\\otimes^{\\mathbf L}_{\\mathbb Z}B_n)$. In particular $M$ is zero if and only if its localized and completed terms are both zero.\n\\end{theorem}\n\\begin{proof}\nAn element of $A$ integral at every $p\\in\\mathcal P$ is an integer: its possible denominator primes all lie in $\\mathcal P$, and integrality removes each of them. This identifies the kernel with the diagonal copy of $\\mathbb Z$.","t term is $R\\varprojlim_n(M\\otimes^{\\mathbf L}_{\\mathbb Z}B_n)$. In particular $M$ is zero if and only if its localized and completed terms are both zero.\n\\end{theorem}\n\\begin{proof}\nAn element of $A$ integral at every $p\\in\\mathcal P$ is an integer: its possible denominator primes all lie in $\\mathcal P$, and integrality removes each of them. This identifies the kernel with the diagonal copy of $\\mathbb Z$.","For surjectivity take $(y_p)\\in G$. Choose $N$ large enough that $D^Ny_p\\in\\mathbb Z_p$ for every $p$. For each $p$, choose the residue of $D^Ny_p$ modulo $p^N$. The ordinary Chinese remainder theorem provides $z\\in\\mathbb Z$ with all these residues. Set $a=z/D^N\\in A$. Then $a-y_p\\in\\mathbb Z_p$ for every $p$, so $(a,(a-y_p)_p)$ maps to $(y_p)_p$. This proves exactness.","The exact sequence gives a cofiber sequence in the derived category. Derived tensor with $M$ is exact, so it gives the displayed homotopy pullback. For a bounded finite free representative of $M$, the tower of reductions has uniform ranks and compatible matrices. The based-tower theorem identifies its derived limit with $M\\otimes B$. Finally if the localized and completed terms are zero, so is their generic base change, and the pullback is zero. The converse is immediate.\n\\end{proof}","\\begin{corollary}[Residuals reconstruct through the same arithmetic square]\\label{cor:residual-fracture}\nLet $X$ be a finite coherent event diagram in $\\operatorname{Perf}(\\mathbb Z)$. Every edge residual, square interaction, and initial-to-terminal residual of $X$ satisfies \\cref{thm:arithmetic-fracture}. Its terms are the corresponding residuals of the diagrams obtained by base change to $A,B,G$. Each such integral residual vanishes if and only if both its localized and completed versions vanish.\n\\end{corollary}\n\\begin{proof}\nDerived base change is exact and therefore preserves all the finite cofibers defining those residuals. Apply the arithmetic fracture theorem to the resulting perfect integral complex. Naturality gives the maps on the whole residual diagram, not merely equality of its cohomolog","\\begin{corollary}[Residuals reconstruct through the same arithmetic square]\\label{cor:residual-fracture}\nLet $X$ be a finite coherent event diagram in $\\operatorname{Perf}(\\mathbb Z)$. Every edge residual, square interaction, and initial-to-terminal residual of $X$ satisfies \\cref{thm:arithmetic-fracture}. Its terms are the corresponding residuals of the diagrams obtained by base change to $A,B,G$. Each such integral residual vanishes if and only if both its localized and completed versions vanish.\n\\end{corollary}\n\\begin{proof}\nDerived base change is exact and therefore preserves all the finite cofibers defining those residuals. Apply the arithmetic fracture theorem to the resulting perfect integral complex. Naturality gives the maps on the whole residual diagram, not merely equality of its cohomology dimensions.\n\\end{proof}","\\begin{theorem}[Effective arithmetic lattice gluing]\\label{thm:lattice-gluing}\nLet $V$ be a finite free $A=\\mathbb Z[1/D]$-module of rank $r$. For each $p\\in\\mathcal P$ let $L_p$ be a finite free $\\mathbb Z_p$-module, with an isomorphism\n\\[\n \\theta_p:V\\otimes_A\\mathbb Q_p\\xrightarrow{\\sim}\n L_p\\otimes_{\\mathbb Z_p}\\mathbb Q_p.\n\\]\nThen\n\\[\n \\Lambda=\\{v\\in V:\\theta_p(v)\\in L_p\\text{ for every }p\\in\\mathcal P\\}\n\\]\nis a finite free $\\mathbb Z$-module of rank $r$, and the natural maps identify\n\\[\n \\Lambda[1/D]\\cong V,\\qquad\n \\Lambda\\otimes_{\\mathbb Z}\\mathbb Z_p\\cong L_p.\n\\]\nIt is the unique lattice inside $V$ with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent","\\begin{theorem}[Effective arithmetic lattice gluing]\\label{thm:lattice-gluing}\nLet $V$ be a finite free $A=\\mathbb Z[1/D]$-module of rank $r$. For each $p\\in\\mathcal P$ let $L_p$ be a finite free $\\mathbb Z_p$-module, with an isomorphism\n\\[\n \\theta_p:V\\otimes_A\\mathbb Q_p\\xrightarrow{\\sim}\n L_p\\otimes_{\\mathbb Z_p}\\mathbb Q_p.\n\\]\nThen\n\\[\n \\Lambda=\\{v\\in V:\\theta_p(v)\\in L_p\\text{ for every }p\\in\\mathcal P\\}\n\\]\nis a finite free $\\mathbb Z$-module of rank $r$, and the natural maps identify\n\\[\n \\Lambda[1/D]\\cong V,\\qquad\n \\Lambda\\otimes_{\\mathbb Z}\\mathbb Z_p\\cong L_p.\n\\]\nIt is the unique lattice inside $V$ with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free","\\begin{theorem}[Effective arithmetic lattice gluing]\\label{thm:lattice-gluing}\nLet $V$ be a finite free $A=\\mathbb Z[1/D]$-module of rank $r$. For each $p\\in\\mathcal P$ let $L_p$ be a finite free $\\mathbb Z_p$-module, with an isomorphism\n\\[\n \\theta_p:V\\otimes_A\\mathbb Q_p\\xrightarrow{\\sim}\n L_p\\otimes_{\\mathbb Z_p}\\mathbb Q_p.\n\\]\nThen\n\\[\n \\Lambda=\\{v\\in V:\\theta_p(v)\\in L_p\\text{ for every }p\\in\\mathcal P\\}\n\\]\nis a finite free $\\mathbb Z$-module of rank $r$, and the natural maps identify\n\\[\n \\Lambda[1/D]\\cong V,\\qquad\n \\Lambda\\otimes_{\\mathbb Z}\\mathbb Z_p\\cong L_p.\n\\]\nIt is the unique lattice inside $V$ with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local latt","\\begin{theorem}[Effective arithmetic lattice gluing]\\label{thm:lattice-gluing}\nLet $V$ be a finite free $A=\\mathbb Z[1/D]$-module of rank $r$. For each $p\\in\\mathcal P$ let $L_p$ be a finite free $\\mathbb Z_p$-module, with an isomorphism\n\\[\n \\theta_p:V\\otimes_A\\mathbb Q_p\\xrightarrow{\\sim}\n L_p\\otimes_{\\mathbb Z_p}\\mathbb Q_p.\n\\]\nThen\n\\[\n \\Lambda=\\{v\\in V:\\theta_p(v)\\in L_p\\text{ for every }p\\in\\mathcal P\\}\n\\]\nis a finite free $\\mathbb Z$-module of rank $r$, and the natural maps identify\n\\[\n \\Lambda[1/D]\\cong V,\\qquad\n \\Lambda\\otimes_{\\mathbb Z}\\mathbb Z_p\\cong L_p.\n\\]\nIt is the unique lattice inside $V$ with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic","\\begin{theorem}[Effective arithmetic lattice gluing]\\label{thm:lattice-gluing}\nLet $V$ be a finite free $A=\\mathbb Z[1/D]$-module of rank $r$. For each $p\\in\\mathcal P$ let $L_p$ be a finite free $\\mathbb Z_p$-module, with an isomorphism\n\\[\n \\theta_p:V\\otimes_A\\mathbb Q_p\\xrightarrow{\\sim}\n L_p\\otimes_{\\mathbb Z_p}\\mathbb Q_p.\n\\]\nThen\n\\[\n \\Lambda=\\{v\\in V:\\theta_p(v)\\in L_p\\text{ for every }p\\in\\mathcal P\\}\n\\]\nis a finite free $\\mathbb Z$-module of rank $r$, and the natural maps identify\n\\[\n \\Lambda[1/D]\\cong V,\\qquad\n \\Lambda\\otimes_{\\mathbb Z}\\mathbb Z_p\\cong L_p.\n\\]\nIt is the unique lattice inside $V$ with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identific","\\begin{theorem}[Effective arithmetic lattice gluing]\\label{thm:lattice-gluing}\nLet $V$ be a finite free $A=\\mathbb Z[1/D]$-module of rank $r$. For each $p\\in\\mathcal P$ let $L_p$ be a finite free $\\mathbb Z_p$-module, with an isomorphism\n\\[\n \\theta_p:V\\otimes_A\\mathbb Q_p\\xrightarrow{\\sim}\n L_p\\otimes_{\\mathbb Z_p}\\mathbb Q_p.\n\\]\nThen\n\\[\n \\Lambda=\\{v\\in V:\\theta_p(v)\\in L_p\\text{ for every }p\\in\\mathcal P\\}\n\\]\nis a finite free $\\mathbb Z$-module of rank $r$, and the natural maps identify\n\\[\n \\Lambda[1/D]\\cong V,\\qquad\n \\Lambda\\otimes_{\\mathbb Z}\\mathbb Z_p\\cong L_p.\n\\]\nIt is the unique lattice inside $V$ with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications.\n\\end{theorem}\n\\begin{pr","\\begin{theorem}[Effective arithmetic lattice gluing]\\label{thm:lattice-gluing}\nLet $V$ be a finite free $A=\\mathbb Z[1/D]$-module of rank $r$. For each $p\\in\\mathcal P$ let $L_p$ be a finite free $\\mathbb Z_p$-module, with an isomorphism\n\\[\n \\theta_p:V\\otimes_A\\mathbb Q_p\\xrightarrow{\\sim}\n L_p\\otimes_{\\mathbb Z_p}\\mathbb Q_p.\n\\]\nThen\n\\[\n \\Lambda=\\{v\\in V:\\theta_p(v)\\in L_p\\text{ for every }p\\in\\mathcal P\\}\n\\]\nis a finite free $\\mathbb Z$-module of rank $r$, and the natural maps identify\n\\[\n \\Lambda[1/D]\\cong V,\\qquad\n \\Lambda\\otimes_{\\mathbb Z}\\mathbb Z_p\\cong L_p.\n\\]\nIt is the unique lattice inside $V$ with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications.\n\\end{theorem}\n\\begin{proof}\nChoose a basis $V=A^r$, and let $E_p=\\theta_p^{-1}(L_p)$ be the corresponding lattice in $\\mathbb Q_p^r$. Bounds on","\\begin{theorem}[Effective arithmetic lattice gluing]\\label{thm:lattice-gluing}\nLet $V$ be a finite free $A=\\mathbb Z[1/D]$-module of rank $r$. For each $p\\in\\mathcal P$ let $L_p$ be a finite free $\\mathbb Z_p$-module, with an isomorphism\n\\[\n \\theta_p:V\\otimes_A\\mathbb Q_p\\xrightarrow{\\sim}\n L_p\\otimes_{\\mathbb Z_p}\\mathbb Q_p.\n\\]\nThen\n\\[\n \\Lambda=\\{v\\in V:\\theta_p(v)\\in L_p\\text{ for every }p\\in\\mathcal P\\}\n\\]\nis a finite free $\\mathbb Z$-module of rank $r$, and the natural maps identify\n\\[\n \\Lambda[1/D]\\cong V,\\qquad\n \\Lambda\\otimes_{\\mathbb Z}\\mathbb Z_p\\cong L_p.\n\\]\nIt is the unique lattice inside $V$ with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications.\n\\end{theorem}\n\\begin{proof}\nChoose a basis $V=A^r$, and let $E_p=\\theta_p^{-1}(L_p)$ be the corresponding lattice in $\\mathbb Q_p^r$. Bounds on the entries of $\\theta_p$ and its inverse give a common $N\\geq1$ such t…","etic lattice gluing]\\label{thm:lattice-gluing}\nLet $V$ be a finite free $A=\\mathbb Z[1/D]$-module of rank $r$. For each $p\\in\\mathcal P$ let $L_p$ be a finite free $\\mathbb Z_p$-module, with an isomorphism\n\\[\n \\theta_p:V\\otimes_A\\mathbb Q_p\\xrightarrow{\\sim}\n L_p\\otimes_{\\mathbb Z_p}\\mathbb Q_p.\n\\]\nThen\n\\[\n \\Lambda=\\{v\\in V:\\theta_p(v)\\in L_p\\text{ for every }p\\in\\mathcal P\\}\n\\]\nis a finite free $\\mathbb Z$-module of rank $r$, and the natural maps identify\n\\[\n \\Lambda[1/D]\\cong V,\\qquad\n \\Lambda\\otimes_{\\mathbb Z}\\mathbb Z_p\\cong L_p.\n\\]\nIt is the unique lattice inside $V$ with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications.\n\\end{theorem}\n\\begin{proof}\nChoose a basis $V=A^r$, and let $E_p=\\theta_p^{-1}(L_p)$ be the corresponding lattice in $\\mathbb Q_p^r$. Bounds on the entries of $\\theta_p$ and its inverse give a common $N\\geq1$ such that, with $b=D^N$,\n\\[\n b\\mathbb …","thm:lattice-gluing}\nLet $V$ be a finite free $A=\\mathbb Z[1/D]$-module of rank $r$. For each $p\\in\\mathcal P$ let $L_p$ be a finite free $\\mathbb Z_p$-module, with an isomorphism\n\\[\n \\theta_p:V\\otimes_A\\mathbb Q_p\\xrightarrow{\\sim}\n L_p\\otimes_{\\mathbb Z_p}\\mathbb Q_p.\n\\]\nThen\n\\[\n \\Lambda=\\{v\\in V:\\theta_p(v)\\in L_p\\text{ for every }p\\in\\mathcal P\\}\n\\]\nis a finite free $\\mathbb Z$-module of rank $r$, and the natural maps identify\n\\[\n \\Lambda[1/D]\\cong V,\\qquad\n \\Lambda\\otimes_{\\mathbb Z}\\mathbb Z_p\\cong L_p.\n\\]\nIt is the unique lattice inside $V$ with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications.\n\\end{theorem}\n\\begin{proof}\nChoose a basis $V=A^r$, and let $E_p=\\theta_p^{-1}(L_p)$ be the corresponding lattice in $\\mathbb Q_p^r$. Bounds on the entries of $\\theta_p$ and its inverse give a common $N\\geq1$ such that, with $b=D^N$,\n\\[\n b\\mathbb Z_p^r\\subseteq E_p\\subseteq","ite free $A=\\mathbb Z[1/D]$-module of rank $r$. For each $p\\in\\mathcal P$ let $L_p$ be a finite free $\\mathbb Z_p$-module, with an isomorphism\n\\[\n \\theta_p:V\\otimes_A\\mathbb Q_p\\xrightarrow{\\sim}\n L_p\\otimes_{\\mathbb Z_p}\\mathbb Q_p.\n\\]\nThen\n\\[\n \\Lambda=\\{v\\in V:\\theta_p(v)\\in L_p\\text{ for every }p\\in\\mathcal P\\}\n\\]\nis a finite free $\\mathbb Z$-module of rank $r$, and the natural maps identify\n\\[\n \\Lambda[1/D]\\cong V,\\qquad\n \\Lambda\\otimes_{\\mathbb Z}\\mathbb Z_p\\cong L_p.\n\\]\nIt is the unique lattice inside $V$ with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications.\n\\end{theorem}\n\\begin{proof}\nChoose a basis $V=A^r$, and let $E_p=\\theta_p^{-1}(L_p)$ be the corresponding lattice in $\\mathbb Q_p^r$. Bounds on the entries of $\\theta_p$ and its inverse give a common $N\\geq1$ such that, with $b=D^N$,\n\\[\n b\\mathbb Z_p^r\\subseteq E_p\\subseteq b^{-1}\\mathbb Z_p^r\n \\quad\\text{for…","-module, with an isomorphism\n\\[\n \\theta_p:V\\otimes_A\\mathbb Q_p\\xrightarrow{\\sim}\n L_p\\otimes_{\\mathbb Z_p}\\mathbb Q_p.\n\\]\nThen\n\\[\n \\Lambda=\\{v\\in V:\\theta_p(v)\\in L_p\\text{ for every }p\\in\\mathcal P\\}\n\\]\nis a finite free $\\mathbb Z$-module of rank $r$, and the natural maps identify\n\\[\n \\Lambda[1/D]\\cong V,\\qquad\n \\Lambda\\otimes_{\\mathbb Z}\\mathbb Z_p\\cong L_p.\n\\]\nIt is the unique lattice inside $V$ with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications.\n\\end{theorem}\n\\begin{proof}\nChoose a basis $V=A^r$, and let $E_p=\\theta_p^{-1}(L_p)$ be the corresponding lattice in $\\mathbb Q_p^r$. Bounds on the entries of $\\theta_p$ and its inverse give a common $N\\geq1$ such that, with $b=D^N$,\n\\[\n b\\mathbb Z_p^r\\subseteq E_p\\subseteq b^{-1}\\mathbb Z_p^r\n \\quad\\text{for all }p.\n\\]\nThe intersection defining $\\Lambda$ therefore satisfies\n\\[\n b\\mathbb Z^r\\subseteq\\Lambda\\subseteq b^{-…","bda\\otimes_{\\mathbb Z}\\mathbb Z_p\\cong L_p.\n\\]\nIt is the unique lattice inside $V$ with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications.\n\\end{theorem}\n\\begin{proof}\nChoose a basis $V=A^r$, and let $E_p=\\theta_p^{-1}(L_p)$ be the corresponding lattice in $\\mathbb Q_p^r$. Bounds on the entries of $\\theta_p$ and its inverse give a common $N\\geq1$ such that, with $b=D^N$,\n\\[\n b\\mathbb Z_p^r\\subseteq E_p\\subseteq b^{-1}\\mathbb Z_p^r\n \\quad\\text{for all }p.\n\\]\nThe intersection defining $\\Lambda$ therefore satisfies\n\\[\n b\\mathbb Z^r\\subseteq\\Lambda\\subseteq b^{-1}\\mathbb Z^r.\n\\]\nThe right inclusion uses the same integrality-of-denominators argument as the arithmetic fracture theorem, coordinatewise. Consequently $\\Lambda$ is a finite-rank free abelian lattice of rank $r$, and localizing the bounds at $D$ gives $\\Lambda[1/D]=A^r$.","bb Z}\\mathbb Z_p\\cong L_p.\n\\]\nIt is the unique lattice inside $V$ with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications.\n\\end{theorem}\n\\begin{proof}\nChoose a basis $V=A^r$, and let $E_p=\\theta_p^{-1}(L_p)$ be the corresponding lattice in $\\mathbb Q_p^r$. Bounds on the entries of $\\theta_p$ and its inverse give a common $N\\geq1$ such that, with $b=D^N$,\n\\[\n b\\mathbb Z_p^r\\subseteq E_p\\subseteq b^{-1}\\mathbb Z_p^r\n \\quad\\text{for all }p.\n\\]\nThe intersection defining $\\Lambda$ therefore satisfies\n\\[\n b\\mathbb Z^r\\subseteq\\Lambda\\subseteq b^{-1}\\mathbb Z^r.\n\\]\nThe right inclusion uses the same integrality-of-denominators argument as the arithmetic fracture theorem, coordinatewise. Consequently $\\Lambda$ is a finite-rank free abelian lattice of rank $r$, and localizing the bounds at $D$ gives $\\Lambda[1/D]=A^r$.","side $V$ with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications.\n\\end{theorem}\n\\begin{proof}\nChoose a basis $V=A^r$, and let $E_p=\\theta_p^{-1}(L_p)$ be the corresponding lattice in $\\mathbb Q_p^r$. Bounds on the entries of $\\theta_p$ and its inverse give a common $N\\geq1$ such that, with $b=D^N$,\n\\[\n b\\mathbb Z_p^r\\subseteq E_p\\subseteq b^{-1}\\mathbb Z_p^r\n \\quad\\text{for all }p.\n\\]\nThe intersection defining $\\Lambda$ therefore satisfies\n\\[\n b\\mathbb Z^r\\subseteq\\Lambda\\subseteq b^{-1}\\mathbb Z^r.\n\\]\nThe right inclusion uses the same integrality-of-denominators argument as the arithmetic fracture theorem, coordinatewise. Consequently $\\Lambda$ is a finite-rank free abelian lattice of rank $r$, and localizing the bounds at $D$ gives $\\Lambda[1/D]=A^r$.","pecified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications.\n\\end{theorem}\n\\begin{proof}\nChoose a basis $V=A^r$, and let $E_p=\\theta_p^{-1}(L_p)$ be the corresponding lattice in $\\mathbb Q_p^r$. Bounds on the entries of $\\theta_p$ and its inverse give a common $N\\geq1$ such that, with $b=D^N$,\n\\[\n b\\mathbb Z_p^r\\subseteq E_p\\subseteq b^{-1}\\mathbb Z_p^r\n \\quad\\text{for all }p.\n\\]\nThe intersection defining $\\Lambda$ therefore satisfies\n\\[\n b\\mathbb Z^r\\subseteq\\Lambda\\subseteq b^{-1}\\mathbb Z^r.\n\\]\nThe right inclusion uses the same integrality-of-denominators argument as the arithmetic fracture theorem, coordinatewise. Consequently $\\Lambda$ is a finite-rank free abelian lattice of rank $r$, and localizing the bounds at $D$ gives $\\Lambda[1/D]=A^r$.","luing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications.\n\\end{theorem}\n\\begin{proof}\nChoose a basis $V=A^r$, and let $E_p=\\theta_p^{-1}(L_p)$ be the corresponding lattice in $\\mathbb Q_p^r$. Bounds on the entries of $\\theta_p$ and its inverse give a common $N\\geq1$ such that, with $b=D^N$,\n\\[\n b\\mathbb Z_p^r\\subseteq E_p\\subseteq b^{-1}\\mathbb Z_p^r\n \\quad\\text{for all }p.\n\\]\nThe intersection defining $\\Lambda$ therefore satisfies\n\\[\n b\\mathbb Z^r\\subseteq\\Lambda\\subseteq b^{-1}\\mathbb Z^r.\n\\]\nThe right inclusion uses the same integrality-of-denominators argument as the arithmetic fracture theorem, coordinatewise. Consequently $\\Lambda$ is a finite-rank free abelian lattice of rank $r$, and localizing the bounds at $D$ gives $\\Lambda[1/D]=A^r$.","ely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications.\n\\end{theorem}\n\\begin{proof}\nChoose a basis $V=A^r$, and let $E_p=\\theta_p^{-1}(L_p)$ be the corresponding lattice in $\\mathbb Q_p^r$. Bounds on the entries of $\\theta_p$ and its inverse give a common $N\\geq1$ such that, with $b=D^N$,\n\\[\n b\\mathbb Z_p^r\\subseteq E_p\\subseteq b^{-1}\\mathbb Z_p^r\n \\quad\\text{for all }p.\n\\]\nThe intersection defining $\\Lambda$ therefore satisfies\n\\[\n b\\mathbb Z^r\\subseteq\\Lambda\\subseteq b^{-1}\\mathbb Z^r.\n\\]\nThe right inclusion uses the same integrality-of-denominators argument as the arithmetic fracture theorem, coordinatewise. Consequently $\\Lambda$ is a finite-rank free abelian lattice of rank $r$, and localizing the bounds at $D$ gives $\\Lambda[1/D]=A^r$.","s of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications.\n\\end{theorem}\n\\begin{proof}\nChoose a basis $V=A^r$, and let $E_p=\\theta_p^{-1}(L_p)$ be the corresponding lattice in $\\mathbb Q_p^r$. Bounds on the entries of $\\theta_p$ and its inverse give a common $N\\geq1$ such that, with $b=D^N$,\n\\[\n b\\mathbb Z_p^r\\subseteq E_p\\subseteq b^{-1}\\mathbb Z_p^r\n \\quad\\text{for all }p.\n\\]\nThe intersection defining $\\Lambda$ therefore satisfies\n\\[\n b\\mathbb Z^r\\subseteq\\Lambda\\subseteq b^{-1}\\mathbb Z^r.\n\\]\nThe right inclusion uses the same integrality-of-denominators argument as the arithmetic fracture theorem, coordinatewise. Consequently $\\Lambda$ is a finite-rank free abelian lattice of rank $r$, and localizing the bounds at $D$ gives $\\Lambda[1/D]=A^r$.","l lattices, and specified generic identifications.\n\\end{theorem}\n\\begin{proof}\nChoose a basis $V=A^r$, and let $E_p=\\theta_p^{-1}(L_p)$ be the corresponding lattice in $\\mathbb Q_p^r$. Bounds on the entries of $\\theta_p$ and its inverse give a common $N\\geq1$ such that, with $b=D^N$,\n\\[\n b\\mathbb Z_p^r\\subseteq E_p\\subseteq b^{-1}\\mathbb Z_p^r\n \\quad\\text{for all }p.\n\\]\nThe intersection defining $\\Lambda$ therefore satisfies\n\\[\n b\\mathbb Z^r\\subseteq\\Lambda\\subseteq b^{-1}\\mathbb Z^r.\n\\]\nThe right inclusion uses the same integrality-of-denominators argument as the arithmetic fracture theorem, coordinatewise. Consequently $\\Lambda$ is a finite-rank free abelian lattice of rank $r$, and localizing the bounds at $D$ gives $\\Lambda[1/D]=A^r$.","ic identifications.\n\\end{theorem}\n\\begin{proof}\nChoose a basis $V=A^r$, and let $E_p=\\theta_p^{-1}(L_p)$ be the corresponding lattice in $\\mathbb Q_p^r$. Bounds on the entries of $\\theta_p$ and its inverse give a common $N\\geq1$ such that, with $b=D^N$,\n\\[\n b\\mathbb Z_p^r\\subseteq E_p\\subseteq b^{-1}\\mathbb Z_p^r\n \\quad\\text{for all }p.\n\\]\nThe intersection defining $\\Lambda$ therefore satisfies\n\\[\n b\\mathbb Z^r\\subseteq\\Lambda\\subseteq b^{-1}\\mathbb Z^r.\n\\]\nThe right inclusion uses the same integrality-of-denominators argument as the arithmetic fracture theorem, coordinatewise. Consequently $\\Lambda$ is a finite-rank free abelian lattice of rank $r$, and localizing the bounds at $D$ gives $\\Lambda[1/D]=A^r$.","nverse give a common $N\\geq1$ such that, with $b=D^N$,\n\\[\n b\\mathbb Z_p^r\\subseteq E_p\\subseteq b^{-1}\\mathbb Z_p^r\n \\quad\\text{for all }p.\n\\]\nThe intersection defining $\\Lambda$ therefore satisfies\n\\[\n b\\mathbb Z^r\\subseteq\\Lambda\\subseteq b^{-1}\\mathbb Z^r.\n\\]\nThe right inclusion uses the same integrality-of-denominators argument as the arithmetic fracture theorem, coordinatewise. Consequently $\\Lambda$ is a finite-rank free abelian lattice of rank $r$, and localizing the bounds at $D$ gives $\\Lambda[1/D]=A^r$.","[\n b\\mathbb Z_p^r\\subseteq E_p\\subseteq b^{-1}\\mathbb Z_p^r\n \\quad\\text{for all }p.\n\\]\nThe intersection defining $\\Lambda$ therefore satisfies\n\\[\n b\\mathbb Z^r\\subseteq\\Lambda\\subseteq b^{-1}\\mathbb Z^r.\n\\]\nThe right inclusion uses the same integrality-of-denominators argument as the arithmetic fracture theorem, coordinatewise. Consequently $\\Lambda$ is a finite-rank free abelian lattice of rank $r$, and localizing the bounds at $D$ gives $\\Lambda[1/D]=A^r$.","seteq b^{-1}\\mathbb Z_p^r\n \\quad\\text{for all }p.\n\\]\nThe intersection defining $\\Lambda$ therefore satisfies\n\\[\n b\\mathbb Z^r\\subseteq\\Lambda\\subseteq b^{-1}\\mathbb Z^r.\n\\]\nThe right inclusion uses the same integrality-of-denominators argument as the arithmetic fracture theorem, coordinatewise. Consequently $\\Lambda$ is a finite-rank free abelian lattice of rank $r$, and localizing the bounds at $D$ gives $\\Lambda[1/D]=A^r$.","1}\\mathbb Z_p^r\n \\quad\\text{for all }p.\n\\]\nThe intersection defining $\\Lambda$ therefore satisfies\n\\[\n b\\mathbb Z^r\\subseteq\\Lambda\\subseteq b^{-1}\\mathbb Z^r.\n\\]\nThe right inclusion uses the same integrality-of-denominators argument as the arithmetic fracture theorem, coordinatewise. Consequently $\\Lambda$ is a finite-rank free abelian lattice of rank $r$, and localizing the bounds at $D$ gives $\\Lambda[1/D]=A^r$.","Here is also an explicit finite construction proving the local identifications. Put $F_p=bE_p$, so\n\\[\n b^2\\mathbb Z_p^r\\subseteq F_p\\subseteq\\mathbb Z_p^r.\n\\]\nThe Chinese remainder isomorphism identifies\n\\[\n \\mathbb Z^r/b^2\\mathbb Z^r\\cong\n \\prod_{p\\in\\mathcal P}\\mathbb Z_p^r/b^2\\mathbb Z_p^r.\n\\]\nLet $H\\subseteq\\mathbb Z^r$ be the inverse image of the product of the subgroups $F_p/b^2\\mathbb Z_p^r$. Directly from the definitions, $H=b\\Lambda$. The quotient $\\mathbb Z^r/H$ is the product of the corresponding finite $p$-primary quotients. Tensoring with $\\mathbb Z_p$ retains exactly the $p$ component. Thus $H\\otimes\\mathbb Z_p$ is the inverse image of $F_p/b^2\\mathbb Z_p^r$, namely $F_p$. Dividing by $b$ gives $\\Lambda\\otimes\\mathbb Z_p=E_p$, and $\\theta_p$ identifies it with $L_p$.","struction proving the local identifications. Put $F_p=bE_p$, so\n\\[\n b^2\\mathbb Z_p^r\\subseteq F_p\\subseteq\\mathbb Z_p^r.\n\\]\nThe Chinese remainder isomorphism identifies\n\\[\n \\mathbb Z^r/b^2\\mathbb Z^r\\cong\n \\prod_{p\\in\\mathcal P}\\mathbb Z_p^r/b^2\\mathbb Z_p^r.\n\\]\nLet $H\\subseteq\\mathbb Z^r$ be the inverse image of the product of the subgroups $F_p/b^2\\mathbb Z_p^r$. Directly from the definitions, $H=b\\Lambda$. The quotient $\\mathbb Z^r/H$ is the product of the corresponding finite $p$-primary quotients. Tensoring with $\\mathbb Z_p$ retains exactly the $p$ component. Thus $H\\otimes\\mathbb Z_p$ is the inverse image of $F_p/b^2\\mathbb Z_p^r$, namely $F_p$. Dividing by $b$ gives $\\Lambda\\otimes\\mathbb Z_p=E_p$, and $\\theta_p$ identifies it with $L_p$.","identifications. Put $F_p=bE_p$, so\n\\[\n b^2\\mathbb Z_p^r\\subseteq F_p\\subseteq\\mathbb Z_p^r.\n\\]\nThe Chinese remainder isomorphism identifies\n\\[\n \\mathbb Z^r/b^2\\mathbb Z^r\\cong\n \\prod_{p\\in\\mathcal P}\\mathbb Z_p^r/b^2\\mathbb Z_p^r.\n\\]\nLet $H\\subseteq\\mathbb Z^r$ be the inverse image of the product of the subgroups $F_p/b^2\\mathbb Z_p^r$. Directly from the definitions, $H=b\\Lambda$. The quotient $\\mathbb Z^r/H$ is the product of the corresponding finite $p$-primary quotients. Tensoring with $\\mathbb Z_p$ retains exactly the $p$ component. Thus $H\\otimes\\mathbb Z_p$ is the inverse image of $F_p/b^2\\mathbb Z_p^r$, namely $F_p$. Dividing by $b$ gives $\\Lambda\\otimes\\mathbb Z_p=E_p$, and $\\theta_p$ identifies it with $L_p$.","ubseteq F_p\\subseteq\\mathbb Z_p^r.\n\\]\nThe Chinese remainder isomorphism identifies\n\\[\n \\mathbb Z^r/b^2\\mathbb Z^r\\cong\n \\prod_{p\\in\\mathcal P}\\mathbb Z_p^r/b^2\\mathbb Z_p^r.\n\\]\nLet $H\\subseteq\\mathbb Z^r$ be the inverse image of the product of the subgroups $F_p/b^2\\mathbb Z_p^r$. Directly from the definitions, $H=b\\Lambda$. The quotient $\\mathbb Z^r/H$ is the product of the corresponding finite $p$-primary quotients. Tensoring with $\\mathbb Z_p$ retains exactly the $p$ component. Thus $H\\otimes\\mathbb Z_p$ is the inverse image of $F_p/b^2\\mathbb Z_p^r$, namely $F_p$. Dividing by $b$ gives $\\Lambda\\otimes\\mathbb Z_p=E_p$, and $\\theta_p$ identifies it with $L_p$.","The Chinese remainder isomorphism identifies\n\\[\n \\mathbb Z^r/b^2\\mathbb Z^r\\cong\n \\prod_{p\\in\\mathcal P}\\mathbb Z_p^r/b^2\\mathbb Z_p^r.\n\\]\nLet $H\\subseteq\\mathbb Z^r$ be the inverse image of the product of the subgroups $F_p/b^2\\mathbb Z_p^r$. Directly from the definitions, $H=b\\Lambda$. The quotient $\\mathbb Z^r/H$ is the product of the corresponding finite $p$-primary quotients. Tensoring with $\\mathbb Z_p$ retains exactly the $p$ component. Thus $H\\otimes\\mathbb Z_p$ is the inverse image of $F_p/b^2\\mathbb Z_p^r$, namely $F_p$. Dividing by $b$ gives $\\Lambda\\otimes\\mathbb Z_p=E_p$, and $\\theta_p$ identifies it with $L_p$.","identifies\n\\[\n \\mathbb Z^r/b^2\\mathbb Z^r\\cong\n \\prod_{p\\in\\mathcal P}\\mathbb Z_p^r/b^2\\mathbb Z_p^r.\n\\]\nLet $H\\subseteq\\mathbb Z^r$ be the inverse image of the product of the subgroups $F_p/b^2\\mathbb Z_p^r$. Directly from the definitions, $H=b\\Lambda$. The quotient $\\mathbb Z^r/H$ is the product of the corresponding finite $p$-primary quotients. Tensoring with $\\mathbb Z_p$ retains exactly the $p$ component. Thus $H\\otimes\\mathbb Z_p$ is the inverse image of $F_p/b^2\\mathbb Z_p^r$, namely $F_p$. Dividing by $b$ gives $\\Lambda\\otimes\\mathbb Z_p=E_p$, and $\\theta_p$ identifies it with $L_p$.","bb Z^r/b^2\\mathbb Z^r\\cong\n \\prod_{p\\in\\mathcal P}\\mathbb Z_p^r/b^2\\mathbb Z_p^r.\n\\]\nLet $H\\subseteq\\mathbb Z^r$ be the inverse image of the product of the subgroups $F_p/b^2\\mathbb Z_p^r$. Directly from the definitions, $H=b\\Lambda$. The quotient $\\mathbb Z^r/H$ is the product of the corresponding finite $p$-primary quotients. Tensoring with $\\mathbb Z_p$ retains exactly the $p$ component. Thus $H\\otimes\\mathbb Z_p$ is the inverse image of $F_p/b^2\\mathbb Z_p^r$, namely $F_p$. Dividing by $b$ gives $\\Lambda\\otimes\\mathbb Z_p=E_p$, and $\\theta_p$ identifies it with $L_p$.","l P}\\mathbb Z_p^r/b^2\\mathbb Z_p^r.\n\\]\nLet $H\\subseteq\\mathbb Z^r$ be the inverse image of the product of the subgroups $F_p/b^2\\mathbb Z_p^r$. Directly from the definitions, $H=b\\Lambda$. The quotient $\\mathbb Z^r/H$ is the product of the corresponding finite $p$-primary quotients. Tensoring with $\\mathbb Z_p$ retains exactly the $p$ component. Thus $H\\otimes\\mathbb Z_p$ is the inverse image of $F_p/b^2\\mathbb Z_p^r$, namely $F_p$. Dividing by $b$ gives $\\Lambda\\otimes\\mathbb Z_p=E_p$, and $\\theta_p$ identifies it with $L_p$.","p^r.\n\\]\nLet $H\\subseteq\\mathbb Z^r$ be the inverse image of the product of the subgroups $F_p/b^2\\mathbb Z_p^r$. Directly from the definitions, $H=b\\Lambda$. The quotient $\\mathbb Z^r/H$ is the product of the corresponding finite $p$-primary quotients. Tensoring with $\\mathbb Z_p$ retains exactly the $p$ component. Thus $H\\otimes\\mathbb Z_p$ is the inverse image of $F_p/b^2\\mathbb Z_p^r$, namely $F_p$. Dividing by $b$ gives $\\Lambda\\otimes\\mathbb Z_p=E_p$, and $\\theta_p$ identifies it with $L_p$.","eteq\\mathbb Z^r$ be the inverse image of the product of the subgroups $F_p/b^2\\mathbb Z_p^r$. Directly from the definitions, $H=b\\Lambda$. The quotient $\\mathbb Z^r/H$ is the product of the corresponding finite $p$-primary quotients. Tensoring with $\\mathbb Z_p$ retains exactly the $p$ component. Thus $H\\otimes\\mathbb Z_p$ is the inverse image of $F_p/b^2\\mathbb Z_p^r$, namely $F_p$. Dividing by $b$ gives $\\Lambda\\otimes\\mathbb Z_p=E_p$, and $\\theta_p$ identifies it with $L_p$.","b Z^r$ be the inverse image of the product of the subgroups $F_p/b^2\\mathbb Z_p^r$. Directly from the definitions, $H=b\\Lambda$. The quotient $\\mathbb Z^r/H$ is the product of the corresponding finite $p$-primary quotients. Tensoring with $\\mathbb Z_p$ retains exactly the $p$ component. Thus $H\\otimes\\mathbb Z_p$ is the inverse image of $F_p/b^2\\mathbb Z_p^r$, namely $F_p$. Dividing by $b$ gives $\\Lambda\\otimes\\mathbb Z_p=E_p$, and $\\theta_p$ identifies it with $L_p$.","oduct of the subgroups $F_p/b^2\\mathbb Z_p^r$. Directly from the definitions, $H=b\\Lambda$. The quotient $\\mathbb Z^r/H$ is the product of the corresponding finite $p$-primary quotients. Tensoring with $\\mathbb Z_p$ retains exactly the $p$ component. Thus $H\\otimes\\mathbb Z_p$ is the inverse image of $F_p/b^2\\mathbb Z_p^r$, namely $F_p$. Dividing by $b$ gives $\\Lambda\\otimes\\mathbb Z_p=E_p$, and $\\theta_p$ identifies it with $L_p$.","2\\mathbb Z_p^r$. Directly from the definitions, $H=b\\Lambda$. The quotient $\\mathbb Z^r/H$ is the product of the corresponding finite $p$-primary quotients. Tensoring with $\\mathbb Z_p$ retains exactly the $p$ component. Thus $H\\otimes\\mathbb Z_p$ is the inverse image of $F_p/b^2\\mathbb Z_p^r$, namely $F_p$. Dividing by $b$ gives $\\Lambda\\otimes\\mathbb Z_p=E_p$, and $\\theta_p$ identifies it with $L_p$.","For completeness, tensoring the displayed subgroup inclusion is exact: $\\mathbb Z_p$ is torsion-free over $\\mathbb Z$, so it is flat; alternatively the assertion follows from a Smith diagonal presentation of the finite-index subgroup $H$. The latter proves the same statement using only integer bases.","Any other lattice with the specified generic and local identifications has analogous bounds by powers of $D$. After a common scaling, its finite quotient is determined by the same primary subgroups in the Chinese remainder decomposition. It therefore equals $\\Lambda$. A compatible $A$-linear map $V\\to V'$ sends every $\\theta_p$-integral element to a $\\theta'_p$-integral element and hence restricts to $\\Lambda\\to\\Lambda'$. Conversely a lattice map induces the compatible generic and local maps. Uniqueness follows from the embedding in $V$. These constructions are inverse on objects up to the specified identifications and on morphisms, proving the equivalence. The zero-rank case gives the zero module throughout.\n\\end{proof}","ther lattice with the specified generic and local identifications has analogous bounds by powers of $D$. After a common scaling, its finite quotient is determined by the same primary subgroups in the Chinese remainder decomposition. It therefore equals $\\Lambda$. A compatible $A$-linear map $V\\to V'$ sends every $\\theta_p$-integral element to a $\\theta'_p$-integral element and hence restricts to $\\Lambda\\to\\Lambda'$. Conversely a lattice map induces the compatible generic and local maps. Uniqueness follows from the embedding in $V$. These constructions are inverse on objects up to the specified identifications and on morphisms, proving the equivalence. The zero-rank case gives the zero module throughout.\n\\end{proof}","nite quotient is determined by the same primary subgroups in the Chinese remainder decomposition. It therefore equals $\\Lambda$. A compatible $A$-linear map $V\\to V'$ sends every $\\theta_p$-integral element to a $\\theta'_p$-integral element and hence restricts to $\\Lambda\\to\\Lambda'$. Conversely a lattice map induces the compatible generic and local maps. Uniqueness follows from the embedding in $V$. These constructions are inverse on objects up to the specified identifications and on morphisms, proving the equivalence. The zero-rank case gives the zero module throughout.\n\\end{proof}","\\begin{corollary}[Compatible operators glue with the lattice]\\label{cor:lattice-operators}\nIn \\cref{thm:lattice-gluing}, suppose $f_1,\\ldots,f_s\\in\\operatorname{End}_A(V)$ and $g_{p,1},\\ldots,g_{p,s}\\in\\operatorname{End}_{\\mathbb Z_p}(L_p)$ satisfy the exact intertwining equations\n\\[\n \\theta_p(f_i\\otimes1)=(g_{p,i}\\otimes1)\\theta_p.\n\\]\nThen they induce unique integral endomorphisms of $\\Lambda$. Every specified noncommutative polynomial relation with integer coefficients holding on $V$ holds on $\\Lambda$. If the operators and their inverses are supplied compatibly, they glue as automorphisms.\n\\end{corollary}\n\\begin{proof}\nFor $v\\in\\Lambda$, the displayed identity puts $\\theta_p(f_i v)=g_{p,i}\\theta_p(v)$ in $L_p$ for every $p$. Hence $f_i v\\in\\Lambda$. Restriction is unique because $\\Lambda$ spans $V$ after localization. The polynomial relations hold by rest","\\begin{corollary}[Compatible operators glue with the lattice]\\label{cor:lattice-operators}\nIn \\cref{thm:lattice-gluing}, suppose $f_1,\\ldots,f_s\\in\\operatorname{End}_A(V)$ and $g_{p,1},\\ldots,g_{p,s}\\in\\operatorname{End}_{\\mathbb Z_p}(L_p)$ satisfy the exact intertwining equations\n\\[\n \\theta_p(f_i\\otimes1)=(g_{p,i}\\otimes1)\\theta_p.\n\\]\nThen they induce unique integral endomorphisms of $\\Lambda$. Every specified noncommutative polynomial relation with integer coefficients holding on $V$ holds on $\\Lambda$. If the operators and their inverses are supplied compatibly, they glue as automorphisms.\n\\end{corollary}\n\\begin{proof}\nFor $v\\in\\Lambda$, the displayed identity puts $\\theta_p(f_i v)=g_{p,i}\\theta_p(v)$ in $L_p$ for every $p$. Hence $f_i v\\in\\Lambda$. Restriction is unique because $\\Lambda$ spans $V$ after localization. The polynomial relations hold by restriction from $V$, and applying the argument to the supplied inverses","\\begin{corollary}[Compatible operators glue with the lattice]\\label{cor:lattice-operators}\nIn \\cref{thm:lattice-gluing}, suppose $f_1,\\ldots,f_s\\in\\operatorname{End}_A(V)$ and $g_{p,1},\\ldots,g_{p,s}\\in\\operatorname{End}_{\\mathbb Z_p}(L_p)$ satisfy the exact intertwining equations\n\\[\n \\theta_p(f_i\\otimes1)=(g_{p,i}\\otimes1)\\theta_p.\n\\]\nThen they induce unique integral endomorphisms of $\\Lambda$. Every specified noncommutative polynomial relation with integer coefficients holding on $V$ holds on $\\Lambda$. If the operators and their inverses are supplied compatibly, they glue as automorphisms.\n\\end{corollary}\n\\begin{proof}\nFor $v\\in\\Lambda$, the displayed identity puts $\\theta_p(f_i v)=g_{p,i}\\theta_p(v)$ in $L_p$ for every $p$. Hence $f_i v\\in\\Lambda$. Restriction is unique because $\\Lambda$ spans $V$ after localization. The polynomial relations hold by restriction from $V$, and applying the argument to the supplied inverses gives automorphisms.\n\\end{proof}","s}\nIn \\cref{thm:lattice-gluing}, suppose $f_1,\\ldots,f_s\\in\\operatorname{End}_A(V)$ and $g_{p,1},\\ldots,g_{p,s}\\in\\operatorname{End}_{\\mathbb Z_p}(L_p)$ satisfy the exact intertwining equations\n\\[\n \\theta_p(f_i\\otimes1)=(g_{p,i}\\otimes1)\\theta_p.\n\\]\nThen they induce unique integral endomorphisms of $\\Lambda$. Every specified noncommutative polynomial relation with integer coefficients holding on $V$ holds on $\\Lambda$. If the operators and their inverses are supplied compatibly, they glue as automorphisms.\n\\end{corollary}\n\\begin{proof}\nFor $v\\in\\Lambda$, the displayed identity puts $\\theta_p(f_i v)=g_{p,i}\\theta_p(v)$ in $L_p$ for every $p$. Hence $f_i v\\in\\Lambda$. Restriction is unique because $\\Lambda$ spans $V$ after localization. The polynomial relations hold by restriction from $V$, and applying the argument to the supplied inverses gives automorphisms.\n\\end{proof}","hm:lattice-gluing}, suppose $f_1,\\ldots,f_s\\in\\operatorname{End}_A(V)$ and $g_{p,1},\\ldots,g_{p,s}\\in\\operatorname{End}_{\\mathbb Z_p}(L_p)$ satisfy the exact intertwining equations\n\\[\n \\theta_p(f_i\\otimes1)=(g_{p,i}\\otimes1)\\theta_p.\n\\]\nThen they induce unique integral endomorphisms of $\\Lambda$. Every specified noncommutative polynomial relation with integer coefficients holding on $V$ holds on $\\Lambda$. If the operators and their inverses are supplied compatibly, they glue as automorphisms.\n\\end{corollary}\n\\begin{proof}\nFor $v\\in\\Lambda$, the displayed identity puts $\\theta_p(f_i v)=g_{p,i}\\theta_p(v)$ in $L_p$ for every $p$. Hence $f_i v\\in\\Lambda$. Restriction is unique because $\\Lambda$ spans $V$ after localization. The polynomial relations hold by restriction from $V$, and applying the argument to the supplied inverses gives automorphisms.\n\\end{proof}","L_p)$ satisfy the exact intertwining equations\n\\[\n \\theta_p(f_i\\otimes1)=(g_{p,i}\\otimes1)\\theta_p.\n\\]\nThen they induce unique integral endomorphisms of $\\Lambda$. Every specified noncommutative polynomial relation with integer coefficients holding on $V$ holds on $\\Lambda$. If the operators and their inverses are supplied compatibly, they glue as automorphisms.\n\\end{corollary}\n\\begin{proof}\nFor $v\\in\\Lambda$, the displayed identity puts $\\theta_p(f_i v)=g_{p,i}\\theta_p(v)$ in $L_p$ for every $p$. Hence $f_i v\\in\\Lambda$. Restriction is unique because $\\Lambda$ spans $V$ after localization. The polynomial relations hold by restriction from $V$, and applying the argument to the supplied inverses gives automorphisms.\n\\end{proof}","ons\n\\[\n \\theta_p(f_i\\otimes1)=(g_{p,i}\\otimes1)\\theta_p.\n\\]\nThen they induce unique integral endomorphisms of $\\Lambda$. Every specified noncommutative polynomial relation with integer coefficients holding on $V$ holds on $\\Lambda$. If the operators and their inverses are supplied compatibly, they glue as automorphisms.\n\\end{corollary}\n\\begin{proof}\nFor $v\\in\\Lambda$, the displayed identity puts $\\theta_p(f_i v)=g_{p,i}\\theta_p(v)$ in $L_p$ for every $p$. Hence $f_i v\\in\\Lambda$. Restriction is unique because $\\Lambda$ spans $V$ after localization. The polynomial relations hold by restriction from $V$, and applying the argument to the supplied inverses gives automorphisms.\n\\end{proof}","times1)\\theta_p.\n\\]\nThen they induce unique integral endomorphisms of $\\Lambda$. Every specified noncommutative polynomial relation with integer coefficients holding on $V$ holds on $\\Lambda$. If the operators and their inverses are supplied compatibly, they glue as automorphisms.\n\\end{corollary}\n\\begin{proof}\nFor $v\\in\\Lambda$, the displayed identity puts $\\theta_p(f_i v)=g_{p,i}\\theta_p(v)$ in $L_p$ for every $p$. Hence $f_i v\\in\\Lambda$. Restriction is unique because $\\Lambda$ spans $V$ after localization. The polynomial relations hold by restriction from $V$, and applying the argument to the supplied inverses gives automorphisms.\n\\end{proof}","\\]\nThen they induce unique integral endomorphisms of $\\Lambda$. Every specified noncommutative polynomial relation with integer coefficients holding on $V$ holds on $\\Lambda$. If the operators and their inverses are supplied compatibly, they glue as automorphisms.\n\\end{corollary}\n\\begin{proof}\nFor $v\\in\\Lambda$, the displayed identity puts $\\theta_p(f_i v)=g_{p,i}\\theta_p(v)$ in $L_p$ for every $p$. Hence $f_i v\\in\\Lambda$. Restriction is unique because $\\Lambda$ spans $V$ after localization. The polynomial relations hold by restriction from $V$, and applying the argument to the supplied inverses gives automorphisms.\n\\end{proof}","ey induce unique integral endomorphisms of $\\Lambda$. Every specified noncommutative polynomial relation with integer coefficients holding on $V$ holds on $\\Lambda$. If the operators and their inverses are supplied compatibly, they glue as automorphisms.\n\\end{corollary}\n\\begin{proof}\nFor $v\\in\\Lambda$, the displayed identity puts $\\theta_p(f_i v)=g_{p,i}\\theta_p(v)$ in $L_p$ for every $p$. Hence $f_i v\\in\\Lambda$. Restriction is unique because $\\Lambda$ spans $V$ after localization. The polynomial relations hold by restriction from $V$, and applying the argument to the supplied inverses gives automorphisms.\n\\end{proof}","da$. Every specified noncommutative polynomial relation with integer coefficients holding on $V$ holds on $\\Lambda$. If the operators and their inverses are supplied compatibly, they glue as automorphisms.\n\\end{corollary}\n\\begin{proof}\nFor $v\\in\\Lambda$, the displayed identity puts $\\theta_p(f_i v)=g_{p,i}\\theta_p(v)$ in $L_p$ for every $p$. Hence $f_i v\\in\\Lambda$. Restriction is unique because $\\Lambda$ spans $V$ after localization. The polynomial relations hold by restriction from $V$, and applying the argument to the supplied inverses gives automorphisms.\n\\end{proof}","fied noncommutative polynomial relation with integer coefficients holding on $V$ holds on $\\Lambda$. If the operators and their inverses are supplied compatibly, they glue as automorphisms.\n\\end{corollary}\n\\begin{proof}\nFor $v\\in\\Lambda$, the displayed identity puts $\\theta_p(f_i v)=g_{p,i}\\theta_p(v)$ in $L_p$ for every $p$. Hence $f_i v\\in\\Lambda$. Restriction is unique because $\\Lambda$ spans $V$ after localization. The polynomial relations hold by restriction from $V$, and applying the argument to the supplied inverses gives automorphisms.\n\\end{proof}","on $V$ holds on $\\Lambda$. If the operators and their inverses are supplied compatibly, they glue as automorphisms.\n\\end{corollary}\n\\begin{proof}\nFor $v\\in\\Lambda$, the displayed identity puts $\\theta_p(f_i v)=g_{p,i}\\theta_p(v)$ in $L_p$ for every $p$. Hence $f_i v\\in\\Lambda$. Restriction is unique because $\\Lambda$ spans $V$ after localization. The polynomial relations hold by restriction from $V$, and applying the argument to the supplied inverses gives automorphisms.\n\\end{proof}","Throughout the lattice part, fix a prime $p$, write $O=\\mathbb Z_p$ and $O_n=O/p^nO$, and take $n\\geq1$. A lattice means a finite free $O$-module. The exponent in $Q[p^n]$ denotes annihilation by $p^n$, not a quotient. We distinguish a residual module, a selected element of that module obstructing a particular lift, and a cone in a specified derived category.","For a map $d:V\\to W$ of lattices, put\n\\[\n K=\\ker d,\\qquad Q=\\operatorname{coker}d,\\qquad\n K_n=\\ker(d\\otimes_O O_n).\n\\]\nThe module $K$ is saturated in $V$: if $p^n v\\in K$, then $p^n d(v)=0$ in the torsion-free module $W$, so $v\\in K$. It is therefore a finite free direct summand as an $O$-module, although a splitting is not canonical. Reduction supplies a natural injection $\\alpha_n:K/p^nK\\to K_n$.","\\begin{theorem}[Intrinsic kernel exchange and its boundary]\\label{thm:kernel-bockstein}\nThere is a natural short exact sequence\n\\[\n \\boxed{0\\longrightarrow K/p^nK\\xrightarrow{\\alpha_n}K_n\n \\xrightarrow{\\delta_n}Q[p^n]\\longrightarrow0.}\n\\]\nFor $\\bar v\\in K_n$, choose a lift $v\\in V$. Since $d(v)\\in p^nW$, set\n\\[\n \\delta_n(\\bar v)=\\left[\\frac{d(v)}{p^n}\\right]\\in Q.\n\\]\nThis formula is independent of the lift. For $m\\geq n$, ordinary reduction $r_{m,n}:K_m\\to K_n$ satisfies\n\\[\n \\boxed{\\delta_n r_{m,n}=p^{m-n}\\delta_m.}\n\\]\nConsequently the cokernel of the natural comparison $\\alpha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanish","\\begin{theorem}[Intrinsic kernel exchange and its boundary]\\label{thm:kernel-bockstein}\nThere is a natural short exact sequence\n\\[\n \\boxed{0\\longrightarrow K/p^nK\\xrightarrow{\\alpha_n}K_n\n \\xrightarrow{\\delta_n}Q[p^n]\\longrightarrow0.}\n\\]\nFor $\\bar v\\in K_n$, choose a lift $v\\in V$. Since $d(v)\\in p^nW$, set\n\\[\n \\delta_n(\\bar v)=\\left[\\frac{d(v)}{p^n}\\right]\\in Q.\n\\]\nThis formula is independent of the lift. For $m\\geq n$, ordinary reduction $r_{m,n}:K_m\\to K_n$ satisfies\n\\[\n \\boxed{\\delta_n r_{m,n}=p^{m-n}\\delta_m.}\n\\]\nConsequently the cokernel of the natural comparison $\\alpha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d","\\begin{theorem}[Intrinsic kernel exchange and its boundary]\\label{thm:kernel-bockstein}\nThere is a natural short exact sequence\n\\[\n \\boxed{0\\longrightarrow K/p^nK\\xrightarrow{\\alpha_n}K_n\n \\xrightarrow{\\delta_n}Q[p^n]\\longrightarrow0.}\n\\]\nFor $\\bar v\\in K_n$, choose a lift $v\\in V$. Since $d(v)\\in p^nW$, set\n\\[\n \\delta_n(\\bar v)=\\left[\\frac{d(v)}{p^n}\\right]\\in Q.\n\\]\nThis formula is independent of the lift. For $m\\geq n$, ordinary reduction $r_{m,n}:K_m\\to K_n$ satisfies\n\\[\n \\boxed{\\delta_n r_{m,n}=p^{m-n}\\delta_m.}\n\\]\nConsequently the cokernel of the natural comparison $\\alpha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by","\\begin{theorem}[Intrinsic kernel exchange and its boundary]\\label{thm:kernel-bockstein}\nThere is a natural short exact sequence\n\\[\n \\boxed{0\\longrightarrow K/p^nK\\xrightarrow{\\alpha_n}K_n\n \\xrightarrow{\\delta_n}Q[p^n]\\longrightarrow0.}\n\\]\nFor $\\bar v\\in K_n$, choose a lift $v\\in V$. Since $d(v)\\in p^nW$, set\n\\[\n \\delta_n(\\bar v)=\\left[\\frac{d(v)}{p^n}\\right]\\in Q.\n\\]\nThis formula is independent of the lift. For $m\\geq n$, ordinary reduction $r_{m,n}:K_m\\to K_n$ satisfies\n\\[\n \\boxed{\\delta_n r_{m,n}=p^{m-n}\\delta_m.}\n\\]\nConsequently the cokernel of the natural comparison $\\alpha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This p","\\begin{theorem}[Intrinsic kernel exchange and its boundary]\\label{thm:kernel-bockstein}\nThere is a natural short exact sequence\n\\[\n \\boxed{0\\longrightarrow K/p^nK\\xrightarrow{\\alpha_n}K_n\n \\xrightarrow{\\delta_n}Q[p^n]\\longrightarrow0.}\n\\]\nFor $\\bar v\\in K_n$, choose a lift $v\\in V$. Since $d(v)\\in p^nW$, set\n\\[\n \\delta_n(\\bar v)=\\left[\\frac{d(v)}{p^n}\\right]\\in Q.\n\\]\nThis formula is independent of the lift. For $m\\geq n$, ordinary reduction $r_{m,n}:K_m\\to K_n$ satisfies\n\\[\n \\boxed{\\delta_n r_{m,n}=p^{m-n}\\delta_m.}\n\\]\nConsequently the cokernel of the natural comparison $\\alpha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there","Intrinsic kernel exchange and its boundary]\\label{thm:kernel-bockstein}\nThere is a natural short exact sequence\n\\[\n \\boxed{0\\longrightarrow K/p^nK\\xrightarrow{\\alpha_n}K_n\n \\xrightarrow{\\delta_n}Q[p^n]\\longrightarrow0.}\n\\]\nFor $\\bar v\\in K_n$, choose a lift $v\\in V$. Since $d(v)\\in p^nW$, set\n\\[\n \\delta_n(\\bar v)=\\left[\\frac{d(v)}{p^n}\\right]\\in Q.\n\\]\nThis formula is independent of the lift. For $m\\geq n$, ordinary reduction $r_{m,n}:K_m\\to K_n$ satisfies\n\\[\n \\boxed{\\delta_n r_{m,n}=p^{m-n}\\delta_m.}\n\\]\nConsequently the cokernel of the natural comparison $\\alpha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$…","its boundary]\\label{thm:kernel-bockstein}\nThere is a natural short exact sequence\n\\[\n \\boxed{0\\longrightarrow K/p^nK\\xrightarrow{\\alpha_n}K_n\n \\xrightarrow{\\delta_n}Q[p^n]\\longrightarrow0.}\n\\]\nFor $\\bar v\\in K_n$, choose a lift $v\\in V$. Since $d(v)\\in p^nW$, set\n\\[\n \\delta_n(\\bar v)=\\left[\\frac{d(v)}{p^n}\\right]\\in Q.\n\\]\nThis formula is independent of the lift. For $m\\geq n$, ordinary reduction $r_{m,n}:K_m\\to K_n$ satisfies\n\\[\n \\boxed{\\delta_n r_{m,n}=p^{m-n}\\delta_m.}\n\\]\nConsequently the cokernel of the natural comparison $\\alpha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation pro…","ockstein}\nThere is a natural short exact sequence\n\\[\n \\boxed{0\\longrightarrow K/p^nK\\xrightarrow{\\alpha_n}K_n\n \\xrightarrow{\\delta_n}Q[p^n]\\longrightarrow0.}\n\\]\nFor $\\bar v\\in K_n$, choose a lift $v\\in V$. Since $d(v)\\in p^nW$, set\n\\[\n \\delta_n(\\bar v)=\\left[\\frac{d(v)}{p^n}\\right]\\in Q.\n\\]\nThis formula is independent of the lift. For $m\\geq n$, ordinary reduction $r_{m,n}:K_m\\to K_n$ satisfies\n\\[\n \\boxed{\\delta_n r_{m,n}=p^{m-n}\\delta_m.}\n\\]\nConsequently the cokernel of the natural comparison $\\alpha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.","pha_n}K_n\n \\xrightarrow{\\delta_n}Q[p^n]\\longrightarrow0.}\n\\]\nFor $\\bar v\\in K_n$, choose a lift $v\\in V$. Since $d(v)\\in p^nW$, set\n\\[\n \\delta_n(\\bar v)=\\left[\\frac{d(v)}{p^n}\\right]\\in Q.\n\\]\nThis formula is independent of the lift. For $m\\geq n$, ordinary reduction $r_{m,n}:K_m\\to K_n$ satisfies\n\\[\n \\boxed{\\delta_n r_{m,n}=p^{m-n}\\delta_m.}\n\\]\nConsequently the cokernel of the natural comparison $\\alpha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.","ightarrow{\\delta_n}Q[p^n]\\longrightarrow0.}\n\\]\nFor $\\bar v\\in K_n$, choose a lift $v\\in V$. Since $d(v)\\in p^nW$, set\n\\[\n \\delta_n(\\bar v)=\\left[\\frac{d(v)}{p^n}\\right]\\in Q.\n\\]\nThis formula is independent of the lift. For $m\\geq n$, ordinary reduction $r_{m,n}:K_m\\to K_n$ satisfies\n\\[\n \\boxed{\\delta_n r_{m,n}=p^{m-n}\\delta_m.}\n\\]\nConsequently the cokernel of the natural comparison $\\alpha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.","$d(v)\\in p^nW$, set\n\\[\n \\delta_n(\\bar v)=\\left[\\frac{d(v)}{p^n}\\right]\\in Q.\n\\]\nThis formula is independent of the lift. For $m\\geq n$, ordinary reduction $r_{m,n}:K_m\\to K_n$ satisfies\n\\[\n \\boxed{\\delta_n r_{m,n}=p^{m-n}\\delta_m.}\n\\]\nConsequently the cokernel of the natural comparison $\\alpha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.","ependent of the lift. For $m\\geq n$, ordinary reduction $r_{m,n}:K_m\\to K_n$ satisfies\n\\[\n \\boxed{\\delta_n r_{m,n}=p^{m-n}\\delta_m.}\n\\]\nConsequently the cokernel of the natural comparison $\\alpha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.","t. For $m\\geq n$, ordinary reduction $r_{m,n}:K_m\\to K_n$ satisfies\n\\[\n \\boxed{\\delta_n r_{m,n}=p^{m-n}\\delta_m.}\n\\]\nConsequently the cokernel of the natural comparison $\\alpha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.",":K_m\\to K_n$ satisfies\n\\[\n \\boxed{\\delta_n r_{m,n}=p^{m-n}\\delta_m.}\n\\]\nConsequently the cokernel of the natural comparison $\\alpha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.","\\boxed{\\delta_n r_{m,n}=p^{m-n}\\delta_m.}\n\\]\nConsequently the cokernel of the natural comparison $\\alpha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.","a_n r_{m,n}=p^{m-n}\\delta_m.}\n\\]\nConsequently the cokernel of the natural comparison $\\alpha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.","nel of the natural comparison $\\alpha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.","n $\\alpha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.","ha_n$ is $Q[p^n]$, with these specified transition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.","ransition maps. Its cone, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.","ne, viewing both modules in degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.","n degree zero in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.","in $D(O_n)$, is canonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.","anonically equivalent to that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.","that cokernel in degree zero.\n\\end{theorem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.","rem}\n\\begin{proof}\nIf $v$ is replaced by $v+p^n u$, the displayed quotient changes by $d(u)$, which is zero in $Q$. Moreover $p^n[d(v)/p^n]=[d(v)]=0$. If the class vanishes, write $d(v)/p^n=d(u)$ and replace $v$ by $v-p^n u\\in K$. This proves exactness at $K_n$. Conversely, if $[w]\\in Q[p^n]$, there is $v\\in V$ with $p^n w=d(v)$, so $\\bar v\\in K_n$ maps to $[w]$. Saturation proves injectivity of $\\alpha_n$.","For the transition formula, use the same lift $v$ to represent an element first at level $m$ and then at level $n$. The quotients obey $d(v)/p^n=p^{m-n}d(v)/p^m$. Functoriality for a commuting square between two maps $d$ follows from this same formula. Finally the cone of an injection between degree-zero modules has just its cokernel in degree zero.\n\\end{proof}","This is a direct coefficient-boundary argument, not a new general homological-algebra principle. The modules on the two sides need not be perfect over $O_n$. The operation of taking a kernel is not an exact functor on arbitrary short exact sequences. Thus this theorem by itself does not instantiate the earlier exact-endofunctor assertions.","Choose Smith data for $d$ over the discrete valuation ring $O$. After invertible row and column operations its nonzero entries are $p^{a_1},\\ldots,p^{a_r}$, with $a_i\\geq0$. Units have been absorbed into the basis changes. Let\n\\[\n h=\\max\\{a_1,\\ldots,a_r,0\\}.\n\\]\nEquivalently, $h$ is the least integer with $p^h Q_{\\mathrm{tors}}=0$. If $Q_{\\mathrm{tors}}=0$, set $h=0$.","\\begin{theorem}[Residual profile and sharp finite-resolution lifting]\\label{thm:precision-window}\nLet $v=\\operatorname{rank}_O V$. Then\n\\[\n Q[p^n]\\cong\\bigoplus_{i=1}^r O/p^{\\min(a_i,n)}O,\n \\qquad\n \\ell_O(K_n)=n(v-r)+\\sum_{i=1}^r\\min(a_i,n).\n\\]\nThe splittings require chosen Smith bases and are not canonical. For every $n\\geq1$,\n\\[\n \\boxed{\\operatorname{im}(K_{n+h}\\longrightarrow K_n)\n       =\\operatorname{im}(K/p^nK\\longrightarrow K_n).}\n\\]\nThus a particular solution at precision $n$ is the reduction of an exact $p$-adic solution if and only if it lifts, for the same fixed map $d$, to precision $n+h$. No smaller nonnegative delay works uniformly for all $n$ when $h>0$.","ion-window}\nLet $v=\\operatorname{rank}_O V$. Then\n\\[\n Q[p^n]\\cong\\bigoplus_{i=1}^r O/p^{\\min(a_i,n)}O,\n \\qquad\n \\ell_O(K_n)=n(v-r)+\\sum_{i=1}^r\\min(a_i,n).\n\\]\nThe splittings require chosen Smith bases and are not canonical. For every $n\\geq1$,\n\\[\n \\boxed{\\operatorname{im}(K_{n+h}\\longrightarrow K_n)\n       =\\operatorname{im}(K/p^nK\\longrightarrow K_n).}\n\\]\nThus a particular solution at precision $n$ is the reduction of an exact $p$-adic solution if and only if it lifts, for the same fixed map $d$, to precision $n+h$. No smaller nonnegative delay works uniformly for all $n$ when $h>0$.","rank}_O V$. Then\n\\[\n Q[p^n]\\cong\\bigoplus_{i=1}^r O/p^{\\min(a_i,n)}O,\n \\qquad\n \\ell_O(K_n)=n(v-r)+\\sum_{i=1}^r\\min(a_i,n).\n\\]\nThe splittings require chosen Smith bases and are not canonical. For every $n\\geq1$,\n\\[\n \\boxed{\\operatorname{im}(K_{n+h}\\longrightarrow K_n)\n       =\\operatorname{im}(K/p^nK\\longrightarrow K_n).}\n\\]\nThus a particular solution at precision $n$ is the reduction of an exact $p$-adic solution if and only if it lifts, for the same fixed map $d$, to precision $n+h$. No smaller nonnegative delay works uniformly for all $n$ when $h>0$.","}O,\n \\qquad\n \\ell_O(K_n)=n(v-r)+\\sum_{i=1}^r\\min(a_i,n).\n\\]\nThe splittings require chosen Smith bases and are not canonical. For every $n\\geq1$,\n\\[\n \\boxed{\\operatorname{im}(K_{n+h}\\longrightarrow K_n)\n       =\\operatorname{im}(K/p^nK\\longrightarrow K_n).}\n\\]\nThus a particular solution at precision $n$ is the reduction of an exact $p$-adic solution if and only if it lifts, for the same fixed map $d$, to precision $n+h$. No smaller nonnegative delay works uniformly for all $n$ when $h>0$.","O(K_n)=n(v-r)+\\sum_{i=1}^r\\min(a_i,n).\n\\]\nThe splittings require chosen Smith bases and are not canonical. For every $n\\geq1$,\n\\[\n \\boxed{\\operatorname{im}(K_{n+h}\\longrightarrow K_n)\n       =\\operatorname{im}(K/p^nK\\longrightarrow K_n).}\n\\]\nThus a particular solution at precision $n$ is the reduction of an exact $p$-adic solution if and only if it lifts, for the same fixed map $d$, to precision $n+h$. No smaller nonnegative delay works uniformly for all $n$ when $h>0$.","uire chosen Smith bases and are not canonical. For every $n\\geq1$,\n\\[\n \\boxed{\\operatorname{im}(K_{n+h}\\longrightarrow K_n)\n       =\\operatorname{im}(K/p^nK\\longrightarrow K_n).}\n\\]\nThus a particular solution at precision $n$ is the reduction of an exact $p$-adic solution if and only if it lifts, for the same fixed map $d$, to precision $n+h$. No smaller nonnegative delay works uniformly for all $n$ when $h>0$.","en Smith bases and are not canonical. For every $n\\geq1$,\n\\[\n \\boxed{\\operatorname{im}(K_{n+h}\\longrightarrow K_n)\n       =\\operatorname{im}(K/p^nK\\longrightarrow K_n).}\n\\]\nThus a particular solution at precision $n$ is the reduction of an exact $p$-adic solution if and only if it lifts, for the same fixed map $d$, to precision $n+h$. No smaller nonnegative delay works uniformly for all $n$ when $h>0$.","The residual system $(Q[p^n],p^{m-n})$ is pro-zero, and\n\\[\n K\\xrightarrow{\\sim}\\varprojlim_n K_n.\n\\]\nHere pro-zero means that for each target level some later transition is the zero map. It does not mean that the individual residual modules vanish.\n\\end{theorem}\n\\begin{proof}\nIn Smith coordinates a nonzero diagonal entry imposes $p^{a_i}x_i=0$ modulo $p^n$, giving $\\min(a_i,n)$ units of $O$-length. Each of the $v-r$ zero columns contributes a free $O_n$ coordinate. This proves the formulas.","stem $(Q[p^n],p^{m-n})$ is pro-zero, and\n\\[\n K\\xrightarrow{\\sim}\\varprojlim_n K_n.\n\\]\nHere pro-zero means that for each target level some later transition is the zero map. It does not mean that the individual residual modules vanish.\n\\end{theorem}\n\\begin{proof}\nIn Smith coordinates a nonzero diagonal entry imposes $p^{a_i}x_i=0$ modulo $p^n$, giving $\\min(a_i,n)$ units of $O$-length. Each of the $v-r$ zero columns contributes a free $O_n$ coordinate. This proves the formulas.","nd\n\\[\n K\\xrightarrow{\\sim}\\varprojlim_n K_n.\n\\]\nHere pro-zero means that for each target level some later transition is the zero map. It does not mean that the individual residual modules vanish.\n\\end{theorem}\n\\begin{proof}\nIn Smith coordinates a nonzero diagonal entry imposes $p^{a_i}x_i=0$ modulo $p^n$, giving $\\min(a_i,n)$ units of $O$-length. Each of the $v-r$ zero columns contributes a free $O_n$ coordinate. This proves the formulas.","By \\cref{thm:kernel-bockstein}, the image of an element of $K_{n+h}$ has boundary $p^h\\delta_{n+h}=0$, so lies in the image of $K/p^nK$. The reverse inclusion follows by reducing any exact lift at level $n+h$.","If $c<h$, choose an element of exact order $p^h$ in $Q_{\\mathrm{tors}}$ and take $n\\geq h$. Surjectivity of $\\delta_{n+c}$ supplies a preimage. Its reduction to level $n$ has nonzero boundary after multiplication by $p^c$, so it is not the reduction of an exact solution. This proves sharpness. Multiplication by $p^h$ also makes every transition with gap at least $h$ zero on residuals.","Finally a compatible family of vectors modulo $p^n$ has a unique lift to $V$, since $V$ is finite free and complete. If every reduction solves the equation, its image under $d$ lies in $\\bigcap_n p^nW=0$. This proves the inverse-limit assertion directly, without interchanging an arbitrary kernel and an unspecified completion.\n\\end{proof}","\\begin{remark}[What a finite certificate must contain]\nFor integer input matrices, a verified integer Smith decomposition gives the $p$-adic data by taking the valuations of its nonzero diagonal entries. Verification requires the two change-of-basis matrices to be unimodular and their product with $d$ to equal the displayed diagonal matrix. For arbitrary $p$-adic entries, a finite truncation alone does not certify a claimed zero diagonal entry or an exact rank. Such a claim requires an exact input presentation or a separate rank certificate. The theorem does not supply a universal numerical decision procedure from unbounded-precision oracles.\n\\end{remark}","Let $G$ be a finite group acting $O$-linearly on a lattice $M$. The natural comparison is\n\\[\n \\alpha_{G,n}:M^G/p^nM^G\\longrightarrow(M/p^nM)^G.\n\\]\nThe first path takes invariant vectors and then reduces. The second reduces the representation and then takes invariant vectors. Its direction is fixed by reduction of an invariant vector; no arbitrary scalar enters this map.","\\begin{theorem}[Cohomological identification of the symmetry residual]\\label{thm:invariant-bockstein}\nThere is a natural exact sequence\n\\[\n \\boxed{0\\longrightarrow M^G/p^nM^G\\longrightarrow(M/p^nM)^G\n \\xrightarrow{\\partial_n}H^1(G,M)[p^n]\\longrightarrow0.}\n\\]\nFor a fixed residue class $\\bar m$ and a lift $m\\in M$, the obstruction is represented by the crossed homomorphism\n\\[\n c_{\\bar m}(g)=\\frac{gm-m}{p^n}.\n\\]\nIt vanishes exactly when $\\bar m$ has a $G$-invariant lift in $M$. The group $H^1(G,M)$ is finite and killed by $|G|$. In particular it is killed by $p^e$, where $e=v_p(|G|)$, and the delay $e$ is a valid uniform lifting bound.","tion of the symmetry residual]\\label{thm:invariant-bockstein}\nThere is a natural exact sequence\n\\[\n \\boxed{0\\longrightarrow M^G/p^nM^G\\longrightarrow(M/p^nM)^G\n \\xrightarrow{\\partial_n}H^1(G,M)[p^n]\\longrightarrow0.}\n\\]\nFor a fixed residue class $\\bar m$ and a lift $m\\in M$, the obstruction is represented by the crossed homomorphism\n\\[\n c_{\\bar m}(g)=\\frac{gm-m}{p^n}.\n\\]\nIt vanishes exactly when $\\bar m$ has a $G$-invariant lift in $M$. The group $H^1(G,M)$ is finite and killed by $|G|$. In particular it is killed by $p^e$, where $e=v_p(|G|)$, and the delay $e$ is a valid uniform lifting bound.","metry residual]\\label{thm:invariant-bockstein}\nThere is a natural exact sequence\n\\[\n \\boxed{0\\longrightarrow M^G/p^nM^G\\longrightarrow(M/p^nM)^G\n \\xrightarrow{\\partial_n}H^1(G,M)[p^n]\\longrightarrow0.}\n\\]\nFor a fixed residue class $\\bar m$ and a lift $m\\in M$, the obstruction is represented by the crossed homomorphism\n\\[\n c_{\\bar m}(g)=\\frac{gm-m}{p^n}.\n\\]\nIt vanishes exactly when $\\bar m$ has a $G$-invariant lift in $M$. The group $H^1(G,M)$ is finite and killed by $|G|$. In particular it is killed by $p^e$, where $e=v_p(|G|)$, and the delay $e$ is a valid uniform lifting bound.","thm:invariant-bockstein}\nThere is a natural exact sequence\n\\[\n \\boxed{0\\longrightarrow M^G/p^nM^G\\longrightarrow(M/p^nM)^G\n \\xrightarrow{\\partial_n}H^1(G,M)[p^n]\\longrightarrow0.}\n\\]\nFor a fixed residue class $\\bar m$ and a lift $m\\in M$, the obstruction is represented by the crossed homomorphism\n\\[\n c_{\\bar m}(g)=\\frac{gm-m}{p^n}.\n\\]\nIt vanishes exactly when $\\bar m$ has a $G$-invariant lift in $M$. The group $H^1(G,M)$ is finite and killed by $|G|$. In particular it is killed by $p^e$, where $e=v_p(|G|)$, and the delay $e$ is a valid uniform lifting bound.","ockstein}\nThere is a natural exact sequence\n\\[\n \\boxed{0\\longrightarrow M^G/p^nM^G\\longrightarrow(M/p^nM)^G\n \\xrightarrow{\\partial_n}H^1(G,M)[p^n]\\longrightarrow0.}\n\\]\nFor a fixed residue class $\\bar m$ and a lift $m\\in M$, the obstruction is represented by the crossed homomorphism\n\\[\n c_{\\bar m}(g)=\\frac{gm-m}{p^n}.\n\\]\nIt vanishes exactly when $\\bar m$ has a $G$-invariant lift in $M$. The group $H^1(G,M)$ is finite and killed by $|G|$. In particular it is killed by $p^e$, where $e=v_p(|G|)$, and the delay $e$ is a valid uniform lifting bound.","sequence\n\\[\n \\boxed{0\\longrightarrow M^G/p^nM^G\\longrightarrow(M/p^nM)^G\n \\xrightarrow{\\partial_n}H^1(G,M)[p^n]\\longrightarrow0.}\n\\]\nFor a fixed residue class $\\bar m$ and a lift $m\\in M$, the obstruction is represented by the crossed homomorphism\n\\[\n c_{\\bar m}(g)=\\frac{gm-m}{p^n}.\n\\]\nIt vanishes exactly when $\\bar m$ has a $G$-invariant lift in $M$. The group $H^1(G,M)$ is finite and killed by $|G|$. In particular it is killed by $p^e$, where $e=v_p(|G|)$, and the delay $e$ is a valid uniform lifting bound.","M^G/p^nM^G\\longrightarrow(M/p^nM)^G\n \\xrightarrow{\\partial_n}H^1(G,M)[p^n]\\longrightarrow0.}\n\\]\nFor a fixed residue class $\\bar m$ and a lift $m\\in M$, the obstruction is represented by the crossed homomorphism\n\\[\n c_{\\bar m}(g)=\\frac{gm-m}{p^n}.\n\\]\nIt vanishes exactly when $\\bar m$ has a $G$-invariant lift in $M$. The group $H^1(G,M)$ is finite and killed by $|G|$. In particular it is killed by $p^e$, where $e=v_p(|G|)$, and the delay $e$ is a valid uniform lifting bound.","ngrightarrow(M/p^nM)^G\n \\xrightarrow{\\partial_n}H^1(G,M)[p^n]\\longrightarrow0.}\n\\]\nFor a fixed residue class $\\bar m$ and a lift $m\\in M$, the obstruction is represented by the crossed homomorphism\n\\[\n c_{\\bar m}(g)=\\frac{gm-m}{p^n}.\n\\]\nIt vanishes exactly when $\\bar m$ has a $G$-invariant lift in $M$. The group $H^1(G,M)$ is finite and killed by $|G|$. In particular it is killed by $p^e$, where $e=v_p(|G|)$, and the delay $e$ is a valid uniform lifting bound.","ghtarrow{\\partial_n}H^1(G,M)[p^n]\\longrightarrow0.}\n\\]\nFor a fixed residue class $\\bar m$ and a lift $m\\in M$, the obstruction is represented by the crossed homomorphism\n\\[\n c_{\\bar m}(g)=\\frac{gm-m}{p^n}.\n\\]\nIt vanishes exactly when $\\bar m$ has a $G$-invariant lift in $M$. The group $H^1(G,M)$ is finite and killed by $|G|$. In particular it is killed by $p^e$, where $e=v_p(|G|)$, and the delay $e$ is a valid uniform lifting bound.","More precisely, for\n\\[\n d_G:M\\longrightarrow\\prod_{g\\in G}M,\\qquad\n d_G(m)=(gm-m)_g,\n\\]\nthe torsion subgroup of $\\operatorname{coker}d_G$ is canonically $H^1(G,M)$. Therefore the sharp bound is computed by \\cref{thm:precision-window}, and can be strictly smaller than $e$.\n\\end{theorem}\n\\begin{proof}\nThe formula for $c$ satisfies $c(gh)=g c(h)+c(g)$, because this identity holds before division by $p^n$ in the torsion-free module $M$. Changing $m$ by $p^n b$ changes $c$ by $g b-b$. Hence it defines a cohomology class. The usual long exact coefficient sequence gives the displayed short sequence, but it can also be proved directly. If $p^n[c]=0$, there is $m\\in M$ with $p^n c(g)=gm-m$. Its reduction is invariant and maps to $[c]$. If $[c]=0$, write $c(g)=g b-b$ and observe that","More precisely, for\n\\[\n d_G:M\\longrightarrow\\prod_{g\\in G}M,\\qquad\n d_G(m)=(gm-m)_g,\n\\]\nthe torsion subgroup of $\\operatorname{coker}d_G$ is canonically $H^1(G,M)$. Therefore the sharp bound is computed by \\cref{thm:precision-window}, and can be strictly smaller than $e$.\n\\end{theorem}\n\\begin{proof}\nThe formula for $c$ satisfies $c(gh)=g c(h)+c(g)$, because this identity holds before division by $p^n$ in the torsion-free module $M$. Changing $m$ by $p^n b$ changes $c$ by $g b-b$. Hence it defines a cohomology class. The usual long exact coefficient sequence gives the displayed short sequence, but it can also be proved directly. If $p^n[c]=0$, there is $m\\in M$ with $p^n c(g)=gm-m$. Its reduction is invariant and maps to $[c]$. If $[c]=0$, write $c(g)=g b-b$ and observe that $m-p^n b$ is the required invariant lift. Saturatio","More precisely, for\n\\[\n d_G:M\\longrightarrow\\prod_{g\\in G}M,\\qquad\n d_G(m)=(gm-m)_g,\n\\]\nthe torsion subgroup of $\\operatorname{coker}d_G$ is canonically $H^1(G,M)$. Therefore the sharp bound is computed by \\cref{thm:precision-window}, and can be strictly smaller than $e$.\n\\end{theorem}\n\\begin{proof}\nThe formula for $c$ satisfies $c(gh)=g c(h)+c(g)$, because this identity holds before division by $p^n$ in the torsion-free module $M$. Changing $m$ by $p^n b$ changes $c$ by $g b-b$. Hence it defines a cohomology class. The usual long exact coefficient sequence gives the displayed short sequence, but it can also be proved directly. If $p^n[c]=0$, there is $m\\in M$ with $p^n c(g)=gm-m$. Its reduction is invariant and maps to $[c]$. If $[c]=0$, write $c(g)=g b-b$ and observe that $m-p^n b$ is the required invariant lift. Saturation gives injectivity on the","More precisely, for\n\\[\n d_G:M\\longrightarrow\\prod_{g\\in G}M,\\qquad\n d_G(m)=(gm-m)_g,\n\\]\nthe torsion subgroup of $\\operatorname{coker}d_G$ is canonically $H^1(G,M)$. Therefore the sharp bound is computed by \\cref{thm:precision-window}, and can be strictly smaller than $e$.\n\\end{theorem}\n\\begin{proof}\nThe formula for $c$ satisfies $c(gh)=g c(h)+c(g)$, because this identity holds before division by $p^n$ in the torsion-free module $M$. Changing $m$ by $p^n b$ changes $c$ by $g b-b$. Hence it defines a cohomology class. The usual long exact coefficient sequence gives the displayed short sequence, but it can also be proved directly. If $p^n[c]=0$, there is $m\\in M$ with $p^n c(g)=gm-m$. Its reduction is invariant and maps to $[c]$. If $[c]=0$, write $c(g)=g b-b$ and observe that $m-p^n b$ is the required invariant lift. Saturation gives injectivity on the left.","grightarrow\\prod_{g\\in G}M,\\qquad\n d_G(m)=(gm-m)_g,\n\\]\nthe torsion subgroup of $\\operatorname{coker}d_G$ is canonically $H^1(G,M)$. Therefore the sharp bound is computed by \\cref{thm:precision-window}, and can be strictly smaller than $e$.\n\\end{theorem}\n\\begin{proof}\nThe formula for $c$ satisfies $c(gh)=g c(h)+c(g)$, because this identity holds before division by $p^n$ in the torsion-free module $M$. Changing $m$ by $p^n b$ changes $c$ by $g b-b$. Hence it defines a cohomology class. The usual long exact coefficient sequence gives the displayed short sequence, but it can also be proved directly. If $p^n[c]=0$, there is $m\\in M$ with $p^n c(g)=gm-m$. Its reduction is invariant and maps to $[c]$. If $[c]=0$, write $c(g)=g b-b$ and observe that $m-p^n b$ is the required invariant lift. Saturation gives injectivity on the left.","od_{g\\in G}M,\\qquad\n d_G(m)=(gm-m)_g,\n\\]\nthe torsion subgroup of $\\operatorname{coker}d_G$ is canonically $H^1(G,M)$. Therefore the sharp bound is computed by \\cref{thm:precision-window}, and can be strictly smaller than $e$.\n\\end{theorem}\n\\begin{proof}\nThe formula for $c$ satisfies $c(gh)=g c(h)+c(g)$, because this identity holds before division by $p^n$ in the torsion-free module $M$. Changing $m$ by $p^n b$ changes $c$ by $g b-b$. Hence it defines a cohomology class. The usual long exact coefficient sequence gives the displayed short sequence, but it can also be proved directly. If $p^n[c]=0$, there is $m\\in M$ with $p^n c(g)=gm-m$. Its reduction is invariant and maps to $[c]$. If $[c]=0$, write $c(g)=g b-b$ and observe that $m-p^n b$ is the required invariant lift. Saturation gives injectivity on the left.","n G}M,\\qquad\n d_G(m)=(gm-m)_g,\n\\]\nthe torsion subgroup of $\\operatorname{coker}d_G$ is canonically $H^1(G,M)$. Therefore the sharp bound is computed by \\cref{thm:precision-window}, and can be strictly smaller than $e$.\n\\end{theorem}\n\\begin{proof}\nThe formula for $c$ satisfies $c(gh)=g c(h)+c(g)$, because this identity holds before division by $p^n$ in the torsion-free module $M$. Changing $m$ by $p^n b$ changes $c$ by $g b-b$. Hence it defines a cohomology class. The usual long exact coefficient sequence gives the displayed short sequence, but it can also be proved directly. If $p^n[c]=0$, there is $m\\in M$ with $p^n c(g)=gm-m$. Its reduction is invariant and maps to $[c]$. If $[c]=0$, write $c(g)=g b-b$ and observe that $m-p^n b$ is the required invariant lift. Saturation gives injectivity on the left.","G(m)=(gm-m)_g,\n\\]\nthe torsion subgroup of $\\operatorname{coker}d_G$ is canonically $H^1(G,M)$. Therefore the sharp bound is computed by \\cref{thm:precision-window}, and can be strictly smaller than $e$.\n\\end{theorem}\n\\begin{proof}\nThe formula for $c$ satisfies $c(gh)=g c(h)+c(g)$, because this identity holds before division by $p^n$ in the torsion-free module $M$. Changing $m$ by $p^n b$ changes $c$ by $g b-b$. Hence it defines a cohomology class. The usual long exact coefficient sequence gives the displayed short sequence, but it can also be proved directly. If $p^n[c]=0$, there is $m\\in M$ with $p^n c(g)=gm-m$. Its reduction is invariant and maps to $[c]$. If $[c]=0$, write $c(g)=g b-b$ and observe that $m-p^n b$ is the required invariant lift. Saturation gives injectivity on the left.","m-m)_g,\n\\]\nthe torsion subgroup of $\\operatorname{coker}d_G$ is canonically $H^1(G,M)$. Therefore the sharp bound is computed by \\cref{thm:precision-window}, and can be strictly smaller than $e$.\n\\end{theorem}\n\\begin{proof}\nThe formula for $c$ satisfies $c(gh)=g c(h)+c(g)$, because this identity holds before division by $p^n$ in the torsion-free module $M$. Changing $m$ by $p^n b$ changes $c$ by $g b-b$. Hence it defines a cohomology class. The usual long exact coefficient sequence gives the displayed short sequence, but it can also be proved directly. If $p^n[c]=0$, there is $m\\in M$ with $p^n c(g)=gm-m$. Its reduction is invariant and maps to $[c]$. If $[c]=0$, write $c(g)=g b-b$ and observe that $m-p^n b$ is the required invariant lift. Saturation gives injectivity on the left.","an be strictly smaller than $e$.\n\\end{theorem}\n\\begin{proof}\nThe formula for $c$ satisfies $c(gh)=g c(h)+c(g)$, because this identity holds before division by $p^n$ in the torsion-free module $M$. Changing $m$ by $p^n b$ changes $c$ by $g b-b$. Hence it defines a cohomology class. The usual long exact coefficient sequence gives the displayed short sequence, but it can also be proved directly. If $p^n[c]=0$, there is $m\\in M$ with $p^n c(g)=gm-m$. Its reduction is invariant and maps to $[c]$. If $[c]=0$, write $c(g)=g b-b$ and observe that $m-p^n b$ is the required invariant lift. Saturation gives injectivity on the left.","r than $e$.\n\\end{theorem}\n\\begin{proof}\nThe formula for $c$ satisfies $c(gh)=g c(h)+c(g)$, because this identity holds before division by $p^n$ in the torsion-free module $M$. Changing $m$ by $p^n b$ changes $c$ by $g b-b$. Hence it defines a cohomology class. The usual long exact coefficient sequence gives the displayed short sequence, but it can also be proved directly. If $p^n[c]=0$, there is $m\\in M$ with $p^n c(g)=gm-m$. Its reduction is invariant and maps to $[c]$. If $[c]=0$, write $c(g)=g b-b$ and observe that $m-p^n b$ is the required invariant lift. Saturation gives injectivity on the left.","nd{theorem}\n\\begin{proof}\nThe formula for $c$ satisfies $c(gh)=g c(h)+c(g)$, because this identity holds before division by $p^n$ in the torsion-free module $M$. Changing $m$ by $p^n b$ changes $c$ by $g b-b$. Hence it defines a cohomology class. The usual long exact coefficient sequence gives the displayed short sequence, but it can also be proved directly. If $p^n[c]=0$, there is $m\\in M$ with $p^n c(g)=gm-m$. Its reduction is invariant and maps to $[c]$. If $[c]=0$, write $c(g)=g b-b$ and observe that $m-p^n b$ is the required invariant lift. Saturation gives injectivity on the left.","$c(gh)=g c(h)+c(g)$, because this identity holds before division by $p^n$ in the torsion-free module $M$. Changing $m$ by $p^n b$ changes $c$ by $g b-b$. Hence it defines a cohomology class. The usual long exact coefficient sequence gives the displayed short sequence, but it can also be proved directly. If $p^n[c]=0$, there is $m\\in M$ with $p^n c(g)=gm-m$. Its reduction is invariant and maps to $[c]$. If $[c]=0$, write $c(g)=g b-b$ and observe that $m-p^n b$ is the required invariant lift. Saturation gives injectivity on the left.","(h)+c(g)$, because this identity holds before division by $p^n$ in the torsion-free module $M$. Changing $m$ by $p^n b$ changes $c$ by $g b-b$. Hence it defines a cohomology class. The usual long exact coefficient sequence gives the displayed short sequence, but it can also be proved directly. If $p^n[c]=0$, there is $m\\in M$ with $p^n c(g)=gm-m$. Its reduction is invariant and maps to $[c]$. If $[c]=0$, write $c(g)=g b-b$ and observe that $m-p^n b$ is the required invariant lift. Saturation gives injectivity on the left.","use this identity holds before division by $p^n$ in the torsion-free module $M$. Changing $m$ by $p^n b$ changes $c$ by $g b-b$. Hence it defines a cohomology class. The usual long exact coefficient sequence gives the displayed short sequence, but it can also be proved directly. If $p^n[c]=0$, there is $m\\in M$ with $p^n c(g)=gm-m$. Its reduction is invariant and maps to $[c]$. If $[c]=0$, write $c(g)=g b-b$ and observe that $m-p^n b$ is the required invariant lift. Saturation gives injectivity on the left.","division by $p^n$ in the torsion-free module $M$. Changing $m$ by $p^n b$ changes $c$ by $g b-b$. Hence it defines a cohomology class. The usual long exact coefficient sequence gives the displayed short sequence, but it can also be proved directly. If $p^n[c]=0$, there is $m\\in M$ with $p^n c(g)=gm-m$. Its reduction is invariant and maps to $[c]$. If $[c]=0$, write $c(g)=g b-b$ and observe that $m-p^n b$ is the required invariant lift. Saturation gives injectivity on the left.","For completeness, let $c:G\\to M$ be a cocycle and $b=\\sum_{g\\in G}c(g)$. For $u\\in G$,\n\\[\n ub=b-|G|c(u),\\qquad |G|c(u)=u(-b)-(-b).\n\\]\nThus $|G|[c]=0$. The cocycle and coboundary modules are subquotients of finite free $O$-modules, so $H^1$ is finitely generated and hence finite when annihilated by this nonzero integer.","Write $d^1(c)(g,u)=g c(u)-c(gu)+c(g)$. If a class $[c]$ in $\\operatorname{coker}d_G$ is torsion, then $p^a c=d_G(m)$ for some $a$. Applying $d^1$ gives $p^a d^1(c)=0$, whence $d^1(c)=0$ since the target is torsion-free. Thus its class lies in $H^1$. Conversely the preceding annihilation argument shows that every $H^1$ class is torsion. This proves the canonical identification and the sharp-bound assertion.\n\\end{proof}","The long exact sequence, explicit boundary, and cyclic-group cohomology used here are classical. Our conventions agree with \\cite[Chapter II, 1.9, 1.18--1.21]{MilneCFT2020}. The proof above is included to fix naturality, the coefficient regime, and the transition maps. If $p$ does not divide $|G|$, all these residuals vanish. Divisibility of the group order alone does not imply a nonzero residual: the trivial action on a lattice has $H^1(G,M)=0$.","dary, and cyclic-group cohomology used here are classical. Our conventions agree with \\cite[Chapter II, 1.9, 1.18--1.21]{MilneCFT2020}. The proof above is included to fix naturality, the coefficient regime, and the transition maps. If $p$ does not divide $|G|$, all these residuals vanish. Divisibility of the group order alone does not imply a nonzero residual: the trivial action on a lattice has $H^1(G,M)=0$.","\\begin{theorem}[Certified lifting of equivariant maps]\\label{thm:intertwiner-lifting}\nLet $L$ and $M$ be $G$-lattices and put\n\\[\n V=\\operatorname{Hom}_O(L,M),\\qquad\n (g\\cdot f)=\\rho_M(g)f\\rho_L(g)^{-1}.\n\\]\nThen\n\\[\n \\begin{split}\n 0\\longrightarrow&\\operatorname{Hom}_{O[G]}(L,M)/p^n\n \\longrightarrow\\operatorname{Hom}_{O_n[G]}(L/p^nL,M/p^nM)","\\begin{theorem}[Certified lifting of equivariant maps]\\label{thm:intertwiner-lifting}\nLet $L$ and $M$ be $G$-lattices and put\n\\[\n V=\\operatorname{Hom}_O(L,M),\\qquad\n (g\\cdot f)=\\rho_M(g)f\\rho_L(g)^{-1}.\n\\]\nThen\n\\[\n \\begin{split}\n 0\\longrightarrow&\\operatorname{Hom}_{O[G]}(L,M)/p^n\n \\longrightarrow\\operatorname{Hom}_{O_n[G]}(L/p^nL,M/p^nM)\\\\\n \\longrightarrow&H^1(G,V)[p^n]\\longrightarrow0\n \\end{split}\n\\]\nis exact and natural. A modular equivariant map lifts to an integral equivariant map if and only if its boundary class vanishes. If $p^h$ annihilates the $p$-primary cohomology as in \\cref{thm:precision-window}, this is equivalent to lifting the same map equivariantly to precision $n+h$.","\\longrightarrow&H^1(G,V)[p^n]\\longrightarrow0\n \\end{split}\n\\]\nis exact and natural. A modular equivariant map lifts to an integral equivariant map if and only if its boundary class vanishes. If $p^h$ annihilates the $p$-primary cohomology as in \\cref{thm:precision-window}, this is equivalent to lifting the same map equivariantly to precision $n+h$.","If $L$ and $M$ have the same rank and the given modular map is invertible, every integral lift supplied in this way is an isomorphism. Thus existence of a $G$-equivariant isomorphism at precision $n+h$ implies existence of an integral one whose reduction at precision $n$ is the specified map. The coarser bound $v_p(|G|)$ is always valid.\n\\end{theorem}\n\\begin{proof}\nFinite freeness identifies $V/p^nV$ with the full module of linear maps between the reduced lattices. Its $G$-invariants are exactly the equivariant maps. Apply \\cref{thm:invariant-bockstein,thm:precision-window}. Explicitly, for a lift $\\widetilde f$ the boundary cocycle is\n\\[\n g\\longmapsto\\frac{\\rho_M(g)\\widetilde f\\rho_L(g)^{-1}-\\wi","If $L$ and $M$ have the same rank and the given modular map is invertible, every integral lift supplied in this way is an isomorphism. Thus existence of a $G$-equivariant isomorphism at precision $n+h$ implies existence of an integral one whose reduction at precision $n$ is the specified map. The coarser bound $v_p(|G|)$ is always valid.\n\\end{theorem}\n\\begin{proof}\nFinite freeness identifies $V/p^nV$ with the full module of linear maps between the reduced lattices. Its $G$-invariants are exactly the equivariant maps. Apply \\cref{thm:invariant-bockstein,thm:precision-window}. Explicitly, for a lift $\\widetilde f$ the boundary cocycle is\n\\[\n g\\longmapsto\\frac{\\rho_M(g)\\widetilde f\\rho_L(g)^{-1}-\\widetilde","If $L$ and $M$ have the same rank and the given modular map is invertible, every integral lift supplied in this way is an isomorphism. Thus existence of a $G$-equivariant isomorphism at precision $n+h$ implies existence of an integral one whose reduction at precision $n$ is the specified map. The coarser bound $v_p(|G|)$ is always valid.\n\\end{theorem}\n\\begin{proof}\nFinite freeness identifies $V/p^nV$ with the full module of linear maps between the reduced lattices. Its $G$-invariants are exactly the equivariant maps. Apply \\cref{thm:invariant-bockstein,thm:precision-window}. Explicitly, for a lift $\\widetilde f$ the boundary cocycle is\n\\[\n g\\longmapsto\\frac{\\rho_M(g)\\widetilde f\\rho_L(g)^{-1}-\\widetilde f}{p^n}.\n\\]\nIf this is $g\\cdot b-b$, then $\\widetilde f-p^n b$ is an integral equivariant lift. In the equal-rank case its determinant is a unit","If $L$ and $M$ have the same rank and the given modular map is invertible, every integral lift supplied in this way is an isomorphism. Thus existence of a $G$-equivariant isomorphism at precision $n+h$ implies existence of an integral one whose reduction at precision $n$ is the specified map. The coarser bound $v_p(|G|)$ is always valid.\n\\end{theorem}\n\\begin{proof}\nFinite freeness identifies $V/p^nV$ with the full module of linear maps between the reduced lattices. Its $G$-invariants are exactly the equivariant maps. Apply \\cref{thm:invariant-bockstein,thm:precision-window}. Explicitly, for a lift $\\widetilde f$ the boundary cocycle is\n\\[\n g\\longmapsto\\frac{\\rho_M(g)\\widetilde f\\rho_L(g)^{-1}-\\widetilde f}{p^n}.\n\\]\nIf this is $g\\cdot b-b$, then $\\widetilde f-p^n b$ is an integral equivariant lift. In the equal-rank case its determinant is a unit because it is a unit modulo $p$. Its inver","If $L$ and $M$ have the same rank and the given modular map is invertible, every integral lift supplied in this way is an isomorphism. Thus existence of a $G$-equivariant isomorphism at precision $n+h$ implies existence of an integral one whose reduction at precision $n$ is the specified map. The coarser bound $v_p(|G|)$ is always valid.\n\\end{theorem}\n\\begin{proof}\nFinite freeness identifies $V/p^nV$ with the full module of linear maps between the reduced lattices. Its $G$-invariants are exactly the equivariant maps. Apply \\cref{thm:invariant-bockstein,thm:precision-window}. Explicitly, for a lift $\\widetilde f$ the boundary cocycle is\n\\[\n g\\longmapsto\\frac{\\rho_M(g)\\widetilde f\\rho_L(g)^{-1}-\\widetilde f}{p^n}.\n\\]\nIf this is $g\\cdot b-b$, then $\\widetilde f-p^n b$ is an integral equivariant lift. In the equal-rank case its determinant is a unit because it is a unit modulo $p$. Its inverse is integral and equivariant. No chosen modular intertwiner is repla","If $L$ and $M$ have the same rank and the given modular map is invertible, every integral lift supplied in this way is an isomorphism. Thus existence of a $G$-equivariant isomorphism at precision $n+h$ implies existence of an integral one whose reduction at precision $n$ is the specified map. The coarser bound $v_p(|G|)$ is always valid.\n\\end{theorem}\n\\begin{proof}\nFinite freeness identifies $V/p^nV$ with the full module of linear maps between the reduced lattices. Its $G$-invariants are exactly the equivariant maps. Apply \\cref{thm:invariant-bockstein,thm:precision-window}. Explicitly, for a lift $\\widetilde f$ the boundary cocycle is\n\\[\n g\\longmapsto\\frac{\\rho_M(g)\\widetilde f\\rho_L(g)^{-1}-\\widetilde f}{p^n}.\n\\]\nIf this is $g\\cdot b-b$, then $\\widetilde f-p^n b$ is an integral equivariant lift. In the equal-rank case its determinant is a unit because it is a unit modulo $p$. Its inverse is integral and equivariant. No chosen modular intertwiner is replaced silently by another at the target precision.\n\\e","If $L$ and $M$ have the same rank and the given modular map is invertible, every integral lift supplied in this way is an isomorphism. Thus existence of a $G$-equivariant isomorphism at precision $n+h$ implies existence of an integral one whose reduction at precision $n$ is the specified map. The coarser bound $v_p(|G|)$ is always valid.\n\\end{theorem}\n\\begin{proof}\nFinite freeness identifies $V/p^nV$ with the full module of linear maps between the reduced lattices. Its $G$-invariants are exactly the equivariant maps. Apply \\cref{thm:invariant-bockstein,thm:precision-window}. Explicitly, for a lift $\\widetilde f$ the boundary cocycle is\n\\[\n g\\longmapsto\\frac{\\rho_M(g)\\widetilde f\\rho_L(g)^{-1}-\\widetilde f}{p^n}.\n\\]\nIf this is $g\\cdot b-b$, then $\\widetilde f-p^n b$ is an integral equivariant lift. In the equal-rank case its determinant is a unit because it is a unit modulo $p$. Its inverse is integral and equivariant. No chosen modular intertwiner is replaced silently by another at the target precision.\n\\end{proof}","every integral lift supplied in this way is an isomorphism. Thus existence of a $G$-equivariant isomorphism at precision $n+h$ implies existence of an integral one whose reduction at precision $n$ is the specified map. The coarser bound $v_p(|G|)$ is always valid.\n\\end{theorem}\n\\begin{proof}\nFinite freeness identifies $V/p^nV$ with the full module of linear maps between the reduced lattices. Its $G$-invariants are exactly the equivariant maps. Apply \\cref{thm:invariant-bockstein,thm:precision-window}. Explicitly, for a lift $\\widetilde f$ the boundary cocycle is\n\\[\n g\\longmapsto\\frac{\\rho_M(g)\\widetilde f\\rho_L(g)^{-1}-\\widetilde f}{p^n}.\n\\]\nIf this is $g\\cdot b-b$, then $\\widetilde f-p^n b$ is an integral equivariant lift. In the equal-rank case its determinant is a unit because it is a unit modulo $p$. Its inverse is integral and equivariant. No chosen modular intertwiner is replaced silently by another at the target precision.\n\\end{proof}","ies existence of an integral one whose reduction at precision $n$ is the specified map. The coarser bound $v_p(|G|)$ is always valid.\n\\end{theorem}\n\\begin{proof}\nFinite freeness identifies $V/p^nV$ with the full module of linear maps between the reduced lattices. Its $G$-invariants are exactly the equivariant maps. Apply \\cref{thm:invariant-bockstein,thm:precision-window}. Explicitly, for a lift $\\widetilde f$ the boundary cocycle is\n\\[\n g\\longmapsto\\frac{\\rho_M(g)\\widetilde f\\rho_L(g)^{-1}-\\widetilde f}{p^n}.\n\\]\nIf this is $g\\cdot b-b$, then $\\widetilde f-p^n b$ is an integral equivariant lift. In the equal-rank case its determinant is a unit because it is a unit modulo $p$. Its inverse is integral and equivariant. No chosen modular intertwiner is replaced silently by another at the target precision.\n\\end{proof}","eduction at precision $n$ is the specified map. The coarser bound $v_p(|G|)$ is always valid.\n\\end{theorem}\n\\begin{proof}\nFinite freeness identifies $V/p^nV$ with the full module of linear maps between the reduced lattices. Its $G$-invariants are exactly the equivariant maps. Apply \\cref{thm:invariant-bockstein,thm:precision-window}. Explicitly, for a lift $\\widetilde f$ the boundary cocycle is\n\\[\n g\\longmapsto\\frac{\\rho_M(g)\\widetilde f\\rho_L(g)^{-1}-\\widetilde f}{p^n}.\n\\]\nIf this is $g\\cdot b-b$, then $\\widetilde f-p^n b$ is an integral equivariant lift. In the equal-rank case its determinant is a unit because it is a unit modulo $p$. Its inverse is integral and equivariant. No chosen modular intertwiner is replaced silently by another at the target precision.\n\\end{proof}","d.\n\\end{theorem}\n\\begin{proof}\nFinite freeness identifies $V/p^nV$ with the full module of linear maps between the reduced lattices. Its $G$-invariants are exactly the equivariant maps. Apply \\cref{thm:invariant-bockstein,thm:precision-window}. Explicitly, for a lift $\\widetilde f$ the boundary cocycle is\n\\[\n g\\longmapsto\\frac{\\rho_M(g)\\widetilde f\\rho_L(g)^{-1}-\\widetilde f}{p^n}.\n\\]\nIf this is $g\\cdot b-b$, then $\\widetilde f-p^n b$ is an integral equivariant lift. In the equal-rank case its determinant is a unit because it is a unit modulo $p$. Its inverse is integral and equivariant. No chosen modular intertwiner is replaced silently by another at the target precision.\n\\end{proof}","egin{proof}\nFinite freeness identifies $V/p^nV$ with the full module of linear maps between the reduced lattices. Its $G$-invariants are exactly the equivariant maps. Apply \\cref{thm:invariant-bockstein,thm:precision-window}. Explicitly, for a lift $\\widetilde f$ the boundary cocycle is\n\\[\n g\\longmapsto\\frac{\\rho_M(g)\\widetilde f\\rho_L(g)^{-1}-\\widetilde f}{p^n}.\n\\]\nIf this is $g\\cdot b-b$, then $\\widetilde f-p^n b$ is an integral equivariant lift. In the equal-rank case its determinant is a unit because it is a unit modulo $p$. Its inverse is integral and equivariant. No chosen modular intertwiner is replaced silently by another at the target precision.\n\\end{proof}","s are exactly the equivariant maps. Apply \\cref{thm:invariant-bockstein,thm:precision-window}. Explicitly, for a lift $\\widetilde f$ the boundary cocycle is\n\\[\n g\\longmapsto\\frac{\\rho_M(g)\\widetilde f\\rho_L(g)^{-1}-\\widetilde f}{p^n}.\n\\]\nIf this is $g\\cdot b-b$, then $\\widetilde f-p^n b$ is an integral equivariant lift. In the equal-rank case its determinant is a unit because it is a unit modulo $p$. Its inverse is integral and equivariant. No chosen modular intertwiner is replaced silently by another at the target precision.\n\\end{proof}","Let $m\\geq2$, and put\n\\[\n N_m(t)=1+t+\\cdots+t^{m-1},\\qquad\n M_m=O[t]/(N_m(t)).\n\\]\nThis is free of rank $m-1$. Let a generator $\\sigma$ of $C_m$ act by multiplication by $t$. Since $(t-1)N_m(t)=t^m-1$, this is a genuine cyclic-group action.","\\begin{theorem}[Norm-quotient lattice and transient invariants]\\label{thm:norm-quotient}\nFor this action,\n\\[\n M_m^{C_m}=0,\\qquad\n H^1(C_m,M_m)\\cong O/mO.\n\\]\nConsequently, with $a=v_p(m)$,\n\\[\n (M_m/p^nM_m)^{C_m}\\cong O/p^{\\min(a,n)}O.\n\\]\nUnder the boundary identification with $(O/mO)[p^n]$, reduction from level $s$ to level $n$ is multiplication by $p^{s-n}$. For $a>0$ the sharp uniform delay is $a$. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over $O$.\n\\end{theorem}\n\\begin{proof}\nThe operator $\\sigma-1$ is multiplication by $t-1$. Its cokernel is $O/(N_m(1))=O/mO$. Over $\\mathbb Q_p$, the element $t-1$ is invertible modulo $N_m(t)$ because $N_m(1)=m\\ne0$. Hence its kernel over $O$ is zero. The norm operator is multiplication by $N_m(t)$ and therefore zero on $M_m$.",",\n\\[\n M_m^{C_m}=0,\\qquad\n H^1(C_m,M_m)\\cong O/mO.\n\\]\nConsequently, with $a=v_p(m)$,\n\\[\n (M_m/p^nM_m)^{C_m}\\cong O/p^{\\min(a,n)}O.\n\\]\nUnder the boundary identification with $(O/mO)[p^n]$, reduction from level $s$ to level $n$ is multiplication by $p^{s-n}$. For $a>0$ the sharp uniform delay is $a$. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over $O$.\n\\end{theorem}\n\\begin{proof}\nThe operator $\\sigma-1$ is multiplication by $t-1$. Its cokernel is $O/(N_m(1))=O/mO$. Over $\\mathbb Q_p$, the element $t-1$ is invertible modulo $N_m(t)$ because $N_m(1)=m\\ne0$. Hence its kernel over $O$ is zero. The norm operator is multiplication by $N_m(t)$ and therefore zero on $M_m$.","mO.\n\\]\nConsequently, with $a=v_p(m)$,\n\\[\n (M_m/p^nM_m)^{C_m}\\cong O/p^{\\min(a,n)}O.\n\\]\nUnder the boundary identification with $(O/mO)[p^n]$, reduction from level $s$ to level $n$ is multiplication by $p^{s-n}$. For $a>0$ the sharp uniform delay is $a$. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over $O$.\n\\end{theorem}\n\\begin{proof}\nThe operator $\\sigma-1$ is multiplication by $t-1$. Its cokernel is $O/(N_m(1))=O/mO$. Over $\\mathbb Q_p$, the element $t-1$ is invertible modulo $N_m(t)$ because $N_m(1)=m\\ne0$. Hence its kernel over $O$ is zero. The norm operator is multiplication by $N_m(t)$ and therefore zero on $M_m$.","(m)$,\n\\[\n (M_m/p^nM_m)^{C_m}\\cong O/p^{\\min(a,n)}O.\n\\]\nUnder the boundary identification with $(O/mO)[p^n]$, reduction from level $s$ to level $n$ is multiplication by $p^{s-n}$. For $a>0$ the sharp uniform delay is $a$. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over $O$.\n\\end{theorem}\n\\begin{proof}\nThe operator $\\sigma-1$ is multiplication by $t-1$. Its cokernel is $O/(N_m(1))=O/mO$. Over $\\mathbb Q_p$, the element $t-1$ is invertible modulo $N_m(t)$ because $N_m(1)=m\\ne0$. Hence its kernel over $O$ is zero. The norm operator is multiplication by $N_m(t)$ and therefore zero on $M_m$.","{C_m}\\cong O/p^{\\min(a,n)}O.\n\\]\nUnder the boundary identification with $(O/mO)[p^n]$, reduction from level $s$ to level $n$ is multiplication by $p^{s-n}$. For $a>0$ the sharp uniform delay is $a$. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over $O$.\n\\end{theorem}\n\\begin{proof}\nThe operator $\\sigma-1$ is multiplication by $t-1$. Its cokernel is $O/(N_m(1))=O/mO$. Over $\\mathbb Q_p$, the element $t-1$ is invertible modulo $N_m(t)$ because $N_m(1)=m\\ne0$. Hence its kernel over $O$ is zero. The norm operator is multiplication by $N_m(t)$ and therefore zero on $M_m$.",")}O.\n\\]\nUnder the boundary identification with $(O/mO)[p^n]$, reduction from level $s$ to level $n$ is multiplication by $p^{s-n}$. For $a>0$ the sharp uniform delay is $a$. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over $O$.\n\\end{theorem}\n\\begin{proof}\nThe operator $\\sigma-1$ is multiplication by $t-1$. Its cokernel is $O/(N_m(1))=O/mO$. Over $\\mathbb Q_p$, the element $t-1$ is invertible modulo $N_m(t)$ because $N_m(1)=m\\ne0$. Hence its kernel over $O$ is zero. The norm operator is multiplication by $N_m(t)$ and therefore zero on $M_m$.","identification with $(O/mO)[p^n]$, reduction from level $s$ to level $n$ is multiplication by $p^{s-n}$. For $a>0$ the sharp uniform delay is $a$. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over $O$.\n\\end{theorem}\n\\begin{proof}\nThe operator $\\sigma-1$ is multiplication by $t-1$. Its cokernel is $O/(N_m(1))=O/mO$. Over $\\mathbb Q_p$, the element $t-1$ is invertible modulo $N_m(t)$ because $N_m(1)=m\\ne0$. Hence its kernel over $O$ is zero. The norm operator is multiplication by $N_m(t)$ and therefore zero on $M_m$.","[p^n]$, reduction from level $s$ to level $n$ is multiplication by $p^{s-n}$. For $a>0$ the sharp uniform delay is $a$. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over $O$.\n\\end{theorem}\n\\begin{proof}\nThe operator $\\sigma-1$ is multiplication by $t-1$. Its cokernel is $O/(N_m(1))=O/mO$. Over $\\mathbb Q_p$, the element $t-1$ is invertible modulo $N_m(t)$ because $N_m(1)=m\\ne0$. Hence its kernel over $O$ is zero. The norm operator is multiplication by $N_m(t)$ and therefore zero on $M_m$.","from level $s$ to level $n$ is multiplication by $p^{s-n}$. For $a>0$ the sharp uniform delay is $a$. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over $O$.\n\\end{theorem}\n\\begin{proof}\nThe operator $\\sigma-1$ is multiplication by $t-1$. Its cokernel is $O/(N_m(1))=O/mO$. Over $\\mathbb Q_p$, the element $t-1$ is invertible modulo $N_m(t)$ because $N_m(1)=m\\ne0$. Hence its kernel over $O$ is zero. The norm operator is multiplication by $N_m(t)$ and therefore zero on $M_m$.","lication by $p^{s-n}$. For $a>0$ the sharp uniform delay is $a$. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over $O$.\n\\end{theorem}\n\\begin{proof}\nThe operator $\\sigma-1$ is multiplication by $t-1$. Its cokernel is $O/(N_m(1))=O/mO$. Over $\\mathbb Q_p$, the element $t-1$ is invertible modulo $N_m(t)$ because $N_m(1)=m\\ne0$. Hence its kernel over $O$ is zero. The norm operator is multiplication by $N_m(t)$ and therefore zero on $M_m$.","elay is $a$. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over $O$.\n\\end{theorem}\n\\begin{proof}\nThe operator $\\sigma-1$ is multiplication by $t-1$. Its cokernel is $O/(N_m(1))=O/mO$. Over $\\mathbb Q_p$, the element $t-1$ is invertible modulo $N_m(t)$ because $N_m(1)=m\\ne0$. Hence its kernel over $O$ is zero. The norm operator is multiplication by $N_m(t)$ and therefore zero on $M_m$.","te invariant fails to lift integrally, although the action itself is defined over $O$.\n\\end{theorem}\n\\begin{proof}\nThe operator $\\sigma-1$ is multiplication by $t-1$. Its cokernel is $O/(N_m(1))=O/mO$. Over $\\mathbb Q_p$, the element $t-1$ is invertible modulo $N_m(t)$ because $N_m(1)=m\\ne0$. Hence its kernel over $O$ is zero. The norm operator is multiplication by $N_m(t)$ and therefore zero on $M_m$.","For a cyclic group, a crossed homomorphism is determined by $c(\\sigma)=x$ with norm zero, and a coboundary has $x=(\\sigma-1)y$. This gives $H^1=\\ker N/(\\sigma-1)M_m=O/mO$ directly, agreeing with \\cite[Chapter II, 1.20]{MilneCFT2020}. The remaining assertions follow from the preceding theorems.\n\\end{proof}","For $p=m=3$, in the basis $1,t$,\n\\[\n A=\\begin{pmatrix}0&-1","For $p=m=3$, in the basis $1,t$,\n\\[\n A=\\begin{pmatrix}0&-1\\\\1&-1\\end{pmatrix},\\qquad\n A^3=I,\\qquad \\det(A-I)=3.\n\\]\nThe fixed vectors modulo $3^n$ are exactly\n\\[\n \\left\\{b\\,3^{n-1}\\binom{1}{-1}:b\\in\\{0,1,2\\}\\right\\}.\n\\]\nThe three vectors at level $n+1$ all reduce to zero at level $n$. Thus a nonzero residual appears at every level, while its inverse system has zero limit. This is not persistent nonexistence of the lattice; it is the failure of a particular finite-level vector to be the reduction of an invariant integral vector.","1&-1\\end{pmatrix},\\qquad\n A^3=I,\\qquad \\det(A-I)=3.\n\\]\nThe fixed vectors modulo $3^n$ are exactly\n\\[\n \\left\\{b\\,3^{n-1}\\binom{1}{-1}:b\\in\\{0,1,2\\}\\right\\}.\n\\]\nThe three vectors at level $n+1$ all reduce to zero at level $n$. Thus a nonzero residual appears at every level, while its inverse system has zero limit. This is not persistent nonexistence of the lattice; it is the failure of a particular finite-level vector to be the reduction of an invariant integral vector.","For a commutative ring $R$, a finite group $G$, and $M\\in D(R[G])$, define\n\\[\n \\Gamma_R(M)=\\mathbf R\\operatorname{Hom}_{R[G]}(R,M)\\in D(R).\n\\]\nThis is the derived invariant functor. Derived categories in this section are the enhanced unbounded derived categories. They are stable. Their exact functors preserve finite cofiber sequences but are not asserted to preserve every infinite limit or colimit.","For a ring map $R\\to S$ there is a canonical comparison\n\\[\n \\boxed{\\beta_{R,S,M}:\\Gamma_R(M)\\otimes_R^{\\mathbf L}S\n \\longrightarrow\\Gamma_S(M\\otimes_R^{\\mathbf L}S).}\n\\]\nIt is adjoint to base change of the evaluation map from the object of homotopy invariants, regarded as carrying the trivial group action, to $M$. Its construction is natural in $M$ and uses the actual coefficient map. No exchange weight is selected.","This direction is reduction-after-invariants to invariants-after-reduction. It is the reverse of the optional $A_rC\\to CA_s$ convention used for some earlier signatures. We use the direction supplied by the arithmetic comparison, not an unproved inverse. No earlier orientation-dependent normalization theorem is applied without changing its signature explicitly.","A two-fiber exact model can be written entirely with endofunctors: set\n\\[\n \\mathcal K_R=D(R[G])\\times D(R),\\qquad\n A_R(M,N)=(0,\\Gamma_R(M)),\n\\]\nand let $C_{R,S}$ apply derived coefficient change in both components. The second component of $C_{R,S}A_R\\to A_SC_{R,S}$ is precisely $\\beta$. This places the example in a genuine two-axis exact system, rather than identifying differently typed functors by notation. A fixed ambient universe is understood. A small stable subdiagram can instead be generated by the stated inputs and their images, without making a claim that all outputs are perfect.","\\begin{theorem}[Perfect coefficient changes have zero derived exchange defect]\\label{thm:perfect-invariant-exchange}\nIf $S$ is perfect as an $R$-module, then $\\beta_{R,S,M}$ is an equivalence for every $M\\in D(R[G])$. In particular, reduction $\\mathbb Z_p\\to\\mathbb Z/p^n$ commutes with fully derived invariants.\n\\end{theorem}\n\\begin{proof}\nFor a complex $P\\in D(R)$ with trivial $G$-action, evaluation defines a natural comparison\n\\[\n \\Gamma_R(M)\\otimes_R^{\\mathbf L}P\n \\longrightarrow\\Gamma_R(M\\otimes_R^{\\mathbf L}P).\n\\]\nBoth sides are exact functors of $P$. The right side is exact because homotopy invariants preserve finite limits, and finite limits agree with finite colimits in stable categories. The comparison is the identity for $P=R$. The class of $P$ on which it is an equivalence is closed under finite cofib","\\begin{theorem}[Perfect coefficient changes have zero derived exchange defect]\\label{thm:perfect-invariant-exchange}\nIf $S$ is perfect as an $R$-module, then $\\beta_{R,S,M}$ is an equivalence for every $M\\in D(R[G])$. In particular, reduction $\\mathbb Z_p\\to\\mathbb Z/p^n$ commutes with fully derived invariants.\n\\end{theorem}\n\\begin{proof}\nFor a complex $P\\in D(R)$ with trivial $G$-action, evaluation defines a natural comparison\n\\[\n \\Gamma_R(M)\\otimes_R^{\\mathbf L}P\n \\longrightarrow\\Gamma_R(M\\otimes_R^{\\mathbf L}P).\n\\]\nBoth sides are exact functors of $P$. The right side is exact because homotopy invariants preserve finite limits, and finite limits agree with finite colimits in stable categories. The comparison is the identity for $P=R$. The class of $P$ on which it is an equivalence is closed under finite cofibers, shifts, and retr","\\begin{theorem}[Perfect coefficient changes have zero derived exchange defect]\\label{thm:perfect-invariant-exchange}\nIf $S$ is perfect as an $R$-module, then $\\beta_{R,S,M}$ is an equivalence for every $M\\in D(R[G])$. In particular, reduction $\\mathbb Z_p\\to\\mathbb Z/p^n$ commutes with fully derived invariants.\n\\end{theorem}\n\\begin{proof}\nFor a complex $P\\in D(R)$ with trivial $G$-action, evaluation defines a natural comparison\n\\[\n \\Gamma_R(M)\\otimes_R^{\\mathbf L}P\n \\longrightarrow\\Gamma_R(M\\otimes_R^{\\mathbf L}P).\n\\]\nBoth sides are exact functors of $P$. The right side is exact because homotopy invariants preserve finite limits, and finite limits agree with finite colimits in stable categories. The comparison is the identity for $P=R$. The class of $P$ on which it is an equivalence is closed under finite cofibers, shifts, and retracts, hence contains $\\operat","\\begin{theorem}[Perfect coefficient changes have zero derived exchange defect]\\label{thm:perfect-invariant-exchange}\nIf $S$ is perfect as an $R$-module, then $\\beta_{R,S,M}$ is an equivalence for every $M\\in D(R[G])$. In particular, reduction $\\mathbb Z_p\\to\\mathbb Z/p^n$ commutes with fully derived invariants.\n\\end{theorem}\n\\begin{proof}\nFor a complex $P\\in D(R)$ with trivial $G$-action, evaluation defines a natural comparison\n\\[\n \\Gamma_R(M)\\otimes_R^{\\mathbf L}P\n \\longrightarrow\\Gamma_R(M\\otimes_R^{\\mathbf L}P).\n\\]\nBoth sides are exact functors of $P$. The right side is exact because homotopy invariants preserve finite limits, and finite limits agree with finite colimits in stable categories. The comparison is the identity for $P=R$. The class of $P$ on which it is an equivalence is closed under finite cofibers, shifts, and retracts, hence contains $\\operatorname{Perf}(R)=\\operatorname{Thick}(R)$. F","\\begin{theorem}[Perfect coefficient changes have zero derived exchange defect]\\label{thm:perfect-invariant-exchange}\nIf $S$ is perfect as an $R$-module, then $\\beta_{R,S,M}$ is an equivalence for every $M\\in D(R[G])$. In particular, reduction $\\mathbb Z_p\\to\\mathbb Z/p^n$ commutes with fully derived invariants.\n\\end{theorem}\n\\begin{proof}\nFor a complex $P\\in D(R)$ with trivial $G$-action, evaluation defines a natural comparison\n\\[\n \\Gamma_R(M)\\otimes_R^{\\mathbf L}P\n \\longrightarrow\\Gamma_R(M\\otimes_R^{\\mathbf L}P).\n\\]\nBoth sides are exact functors of $P$. The right side is exact because homotopy invariants preserve finite limits, and finite limits agree with finite colimits in stable categories. The comparison is the identity for $P=R$. The class of $P$ on which it is an equivalence is closed under finite cofibers, shifts, and retracts, hence contains $\\operatorname{Perf}(R)=\\operatorname{Thick}(R)$. For $P=S$, the right side has its natural $S$-module stru","\\begin{theorem}[Perfect coefficient changes have zero derived exchange defect]\\label{thm:perfect-invariant-exchange}\nIf $S$ is perfect as an $R$-module, then $\\beta_{R,S,M}$ is an equivalence for every $M\\in D(R[G])$. In particular, reduction $\\mathbb Z_p\\to\\mathbb Z/p^n$ commutes with fully derived invariants.\n\\end{theorem}\n\\begin{proof}\nFor a complex $P\\in D(R)$ with trivial $G$-action, evaluation defines a natural comparison\n\\[\n \\Gamma_R(M)\\otimes_R^{\\mathbf L}P\n \\longrightarrow\\Gamma_R(M\\otimes_R^{\\mathbf L}P).\n\\]\nBoth sides are exact functors of $P$. The right side is exact because homotopy invariants preserve finite limits, and finite limits agree with finite colimits in stable categories. The comparison is the identity for $P=R$. The class of $P$ on which it is an equivalence is closed under finite cofibers, shifts, and retracts, hence contains $\\operatorname{Perf}(R)=\\operatorname{Thick}(R)$. For $P=S$, the right side has its natural $S$-module structure and is the stated derived invariant object over $S$.","t changes have zero derived exchange defect]\\label{thm:perfect-invariant-exchange}\nIf $S$ is perfect as an $R$-module, then $\\beta_{R,S,M}$ is an equivalence for every $M\\in D(R[G])$. In particular, reduction $\\mathbb Z_p\\to\\mathbb Z/p^n$ commutes with fully derived invariants.\n\\end{theorem}\n\\begin{proof}\nFor a complex $P\\in D(R)$ with trivial $G$-action, evaluation defines a natural comparison\n\\[\n \\Gamma_R(M)\\otimes_R^{\\mathbf L}P\n \\longrightarrow\\Gamma_R(M\\otimes_R^{\\mathbf L}P).\n\\]\nBoth sides are exact functors of $P$. The right side is exact because homotopy invariants preserve finite limits, and finite limits agree with finite colimits in stable categories. The comparison is the identity for $P=R$. The class of $P$ on which it is an equivalence is closed under finite cofibers, shifts, and retracts, hence contains $\\operatorname{Perf}(R)=\\operatorname{Thick}(R)$. For $P=S$, the right side has its natural $S$-module structure and is the stated derived invariant object over $S$.","$\\beta_{R,S,M}$ is an equivalence for every $M\\in D(R[G])$. In particular, reduction $\\mathbb Z_p\\to\\mathbb Z/p^n$ commutes with fully derived invariants.\n\\end{theorem}\n\\begin{proof}\nFor a complex $P\\in D(R)$ with trivial $G$-action, evaluation defines a natural comparison\n\\[\n \\Gamma_R(M)\\otimes_R^{\\mathbf L}P\n \\longrightarrow\\Gamma_R(M\\otimes_R^{\\mathbf L}P).\n\\]\nBoth sides are exact functors of $P$. The right side is exact because homotopy invariants preserve finite limits, and finite limits agree with finite colimits in stable categories. The comparison is the identity for $P=R$. The class of $P$ on which it is an equivalence is closed under finite cofibers, shifts, and retracts, hence contains $\\operatorname{Perf}(R)=\\operatorname{Thick}(R)$. For $P=S$, the right side has its natural $S$-module structure and is the stated derived invariant object over $S$.","For a complex $P\\in D(R)$ with trivial $G$-action, evaluation defines a natural comparison\n\\[\n \\Gamma_R(M)\\otimes_R^{\\mathbf L}P\n \\longrightarrow\\Gamma_R(M\\otimes_R^{\\mathbf L}P).\n\\]\nBoth sides are exact functors of $P$. The right side is exact because homotopy invariants preserve finite limits, and finite limits agree with finite colimits in stable categories. The comparison is the identity for $P=R$. The class of $P$ on which it is an equivalence is closed under finite cofibers, shifts, and retracts, hence contains $\\operatorname{Perf}(R)=\\operatorname{Thick}(R)$. For $P=S$, the right side has its natural $S$-module structure and is the stated derived invariant object over $S$.","D(R)$ with trivial $G$-action, evaluation defines a natural comparison\n\\[\n \\Gamma_R(M)\\otimes_R^{\\mathbf L}P\n \\longrightarrow\\Gamma_R(M\\otimes_R^{\\mathbf L}P).\n\\]\nBoth sides are exact functors of $P$. The right side is exact because homotopy invariants preserve finite limits, and finite limits agree with finite colimits in stable categories. The comparison is the identity for $P=R$. The class of $P$ on which it is an equivalence is closed under finite cofibers, shifts, and retracts, hence contains $\\operatorname{Perf}(R)=\\operatorname{Thick}(R)$. For $P=S$, the right side has its natural $S$-module structure and is the stated derived invariant object over $S$.","L}P\n \\longrightarrow\\Gamma_R(M\\otimes_R^{\\mathbf L}P).\n\\]\nBoth sides are exact functors of $P$. The right side is exact because homotopy invariants preserve finite limits, and finite limits agree with finite colimits in stable categories. The comparison is the identity for $P=R$. The class of $P$ on which it is an equivalence is closed under finite cofibers, shifts, and retracts, hence contains $\\operatorname{Perf}(R)=\\operatorname{Thick}(R)$. For $P=S$, the right side has its natural $S$-module structure and is the stated derived invariant object over $S$.",".\n\\]\nBoth sides are exact functors of $P$. The right side is exact because homotopy invariants preserve finite limits, and finite limits agree with finite colimits in stable categories. The comparison is the identity for $P=R$. The class of $P$ on which it is an equivalence is closed under finite cofibers, shifts, and retracts, hence contains $\\operatorname{Perf}(R)=\\operatorname{Thick}(R)$. For $P=S$, the right side has its natural $S$-module structure and is the stated derived invariant object over $S$.","$P$. The right side is exact because homotopy invariants preserve finite limits, and finite limits agree with finite colimits in stable categories. The comparison is the identity for $P=R$. The class of $P$ on which it is an equivalence is closed under finite cofibers, shifts, and retracts, hence contains $\\operatorname{Perf}(R)=\\operatorname{Thick}(R)$. For $P=S$, the right side has its natural $S$-module structure and is the stated derived invariant object over $S$.","ite limits, and finite limits agree with finite colimits in stable categories. The comparison is the identity for $P=R$. The class of $P$ on which it is an equivalence is closed under finite cofibers, shifts, and retracts, hence contains $\\operatorname{Perf}(R)=\\operatorname{Thick}(R)$. For $P=S$, the right side has its natural $S$-module structure and is the stated derived invariant object over $S$.","Finally $\\mathbb Z/p^n$ has the two-term finite free resolution $[\\mathbb Z_p\\xrightarrow{p^n}\\mathbb Z_p]$. It is perfect over $\\mathbb Z_p$. This uses the ordinary derived tensor construction, not termwise tensor of an arbitrary unbounded non-K-flat complex \\cite{StacksDerivedTensor}.\n\\end{proof}","For a lattice the earlier nonzero $H^1[p^n]$ therefore records the effect of extracting degree-zero invariants, not a failure of this fully derived comparison over $\\mathbb Z_p$. This positive theorem rules out a tempting but incorrect source of a new exact residual. It also gives an information-preserving way to handle that particular truncation defect: retain derived invariant data rather than require degree-zero invariants to commute with reduction.","The perfectness hypothesis cannot be removed. Fix an odd prime $p$, let\n\\[\n R=\\mathbb Z/p^2,\\qquad k=\\mathbb F_p,\\qquad G=C_p,\n\\]\nand choose a matrix $J\\in M_d(\\mathbb Z)$ whose reduction modulo $p$ is invertible. Put $M=R^d$ and let the chosen generator act by\n\\[\n A=I+pJ\\pmod{p^2}.\n\\]\nFor $d=2$, the choice $J=\\left(\\begin{smallmatrix}0&1","The perfectness hypothesis cannot be removed. Fix an odd prime $p$, let\n\\[\n R=\\mathbb Z/p^2,\\qquad k=\\mathbb F_p,\\qquad G=C_p,\n\\]\nand choose a matrix $J\\in M_d(\\mathbb Z)$ whose reduction modulo $p$ is invertible. Put $M=R^d$ and let the chosen generator act by\n\\[\n A=I+pJ\\pmod{p^2}.\n\\]\nFor $d=2$, the choice $J=\\left(\\begin{smallmatrix}0&1\\\\1&0\\end{smallmatrix}\\right)$ makes this action non-scalar. It mixes the two coordinates. The group action is input, but the exchange map remains the canonical $\\beta$, not a scalar multiple of another comparison.","1&0\\end{smallmatrix}\\right)$ makes this action non-scalar. It mixes the two coordinates. The group action is input, but the exchange map remains the canonical $\\beta$, not a scalar multiple of another comparison.","\\begin{theorem}[Unbounded intrinsic derived exchange defect]\\label{thm:nonflat-invariant-exchange}\nFor the preceding data, $A^p=I$ and the norm is $N=I+A+\\cdots+A^{p-1}=pI$. Moreover,\n\\[\n \\Gamma_R(M)\\simeq pM[0]\\cong k^d[0].\n\\]\nLet $U=\\Gamma_R(M)\\otimes_R^{\\mathbf L}k$ and $V=\\Gamma_k(M\\otimes_R^{\\mathbf L}k)$. Their cohomology is\n\\[\n H^i(U)\\cong\\begin{cases}k^d&i\\leq0,","\\begin{theorem}[Unbounded intrinsic derived exchange defect]\\label{thm:nonflat-invariant-exchange}\nFor the preceding data, $A^p=I$ and the norm is $N=I+A+\\cdots+A^{p-1}=pI$. Moreover,\n\\[\n \\Gamma_R(M)\\simeq pM[0]\\cong k^d[0].\n\\]\nLet $U=\\Gamma_R(M)\\otimes_R^{\\mathbf L}k$ and $V=\\Gamma_k(M\\otimes_R^{\\mathbf L}k)$. Their cohomology is\n\\[\n H^i(U)\\cong\\begin{cases}k^d&i\\leq0,\\\\0&i>0,\\end{cases}\n \\qquad\n H^i(V)\\cong\\begin{cases}0&i<0,\\\\k^d&i\\geq0.\\end{cases}\n\\]\nThe natural comparison $\\beta:U\\to V$ is zero on $H^0$. Its functor-intrinsic cofiber satisfies\n\\[\n \\boxed{H^i\\Cofib(\\beta)\\cong k^d\\quad\\text{for every }i\\in\\mathbb Z.}\n\\]\nBoth composites defining $\\beta$ are exact. The nonzero residual is therefore a genuine defect between fully derived exact functors, not solely a degree-zero truncation effect.\n\\end{theorem}\n\\begin{proof}\nThe binomial expansion gives $(I+pJ)^p=I$ modulo $p^2$. Also $A^i=I+ipJ$ modulo $p^2$, so\n\\[\n N=pI+p\\frac{p(p-1)}2J=pI\\pmod{p^2},\n\\]\nusing that $p$ is odd. The standard cyclic free resolution of the trivial $R[G]$-module gives the cochain complex\n\\[\n M\\xrightarro…","k^d&i\\geq0.\\end{cases}\n\\]\nThe natural comparison $\\beta:U\\to V$ is zero on $H^0$. Its functor-intrinsic cofiber satisfies\n\\[\n \\boxed{H^i\\Cofib(\\beta)\\cong k^d\\quad\\text{for every }i\\in\\mathbb Z.}\n\\]\nBoth composites defining $\\beta$ are exact. The nonzero residual is therefore a genuine defect between fully derived exact functors, not solely a degree-zero truncation effect.\n\\end{theorem}\n\\begin{proof}\nThe binomial expansion gives $(I+pJ)^p=I$ modulo $p^2$. Also $A^i=I+ipJ$ modulo $p^2$, so\n\\[\n N=pI+p\\frac{p(p-1)}2J=pI\\pmod{p^2},\n\\]\nusing that $p$ is odd. The standard cyclic free resolution of the trivial $R[G]$-module gives the cochain complex\n\\[\n M\\xrightarrow{pJ}M\\xrightarrow{pI}M\n \\xrightarrow{pJ}M\\xrightarrow{pI}\\cdots\n\\]\nin nonnegative degrees. The","k^d&i\\geq0.\\end{cases}\n\\]\nThe natural comparison $\\beta:U\\to V$ is zero on $H^0$. Its functor-intrinsic cofiber satisfies\n\\[\n \\boxed{H^i\\Cofib(\\beta)\\cong k^d\\quad\\text{for every }i\\in\\mathbb Z.}\n\\]\nBoth composites defining $\\beta$ are exact. The nonzero residual is therefore a genuine defect between fully derived exact functors, not solely a degree-zero truncation effect.\n\\end{theorem}\n\\begin{proof}\nThe binomial expansion gives $(I+pJ)^p=I$ modulo $p^2$. Also $A^i=I+ipJ$ modulo $p^2$, so\n\\[\n N=pI+p\\frac{p(p-1)}2J=pI\\pmod{p^2},\n\\]\nusing that $p$ is odd. The standard cyclic free resolution of the trivial $R[G]$-module gives the cochain complex\n\\[\n M\\xrightarrow{pJ}M\\xrightarrow{pI}M\n \\xrightarrow{pJ}M\\xrightarrow{pI}\\cdots\n\\]\nin nonnegative degrees. The cyclic resolutio","k^d&i\\geq0.\\end{cases}\n\\]\nThe natural comparison $\\beta:U\\to V$ is zero on $H^0$. Its functor-intrinsic cofiber satisfies\n\\[\n \\boxed{H^i\\Cofib(\\beta)\\cong k^d\\quad\\text{for every }i\\in\\mathbb Z.}\n\\]\nBoth composites defining $\\beta$ are exact. The nonzero residual is therefore a genuine defect between fully derived exact functors, not solely a degree-zero truncation effect.\n\\end{theorem}\n\\begin{proof}\nThe binomial expansion gives $(I+pJ)^p=I$ modulo $p^2$. Also $A^i=I+ipJ$ modulo $p^2$, so\n\\[\n N=pI+p\\frac{p(p-1)}2J=pI\\pmod{p^2},\n\\]\nusing that $p$ is odd. The standard cyclic free resolution of the trivial $R[G]$-module gives the cochain complex\n\\[\n M\\xrightarrow{pJ}M\\xrightarrow{pI}M\n \\xrightarrow{pJ}M\\xrightarrow{pI}\\cdots\n\\]\nin nonnegative degrees. The cyclic resolution is obtained from the integral one, whose exact sequences split as abelian groups, so it remains a free resolution a","k^d&i\\geq0.\\end{cases}\n\\]\nThe natural comparison $\\beta:U\\to V$ is zero on $H^0$. Its functor-intrinsic cofiber satisfies\n\\[\n \\boxed{H^i\\Cofib(\\beta)\\cong k^d\\quad\\text{for every }i\\in\\mathbb Z.}\n\\]\nBoth composites defining $\\beta$ are exact. The nonzero residual is therefore a genuine defect between fully derived exact functors, not solely a degree-zero truncation effect.\n\\end{theorem}\n\\begin{proof}\nThe binomial expansion gives $(I+pJ)^p=I$ modulo $p^2$. Also $A^i=I+ipJ$ modulo $p^2$, so\n\\[\n N=pI+p\\frac{p(p-1)}2J=pI\\pmod{p^2},\n\\]\nusing that $p$ is odd. The standard cyclic free resolution of the trivial $R[G]$-module gives the cochain complex\n\\[\n M\\xrightarrow{pJ}M\\xrightarrow{pI}M\n \\xrightarrow{pJ}M\\xrightarrow{pI}\\cdots\n\\]\nin nonnegative degrees. The cyclic resolution is obtained from the integral one, whose exact sequences split as abelian groups, so it remains a free resolution after coefficient change. Its alter","ral comparison $\\beta:U\\to V$ is zero on $H^0$. Its functor-intrinsic cofiber satisfies\n\\[\n \\boxed{H^i\\Cofib(\\beta)\\cong k^d\\quad\\text{for every }i\\in\\mathbb Z.}\n\\]\nBoth composites defining $\\beta$ are exact. The nonzero residual is therefore a genuine defect between fully derived exact functors, not solely a degree-zero truncation effect.\n\\end{theorem}\n\\begin{proof}\nThe binomial expansion gives $(I+pJ)^p=I$ modulo $p^2$. Also $A^i=I+ipJ$ modulo $p^2$, so\n\\[\n N=pI+p\\frac{p(p-1)}2J=pI\\pmod{p^2},\n\\]\nusing that $p$ is odd. The standard cyclic free resolution of the trivial $R[G]$-module gives the cochain complex\n\\[\n M\\xrightarrow{pJ}M\\xrightarrow{pI}M\n \\xrightarrow{pJ}M\\xrightarrow{pI}\\cdots\n\\]\nin nonnegative degrees. The cyclic resolution is obtained from the integral one, whose exact sequences split as abelian groups, so it remains a free resolution after coefficient change. Its alternating difference and norm maps agree with \\cite[Chapter II, 1.20 and 3.4]{MilneCFT2020}.","a:U\\to V$ is zero on $H^0$. Its functor-intrinsic cofiber satisfies\n\\[\n \\boxed{H^i\\Cofib(\\beta)\\cong k^d\\quad\\text{for every }i\\in\\mathbb Z.}\n\\]\nBoth composites defining $\\beta$ are exact. The nonzero residual is therefore a genuine defect between fully derived exact functors, not solely a degree-zero truncation effect.\n\\end{theorem}\n\\begin{proof}\nThe binomial expansion gives $(I+pJ)^p=I$ modulo $p^2$. Also $A^i=I+ipJ$ modulo $p^2$, so\n\\[\n N=pI+p\\frac{p(p-1)}2J=pI\\pmod{p^2},\n\\]\nusing that $p$ is odd. The standard cyclic free resolution of the trivial $R[G]$-module gives the cochain complex\n\\[\n M\\xrightarrow{pJ}M\\xrightarrow{pI}M\n \\xrightarrow{pJ}M\\xrightarrow{pI}\\cdots\n\\]\nin nonnegative degrees. The cyclic resolution is obtained from the integral one, whose exact sequences split as abelian groups, so it remains a free resolution after coefficient change. Its alternating difference and norm maps agree with \\cite[Chapter II, 1.20 and 3.4]{MilneCFT2020}.","zero on $H^0$. Its functor-intrinsic cofiber satisfies\n\\[\n \\boxed{H^i\\Cofib(\\beta)\\cong k^d\\quad\\text{for every }i\\in\\mathbb Z.}\n\\]\nBoth composites defining $\\beta$ are exact. The nonzero residual is therefore a genuine defect between fully derived exact functors, not solely a degree-zero truncation effect.\n\\end{theorem}\n\\begin{proof}\nThe binomial expansion gives $(I+pJ)^p=I$ modulo $p^2$. Also $A^i=I+ipJ$ modulo $p^2$, so\n\\[\n N=pI+p\\frac{p(p-1)}2J=pI\\pmod{p^2},\n\\]\nusing that $p$ is odd. The standard cyclic free resolution of the trivial $R[G]$-module gives the cochain complex\n\\[\n M\\xrightarrow{pJ}M\\xrightarrow{pI}M\n \\xrightarrow{pJ}M\\xrightarrow{pI}\\cdots\n\\]\nin nonnegative degrees. The cyclic resolution is obtained from the integral one, whose exact sequences split as abelian groups, so it remains a free resolution after coefficient change. Its alternating difference and norm maps agree with \\cite[Chapter II, 1.20 and 3.4]{MilneCFT2020}.","functor-intrinsic cofiber satisfies\n\\[\n \\boxed{H^i\\Cofib(\\beta)\\cong k^d\\quad\\text{for every }i\\in\\mathbb Z.}\n\\]\nBoth composites defining $\\beta$ are exact. The nonzero residual is therefore a genuine defect between fully derived exact functors, not solely a degree-zero truncation effect.\n\\end{theorem}\n\\begin{proof}\nThe binomial expansion gives $(I+pJ)^p=I$ modulo $p^2$. Also $A^i=I+ipJ$ modulo $p^2$, so\n\\[\n N=pI+p\\frac{p(p-1)}2J=pI\\pmod{p^2},\n\\]\nusing that $p$ is odd. The standard cyclic free resolution of the trivial $R[G]$-module gives the cochain complex\n\\[\n M\\xrightarrow{pJ}M\\xrightarrow{pI}M\n \\xrightarrow{pJ}M\\xrightarrow{pI}\\cdots\n\\]\nin nonnegative degrees. The cyclic resolution is obtained from the integral one, whose exact sequences split as abelian groups, so it remains a free resolution after coefficient change. Its alternating difference and norm maps agree with \\cite[Chapter II, 1.20 and 3.4]{MilneCFT2020}.","trinsic cofiber satisfies\n\\[\n \\boxed{H^i\\Cofib(\\beta)\\cong k^d\\quad\\text{for every }i\\in\\mathbb Z.}\n\\]\nBoth composites defining $\\beta$ are exact. The nonzero residual is therefore a genuine defect between fully derived exact functors, not solely a degree-zero truncation effect.\n\\end{theorem}\n\\begin{proof}\nThe binomial expansion gives $(I+pJ)^p=I$ modulo $p^2$. Also $A^i=I+ipJ$ modulo $p^2$, so\n\\[\n N=pI+p\\frac{p(p-1)}2J=pI\\pmod{p^2},\n\\]\nusing that $p$ is odd. The standard cyclic free resolution of the trivial $R[G]$-module gives the cochain complex\n\\[\n M\\xrightarrow{pJ}M\\xrightarrow{pI}M\n \\xrightarrow{pJ}M\\xrightarrow{pI}\\cdots\n\\]\nin nonnegative degrees. The cyclic resolution is obtained from the integral one, whose exact sequences split as abelian groups, so it remains a free resolution after coefficient change. Its alternating difference and norm maps agree with \\cite[Chapter II, 1.20 and 3.4]{MilneCFT2020}.","\\cong k^d\\quad\\text{for every }i\\in\\mathbb Z.}\n\\]\nBoth composites defining $\\beta$ are exact. The nonzero residual is therefore a genuine defect between fully derived exact functors, not solely a degree-zero truncation effect.\n\\end{theorem}\n\\begin{proof}\nThe binomial expansion gives $(I+pJ)^p=I$ modulo $p^2$. Also $A^i=I+ipJ$ modulo $p^2$, so\n\\[\n N=pI+p\\frac{p(p-1)}2J=pI\\pmod{p^2},\n\\]\nusing that $p$ is odd. The standard cyclic free resolution of the trivial $R[G]$-module gives the cochain complex\n\\[\n M\\xrightarrow{pJ}M\\xrightarrow{pI}M\n \\xrightarrow{pJ}M\\xrightarrow{pI}\\cdots\n\\]\nin nonnegative degrees. The cyclic resolution is obtained from the integral one, whose exact sequences split as abelian groups, so it remains a free resolution after coefficient change. Its alternating difference and norm maps agree with \\cite[Chapter II, 1.20 and 3.4]{MilneCFT2020}.","es defining $\\beta$ are exact. The nonzero residual is therefore a genuine defect between fully derived exact functors, not solely a degree-zero truncation effect.\n\\end{theorem}\n\\begin{proof}\nThe binomial expansion gives $(I+pJ)^p=I$ modulo $p^2$. Also $A^i=I+ipJ$ modulo $p^2$, so\n\\[\n N=pI+p\\frac{p(p-1)}2J=pI\\pmod{p^2},\n\\]\nusing that $p$ is odd. The standard cyclic free resolution of the trivial $R[G]$-module gives the cochain complex\n\\[\n M\\xrightarrow{pJ}M\\xrightarrow{pI}M\n \\xrightarrow{pJ}M\\xrightarrow{pI}\\cdots\n\\]\nin nonnegative degrees. The cyclic resolution is obtained from the integral one, whose exact sequences split as abelian groups, so it remains a free resolution after coefficient change. Its alternating difference and norm maps agree with \\cite[Chapter II, 1.20 and 3.4]{MilneCFT2020}.","o residual is therefore a genuine defect between fully derived exact functors, not solely a degree-zero truncation effect.\n\\end{theorem}\n\\begin{proof}\nThe binomial expansion gives $(I+pJ)^p=I$ modulo $p^2$. Also $A^i=I+ipJ$ modulo $p^2$, so\n\\[\n N=pI+p\\frac{p(p-1)}2J=pI\\pmod{p^2},\n\\]\nusing that $p$ is odd. The standard cyclic free resolution of the trivial $R[G]$-module gives the cochain complex\n\\[\n M\\xrightarrow{pJ}M\\xrightarrow{pI}M\n \\xrightarrow{pJ}M\\xrightarrow{pI}\\cdots\n\\]\nin nonnegative degrees. The cyclic resolution is obtained from the integral one, whose exact sequences split as abelian groups, so it remains a free resolution after coefficient change. Its alternating difference and norm maps agree with \\cite[Chapter II, 1.20 and 3.4]{MilneCFT2020}.","Since $J$ is invertible over $R$, each displayed differential has image and kernel $pM$. All positive cohomology vanishes and $H^0=pM$. Inclusion of that kernel in degree zero is a quasi-isomorphism $pM[0]\\to\\Gamma_R(M)$.","The $R$-module $k$ has an infinite free resolution, in nonpositive cochain degrees, with each differential multiplication by $p$. It is exact because $\\ker(p:R\\to R)=pR=\\operatorname{im}(p)$. Tensoring this bounded-above free resolution with $k$ gives zero differentials. Since $pM\\cong k^d$, this computes $H^i(U)$ as claimed.","On the other side $M$ is free over $R$, so $M\\otimes_R^{\\mathbf L}k=k^d[0]$. Its action is trivial. In the cyclic cochain resolution over $k$, both $A-I$ and the norm are zero. This gives the displayed cohomology of $V$.","The evaluation map on degree-zero invariants is the inclusion $pM\\hookrightarrow M$. On tensoring with $k$ its degree-zero map is zero. Evaluation on $H^0(V)=(k^d)^G=k^d$ is an isomorphism. Naturality of evaluation defining $\\beta$ therefore proves $H^0(\\beta)=0$. The long exact cohomology sequence of its cofiber now gives $k^d$ in every degree: the groups in negative degrees come from $H^{i+1}(U)$, those in positive degrees from $H^i(V)$, and the two middle groups use the zero degree-zero comparison.","nvariants is the inclusion $pM\\hookrightarrow M$. On tensoring with $k$ its degree-zero map is zero. Evaluation on $H^0(V)=(k^d)^G=k^d$ is an isomorphism. Naturality of evaluation defining $\\beta$ therefore proves $H^0(\\beta)=0$. The long exact cohomology sequence of its cofiber now gives $k^d$ in every degree: the groups in negative degrees come from $H^{i+1}(U)$, those in positive degrees from $H^i(V)$, and the two middle groups use the zero degree-zero comparison.","Exactness of derived coefficient change and derived invariants proves exactness of both composites. Their canonical natural transformation has a functorial cofiber in the stable enhancement. No illicit termwise tensor of the nonnegative invariant complex is used in computing $U$.\n\\end{proof}","\\begin{example}[The two-dimensional $\\mathbb Z/9$ instance]\\label{ex:nine-invariants}\nTake\n\\[\n R=\\mathbb Z/9,\\qquad\n A=\\begin{pmatrix}1&3","\\begin{example}[The two-dimensional $\\mathbb Z/9$ instance]\\label{ex:nine-invariants}\nTake\n\\[\n R=\\mathbb Z/9,\\qquad\n A=\\begin{pmatrix}1&3\\\\3&1\\end{pmatrix},\\qquad G=C_3.\n\\]\nThen $A^3=I$ over $R$, $I+A+A^2=3I$, and $M^G=3M$. The invariant cochain complex is quasi-isomorphic to $(\\mathbb F_3)^2$ in degree zero. Derived reduction of that invariant object has cohomology in every nonpositive degree, whereas taking derived invariants after reduction has cohomology in every nonnegative degree. The comparison cone has $(\\mathbb F_3)^2$ in every degree. Ordinary tensor of the original nonnegative complex would incorrectly reproduce only the right-hand answer.\n\\end{example}","3&1\\end{pmatrix},\\qquad G=C_3.\n\\]\nThen $A^3=I$ over $R$, $I+A+A^2=3I$, and $M^G=3M$. The invariant cochain complex is quasi-isomorphic to $(\\mathbb F_3)^2$ in degree zero. Derived reduction of that invariant object has cohomology in every nonpositive degree, whereas taking derived invariants after reduction has cohomology in every nonnegative degree. The comparison cone has $(\\mathbb F_3)^2$ in every degree. Ordinary tensor of the original nonnegative complex would incorrectly reproduce only the right-hand answer.\n\\end{example}","\\begin{corollary}[Erasing this residual loses the coefficient unit]\\label{cor:intrinsic-unit-loss}\nLet $\\Delta=\\Cofib(\\beta)$ in \\cref{thm:nonflat-invariant-exchange}. Any exact functor out of $D(k)$ that annihilates $\\Delta$ annihilates every perfect $k$-complex. If that functor also preserves all coproducts, it annihilates every object of $D(k)$.\n\\end{corollary}\n\\begin{proof}\nEvery complex of vector spaces admits a generally noncanonical splitting into its cohomology and contractible pieces, obtained by choosing complements to boundaries inside cycles and to cycles inside each degree. Since $H^0(\\Delta)=k^d\\ne0$, the unit $k[0]$ is a retract of $\\Delta$ in $D(k)$. Exact functors preserve retracts, finite sums, shifts, and cofibers, so annihilating $\\Delta$ annihilates $\\operatorname{Thick}(k)=\\operatorname{Perf","\\begin{corollary}[Erasing this residual loses the coefficient unit]\\label{cor:intrinsic-unit-loss}\nLet $\\Delta=\\Cofib(\\beta)$ in \\cref{thm:nonflat-invariant-exchange}. Any exact functor out of $D(k)$ that annihilates $\\Delta$ annihilates every perfect $k$-complex. If that functor also preserves all coproducts, it annihilates every object of $D(k)$.\n\\end{corollary}\n\\begin{proof}\nEvery complex of vector spaces admits a generally noncanonical splitting into its cohomology and contractible pieces, obtained by choosing complements to boundaries inside cycles and to cycles inside each degree. Since $H^0(\\Delta)=k^d\\ne0$, the unit $k[0]$ is a retract of $\\Delta$ in $D(k)$. Exact functors preserve retracts, finite sums, shifts, and cofibers, so annihilating $\\Delta$ annihilates $\\operatorname{Thick}(k)=\\operatorname{Perf}(k)$. If coproducts are preserved, every split complex is a coproduct of","\\begin{corollary}[Erasing this residual loses the coefficient unit]\\label{cor:intrinsic-unit-loss}\nLet $\\Delta=\\Cofib(\\beta)$ in \\cref{thm:nonflat-invariant-exchange}. Any exact functor out of $D(k)$ that annihilates $\\Delta$ annihilates every perfect $k$-complex. If that functor also preserves all coproducts, it annihilates every object of $D(k)$.\n\\end{corollary}\n\\begin{proof}\nEvery complex of vector spaces admits a generally noncanonical splitting into its cohomology and contractible pieces, obtained by choosing complements to boundaries inside cycles and to cycles inside each degree. Since $H^0(\\Delta)=k^d\\ne0$, the unit $k[0]$ is a retract of $\\Delta$ in $D(k)$. Exact functors preserve retracts, finite sums, shifts, and cofibers, so annihilating $\\Delta$ annihilates $\\operatorname{Thick}(k)=\\operatorname{Perf}(k)$. If coproducts are preserved, every split complex is a coproduct of shifts of vector spaces an","\\begin{corollary}[Erasing this residual loses the coefficient unit]\\label{cor:intrinsic-unit-loss}\nLet $\\Delta=\\Cofib(\\beta)$ in \\cref{thm:nonflat-invariant-exchange}. Any exact functor out of $D(k)$ that annihilates $\\Delta$ annihilates every perfect $k$-complex. If that functor also preserves all coproducts, it annihilates every object of $D(k)$.\n\\end{corollary}\n\\begin{proof}\nEvery complex of vector spaces admits a generally noncanonical splitting into its cohomology and contractible pieces, obtained by choosing complements to boundaries inside cycles and to cycles inside each degree. Since $H^0(\\Delta)=k^d\\ne0$, the unit $k[0]$ is a retract of $\\Delta$ in $D(k)$. Exact functors preserve retracts, finite sums, shifts, and cofibers, so annihilating $\\Delta$ annihilates $\\operatorname{Thick}(k)=\\operatorname{Perf}(k)$. If coproducts are preserved, every split complex is a coproduct of shifts of vector spaces and every vector space is a copr","\\begin{corollary}[Erasing this residual loses the coefficient unit]\\label{cor:intrinsic-unit-loss}\nLet $\\Delta=\\Cofib(\\beta)$ in \\cref{thm:nonflat-invariant-exchange}. Any exact functor out of $D(k)$ that annihilates $\\Delta$ annihilates every perfect $k$-complex. If that functor also preserves all coproducts, it annihilates every object of $D(k)$.\n\\end{corollary}\n\\begin{proof}\nEvery complex of vector spaces admits a generally noncanonical splitting into its cohomology and contractible pieces, obtained by choosing complements to boundaries inside cycles and to cycles inside each degree. Since $H^0(\\Delta)=k^d\\ne0$, the unit $k[0]$ is a retract of $\\Delta$ in $D(k)$. Exact functors preserve retracts, finite sums, shifts, and cofibers, so annihilating $\\Delta$ annihilates $\\operatorname{Thick}(k)=\\operatorname{Perf}(k)$. If coproducts are preserved, every split complex is a coproduct of shifts of vector spaces and every vector space is a coproduct of copies of $k$. This proves the stronger conclusion.\n\\end{proof}","hat annihilates $\\Delta$ annihilates every perfect $k$-complex. If that functor also preserves all coproducts, it annihilates every object of $D(k)$.\n\\end{corollary}\n\\begin{proof}\nEvery complex of vector spaces admits a generally noncanonical splitting into its cohomology and contractible pieces, obtained by choosing complements to boundaries inside cycles and to cycles inside each degree. Since $H^0(\\Delta)=k^d\\ne0$, the unit $k[0]$ is a retract of $\\Delta$ in $D(k)$. Exact functors preserve retracts, finite sums, shifts, and cofibers, so annihilating $\\Delta$ annihilates $\\operatorname{Thick}(k)=\\operatorname{Perf}(k)$. If coproducts are preserved, every split complex is a coproduct of shifts of vector spaces and every vector space is a coproduct of copies of $k$. This proves the stronger conclusion.\n\\end{proof}","ates every perfect $k$-complex. If that functor also preserves all coproducts, it annihilates every object of $D(k)$.\n\\end{corollary}\n\\begin{proof}\nEvery complex of vector spaces admits a generally noncanonical splitting into its cohomology and contractible pieces, obtained by choosing complements to boundaries inside cycles and to cycles inside each degree. Since $H^0(\\Delta)=k^d\\ne0$, the unit $k[0]$ is a retract of $\\Delta$ in $D(k)$. Exact functors preserve retracts, finite sums, shifts, and cofibers, so annihilating $\\Delta$ annihilates $\\operatorname{Thick}(k)=\\operatorname{Perf}(k)$. If coproducts are preserved, every split complex is a coproduct of shifts of vector spaces and every vector space is a coproduct of copies of $k$. This proves the stronger conclusion.\n\\end{proof}","complex. If that functor also preserves all coproducts, it annihilates every object of $D(k)$.\n\\end{corollary}\n\\begin{proof}\nEvery complex of vector spaces admits a generally noncanonical splitting into its cohomology and contractible pieces, obtained by choosing complements to boundaries inside cycles and to cycles inside each degree. Since $H^0(\\Delta)=k^d\\ne0$, the unit $k[0]$ is a retract of $\\Delta$ in $D(k)$. Exact functors preserve retracts, finite sums, shifts, and cofibers, so annihilating $\\Delta$ annihilates $\\operatorname{Thick}(k)=\\operatorname{Perf}(k)$. If coproducts are preserved, every split complex is a coproduct of shifts of vector spaces and every vector space is a coproduct of copies of $k$. This proves the stronger conclusion.\n\\end{proof}","that functor also preserves all coproducts, it annihilates every object of $D(k)$.\n\\end{corollary}\n\\begin{proof}\nEvery complex of vector spaces admits a generally noncanonical splitting into its cohomology and contractible pieces, obtained by choosing complements to boundaries inside cycles and to cycles inside each degree. Since $H^0(\\Delta)=k^d\\ne0$, the unit $k[0]$ is a retract of $\\Delta$ in $D(k)$. Exact functors preserve retracts, finite sums, shifts, and cofibers, so annihilating $\\Delta$ annihilates $\\operatorname{Thick}(k)=\\operatorname{Perf}(k)$. If coproducts are preserved, every split complex is a coproduct of shifts of vector spaces and every vector space is a coproduct of copies of $k$. This proves the stronger conclusion.\n\\end{proof}","lary}\n\\begin{proof}\nEvery complex of vector spaces admits a generally noncanonical splitting into its cohomology and contractible pieces, obtained by choosing complements to boundaries inside cycles and to cycles inside each degree. Since $H^0(\\Delta)=k^d\\ne0$, the unit $k[0]$ is a retract of $\\Delta$ in $D(k)$. Exact functors preserve retracts, finite sums, shifts, and cofibers, so annihilating $\\Delta$ annihilates $\\operatorname{Thick}(k)=\\operatorname{Perf}(k)$. If coproducts are preserved, every split complex is a coproduct of shifts of vector spaces and every vector space is a coproduct of copies of $k$. This proves the stronger conclusion.\n\\end{proof}","very complex of vector spaces admits a generally noncanonical splitting into its cohomology and contractible pieces, obtained by choosing complements to boundaries inside cycles and to cycles inside each degree. Since $H^0(\\Delta)=k^d\\ne0$, the unit $k[0]$ is a retract of $\\Delta$ in $D(k)$. Exact functors preserve retracts, finite sums, shifts, and cofibers, so annihilating $\\Delta$ annihilates $\\operatorname{Thick}(k)=\\operatorname{Perf}(k)$. If coproducts are preserved, every split complex is a coproduct of shifts of vector spaces and every vector space is a coproduct of copies of $k$. This proves the stronger conclusion.\n\\end{proof}","inside each degree. Since $H^0(\\Delta)=k^d\\ne0$, the unit $k[0]$ is a retract of $\\Delta$ in $D(k)$. Exact functors preserve retracts, finite sums, shifts, and cofibers, so annihilating $\\Delta$ annihilates $\\operatorname{Thick}(k)=\\operatorname{Perf}(k)$. If coproducts are preserved, every split complex is a coproduct of shifts of vector spaces and every vector space is a coproduct of copies of $k$. This proves the stronger conclusion.\n\\end{proof}","Thus exactness alone does not make the comparison invertible, and existence of its cofiber does not make a quotient repair harmless. The arithmetic coefficient module $k$ is not perfect over $R$. Also $\\Gamma_R(M)\\simeq k^d$ is not perfect over $R$, and the resulting residual is unbounded over $k$. These facts explain exactly why neither the preceding perfect-base-change theorem nor a blanket perfect-fiber model applies.","\\begin{theorem}[A one-step representation-lifting obstruction]\\label{thm:one-step-group-lift}\nIn \\cref{thm:nonflat-invariant-exchange}, the representation of $C_p$ on $(\\mathbb Z/p^2)^d$ has no lift to a representation on $(\\mathbb Z/p^3)^d$ with that reduction. More explicitly, every matrix lift $\\widetilde A$ obeys\n\\[\n \\boxed{\\widetilde A^p-I\\equiv p^2J\\not\\equiv0\\pmod{p^3}.}\n\\]\nThis obstruction is unchanged by the choice of the lift. There is also no finite free $\\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish mod","\\begin{theorem}[A one-step representation-lifting obstruction]\\label{thm:one-step-group-lift}\nIn \\cref{thm:nonflat-invariant-exchange}, the representation of $C_p$ on $(\\mathbb Z/p^2)^d$ has no lift to a representation on $(\\mathbb Z/p^3)^d$ with that reduction. More explicitly, every matrix lift $\\widetilde A$ obeys\n\\[\n \\boxed{\\widetilde A^p-I\\equiv p^2J\\not\\equiv0\\pmod{p^3}.}\n\\]\nThis obstruction is unchanged by the choice of the lift. There is also no finite free $\\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last ter","\\begin{theorem}[A one-step representation-lifting obstruction]\\label{thm:one-step-group-lift}\nIn \\cref{thm:nonflat-invariant-exchange}, the representation of $C_p$ on $(\\mathbb Z/p^2)^d$ has no lift to a representation on $(\\mathbb Z/p^3)^d$ with that reduction. More explicitly, every matrix lift $\\widetilde A$ obeys\n\\[\n \\boxed{\\widetilde A^p-I\\equiv p^2J\\not\\equiv0\\pmod{p^3}.}\n\\]\nThis obstruction is unchanged by the choice of the lift. There is also no finite free $\\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is in","\\begin{theorem}[A one-step representation-lifting obstruction]\\label{thm:one-step-group-lift}\nIn \\cref{thm:nonflat-invariant-exchange}, the representation of $C_p$ on $(\\mathbb Z/p^2)^d$ has no lift to a representation on $(\\mathbb Z/p^3)^d$ with that reduction. More explicitly, every matrix lift $\\widetilde A$ obeys\n\\[\n \\boxed{\\widetilde A^p-I\\equiv p^2J\\not\\equiv0\\pmod{p^3}.}\n\\]\nThis obstruction is unchanged by the choice of the lift. There is also no finite free $\\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation","\\begin{theorem}[A one-step representation-lifting obstruction]\\label{thm:one-step-group-lift}\nIn \\cref{thm:nonflat-invariant-exchange}, the representation of $C_p$ on $(\\mathbb Z/p^2)^d$ has no lift to a representation on $(\\mathbb Z/p^3)^d$ with that reduction. More explicitly, every matrix lift $\\widetilde A$ obeys\n\\[\n \\boxed{\\widetilde A^p-I\\equiv p^2J\\not\\equiv0\\pmod{p^3}.}\n\\]\nThis obstruction is unchanged by the choice of the lift. There is also no finite free $\\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","m:one-step-group-lift}\nIn \\cref{thm:nonflat-invariant-exchange}, the representation of $C_p$ on $(\\mathbb Z/p^2)^d$ has no lift to a representation on $(\\mathbb Z/p^3)^d$ with that reduction. More explicitly, every matrix lift $\\widetilde A$ obeys\n\\[\n \\boxed{\\widetilde A^p-I\\equiv p^2J\\not\\equiv0\\pmod{p^3}.}\n\\]\nThis obstruction is unchanged by the choice of the lift. There is also no finite free $\\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","thbb Z/p^3)^d$ with that reduction. More explicitly, every matrix lift $\\widetilde A$ obeys\n\\[\n \\boxed{\\widetilde A^p-I\\equiv p^2J\\not\\equiv0\\pmod{p^3}.}\n\\]\nThis obstruction is unchanged by the choice of the lift. There is also no finite free $\\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","that reduction. More explicitly, every matrix lift $\\widetilde A$ obeys\n\\[\n \\boxed{\\widetilde A^p-I\\equiv p^2J\\not\\equiv0\\pmod{p^3}.}\n\\]\nThis obstruction is unchanged by the choice of the lift. There is also no finite free $\\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","tion. More explicitly, every matrix lift $\\widetilde A$ obeys\n\\[\n \\boxed{\\widetilde A^p-I\\equiv p^2J\\not\\equiv0\\pmod{p^3}.}\n\\]\nThis obstruction is unchanged by the choice of the lift. There is also no finite free $\\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","etilde A$ obeys\n\\[\n \\boxed{\\widetilde A^p-I\\equiv p^2J\\not\\equiv0\\pmod{p^3}.}\n\\]\nThis obstruction is unchanged by the choice of the lift. There is also no finite free $\\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","{\\widetilde A^p-I\\equiv p^2J\\not\\equiv0\\pmod{p^3}.}\n\\]\nThis obstruction is unchanged by the choice of the lift. There is also no finite free $\\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","\\equiv p^2J\\not\\equiv0\\pmod{p^3}.}\n\\]\nThis obstruction is unchanged by the choice of the lift. There is also no finite free $\\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","\\not\\equiv0\\pmod{p^3}.}\n\\]\nThis obstruction is unchanged by the choice of the lift. There is also no finite free $\\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","is obstruction is unchanged by the choice of the lift. There is also no finite free $\\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","by the choice of the lift. There is also no finite free $\\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","hoice of the lift. There is also no finite free $\\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","the lift. There is also no finite free $\\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","ee $\\mathbb Z_p$-representation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","esentation with this reduction, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","on, even up to isomorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","somorphism of the reduced representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","representation.\n\\end{theorem}\n\\begin{proof}\nEvery lift has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","has the form $I+pJ+p^2B$ modulo $p^3$. Put $X=pJ+p^2B$ and use the binomial identity for $I+X$. The linear term is $pX=p^2J$ modulo $p^3$. For $2\\leq j<p$, the coefficient $\\binom pj$ is divisible by $p$ and $X^j$ by $p^j$, so those terms vanish modulo $p^3$. The last term $X^p$ also vanishes because $p\\geq3$. Since $J$ is invertible modulo $p$, the surviving term is nonzero. Hence the relation $\\sigma^p=1$ cannot hold at the next precision.","A $\\mathbb Z_p$-lift would reduce to such a level-three lift, which is impossible. Finally an invertible change of basis modulo $p^2$ lifts to an invertible matrix modulo $p^3$. Conjugating by that lift reduces an alleged representation lifting only up to isomorphism to the already refuted matrix-lifting problem.\n\\end{proof}","This is a concrete relation obstruction for a prescribed deformation problem. It is not asserted to classify arbitrary deformation obstructions or to identify the entire unbounded exchange cone with an $H^2$ obstruction group. In contrast, the norm-quotient actions in \\cref{thm:norm-quotient} genuinely exist over $\\mathbb Z_p$ at all precisions, even though some of their finite-level invariant vectors do not lift. These are different failures.","\\begin{theorem}[Lifted-input exchange and a necessary obstruction to lifting]\\label{thm:lifted-input-exchange}\nLet $O\\to R\\to S$ be maps of commutative rings. Assume $R$ and $S$ are perfect as $O$-modules. For every $N\\in D(O[G])$, put $M=N\\otimes_O^{\\mathbf L}R$. Then\n\\[\n \\boxed{\\beta_{R,S,M}:\\Gamma_R(M)\\otimes_R^{\\mathbf L}S\n \\xrightarrow{\\sim}\\Gamma_S(M\\otimes_R^{\\mathbf L}S).}\n\\]\nNo perfectness of $S$ over $R$ is required. Consequently, if $\\Cofib(\\beta_{R,S,M})$ is nonzero, there is no $N\\in D(O[G])$ together with an equivalence $N\\otimes_O^{\\mathbf L}R\\simeq M$.","{theorem}[Lifted-input exchange and a necessary obstruction to lifting]\\label{thm:lifted-input-exchange}\nLet $O\\to R\\to S$ be maps of commutative rings. Assume $R$ and $S$ are perfect as $O$-modules. For every $N\\in D(O[G])$, put $M=N\\otimes_O^{\\mathbf L}R$. Then\n\\[\n \\boxed{\\beta_{R,S,M}:\\Gamma_R(M)\\otimes_R^{\\mathbf L}S\n \\xrightarrow{\\sim}\\Gamma_S(M\\otimes_R^{\\mathbf L}S).}\n\\]\nNo perfectness of $S$ over $R$ is required. Consequently, if $\\Cofib(\\beta_{R,S,M})$ is nonzero, there is no $N\\in D(O[G])$ together with an equivalence $N\\otimes_O^{\\mathbf L}R\\simeq M$.","[Lifted-input exchange and a necessary obstruction to lifting]\\label{thm:lifted-input-exchange}\nLet $O\\to R\\to S$ be maps of commutative rings. Assume $R$ and $S$ are perfect as $O$-modules. For every $N\\in D(O[G])$, put $M=N\\otimes_O^{\\mathbf L}R$. Then\n\\[\n \\boxed{\\beta_{R,S,M}:\\Gamma_R(M)\\otimes_R^{\\mathbf L}S\n \\xrightarrow{\\sim}\\Gamma_S(M\\otimes_R^{\\mathbf L}S).}\n\\]\nNo perfectness of $S$ over $R$ is required. Consequently, if $\\Cofib(\\beta_{R,S,M})$ is nonzero, there is no $N\\in D(O[G])$ together with an equivalence $N\\otimes_O^{\\mathbf L}R\\simeq M$.","sary obstruction to lifting]\\label{thm:lifted-input-exchange}\nLet $O\\to R\\to S$ be maps of commutative rings. Assume $R$ and $S$ are perfect as $O$-modules. For every $N\\in D(O[G])$, put $M=N\\otimes_O^{\\mathbf L}R$. Then\n\\[\n \\boxed{\\beta_{R,S,M}:\\Gamma_R(M)\\otimes_R^{\\mathbf L}S\n \\xrightarrow{\\sim}\\Gamma_S(M\\otimes_R^{\\mathbf L}S).}\n\\]\nNo perfectness of $S$ over $R$ is required. Consequently, if $\\Cofib(\\beta_{R,S,M})$ is nonzero, there is no $N\\in D(O[G])$ together with an equivalence $N\\otimes_O^{\\mathbf L}R\\simeq M$.","put-exchange}\nLet $O\\to R\\to S$ be maps of commutative rings. Assume $R$ and $S$ are perfect as $O$-modules. For every $N\\in D(O[G])$, put $M=N\\otimes_O^{\\mathbf L}R$. Then\n\\[\n \\boxed{\\beta_{R,S,M}:\\Gamma_R(M)\\otimes_R^{\\mathbf L}S\n \\xrightarrow{\\sim}\\Gamma_S(M\\otimes_R^{\\mathbf L}S).}\n\\]\nNo perfectness of $S$ over $R$ is required. Consequently, if $\\Cofib(\\beta_{R,S,M})$ is nonzero, there is no $N\\in D(O[G])$ together with an equivalence $N\\otimes_O^{\\mathbf L}R\\simeq M$.","ommutative rings. Assume $R$ and $S$ are perfect as $O$-modules. For every $N\\in D(O[G])$, put $M=N\\otimes_O^{\\mathbf L}R$. Then\n\\[\n \\boxed{\\beta_{R,S,M}:\\Gamma_R(M)\\otimes_R^{\\mathbf L}S\n \\xrightarrow{\\sim}\\Gamma_S(M\\otimes_R^{\\mathbf L}S).}\n\\]\nNo perfectness of $S$ over $R$ is required. Consequently, if $\\Cofib(\\beta_{R,S,M})$ is nonzero, there is no $N\\in D(O[G])$ together with an equivalence $N\\otimes_O^{\\mathbf L}R\\simeq M$.","For $O=\\mathbb Z_p$, $R=\\mathbb Z/p^m$, and $S=\\mathbb Z/p^n$ with $m\\geq n\\geq1$, this applies at every pair of levels. It applies in particular to reductions of a supplied integral $G$-lattice.\n\\end{theorem}\n\\begin{proof}\nThe canonical base-change comparisons obey transitivity. After the tensor associativities, the composite\n\\[\n \\begin{split}\n \\Gamma_O(N)\\otimes_O^{\\mathbf L}S\n &\\xrightarrow{\\beta_{O,R,N}\\otimes_R S}\n \\Gamma_R(N\\otimes_O^{\\mathbf L}R)\\otimes_R^{\\mathbf L}S","For $O=\\mathbb Z_p$, $R=\\mathbb Z/p^m$, and $S=\\mathbb Z/p^n$ with $m\\geq n\\geq1$, this applies at every pair of levels. It applies in particular to reductions of a supplied integral $G$-lattice.\n\\end{theorem}\n\\begin{proof}\nThe canonical base-change comparisons obey transitivity. After the tensor associativities, the composite\n\\[\n \\begin{split}\n \\Gamma_O(N)\\otimes_O^{\\mathbf L}S\n &\\xrightarrow{\\beta_{O,R,N}\\otimes_R S}\n \\Gamma_R(N\\otimes_O^{\\mathbf L}R)\\otimes_R^{\\mathbf L}S\\\\\n &\\xrightarrow{\\beta_{R,S,M}}\n \\Gamma_S(N\\otimes_O^{\\mathbf L}S)\n \\end{split}\n\\]\nis $\\beta_{O,S,N}$. To check this equality with its natural coherence, take the adjoints defining the comparisons: both are base change of the same evaluation map from homotopy invariants to $N$. The coefficient extension adjunctions compose by derived tensor associativity. Their mates therefore agree. No arbitrary homotopy-category cone is chosen in this argument.","&\\xrightarrow{\\beta_{R,S,M}}\n \\Gamma_S(N\\otimes_O^{\\mathbf L}S)\n \\end{split}\n\\]\nis $\\beta_{O,S,N}$. To check this equality with its natural coherence, take the adjoints defining the comparisons: both are base change of the same evaluation map from homotopy invariants to $N$. The coefficient extension adjunctions compose by derived tensor associativity. Their mates therefore agree. No arbitrary homotopy-category cone is chosen in this argument.","By \\cref{thm:perfect-invariant-exchange}, $\\beta_{O,R,N}$ and $\\beta_{O,S,N}$ are equivalences. Tensoring preserves an equivalence, so the two-out-of-three property proves that $\\beta_{R,S,M}$ is an equivalence. Naturality makes the conclusion invariant under a specified equivalence between $M$ and a lifted object. The obstruction assertion is the contrapositive. For the arithmetic specialization, both quotients have two-term finite free resolutions over $\\mathbb Z_p$.\n\\end{proof}","d $\\beta_{O,S,N}$ are equivalences. Tensoring preserves an equivalence, so the two-out-of-three property proves that $\\beta_{R,S,M}$ is an equivalence. Naturality makes the conclusion invariant under a specified equivalence between $M$ and a lifted object. The obstruction assertion is the contrapositive. For the arithmetic specialization, both quotients have two-term finite free resolutions over $\\mathbb Z_p$.\n\\end{proof}","The cofiber of $\\beta_{R,S,-}$ is an exact functor. Its vanishing class is therefore thick and contains every coefficient-lifted input, as well as the thick closure of those inputs. The example in \\cref{thm:nonflat-invariant-exchange} lies outside that class. This is a functor-intrinsic necessary lifting obstruction, not merely a failed scalar equality.","There are now four precisely separated regimes.\n\\begin{enumerate}\n\\item For a supplied integral lattice, ordinary invariants and reduction can differ. The residual is $H^1[p^n]$, and a finite amount of additional precision detects and repairs liftability of a specified vector or intertwiner without taking a destructive quotient.\n\\item For a perfect coefficient change, fully derived invariants and reduction agree. Passing to derived data resolves the degree-zero truncation problem in this regime.\n\\item A supplied coefficient lift over a common ring makes the intermediate comparison exact when the two coefficient quotients are perfect over that common ring. Nonvanishing of the comparison cone is therefore a necessary obstruction to such a lift.\n\\item For the exhibited nonperfect change $\\mathbb Z/p^2\\to\\mathbb F_p$, the fully derived canonical exchange itself has a","and reduction agree. Passing to derived data resolves the degree-zero truncation problem in this regime.\n\\item A supplied coefficient lift over a common ring makes the intermediate comparison exact when the two coefficient quotients are perfect over that common ring. Nonvanishing of the comparison cone is therefore a necessary obstruction to such a lift.\n\\item For the exhibited nonperfect change $\\mathbb Z/p^2\\to\\mathbb F_p$, the fully derived canonical exchange itself has a nonzero, unbounded cofiber. Killing that cofiber destroys the target's perfect coefficient theory. Its input representation also fails to extend one coefficient level further.\n\\end{enumerate}\nThese statements distinguish intrinsic exchange from actual existence. The following section refutes sufficiency of the single comparison-cone test and proves a finite necessary-and-sufficient lifting criterion for a fixed finite group. That correction preserves the requested finite-level representation without killing the coefficient unit. It does not close unrestricted profinite-tower existence or the Langlands-facing glob…","This section closes a specified version of the sufficient-lifting question left by K61. The source group $G$ is fixed and finite, the rank $d$ is fixed, the coefficient ring is $\\mathbb Z_p$, and every lift must satisfy the same group multiplication table. We first exhibit a zero-exchange-defect representation that nevertheless does not lift. We then give a complete one-step obstruction and an explicit finite horizon after which convergent correction constructs an integral action. None of these assertions concerns arbitrary continuous representations of a changing system of finite Galois quotients.","\\paragraph{Attribution.}\nThe one-step obstruction is the classical group-extension class in $H^2$, with the standard inhomogeneous conventions of \\cite[Chapter II, 1.17--1.19]{MilneCFT2020}. Finite-group ultrametric stability and the quantitative bound used below are known: \\cite[Lemma 7.3, Proposition 7.4, Example 7.5]{FournierFacio2026} supply the same valuation threshold, in a broader setting. We provide a direct matrix-averaging proof, translate it into an exact finite lifting criterion, and connect it to the present residual calculus. Historical novelty is not claimed for that bound or the deformation-theoretic mechanism.","Put $R_n=\\mathbb Z/p^n$, $k=\\mathbb F_p$, and fix\n\\[\n \\rho_n:G\\longrightarrow\\operatorname{GL}_d(R_n),\\qquad n\\geq1.\n\\]\nLet $V=\\operatorname{End}_k(k^d)$ with action\n\\[\n g\\cdot X=\\bar\\rho(g)X\\bar\\rho(g)^{-1},\n \\qquad \\bar\\rho=\\rho_n\\bmod p.\n\\]\nChoose matrices $A_g\\in\\operatorname{GL}_d(R_{n+1})$ lifting $\\rho_n(g)$, with $A_1=I$. They need not constitute a representation. Define the normalized relation residual\n\\[\n c_A(g,h)=\\frac{A_gA_hA_{gh}^{-1}-I}{p^n}\\pmod p\\quad\\in V.\n\\]\nDivision here identifies the square-zero ideal $p^nR_{n+1}$ with $k$. It is not inversion of $p$ in $R_{n+1}$. The square-zero assertion uses $2n\\geq n+1$.","Z/p^n$, $k=\\mathbb F_p$, and fix\n\\[\n \\rho_n:G\\longrightarrow\\operatorname{GL}_d(R_n),\\qquad n\\geq1.\n\\]\nLet $V=\\operatorname{End}_k(k^d)$ with action\n\\[\n g\\cdot X=\\bar\\rho(g)X\\bar\\rho(g)^{-1},\n \\qquad \\bar\\rho=\\rho_n\\bmod p.\n\\]\nChoose matrices $A_g\\in\\operatorname{GL}_d(R_{n+1})$ lifting $\\rho_n(g)$, with $A_1=I$. They need not constitute a representation. Define the normalized relation residual\n\\[\n c_A(g,h)=\\frac{A_gA_hA_{gh}^{-1}-I}{p^n}\\pmod p\\quad\\in V.\n\\]\nDivision here identifies the square-zero ideal $p^nR_{n+1}$ with $k$. It is not inversion of $p$ in $R_{n+1}$. The square-zero assertion uses $2n\\geq n+1$.","peratorname{End}_k(k^d)$ with action\n\\[\n g\\cdot X=\\bar\\rho(g)X\\bar\\rho(g)^{-1},\n \\qquad \\bar\\rho=\\rho_n\\bmod p.\n\\]\nChoose matrices $A_g\\in\\operatorname{GL}_d(R_{n+1})$ lifting $\\rho_n(g)$, with $A_1=I$. They need not constitute a representation. Define the normalized relation residual\n\\[\n c_A(g,h)=\\frac{A_gA_hA_{gh}^{-1}-I}{p^n}\\pmod p\\quad\\in V.\n\\]\nDivision here identifies the square-zero ideal $p^nR_{n+1}$ with $k$. It is not inversion of $p$ in $R_{n+1}$. The square-zero assertion uses $2n\\geq n+1$.","(k^d)$ with action\n\\[\n g\\cdot X=\\bar\\rho(g)X\\bar\\rho(g)^{-1},\n \\qquad \\bar\\rho=\\rho_n\\bmod p.\n\\]\nChoose matrices $A_g\\in\\operatorname{GL}_d(R_{n+1})$ lifting $\\rho_n(g)$, with $A_1=I$. They need not constitute a representation. Define the normalized relation residual\n\\[\n c_A(g,h)=\\frac{A_gA_hA_{gh}^{-1}-I}{p^n}\\pmod p\\quad\\in V.\n\\]\nDivision here identifies the square-zero ideal $p^nR_{n+1}$ with $k$. It is not inversion of $p$ in $R_{n+1}$. The square-zero assertion uses $2n\\geq n+1$.","t X=\\bar\\rho(g)X\\bar\\rho(g)^{-1},\n \\qquad \\bar\\rho=\\rho_n\\bmod p.\n\\]\nChoose matrices $A_g\\in\\operatorname{GL}_d(R_{n+1})$ lifting $\\rho_n(g)$, with $A_1=I$. They need not constitute a representation. Define the normalized relation residual\n\\[\n c_A(g,h)=\\frac{A_gA_hA_{gh}^{-1}-I}{p^n}\\pmod p\\quad\\in V.\n\\]\nDivision here identifies the square-zero ideal $p^nR_{n+1}$ with $k$. It is not inversion of $p$ in $R_{n+1}$. The square-zero assertion uses $2n\\geq n+1$.","r\\rho(g)X\\bar\\rho(g)^{-1},\n \\qquad \\bar\\rho=\\rho_n\\bmod p.\n\\]\nChoose matrices $A_g\\in\\operatorname{GL}_d(R_{n+1})$ lifting $\\rho_n(g)$, with $A_1=I$. They need not constitute a representation. Define the normalized relation residual\n\\[\n c_A(g,h)=\\frac{A_gA_hA_{gh}^{-1}-I}{p^n}\\pmod p\\quad\\in V.\n\\]\nDivision here identifies the square-zero ideal $p^nR_{n+1}$ with $k$. It is not inversion of $p$ in $R_{n+1}$. The square-zero assertion uses $2n\\geq n+1$.","o=\\rho_n\\bmod p.\n\\]\nChoose matrices $A_g\\in\\operatorname{GL}_d(R_{n+1})$ lifting $\\rho_n(g)$, with $A_1=I$. They need not constitute a representation. Define the normalized relation residual\n\\[\n c_A(g,h)=\\frac{A_gA_hA_{gh}^{-1}-I}{p^n}\\pmod p\\quad\\in V.\n\\]\nDivision here identifies the square-zero ideal $p^nR_{n+1}$ with $k$. It is not inversion of $p$ in $R_{n+1}$. The square-zero assertion uses $2n\\geq n+1$.","For a normalized $1$-cochain $b:G\\to V$, set\n\\[\n (\\delta b)(g,h)=b(g)+g\\cdot b(h)-b(gh).\n\\]\nFor a normalized $2$-cochain set\n\\[\n (\\delta c)(g,h,t)=g\\cdot c(h,t)-c(gh,t)+c(g,ht)-c(g,h).\n\\]\nA cochain is normalized when it vanishes whenever an argument is the identity.","\\begin{theorem}[Complete one-step relation obstruction]\\label{thm:relation-H2}\nThe residual $c_A$ is a $2$-cocycle. Its cohomology class\n\\[\n o_n(\\rho_n)=[c_A]\\in H^2(G,V)\n\\]\nis independent of the chosen matrix lifts. Moreover,\n\\[\n \\boxed{o_n(\\rho_n)=0\n \\quad\\Longleftrightarrow\\quad\n \\rho_n\\text{ lifts to }\\operatorname{GL}_d(R_{n+1}).}\n\\]\nMore precisely, all lifts with the fixed reduction have the form\n\\[\n A_g^b=(I+p^n b(g))A_g,\n\\]\nwhere representatives of $b(g)$ may be chosen arbitrarily, and the necessary and sufficient equation is\n\\[\n \\boxed{\\delta b=-c_A.}\n\\]\n\\end{theorem}\n\\begin{proof}\nAssociativity of the threefold product gives the exact identity\n\\[\n (I+p^nc_A(g,h))(I+p^nc_A(gh,t))\n = (I+p^n g\\cdot c_A(h,t))(I+p^nc_A(g,ht))\n\\]\nin $R_{n+1}$, where the conjugation can be reduced m","\\begin{theorem}[Complete one-step relation obstruction]\\label{thm:relation-H2}\nThe residual $c_A$ is a $2$-cocycle. Its cohomology class\n\\[\n o_n(\\rho_n)=[c_A]\\in H^2(G,V)\n\\]\nis independent of the chosen matrix lifts. Moreover,\n\\[\n \\boxed{o_n(\\rho_n)=0\n \\quad\\Longleftrightarrow\\quad\n \\rho_n\\text{ lifts to }\\operatorname{GL}_d(R_{n+1}).}\n\\]\nMore precisely, all lifts with the fixed reduction have the form\n\\[\n A_g^b=(I+p^n b(g))A_g,\n\\]\nwhere representatives of $b(g)$ may be chosen arbitrarily, and the necessary and sufficient equation is\n\\[\n \\boxed{\\delta b=-c_A.}\n\\]\n\\end{theorem}\n\\begin{proof}\nAssociativity of the threefold product gives the exact identity\n\\[\n (I+p^nc_A(g,h))(I+p^nc_A(gh,t))\n = (I+p^n g\\cdot c_A(h,t))(I+p^nc_A(g,ht))\n\\]\nin $R_{n+1}$, where the conjugation can be reduced modulo $p$","\\begin{theorem}[Complete one-step relation obstruction]\\label{thm:relation-H2}\nThe residual $c_A$ is a $2$-cocycle. Its cohomology class\n\\[\n o_n(\\rho_n)=[c_A]\\in H^2(G,V)\n\\]\nis independent of the chosen matrix lifts. Moreover,\n\\[\n \\boxed{o_n(\\rho_n)=0\n \\quad\\Longleftrightarrow\\quad\n \\rho_n\\text{ lifts to }\\operatorname{GL}_d(R_{n+1}).}\n\\]\nMore precisely, all lifts with the fixed reduction have the form\n\\[\n A_g^b=(I+p^n b(g))A_g,\n\\]\nwhere representatives of $b(g)$ may be chosen arbitrarily, and the necessary and sufficient equation is\n\\[\n \\boxed{\\delta b=-c_A.}\n\\]\n\\end{theorem}\n\\begin{proof}\nAssociativity of the threefold product gives the exact identity\n\\[\n (I+p^nc_A(g,h))(I+p^nc_A(gh,t))\n = (I+p^n g\\cdot c_A(h,t))(I+p^nc_A(g,ht))\n\\]\nin $R_{n+1}$, where the conjugation can be reduced modulo $p$. All products of two $p^n$ terms vanish. Dividing the resulting l","\\begin{theorem}[Complete one-step relation obstruction]\\label{thm:relation-H2}\nThe residual $c_A$ is a $2$-cocycle. Its cohomology class\n\\[\n o_n(\\rho_n)=[c_A]\\in H^2(G,V)\n\\]\nis independent of the chosen matrix lifts. Moreover,\n\\[\n \\boxed{o_n(\\rho_n)=0\n \\quad\\Longleftrightarrow\\quad\n \\rho_n\\text{ lifts to }\\operatorname{GL}_d(R_{n+1}).}\n\\]\nMore precisely, all lifts with the fixed reduction have the form\n\\[\n A_g^b=(I+p^n b(g))A_g,\n\\]\nwhere representatives of $b(g)$ may be chosen arbitrarily, and the necessary and sufficient equation is\n\\[\n \\boxed{\\delta b=-c_A.}\n\\]\n\\end{theorem}\n\\begin{proof}\nAssociativity of the threefold product gives the exact identity\n\\[\n (I+p^nc_A(g,h))(I+p^nc_A(gh,t))\n = (I+p^n g\\cdot c_A(h,t))(I+p^nc_A(g,ht))\n\\]\nin $R_{n+1}$, where the conjugation can be reduced modulo $p$. All products of two $p^n$ terms vanish. Dividing the resulting linear identity by $p^n$ gives $\\delta c_A=0$.","heorem}[Complete one-step relation obstruction]\\label{thm:relation-H2}\nThe residual $c_A$ is a $2$-cocycle. Its cohomology class\n\\[\n o_n(\\rho_n)=[c_A]\\in H^2(G,V)\n\\]\nis independent of the chosen matrix lifts. Moreover,\n\\[\n \\boxed{o_n(\\rho_n)=0\n \\quad\\Longleftrightarrow\\quad\n \\rho_n\\text{ lifts to }\\operatorname{GL}_d(R_{n+1}).}\n\\]\nMore precisely, all lifts with the fixed reduction have the form\n\\[\n A_g^b=(I+p^n b(g))A_g,\n\\]\nwhere representatives of $b(g)$ may be chosen arbitrarily, and the necessary and sufficient equation is\n\\[\n \\boxed{\\delta b=-c_A.}\n\\]\n\\end{theorem}\n\\begin{proof}\nAssociativity of the threefold product gives the exact identity\n\\[\n (I+p^nc_A(g,h))(I+p^nc_A(gh,t))\n = (I+p^n g\\cdot c_A(h,t))(I+p^nc_A(g,ht))\n\\]\nin $R_{n+1}$, where the conjugation can be reduced modulo $p$. All products of two $p^n$ terms vanish. Dividing the resulting linear identity by $p^n$ gives $\\delta c_A=0$.","thm:relation-H2}\nThe residual $c_A$ is a $2$-cocycle. Its cohomology class\n\\[\n o_n(\\rho_n)=[c_A]\\in H^2(G,V)\n\\]\nis independent of the chosen matrix lifts. Moreover,\n\\[\n \\boxed{o_n(\\rho_n)=0\n \\quad\\Longleftrightarrow\\quad\n \\rho_n\\text{ lifts to }\\operatorname{GL}_d(R_{n+1}).}\n\\]\nMore precisely, all lifts with the fixed reduction have the form\n\\[\n A_g^b=(I+p^n b(g))A_g,\n\\]\nwhere representatives of $b(g)$ may be chosen arbitrarily, and the necessary and sufficient equation is\n\\[\n \\boxed{\\delta b=-c_A.}\n\\]\n\\end{theorem}\n\\begin{proof}\nAssociativity of the threefold product gives the exact identity\n\\[\n (I+p^nc_A(g,h))(I+p^nc_A(gh,t))\n = (I+p^n g\\cdot c_A(h,t))(I+p^nc_A(g,ht))\n\\]\nin $R_{n+1}$, where the conjugation can be reduced modulo $p$. All products of two $p^n$ terms vanish. Dividing the resulting linear identity by $p^n$ gives $\\delta c_A=0$.","_n(\\rho_n)=[c_A]\\in H^2(G,V)\n\\]\nis independent of the chosen matrix lifts. Moreover,\n\\[\n \\boxed{o_n(\\rho_n)=0\n \\quad\\Longleftrightarrow\\quad\n \\rho_n\\text{ lifts to }\\operatorname{GL}_d(R_{n+1}).}\n\\]\nMore precisely, all lifts with the fixed reduction have the form\n\\[\n A_g^b=(I+p^n b(g))A_g,\n\\]\nwhere representatives of $b(g)$ may be chosen arbitrarily, and the necessary and sufficient equation is\n\\[\n \\boxed{\\delta b=-c_A.}\n\\]\n\\end{theorem}\n\\begin{proof}\nAssociativity of the threefold product gives the exact identity\n\\[\n (I+p^nc_A(g,h))(I+p^nc_A(gh,t))\n = (I+p^n g\\cdot c_A(h,t))(I+p^nc_A(g,ht))\n\\]\nin $R_{n+1}$, where the conjugation can be reduced modulo $p$. All products of two $p^n$ terms vanish. Dividing the resulting linear identity by $p^n$ gives $\\delta c_A=0$.","eftrightarrow\\quad\n \\rho_n\\text{ lifts to }\\operatorname{GL}_d(R_{n+1}).}\n\\]\nMore precisely, all lifts with the fixed reduction have the form\n\\[\n A_g^b=(I+p^n b(g))A_g,\n\\]\nwhere representatives of $b(g)$ may be chosen arbitrarily, and the necessary and sufficient equation is\n\\[\n \\boxed{\\delta b=-c_A.}\n\\]\n\\end{theorem}\n\\begin{proof}\nAssociativity of the threefold product gives the exact identity\n\\[\n (I+p^nc_A(g,h))(I+p^nc_A(gh,t))\n = (I+p^n g\\cdot c_A(h,t))(I+p^nc_A(g,ht))\n\\]\nin $R_{n+1}$, where the conjugation can be reduced modulo $p$. All products of two $p^n$ terms vanish. Dividing the resulting linear identity by $p^n$ gives $\\delta c_A=0$.","ecisely, all lifts with the fixed reduction have the form\n\\[\n A_g^b=(I+p^n b(g))A_g,\n\\]\nwhere representatives of $b(g)$ may be chosen arbitrarily, and the necessary and sufficient equation is\n\\[\n \\boxed{\\delta b=-c_A.}\n\\]\n\\end{theorem}\n\\begin{proof}\nAssociativity of the threefold product gives the exact identity\n\\[\n (I+p^nc_A(g,h))(I+p^nc_A(gh,t))\n = (I+p^n g\\cdot c_A(h,t))(I+p^nc_A(g,ht))\n\\]\nin $R_{n+1}$, where the conjugation can be reduced modulo $p$. All products of two $p^n$ terms vanish. Dividing the resulting linear identity by $p^n$ gives $\\delta c_A=0$.","rm\n\\[\n A_g^b=(I+p^n b(g))A_g,\n\\]\nwhere representatives of $b(g)$ may be chosen arbitrarily, and the necessary and sufficient equation is\n\\[\n \\boxed{\\delta b=-c_A.}\n\\]\n\\end{theorem}\n\\begin{proof}\nAssociativity of the threefold product gives the exact identity\n\\[\n (I+p^nc_A(g,h))(I+p^nc_A(gh,t))\n = (I+p^n g\\cdot c_A(h,t))(I+p^nc_A(g,ht))\n\\]\nin $R_{n+1}$, where the conjugation can be reduced modulo $p$. All products of two $p^n$ terms vanish. Dividing the resulting linear identity by $p^n$ gives $\\delta c_A=0$.","A_g,\n\\]\nwhere representatives of $b(g)$ may be chosen arbitrarily, and the necessary and sufficient equation is\n\\[\n \\boxed{\\delta b=-c_A.}\n\\]\n\\end{theorem}\n\\begin{proof}\nAssociativity of the threefold product gives the exact identity\n\\[\n (I+p^nc_A(g,h))(I+p^nc_A(gh,t))\n = (I+p^n g\\cdot c_A(h,t))(I+p^nc_A(g,ht))\n\\]\nin $R_{n+1}$, where the conjugation can be reduced modulo $p$. All products of two $p^n$ terms vanish. Dividing the resulting linear identity by $p^n$ gives $\\delta c_A=0$.","ily, and the necessary and sufficient equation is\n\\[\n \\boxed{\\delta b=-c_A.}\n\\]\n\\end{theorem}\n\\begin{proof}\nAssociativity of the threefold product gives the exact identity\n\\[\n (I+p^nc_A(g,h))(I+p^nc_A(gh,t))\n = (I+p^n g\\cdot c_A(h,t))(I+p^nc_A(g,ht))\n\\]\nin $R_{n+1}$, where the conjugation can be reduced modulo $p$. All products of two $p^n$ terms vanish. Dividing the resulting linear identity by $p^n$ gives $\\delta c_A=0$.","necessary and sufficient equation is\n\\[\n \\boxed{\\delta b=-c_A.}\n\\]\n\\end{theorem}\n\\begin{proof}\nAssociativity of the threefold product gives the exact identity\n\\[\n (I+p^nc_A(g,h))(I+p^nc_A(gh,t))\n = (I+p^n g\\cdot c_A(h,t))(I+p^nc_A(g,ht))\n\\]\nin $R_{n+1}$, where the conjugation can be reduced modulo $p$. All products of two $p^n$ terms vanish. Dividing the resulting linear identity by $p^n$ gives $\\delta c_A=0$.","Any alternative matrix lift is uniquely of the form $(I+p^nb(g))A_g$, for a normalized cochain $b$. Expanding its multiplicativity residual and discarding square-zero terms gives\n\\[\n c_{A^b}=c_A+\\delta b.\n\\]\nThus the class is unchanged and the new matrices satisfy every group relation exactly if and only if $\\delta b=-c_A$. Such a cochain exists precisely when $[c_A]=0$. Invertibility is automatic because $I+p^nb(g)$ has inverse $I-p^nb(g)$ at this level.\n\\end{proof}","ative matrix lift is uniquely of the form $(I+p^nb(g))A_g$, for a normalized cochain $b$. Expanding its multiplicativity residual and discarding square-zero terms gives\n\\[\n c_{A^b}=c_A+\\delta b.\n\\]\nThus the class is unchanged and the new matrices satisfy every group relation exactly if and only if $\\delta b=-c_A$. Such a cochain exists precisely when $[c_A]=0$. Invertibility is automatic because $I+p^nb(g)$ has inverse $I-p^nb(g)$ at this level.\n\\end{proof}","quely of the form $(I+p^nb(g))A_g$, for a normalized cochain $b$. Expanding its multiplicativity residual and discarding square-zero terms gives\n\\[\n c_{A^b}=c_A+\\delta b.\n\\]\nThus the class is unchanged and the new matrices satisfy every group relation exactly if and only if $\\delta b=-c_A$. Such a cochain exists precisely when $[c_A]=0$. Invertibility is automatic because $I+p^nb(g)$ has inverse $I-p^nb(g)$ at this level.\n\\end{proof}","\\begin{proposition}[Repair ambiguity, conjugacy, and finite certificates]\\label{prop:repair-H1}\nIf the one-step lift set is nonempty, its framed lifts form a torsor for $Z^1(G,V)$. Conjugation by matrices $I+p^nX$, $X\\in V$, quotients this torsor by coboundaries. Hence the strict-conjugacy classes form a torsor for $H^1(G,V)$, and the stabilizer of a lift under this conjugation is $H^0(G,V)$ as an additive group.","For a fixed multiplication table and residual action, deciding existence is a finite linear problem over $k$. A solution $b$ is a positive certificate. A linear functional $\\lambda$ on normalized $2$-cochains with\n\\[\n \\lambda\\circ\\delta=0,\\qquad \\lambda(c_A)\\ne0\n\\]\nis a negative certificate.\n\\end{proposition}\n\\begin{proof}\nIf $b_0$ is one solution, every other solution is $b_0+z$ with $\\delta z=0$. Conjugating the matrices by $I+p^nX$ replaces $b(g)$ by $b(g)+X-g\\cdot X$. These are exactly the negatives of the usual $0$-coboundaries and yield the same quotient subgroup. Conjugation fixes the action precisely when $X$ is invariant. The cochain spaces have dimensions $(|G|-1)d^2$ and $(|G|-1)^2d^2$. Gaussian elimination decides whether $-c_A$ is in the image of $\\delta$. If it is not, elementary li","For a fixed multiplication table and residual action, deciding existence is a finite linear problem over $k$. A solution $b$ is a positive certificate. A linear functional $\\lambda$ on normalized $2$-cochains with\n\\[\n \\lambda\\circ\\delta=0,\\qquad \\lambda(c_A)\\ne0\n\\]\nis a negative certificate.\n\\end{proposition}\n\\begin{proof}\nIf $b_0$ is one solution, every other solution is $b_0+z$ with $\\delta z=0$. Conjugating the matrices by $I+p^nX$ replaces $b(g)$ by $b(g)+X-g\\cdot X$. These are exactly the negatives of the usual $0$-coboundaries and yield the same quotient subgroup. Conjugation fixes the action precisely when $X$ is invariant. The cochain spaces have dimensions $(|G|-1)d^2$ and $(|G|-1)^2d^2$. Gaussian elimination decides whether $-c_A$ is in the image of $\\delta$. If it is not, elementary linear algebra sup","For a fixed multiplication table and residual action, deciding existence is a finite linear problem over $k$. A solution $b$ is a positive certificate. A linear functional $\\lambda$ on normalized $2$-cochains with\n\\[\n \\lambda\\circ\\delta=0,\\qquad \\lambda(c_A)\\ne0\n\\]\nis a negative certificate.\n\\end{proposition}\n\\begin{proof}\nIf $b_0$ is one solution, every other solution is $b_0+z$ with $\\delta z=0$. Conjugating the matrices by $I+p^nX$ replaces $b(g)$ by $b(g)+X-g\\cdot X$. These are exactly the negatives of the usual $0$-coboundaries and yield the same quotient subgroup. Conjugation fixes the action precisely when $X$ is invariant. The cochain spaces have dimensions $(|G|-1)d^2$ and $(|G|-1)^2d^2$. Gaussian elimination decides whether $-c_A$ is in the image of $\\delta$. If it is not, elementary linear algebra supplies a functional annihilating that image but not $c_A$.","For a fixed multiplication table and residual action, deciding existence is a finite linear problem over $k$. A solution $b$ is a positive certificate. A linear functional $\\lambda$ on normalized $2$-cochains with\n\\[\n \\lambda\\circ\\delta=0,\\qquad \\lambda(c_A)\\ne0\n\\]\nis a negative certificate.\n\\end{proposition}\n\\begin{proof}\nIf $b_0$ is one solution, every other solution is $b_0+z$ with $\\delta z=0$. Conjugating the matrices by $I+p^nX$ replaces $b(g)$ by $b(g)+X-g\\cdot X$. These are exactly the negatives of the usual $0$-coboundaries and yield the same quotient subgroup. Conjugation fixes the action precisely when $X$ is invariant. The cochain spaces have dimensions $(|G|-1)d^2$ and $(|G|-1)^2d^2$. Gaussian elimination decides whether $-c_A$ is in the image of $\\delta$. If it is not, elementary linear algebra supplies a functional annihilating that image but not $c_A$.\n\\end{proof}","e and residual action, deciding existence is a finite linear problem over $k$. A solution $b$ is a positive certificate. A linear functional $\\lambda$ on normalized $2$-cochains with\n\\[\n \\lambda\\circ\\delta=0,\\qquad \\lambda(c_A)\\ne0\n\\]\nis a negative certificate.\n\\end{proposition}\n\\begin{proof}\nIf $b_0$ is one solution, every other solution is $b_0+z$ with $\\delta z=0$. Conjugating the matrices by $I+p^nX$ replaces $b(g)$ by $b(g)+X-g\\cdot X$. These are exactly the negatives of the usual $0$-coboundaries and yield the same quotient subgroup. Conjugation fixes the action precisely when $X$ is invariant. The cochain spaces have dimensions $(|G|-1)d^2$ and $(|G|-1)^2d^2$. Gaussian elimination decides whether $-c_A$ is in the image of $\\delta$. If it is not, elementary linear algebra supplies a functional annihilating that image but not $c_A$.\n\\end{proof}","ion, deciding existence is a finite linear problem over $k$. A solution $b$ is a positive certificate. A linear functional $\\lambda$ on normalized $2$-cochains with\n\\[\n \\lambda\\circ\\delta=0,\\qquad \\lambda(c_A)\\ne0\n\\]\nis a negative certificate.\n\\end{proposition}\n\\begin{proof}\nIf $b_0$ is one solution, every other solution is $b_0+z$ with $\\delta z=0$. Conjugating the matrices by $I+p^nX$ replaces $b(g)$ by $b(g)+X-g\\cdot X$. These are exactly the negatives of the usual $0$-coboundaries and yield the same quotient subgroup. Conjugation fixes the action precisely when $X$ is invariant. The cochain spaces have dimensions $(|G|-1)d^2$ and $(|G|-1)^2d^2$. Gaussian elimination decides whether $-c_A$ is in the image of $\\delta$. If it is not, elementary linear algebra supplies a functional annihilating that image but not $c_A$.\n\\end{proof}","ing existence is a finite linear problem over $k$. A solution $b$ is a positive certificate. A linear functional $\\lambda$ on normalized $2$-cochains with\n\\[\n \\lambda\\circ\\delta=0,\\qquad \\lambda(c_A)\\ne0\n\\]\nis a negative certificate.\n\\end{proposition}\n\\begin{proof}\nIf $b_0$ is one solution, every other solution is $b_0+z$ with $\\delta z=0$. Conjugating the matrices by $I+p^nX$ replaces $b(g)$ by $b(g)+X-g\\cdot X$. These are exactly the negatives of the usual $0$-coboundaries and yield the same quotient subgroup. Conjugation fixes the action precisely when $X$ is invariant. The cochain spaces have dimensions $(|G|-1)d^2$ and $(|G|-1)^2d^2$. Gaussian elimination decides whether $-c_A$ is in the image of $\\delta$. If it is not, elementary linear algebra supplies a functional annihilating that image but not $c_A$.\n\\end{proof}","$ is a positive certificate. A linear functional $\\lambda$ on normalized $2$-cochains with\n\\[\n \\lambda\\circ\\delta=0,\\qquad \\lambda(c_A)\\ne0\n\\]\nis a negative certificate.\n\\end{proposition}\n\\begin{proof}\nIf $b_0$ is one solution, every other solution is $b_0+z$ with $\\delta z=0$. Conjugating the matrices by $I+p^nX$ replaces $b(g)$ by $b(g)+X-g\\cdot X$. These are exactly the negatives of the usual $0$-coboundaries and yield the same quotient subgroup. Conjugation fixes the action precisely when $X$ is invariant. The cochain spaces have dimensions $(|G|-1)d^2$ and $(|G|-1)^2d^2$. Gaussian elimination decides whether $-c_A$ is in the image of $\\delta$. If it is not, elementary linear algebra supplies a functional annihilating that image but not $c_A$.\n\\end{proof}","bda\\circ\\delta=0,\\qquad \\lambda(c_A)\\ne0\n\\]\nis a negative certificate.\n\\end{proposition}\n\\begin{proof}\nIf $b_0$ is one solution, every other solution is $b_0+z$ with $\\delta z=0$. Conjugating the matrices by $I+p^nX$ replaces $b(g)$ by $b(g)+X-g\\cdot X$. These are exactly the negatives of the usual $0$-coboundaries and yield the same quotient subgroup. Conjugation fixes the action precisely when $X$ is invariant. The cochain spaces have dimensions $(|G|-1)d^2$ and $(|G|-1)^2d^2$. Gaussian elimination decides whether $-c_A$ is in the image of $\\delta$. If it is not, elementary linear algebra supplies a functional annihilating that image but not $c_A$.\n\\end{proof}","tive certificate.\n\\end{proposition}\n\\begin{proof}\nIf $b_0$ is one solution, every other solution is $b_0+z$ with $\\delta z=0$. Conjugating the matrices by $I+p^nX$ replaces $b(g)$ by $b(g)+X-g\\cdot X$. These are exactly the negatives of the usual $0$-coboundaries and yield the same quotient subgroup. Conjugation fixes the action precisely when $X$ is invariant. The cochain spaces have dimensions $(|G|-1)d^2$ and $(|G|-1)^2d^2$. Gaussian elimination decides whether $-c_A$ is in the image of $\\delta$. If it is not, elementary linear algebra supplies a functional annihilating that image but not $c_A$.\n\\end{proof}","\\end{proposition}\n\\begin{proof}\nIf $b_0$ is one solution, every other solution is $b_0+z$ with $\\delta z=0$. Conjugating the matrices by $I+p^nX$ replaces $b(g)$ by $b(g)+X-g\\cdot X$. These are exactly the negatives of the usual $0$-coboundaries and yield the same quotient subgroup. Conjugation fixes the action precisely when $X$ is invariant. The cochain spaces have dimensions $(|G|-1)d^2$ and $(|G|-1)^2d^2$. Gaussian elimination decides whether $-c_A$ is in the image of $\\delta$. If it is not, elementary linear algebra supplies a functional annihilating that image but not $c_A$.\n\\end{proof}","tion, every other solution is $b_0+z$ with $\\delta z=0$. Conjugating the matrices by $I+p^nX$ replaces $b(g)$ by $b(g)+X-g\\cdot X$. These are exactly the negatives of the usual $0$-coboundaries and yield the same quotient subgroup. Conjugation fixes the action precisely when $X$ is invariant. The cochain spaces have dimensions $(|G|-1)d^2$ and $(|G|-1)^2d^2$. Gaussian elimination decides whether $-c_A$ is in the image of $\\delta$. If it is not, elementary linear algebra supplies a functional annihilating that image but not $c_A$.\n\\end{proof}","is $b_0+z$ with $\\delta z=0$. Conjugating the matrices by $I+p^nX$ replaces $b(g)$ by $b(g)+X-g\\cdot X$. These are exactly the negatives of the usual $0$-coboundaries and yield the same quotient subgroup. Conjugation fixes the action precisely when $X$ is invariant. The cochain spaces have dimensions $(|G|-1)d^2$ and $(|G|-1)^2d^2$. Gaussian elimination decides whether $-c_A$ is in the image of $\\delta$. If it is not, elementary linear algebra supplies a functional annihilating that image but not $c_A$.\n\\end{proof}",". These are exactly the negatives of the usual $0$-coboundaries and yield the same quotient subgroup. Conjugation fixes the action precisely when $X$ is invariant. The cochain spaces have dimensions $(|G|-1)d^2$ and $(|G|-1)^2d^2$. Gaussian elimination decides whether $-c_A$ is in the image of $\\delta$. If it is not, elementary linear algebra supplies a functional annihilating that image but not $c_A$.\n\\end{proof}","\\begin{theorem}[A central sign hides the obstruction from derived invariants]\\label{thm:blind-exchange}\nLet $p$ be odd, $R=\\mathbb Z/p^2$, and let $J\\in M_d(\\mathbb Z)$ be invertible modulo $p$. On $M=R^d$ let\n\\[\n G=C_p\\times C_2=\\langle\\sigma,\\tau\\rangle,\n \\qquad \\sigma\\mapsto I+pJ,\\quad \\tau\\mapsto-I.\n\\]\nFor the canonical comparison of \\cref{thm:lifted-input-exchange},\n\\[\n \\Gamma_R(M)\\simeq0,\n \\qquad \\Gamma_k(M\\otimes_R^{\\mathbf L}k)\\simeq0,\n \\qquad \\Cofib(\\beta_{R,k,M})\\simeq0.\n\\]\nNevertheless this representation does not lift to $\\mathbb Z/p^3$, and hence does not lift as a finite free $\\mathbb Z_p$-representation of this fixed group.\n\\end{theorem}\n\\begin{proof}\nThe two displayed matrices commute and satisfy the defining relations. Since $2$ is invertible in $R$, the central orthogonal idempoten","\\begin{theorem}[A central sign hides the obstruction from derived invariants]\\label{thm:blind-exchange}\nLet $p$ be odd, $R=\\mathbb Z/p^2$, and let $J\\in M_d(\\mathbb Z)$ be invertible modulo $p$. On $M=R^d$ let\n\\[\n G=C_p\\times C_2=\\langle\\sigma,\\tau\\rangle,\n \\qquad \\sigma\\mapsto I+pJ,\\quad \\tau\\mapsto-I.\n\\]\nFor the canonical comparison of \\cref{thm:lifted-input-exchange},\n\\[\n \\Gamma_R(M)\\simeq0,\n \\qquad \\Gamma_k(M\\otimes_R^{\\mathbf L}k)\\simeq0,\n \\qquad \\Cofib(\\beta_{R,k,M})\\simeq0.\n\\]\nNevertheless this representation does not lift to $\\mathbb Z/p^3$, and hence does not lift as a finite free $\\mathbb Z_p$-representation of this fixed group.\n\\end{theorem}\n\\begin{proof}\nThe two displayed matrices commute and satisfy the defining relations. Since $2$ is invertible in $R$, the central orthogonal idempotents\n\\[\n e_+=(1+\\tau)/2,\\qqu","\\begin{theorem}[A central sign hides the obstruction from derived invariants]\\label{thm:blind-exchange}\nLet $p$ be odd, $R=\\mathbb Z/p^2$, and let $J\\in M_d(\\mathbb Z)$ be invertible modulo $p$. On $M=R^d$ let\n\\[\n G=C_p\\times C_2=\\langle\\sigma,\\tau\\rangle,\n \\qquad \\sigma\\mapsto I+pJ,\\quad \\tau\\mapsto-I.\n\\]\nFor the canonical comparison of \\cref{thm:lifted-input-exchange},\n\\[\n \\Gamma_R(M)\\simeq0,\n \\qquad \\Gamma_k(M\\otimes_R^{\\mathbf L}k)\\simeq0,\n \\qquad \\Cofib(\\beta_{R,k,M})\\simeq0.\n\\]\nNevertheless this representation does not lift to $\\mathbb Z/p^3$, and hence does not lift as a finite free $\\mathbb Z_p$-representation of this fixed group.\n\\end{theorem}\n\\begin{proof}\nThe two displayed matrices commute and satisfy the defining relations. Since $2$ is invertible in $R$, the central orthogonal idempotents\n\\[\n e_+=(1+\\tau)/2,\\qquad e_-=(1-\\tau)/2\n\\]\nsplit $R[G","\\begin{theorem}[A central sign hides the obstruction from derived invariants]\\label{thm:blind-exchange}\nLet $p$ be odd, $R=\\mathbb Z/p^2$, and let $J\\in M_d(\\mathbb Z)$ be invertible modulo $p$. On $M=R^d$ let\n\\[\n G=C_p\\times C_2=\\langle\\sigma,\\tau\\rangle,\n \\qquad \\sigma\\mapsto I+pJ,\\quad \\tau\\mapsto-I.\n\\]\nFor the canonical comparison of \\cref{thm:lifted-input-exchange},\n\\[\n \\Gamma_R(M)\\simeq0,\n \\qquad \\Gamma_k(M\\otimes_R^{\\mathbf L}k)\\simeq0,\n \\qquad \\Cofib(\\beta_{R,k,M})\\simeq0.\n\\]\nNevertheless this representation does not lift to $\\mathbb Z/p^3$, and hence does not lift as a finite free $\\mathbb Z_p$-representation of this fixed group.\n\\end{theorem}\n\\begin{proof}\nThe two displayed matrices commute and satisfy the defining relations. Since $2$ is invertible in $R$, the central orthogonal idempotents\n\\[\n e_+=(1+\\tau)/2,\\qquad e_-=(1-\\tau)/2\n\\]\nsplit $R[G]$ into a product of ring","\\begin{theorem}[A central sign hides the obstruction from derived invariants]\\label{thm:blind-exchange}\nLet $p$ be odd, $R=\\mathbb Z/p^2$, and let $J\\in M_d(\\mathbb Z)$ be invertible modulo $p$. On $M=R^d$ let\n\\[\n G=C_p\\times C_2=\\langle\\sigma,\\tau\\rangle,\n \\qquad \\sigma\\mapsto I+pJ,\\quad \\tau\\mapsto-I.\n\\]\nFor the canonical comparison of \\cref{thm:lifted-input-exchange},\n\\[\n \\Gamma_R(M)\\simeq0,\n \\qquad \\Gamma_k(M\\otimes_R^{\\mathbf L}k)\\simeq0,\n \\qquad \\Cofib(\\beta_{R,k,M})\\simeq0.\n\\]\nNevertheless this representation does not lift to $\\mathbb Z/p^3$, and hence does not lift as a finite free $\\mathbb Z_p$-representation of this fixed group.\n\\end{theorem}\n\\begin{proof}\nThe two displayed matrices commute and satisfy the defining relations. Since $2$ is invertible in $R$, the central orthogonal idempotents\n\\[\n e_+=(1+\\tau)/2,\\qquad e_-=(1-\\tau)/2\n\\]\nsplit $R[G]$ into a product of rings. The trivi","\\begin{theorem}[A central sign hides the obstruction from derived invariants]\\label{thm:blind-exchange}\nLet $p$ be odd, $R=\\mathbb Z/p^2$, and let $J\\in M_d(\\mathbb Z)$ be invertible modulo $p$. On $M=R^d$ let\n\\[\n G=C_p\\times C_2=\\langle\\sigma,\\tau\\rangle,\n \\qquad \\sigma\\mapsto I+pJ,\\quad \\tau\\mapsto-I.\n\\]\nFor the canonical comparison of \\cref{thm:lifted-input-exchange},\n\\[\n \\Gamma_R(M)\\simeq0,\n \\qquad \\Gamma_k(M\\otimes_R^{\\mathbf L}k)\\simeq0,\n \\qquad \\Cofib(\\beta_{R,k,M})\\simeq0.\n\\]\nNevertheless this representation does not lift to $\\mathbb Z/p^3$, and hence does not lift as a finite free $\\mathbb Z_p$-representation of this fixed group.\n\\end{theorem}\n\\begin{proof}\nThe two displayed matrices commute and satisfy the defining relations. Since $2$ is invertible in $R$, the central orthogonal idempotents\n\\[\n e_+=(1+\\tau)/2,\\qquad e_-=(1-\\tau)/2\n\\]\nsplit $R[G]$ into a product of rings. The trivial module $R$ is in the plus factor and $M$ in the minus factor. Every derived Hom between the two fa","\\begin{theorem}[A central sign hides the obstruction from derived invariants]\\label{thm:blind-exchange}\nLet $p$ be odd, $R=\\mathbb Z/p^2$, and let $J\\in M_d(\\mathbb Z)$ be invertible modulo $p$. On $M=R^d$ let\n\\[\n G=C_p\\times C_2=\\langle\\sigma,\\tau\\rangle,\n \\qquad \\sigma\\mapsto I+pJ,\\quad \\tau\\mapsto-I.\n\\]\nFor the canonical comparison of \\cref{thm:lifted-input-exchange},\n\\[\n \\Gamma_R(M)\\simeq0,\n \\qquad \\Gamma_k(M\\otimes_R^{\\mathbf L}k)\\simeq0,\n \\qquad \\Cofib(\\beta_{R,k,M})\\simeq0.\n\\]\nNevertheless this representation does not lift to $\\mathbb Z/p^3$, and hence does not lift as a finite free $\\mathbb Z_p$-representation of this fixed group.\n\\end{theorem}\n\\begin{proof}\nThe two displayed matrices commute and satisfy the defining relations. Since $2$ is invertible in $R$, the central orthogonal idempotents\n\\[\n e_+=(1+\\tau)/2,\\qquad e_-=(1-\\tau)/2\n\\]\nsplit $R[G]$ into a product of rings. The trivial module $R$ is in the plus factor and $M$ in the minus factor. Every derived Hom between the two factors is zero, as is seen by resolving within either direct factor. The same decomposition pers…","nd let $J\\in M_d(\\mathbb Z)$ be invertible modulo $p$. On $M=R^d$ let\n\\[\n G=C_p\\times C_2=\\langle\\sigma,\\tau\\rangle,\n \\qquad \\sigma\\mapsto I+pJ,\\quad \\tau\\mapsto-I.\n\\]\nFor the canonical comparison of \\cref{thm:lifted-input-exchange},\n\\[\n \\Gamma_R(M)\\simeq0,\n \\qquad \\Gamma_k(M\\otimes_R^{\\mathbf L}k)\\simeq0,\n \\qquad \\Cofib(\\beta_{R,k,M})\\simeq0.\n\\]\nNevertheless this representation does not lift to $\\mathbb Z/p^3$, and hence does not lift as a finite free $\\mathbb Z_p$-representation of this fixed group.\n\\end{theorem}\n\\begin{proof}\nThe two displayed matrices commute and satisfy the defining relations. Since $2$ is invertible in $R$, the central orthogonal idempotents\n\\[\n e_+=(1+\\tau)/2,\\qquad e_-=(1-\\tau)/2\n\\]\nsplit $R[G]$ into a product of rings. The trivial module $R$ is in the plus factor and $M$ in the minus factor. Every derived Hom between the two factors is zero, as is seen by resolving within either direct factor. The same decomposition persists after reduction to $k$. Since $M$ is free over $R$, its derived reduction is its ordinary reduction. Both endpoints of $\\beta$ therefor…","$M=R^d$ let\n\\[\n G=C_p\\times C_2=\\langle\\sigma,\\tau\\rangle,\n \\qquad \\sigma\\mapsto I+pJ,\\quad \\tau\\mapsto-I.\n\\]\nFor the canonical comparison of \\cref{thm:lifted-input-exchange},\n\\[\n \\Gamma_R(M)\\simeq0,\n \\qquad \\Gamma_k(M\\otimes_R^{\\mathbf L}k)\\simeq0,\n \\qquad \\Cofib(\\beta_{R,k,M})\\simeq0.\n\\]\nNevertheless this representation does not lift to $\\mathbb Z/p^3$, and hence does not lift as a finite free $\\mathbb Z_p$-representation of this fixed group.\n\\end{theorem}\n\\begin{proof}\nThe two displayed matrices commute and satisfy the defining relations. Since $2$ is invertible in $R$, the central orthogonal idempotents\n\\[\n e_+=(1+\\tau)/2,\\qquad e_-=(1-\\tau)/2\n\\]\nsplit $R[G]$ into a product of rings. The trivial module $R$ is in the plus factor and $M$ in the minus factor. Every derived Hom between the two factors is zero, as is seen by resolving within either direct factor. The same decomposition persists after reduction to $k$. Since $M$ is free over $R$, its derived reduction is its ordinary reduction. Both endpoints of $\\beta$ therefore vanish.","ted-input-exchange},\n\\[\n \\Gamma_R(M)\\simeq0,\n \\qquad \\Gamma_k(M\\otimes_R^{\\mathbf L}k)\\simeq0,\n \\qquad \\Cofib(\\beta_{R,k,M})\\simeq0.\n\\]\nNevertheless this representation does not lift to $\\mathbb Z/p^3$, and hence does not lift as a finite free $\\mathbb Z_p$-representation of this fixed group.\n\\end{theorem}\n\\begin{proof}\nThe two displayed matrices commute and satisfy the defining relations. Since $2$ is invertible in $R$, the central orthogonal idempotents\n\\[\n e_+=(1+\\tau)/2,\\qquad e_-=(1-\\tau)/2\n\\]\nsplit $R[G]$ into a product of rings. The trivial module $R$ is in the plus factor and $M$ in the minus factor. Every derived Hom between the two factors is zero, as is seen by resolving within either direct factor. The same decomposition persists after reduction to $k$. Since $M$ is free over $R$, its derived reduction is its ordinary reduction. Both endpoints of $\\beta$ therefore vanish.","\\[\n \\Gamma_R(M)\\simeq0,\n \\qquad \\Gamma_k(M\\otimes_R^{\\mathbf L}k)\\simeq0,\n \\qquad \\Cofib(\\beta_{R,k,M})\\simeq0.\n\\]\nNevertheless this representation does not lift to $\\mathbb Z/p^3$, and hence does not lift as a finite free $\\mathbb Z_p$-representation of this fixed group.\n\\end{theorem}\n\\begin{proof}\nThe two displayed matrices commute and satisfy the defining relations. Since $2$ is invertible in $R$, the central orthogonal idempotents\n\\[\n e_+=(1+\\tau)/2,\\qquad e_-=(1-\\tau)/2\n\\]\nsplit $R[G]$ into a product of rings. The trivial module $R$ is in the plus factor and $M$ in the minus factor. Every derived Hom between the two factors is zero, as is seen by resolving within either direct factor. The same decomposition persists after reduction to $k$. Since $M$ is free over $R$, its derived reduction is its ordinary reduction. Both endpoints of $\\beta$ therefore vanish.","M\\otimes_R^{\\mathbf L}k)\\simeq0,\n \\qquad \\Cofib(\\beta_{R,k,M})\\simeq0.\n\\]\nNevertheless this representation does not lift to $\\mathbb Z/p^3$, and hence does not lift as a finite free $\\mathbb Z_p$-representation of this fixed group.\n\\end{theorem}\n\\begin{proof}\nThe two displayed matrices commute and satisfy the defining relations. Since $2$ is invertible in $R$, the central orthogonal idempotents\n\\[\n e_+=(1+\\tau)/2,\\qquad e_-=(1-\\tau)/2\n\\]\nsplit $R[G]$ into a product of rings. The trivial module $R$ is in the plus factor and $M$ in the minus factor. Every derived Hom between the two factors is zero, as is seen by resolving within either direct factor. The same decomposition persists after reduction to $k$. Since $M$ is free over $R$, its derived reduction is its ordinary reduction. Both endpoints of $\\beta$ therefore vanish.","beta_{R,k,M})\\simeq0.\n\\]\nNevertheless this representation does not lift to $\\mathbb Z/p^3$, and hence does not lift as a finite free $\\mathbb Z_p$-representation of this fixed group.\n\\end{theorem}\n\\begin{proof}\nThe two displayed matrices commute and satisfy the defining relations. Since $2$ is invertible in $R$, the central orthogonal idempotents\n\\[\n e_+=(1+\\tau)/2,\\qquad e_-=(1-\\tau)/2\n\\]\nsplit $R[G]$ into a product of rings. The trivial module $R$ is in the plus factor and $M$ in the minus factor. Every derived Hom between the two factors is zero, as is seen by resolving within either direct factor. The same decomposition persists after reduction to $k$. Since $M$ is free over $R$, its derived reduction is its ordinary reduction. Both endpoints of $\\beta$ therefore vanish.","tation does not lift to $\\mathbb Z/p^3$, and hence does not lift as a finite free $\\mathbb Z_p$-representation of this fixed group.\n\\end{theorem}\n\\begin{proof}\nThe two displayed matrices commute and satisfy the defining relations. Since $2$ is invertible in $R$, the central orthogonal idempotents\n\\[\n e_+=(1+\\tau)/2,\\qquad e_-=(1-\\tau)/2\n\\]\nsplit $R[G]$ into a product of rings. The trivial module $R$ is in the plus factor and $M$ in the minus factor. Every derived Hom between the two factors is zero, as is seen by resolving within either direct factor. The same decomposition persists after reduction to $k$. Since $M$ is free over $R$, its derived reduction is its ordinary reduction. Both endpoints of $\\beta$ therefore vanish.","bb Z/p^3$, and hence does not lift as a finite free $\\mathbb Z_p$-representation of this fixed group.\n\\end{theorem}\n\\begin{proof}\nThe two displayed matrices commute and satisfy the defining relations. Since $2$ is invertible in $R$, the central orthogonal idempotents\n\\[\n e_+=(1+\\tau)/2,\\qquad e_-=(1-\\tau)/2\n\\]\nsplit $R[G]$ into a product of rings. The trivial module $R$ is in the plus factor and $M$ in the minus factor. Every derived Hom between the two factors is zero, as is seen by resolving within either direct factor. The same decomposition persists after reduction to $k$. Since $M$ is free over $R$, its derived reduction is its ordinary reduction. Both endpoints of $\\beta$ therefore vanish.","defining relations. Since $2$ is invertible in $R$, the central orthogonal idempotents\n\\[\n e_+=(1+\\tau)/2,\\qquad e_-=(1-\\tau)/2\n\\]\nsplit $R[G]$ into a product of rings. The trivial module $R$ is in the plus factor and $M$ in the minus factor. Every derived Hom between the two factors is zero, as is seen by resolving within either direct factor. The same decomposition persists after reduction to $k$. Since $M$ is free over $R$, its derived reduction is its ordinary reduction. Both endpoints of $\\beta$ therefore vanish.","elations. Since $2$ is invertible in $R$, the central orthogonal idempotents\n\\[\n e_+=(1+\\tau)/2,\\qquad e_-=(1-\\tau)/2\n\\]\nsplit $R[G]$ into a product of rings. The trivial module $R$ is in the plus factor and $M$ in the minus factor. Every derived Hom between the two factors is zero, as is seen by resolving within either direct factor. The same decomposition persists after reduction to $k$. Since $M$ is free over $R$, its derived reduction is its ordinary reduction. Both endpoints of $\\beta$ therefore vanish.","2$ is invertible in $R$, the central orthogonal idempotents\n\\[\n e_+=(1+\\tau)/2,\\qquad e_-=(1-\\tau)/2\n\\]\nsplit $R[G]$ into a product of rings. The trivial module $R$ is in the plus factor and $M$ in the minus factor. Every derived Hom between the two factors is zero, as is seen by resolving within either direct factor. The same decomposition persists after reduction to $k$. Since $M$ is free over $R$, its derived reduction is its ordinary reduction. Both endpoints of $\\beta$ therefore vanish.","tau)/2,\\qquad e_-=(1-\\tau)/2\n\\]\nsplit $R[G]$ into a product of rings. The trivial module $R$ is in the plus factor and $M$ in the minus factor. Every derived Hom between the two factors is zero, as is seen by resolving within either direct factor. The same decomposition persists after reduction to $k$. Since $M$ is free over $R$, its derived reduction is its ordinary reduction. Both endpoints of $\\beta$ therefore vanish.","Any lift of this $G$-action would restrict to a lift of its $C_p$-action. By \\cref{thm:one-step-group-lift}, each lift of $I+pJ$ has $p$th power $I+p^2J\\ne I$ modulo $p^3$. This is impossible.\n\\end{proof}","The example refutes sufficiency of the single $\\Gamma$-comparison test in the finite-group class. It does not merely leave the converse unproved. The loss is a loss of detection: a functor can see none of an input while the input still has inconsistent lifting relations. In contrast, conjugation by $-I$ is trivial on $\\operatorname{End}_k(k^d)$, so the adjoint relation obstruction is not hidden by this sign. Twisting by the integral sign character also restores the earlier nonzero exchange example. Neither observation asserts that a finite family of such probe comparisons classifies all lifting obstructions.","Let $N=|G|$ and write $e=v_p(N)$. For a matrix $X$ over $\\mathbb Q_p$, the assertion $v_p(X)\\geq a$ means every entry lies in $p^a\\mathbb Z_p$. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let $A:G\\to\\operatorname{GL}_d(\\mathbb Z_p)$ be a normalized map, not necessarily a homomorphism. Define\n\\[\n E(g,h)=A_gA_hA_{gh}^{-1}-I,\n \\qquad \\alpha_g(X)=A_gXA_g^{-1}.\n\\]\nSuppose $E(g,h)\\in p^mM_d(\\mathbb Z_p)$ for every pair. The operators $\\alpha_g$ are not assumed to give a group action. This distinction is essential before the repair has succeeded.","=v_p(N)$. For a matrix $X$ over $\\mathbb Q_p$, the assertion $v_p(X)\\geq a$ means every entry lies in $p^a\\mathbb Z_p$. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let $A:G\\to\\operatorname{GL}_d(\\mathbb Z_p)$ be a normalized map, not necessarily a homomorphism. Define\n\\[\n E(g,h)=A_gA_hA_{gh}^{-1}-I,\n \\qquad \\alpha_g(X)=A_gXA_g^{-1}.\n\\]\nSuppose $E(g,h)\\in p^mM_d(\\mathbb Z_p)$ for every pair. The operators $\\alpha_g$ are not assumed to give a group action. This distinction is essential before the repair has succeeded.","v_p(X)\\geq a$ means every entry lies in $p^a\\mathbb Z_p$. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let $A:G\\to\\operatorname{GL}_d(\\mathbb Z_p)$ be a normalized map, not necessarily a homomorphism. Define\n\\[\n E(g,h)=A_gA_hA_{gh}^{-1}-I,\n \\qquad \\alpha_g(X)=A_gXA_g^{-1}.\n\\]\nSuppose $E(g,h)\\in p^mM_d(\\mathbb Z_p)$ for every pair. The operators $\\alpha_g$ are not assumed to give a group action. This distinction is essential before the repair has succeeded.","\\begin{lemma}[Averaged correction with controlled precision loss]\\label{lem:quadratic-repair}\nIf $m>e$, define in $M_d(\\mathbb Q_p)$\n\\[\n b(g)=\\frac1N\\sum_{t\\in G}E(g,t),\\qquad\n A'_g=(I-b(g))A_g.\n\\]\nThen $b(g)\\in p^{m-e}M_d(\\mathbb Z_p)$, $b(1)=0$, and $A'$ is a normalized map into $\\operatorname{GL}_d(\\mathbb Z_p)$. Its defect satisfies\n\\[\n \\boxed{A'_gA'_h(A'_{gh})^{-1}-I\\in\n p^{2m-2e}M_d(\\mathbb Z_p).}\n\\]\nAlso $A'_g-A_g\\in p^{m-e}M_d(\\mathbb Z_p)$.\n\\end{lemma}\n\\begin{proof}\nDivision by $N=p^eu$, $u\\in\\mathbb Z_p^\\times$, loses at most $e$ digits, proving the bound on $b$. Normalization gives $b(1)=0$. Because $m-e\\geq1$, each $I-b(g)$ is invertible.","eraged correction with controlled precision loss]\\label{lem:quadratic-repair}\nIf $m>e$, define in $M_d(\\mathbb Q_p)$\n\\[\n b(g)=\\frac1N\\sum_{t\\in G}E(g,t),\\qquad\n A'_g=(I-b(g))A_g.\n\\]\nThen $b(g)\\in p^{m-e}M_d(\\mathbb Z_p)$, $b(1)=0$, and $A'$ is a normalized map into $\\operatorname{GL}_d(\\mathbb Z_p)$. Its defect satisfies\n\\[\n \\boxed{A'_gA'_h(A'_{gh})^{-1}-I\\in\n p^{2m-2e}M_d(\\mathbb Z_p).}\n\\]\nAlso $A'_g-A_g\\in p^{m-e}M_d(\\mathbb Z_p)$.\n\\end{lemma}\n\\begin{proof}\nDivision by $N=p^eu$, $u\\in\\mathbb Z_p^\\times$, loses at most $e$ digits, proving the bound on $b$. Normalization gives $b(1)=0$. Because $m-e\\geq1$, each $I-b(g)$ is invertible.","If $m>e$, define in $M_d(\\mathbb Q_p)$\n\\[\n b(g)=\\frac1N\\sum_{t\\in G}E(g,t),\\qquad\n A'_g=(I-b(g))A_g.\n\\]\nThen $b(g)\\in p^{m-e}M_d(\\mathbb Z_p)$, $b(1)=0$, and $A'$ is a normalized map into $\\operatorname{GL}_d(\\mathbb Z_p)$. Its defect satisfies\n\\[\n \\boxed{A'_gA'_h(A'_{gh})^{-1}-I\\in\n p^{2m-2e}M_d(\\mathbb Z_p).}\n\\]\nAlso $A'_g-A_g\\in p^{m-e}M_d(\\mathbb Z_p)$.\n\\end{lemma}\n\\begin{proof}\nDivision by $N=p^eu$, $u\\in\\mathbb Z_p^\\times$, loses at most $e$ digits, proving the bound on $b$. Normalization gives $b(1)=0$. Because $m-e\\geq1$, each $I-b(g)$ is invertible.","define in $M_d(\\mathbb Q_p)$\n\\[\n b(g)=\\frac1N\\sum_{t\\in G}E(g,t),\\qquad\n A'_g=(I-b(g))A_g.\n\\]\nThen $b(g)\\in p^{m-e}M_d(\\mathbb Z_p)$, $b(1)=0$, and $A'$ is a normalized map into $\\operatorname{GL}_d(\\mathbb Z_p)$. Its defect satisfies\n\\[\n \\boxed{A'_gA'_h(A'_{gh})^{-1}-I\\in\n p^{2m-2e}M_d(\\mathbb Z_p).}\n\\]\nAlso $A'_g-A_g\\in p^{m-e}M_d(\\mathbb Z_p)$.\n\\end{lemma}\n\\begin{proof}\nDivision by $N=p^eu$, $u\\in\\mathbb Z_p^\\times$, loses at most $e$ digits, proving the bound on $b$. Normalization gives $b(1)=0$. Because $m-e\\geq1$, each $I-b(g)$ is invertible.","frac1N\\sum_{t\\in G}E(g,t),\\qquad\n A'_g=(I-b(g))A_g.\n\\]\nThen $b(g)\\in p^{m-e}M_d(\\mathbb Z_p)$, $b(1)=0$, and $A'$ is a normalized map into $\\operatorname{GL}_d(\\mathbb Z_p)$. Its defect satisfies\n\\[\n \\boxed{A'_gA'_h(A'_{gh})^{-1}-I\\in\n p^{2m-2e}M_d(\\mathbb Z_p).}\n\\]\nAlso $A'_g-A_g\\in p^{m-e}M_d(\\mathbb Z_p)$.\n\\end{lemma}\n\\begin{proof}\nDivision by $N=p^eu$, $u\\in\\mathbb Z_p^\\times$, loses at most $e$ digits, proving the bound on $b$. Normalization gives $b(1)=0$. Because $m-e\\geq1$, each $I-b(g)$ is invertible.","A'_g=(I-b(g))A_g.\n\\]\nThen $b(g)\\in p^{m-e}M_d(\\mathbb Z_p)$, $b(1)=0$, and $A'$ is a normalized map into $\\operatorname{GL}_d(\\mathbb Z_p)$. Its defect satisfies\n\\[\n \\boxed{A'_gA'_h(A'_{gh})^{-1}-I\\in\n p^{2m-2e}M_d(\\mathbb Z_p).}\n\\]\nAlso $A'_g-A_g\\in p^{m-e}M_d(\\mathbb Z_p)$.\n\\end{lemma}\n\\begin{proof}\nDivision by $N=p^eu$, $u\\in\\mathbb Z_p^\\times$, loses at most $e$ digits, proving the bound on $b$. Normalization gives $b(1)=0$. Because $m-e\\geq1$, each $I-b(g)$ is invertible.","n $b(g)\\in p^{m-e}M_d(\\mathbb Z_p)$, $b(1)=0$, and $A'$ is a normalized map into $\\operatorname{GL}_d(\\mathbb Z_p)$. Its defect satisfies\n\\[\n \\boxed{A'_gA'_h(A'_{gh})^{-1}-I\\in\n p^{2m-2e}M_d(\\mathbb Z_p).}\n\\]\nAlso $A'_g-A_g\\in p^{m-e}M_d(\\mathbb Z_p)$.\n\\end{lemma}\n\\begin{proof}\nDivision by $N=p^eu$, $u\\in\\mathbb Z_p^\\times$, loses at most $e$ digits, proving the bound on $b$. Normalization gives $b(1)=0$. Because $m-e\\geq1$, each $I-b(g)$ is invertible.","M_d(\\mathbb Z_p)$, $b(1)=0$, and $A'$ is a normalized map into $\\operatorname{GL}_d(\\mathbb Z_p)$. Its defect satisfies\n\\[\n \\boxed{A'_gA'_h(A'_{gh})^{-1}-I\\in\n p^{2m-2e}M_d(\\mathbb Z_p).}\n\\]\nAlso $A'_g-A_g\\in p^{m-e}M_d(\\mathbb Z_p)$.\n\\end{lemma}\n\\begin{proof}\nDivision by $N=p^eu$, $u\\in\\mathbb Z_p^\\times$, loses at most $e$ digits, proving the bound on $b$. Normalization gives $b(1)=0$. Because $m-e\\geq1$, each $I-b(g)$ is invertible.",", $b(1)=0$, and $A'$ is a normalized map into $\\operatorname{GL}_d(\\mathbb Z_p)$. Its defect satisfies\n\\[\n \\boxed{A'_gA'_h(A'_{gh})^{-1}-I\\in\n p^{2m-2e}M_d(\\mathbb Z_p).}\n\\]\nAlso $A'_g-A_g\\in p^{m-e}M_d(\\mathbb Z_p)$.\n\\end{lemma}\n\\begin{proof}\nDivision by $N=p^eu$, $u\\in\\mathbb Z_p^\\times$, loses at most $e$ digits, proving the bound on $b$. Normalization gives $b(1)=0$. Because $m-e\\geq1$, each $I-b(g)$ is invertible.","Associativity gives an identity without assuming an exact adjoint action:\n\\[\n (I+E(g,h))(I+E(gh,t))\n = (I+\\alpha_g E(h,t))(I+E(g,ht)).\n\\]\nConsequently\n\\[\n E(g,h)+E(gh,t)-\\alpha_g E(h,t)-E(g,ht)\n \\in p^{2m}M_d(\\mathbb Z_p).\n\\]\nSum over $t$ and divide by $N$. The permutation $t\\mapsto ht$ identifies the last average with $b(g)$, giving\n\\[\n E(g,h)-b(g)-\\alpha_g b(h)+b(gh)\n \\in p^{2m-e}M_d(\\mathbb Z_p).\n\\]\nPut $a=m-e>0$. The repaired multiplicativity product is exactly\n\\[\n (I-b(g))(I-\\alpha_g b(h))(I+E(g,h))(I-b(gh))^{-1}.\n\\]\nModulo $p^{2a}$ its difference from $I$ is the linear expression just displayed: products of two corrections vanish at that precision, and $(I-b)^{-1}\\equiv I+b\\pmod{p^{2a}}$. That expression is divisible by $p^{2m-e}$, which is contained in $p^{2a}$. This proves the new defect bound. The change in $A_g$ i","Associativity gives an identity without assuming an exact adjoint action:\n\\[\n (I+E(g,h))(I+E(gh,t))\n = (I+\\alpha_g E(h,t))(I+E(g,ht)).\n\\]\nConsequently\n\\[\n E(g,h)+E(gh,t)-\\alpha_g E(h,t)-E(g,ht)\n \\in p^{2m}M_d(\\mathbb Z_p).\n\\]\nSum over $t$ and divide by $N$. The permutation $t\\mapsto ht$ identifies the last average with $b(g)$, giving\n\\[\n E(g,h)-b(g)-\\alpha_g b(h)+b(gh)\n \\in p^{2m-e}M_d(\\mathbb Z_p).\n\\]\nPut $a=m-e>0$. The repaired multiplicativity product is exactly\n\\[\n (I-b(g))(I-\\alpha_g b(h))(I+E(g,h))(I-b(gh))^{-1}.\n\\]\nModulo $p^{2a}$ its difference from $I$ is the linear expression just displayed: products of two corrections vanish at that precision, and $(I-b)^{-1}\\equiv I+b\\pmod{p^{2a}}$. That expression is divisible by $p^{2m-e}$, which is contained in $p^{2a}$. This proves the new defect bound. The change in $A_g$ is $-b(g)A_g$ and has the asserted precision.\n\\end{proof}","ty gives an identity without assuming an exact adjoint action:\n\\[\n (I+E(g,h))(I+E(gh,t))\n = (I+\\alpha_g E(h,t))(I+E(g,ht)).\n\\]\nConsequently\n\\[\n E(g,h)+E(gh,t)-\\alpha_g E(h,t)-E(g,ht)\n \\in p^{2m}M_d(\\mathbb Z_p).\n\\]\nSum over $t$ and divide by $N$. The permutation $t\\mapsto ht$ identifies the last average with $b(g)$, giving\n\\[\n E(g,h)-b(g)-\\alpha_g b(h)+b(gh)\n \\in p^{2m-e}M_d(\\mathbb Z_p).\n\\]\nPut $a=m-e>0$. The repaired multiplicativity product is exactly\n\\[\n (I-b(g))(I-\\alpha_g b(h))(I+E(g,h))(I-b(gh))^{-1}.\n\\]\nModulo $p^{2a}$ its difference from $I$ is the linear expression just displayed: products of two corrections vanish at that precision, and $(I-b)^{-1}\\equiv I+b\\pmod{p^{2a}}$. That expression is divisible by $p^{2m-e}$, which is contained in $p^{2a}$. This proves the new defect bound. The change in $A_g$ is $-b(g)A_g$ and has the asserted precision.\n\\end{proof}",":\n\\[\n (I+E(g,h))(I+E(gh,t))\n = (I+\\alpha_g E(h,t))(I+E(g,ht)).\n\\]\nConsequently\n\\[\n E(g,h)+E(gh,t)-\\alpha_g E(h,t)-E(g,ht)\n \\in p^{2m}M_d(\\mathbb Z_p).\n\\]\nSum over $t$ and divide by $N$. The permutation $t\\mapsto ht$ identifies the last average with $b(g)$, giving\n\\[\n E(g,h)-b(g)-\\alpha_g b(h)+b(gh)\n \\in p^{2m-e}M_d(\\mathbb Z_p).\n\\]\nPut $a=m-e>0$. The repaired multiplicativity product is exactly\n\\[\n (I-b(g))(I-\\alpha_g b(h))(I+E(g,h))(I-b(gh))^{-1}.\n\\]\nModulo $p^{2a}$ its difference from $I$ is the linear expression just displayed: products of two corrections vanish at that precision, and $(I-b)^{-1}\\equiv I+b\\pmod{p^{2a}}$. That expression is divisible by $p^{2m-e}$, which is contained in $p^{2a}$. This proves the new defect bound. The change in $A_g$ is $-b(g)A_g$ and has the asserted precision.\n\\end{proof}","]\nConsequently\n\\[\n E(g,h)+E(gh,t)-\\alpha_g E(h,t)-E(g,ht)\n \\in p^{2m}M_d(\\mathbb Z_p).\n\\]\nSum over $t$ and divide by $N$. The permutation $t\\mapsto ht$ identifies the last average with $b(g)$, giving\n\\[\n E(g,h)-b(g)-\\alpha_g b(h)+b(gh)\n \\in p^{2m-e}M_d(\\mathbb Z_p).\n\\]\nPut $a=m-e>0$. The repaired multiplicativity product is exactly\n\\[\n (I-b(g))(I-\\alpha_g b(h))(I+E(g,h))(I-b(gh))^{-1}.\n\\]\nModulo $p^{2a}$ its difference from $I$ is the linear expression just displayed: products of two corrections vanish at that precision, and $(I-b)^{-1}\\equiv I+b\\pmod{p^{2a}}$. That expression is divisible by $p^{2m-e}$, which is contained in $p^{2a}$. This proves the new defect bound. The change in $A_g$ is $-b(g)A_g$ and has the asserted precision.\n\\end{proof}","h,t)-\\alpha_g E(h,t)-E(g,ht)\n \\in p^{2m}M_d(\\mathbb Z_p).\n\\]\nSum over $t$ and divide by $N$. The permutation $t\\mapsto ht$ identifies the last average with $b(g)$, giving\n\\[\n E(g,h)-b(g)-\\alpha_g b(h)+b(gh)\n \\in p^{2m-e}M_d(\\mathbb Z_p).\n\\]\nPut $a=m-e>0$. The repaired multiplicativity product is exactly\n\\[\n (I-b(g))(I-\\alpha_g b(h))(I+E(g,h))(I-b(gh))^{-1}.\n\\]\nModulo $p^{2a}$ its difference from $I$ is the linear expression just displayed: products of two corrections vanish at that precision, and $(I-b)^{-1}\\equiv I+b\\pmod{p^{2a}}$. That expression is divisible by $p^{2m-e}$, which is contained in $p^{2a}$. This proves the new defect bound. The change in $A_g$ is $-b(g)A_g$ and has the asserted precision.\n\\end{proof}","ation $t\\mapsto ht$ identifies the last average with $b(g)$, giving\n\\[\n E(g,h)-b(g)-\\alpha_g b(h)+b(gh)\n \\in p^{2m-e}M_d(\\mathbb Z_p).\n\\]\nPut $a=m-e>0$. The repaired multiplicativity product is exactly\n\\[\n (I-b(g))(I-\\alpha_g b(h))(I+E(g,h))(I-b(gh))^{-1}.\n\\]\nModulo $p^{2a}$ its difference from $I$ is the linear expression just displayed: products of two corrections vanish at that precision, and $(I-b)^{-1}\\equiv I+b\\pmod{p^{2a}}$. That expression is divisible by $p^{2m-e}$, which is contained in $p^{2a}$. This proves the new defect bound. The change in $A_g$ is $-b(g)A_g$ and has the asserted precision.\n\\end{proof}","[\n E(g,h)-b(g)-\\alpha_g b(h)+b(gh)\n \\in p^{2m-e}M_d(\\mathbb Z_p).\n\\]\nPut $a=m-e>0$. The repaired multiplicativity product is exactly\n\\[\n (I-b(g))(I-\\alpha_g b(h))(I+E(g,h))(I-b(gh))^{-1}.\n\\]\nModulo $p^{2a}$ its difference from $I$ is the linear expression just displayed: products of two corrections vanish at that precision, and $(I-b)^{-1}\\equiv I+b\\pmod{p^{2a}}$. That expression is divisible by $p^{2m-e}$, which is contained in $p^{2a}$. This proves the new defect bound. The change in $A_g$ is $-b(g)A_g$ and has the asserted precision.\n\\end{proof}","\\in p^{2m-e}M_d(\\mathbb Z_p).\n\\]\nPut $a=m-e>0$. The repaired multiplicativity product is exactly\n\\[\n (I-b(g))(I-\\alpha_g b(h))(I+E(g,h))(I-b(gh))^{-1}.\n\\]\nModulo $p^{2a}$ its difference from $I$ is the linear expression just displayed: products of two corrections vanish at that precision, and $(I-b)^{-1}\\equiv I+b\\pmod{p^{2a}}$. That expression is divisible by $p^{2m-e}$, which is contained in $p^{2a}$. This proves the new defect bound. The change in $A_g$ is $-b(g)A_g$ and has the asserted precision.\n\\end{proof}","ired multiplicativity product is exactly\n\\[\n (I-b(g))(I-\\alpha_g b(h))(I+E(g,h))(I-b(gh))^{-1}.\n\\]\nModulo $p^{2a}$ its difference from $I$ is the linear expression just displayed: products of two corrections vanish at that precision, and $(I-b)^{-1}\\equiv I+b\\pmod{p^{2a}}$. That expression is divisible by $p^{2m-e}$, which is contained in $p^{2a}$. This proves the new defect bound. The change in $A_g$ is $-b(g)A_g$ and has the asserted precision.\n\\end{proof}","iplicativity product is exactly\n\\[\n (I-b(g))(I-\\alpha_g b(h))(I+E(g,h))(I-b(gh))^{-1}.\n\\]\nModulo $p^{2a}$ its difference from $I$ is the linear expression just displayed: products of two corrections vanish at that precision, and $(I-b)^{-1}\\equiv I+b\\pmod{p^{2a}}$. That expression is divisible by $p^{2m-e}$, which is contained in $p^{2a}$. This proves the new defect bound. The change in $A_g$ is $-b(g)A_g$ and has the asserted precision.\n\\end{proof}","\\begin{theorem}[Convergent finite-group action repair]\\label{thm:convergent-action-repair}\nIf the initial defect depth satisfies $m>2e$, the preceding averaging operation iterates to a representation\n\\[\n \\rho:G\\longrightarrow\\operatorname{GL}_d(\\mathbb Z_p)\n \\quad\\text{with}\\quad\n \\rho(g)\\equiv A_g\\pmod{p^{m-e}}.\n\\]\nAfter $j$ iterations the certified defect depth is\n\\[\n m_j=2e+2^j(m-2e),\n\\]\nand the next change has depth at least $m_j-e$. In particular, the sequence converges uniformly on $G$ and all prescribed multiplication relations hold exactly in the limit.\n\\end{theorem}\n\\begin{proof}\nSet $m_0=m$ and $m_{j+1}=2m_j-2e$. Induction gives the displayed formula, so $m_j>2e$ at every stage and $m_j\\to\\infty$. The correction lemma applies repeatedly. Differences of consecutive matrices have entry valuations tending to infinit","\\begin{theorem}[Convergent finite-group action repair]\\label{thm:convergent-action-repair}\nIf the initial defect depth satisfies $m>2e$, the preceding averaging operation iterates to a representation\n\\[\n \\rho:G\\longrightarrow\\operatorname{GL}_d(\\mathbb Z_p)\n \\quad\\text{with}\\quad\n \\rho(g)\\equiv A_g\\pmod{p^{m-e}}.\n\\]\nAfter $j$ iterations the certified defect depth is\n\\[\n m_j=2e+2^j(m-2e),\n\\]\nand the next change has depth at least $m_j-e$. In particular, the sequence converges uniformly on $G$ and all prescribed multiplication relations hold exactly in the limit.\n\\end{theorem}\n\\begin{proof}\nSet $m_0=m$ and $m_{j+1}=2m_j-2e$. Induction gives the displayed formula, so $m_j>2e$ at every stage and $m_j\\to\\infty$. The correction lemma applies repeatedly. Differences of consecutive matrices have entry valuations tending to infinity, hence their telescoping sums converge in the complete ring $M_d(\\mathbb Z_p)$. Every correction preserves reduction modulo $p$, so the limits remain invertible. Their reduction m","\\begin{theorem}[Convergent finite-group action repair]\\label{thm:convergent-action-repair}\nIf the initial defect depth satisfies $m>2e$, the preceding averaging operation iterates to a representation\n\\[\n \\rho:G\\longrightarrow\\operatorname{GL}_d(\\mathbb Z_p)\n \\quad\\text{with}\\quad\n \\rho(g)\\equiv A_g\\pmod{p^{m-e}}.\n\\]\nAfter $j$ iterations the certified defect depth is\n\\[\n m_j=2e+2^j(m-2e),\n\\]\nand the next change has depth at least $m_j-e$. In particular, the sequence converges uniformly on $G$ and all prescribed multiplication relations hold exactly in the limit.\n\\end{theorem}\n\\begin{proof}\nSet $m_0=m$ and $m_{j+1}=2m_j-2e$. Induction gives the displayed formula, so $m_j>2e$ at every stage and $m_j\\to\\infty$. The correction lemma applies repeatedly. Differences of consecutive matrices have entry valuations tending to infinity, hence their telescoping sums converge in the complete ring $M_d(\\mathbb Z_p)$. Every correction preserves reduction modulo $p$, so the limits remain invertible. Their reduction modulo $p^{m","\\begin{theorem}[Convergent finite-group action repair]\\label{thm:convergent-action-repair}\nIf the initial defect depth satisfies $m>2e$, the preceding averaging operation iterates to a representation\n\\[\n \\rho:G\\longrightarrow\\operatorname{GL}_d(\\mathbb Z_p)\n \\quad\\text{with}\\quad\n \\rho(g)\\equiv A_g\\pmod{p^{m-e}}.\n\\]\nAfter $j$ iterations the certified defect depth is\n\\[\n m_j=2e+2^j(m-2e),\n\\]\nand the next change has depth at least $m_j-e$. In particular, the sequence converges uniformly on $G$ and all prescribed multiplication relations hold exactly in the limit.\n\\end{theorem}\n\\begin{proof}\nSet $m_0=m$ and $m_{j+1}=2m_j-2e$. Induction gives the displayed formula, so $m_j>2e$ at every stage and $m_j\\to\\infty$. The correction lemma applies repeatedly. Differences of consecutive matrices have entry valuations tending to infinity, hence their telescoping sums converge in the complete ring $M_d(\\mathbb Z_p)$. Every correction preserves reduction modulo $p$, so the limits remain invertible. Their reduction modulo $p^{m-e}$ is the original one.","group action repair]\\label{thm:convergent-action-repair}\nIf the initial defect depth satisfies $m>2e$, the preceding averaging operation iterates to a representation\n\\[\n \\rho:G\\longrightarrow\\operatorname{GL}_d(\\mathbb Z_p)\n \\quad\\text{with}\\quad\n \\rho(g)\\equiv A_g\\pmod{p^{m-e}}.\n\\]\nAfter $j$ iterations the certified defect depth is\n\\[\n m_j=2e+2^j(m-2e),\n\\]\nand the next change has depth at least $m_j-e$. In particular, the sequence converges uniformly on $G$ and all prescribed multiplication relations hold exactly in the limit.\n\\end{theorem}\n\\begin{proof}\nSet $m_0=m$ and $m_{j+1}=2m_j-2e$. Induction gives the displayed formula, so $m_j>2e$ at every stage and $m_j\\to\\infty$. The correction lemma applies repeatedly. Differences of consecutive matrices have entry valuations tending to infinity, hence their telescoping sums converge in the complete ring $M_d(\\mathbb Z_p)$. Every correction preserves reduction modulo $p$, so the limits remain invertible. Their reduction modulo $p^{m-e}$ is the original one.","the initial defect depth satisfies $m>2e$, the preceding averaging operation iterates to a representation\n\\[\n \\rho:G\\longrightarrow\\operatorname{GL}_d(\\mathbb Z_p)\n \\quad\\text{with}\\quad\n \\rho(g)\\equiv A_g\\pmod{p^{m-e}}.\n\\]\nAfter $j$ iterations the certified defect depth is\n\\[\n m_j=2e+2^j(m-2e),\n\\]\nand the next change has depth at least $m_j-e$. In particular, the sequence converges uniformly on $G$ and all prescribed multiplication relations hold exactly in the limit.\n\\end{theorem}\n\\begin{proof}\nSet $m_0=m$ and $m_{j+1}=2m_j-2e$. Induction gives the displayed formula, so $m_j>2e$ at every stage and $m_j\\to\\infty$. The correction lemma applies repeatedly. Differences of consecutive matrices have entry valuations tending to infinity, hence their telescoping sums converge in the complete ring $M_d(\\mathbb Z_p)$. Every correction preserves reduction modulo $p$, so the limits remain invertible. Their reduction modulo $p^{m-e}$ is the original one.","[\n \\rho:G\\longrightarrow\\operatorname{GL}_d(\\mathbb Z_p)\n \\quad\\text{with}\\quad\n \\rho(g)\\equiv A_g\\pmod{p^{m-e}}.\n\\]\nAfter $j$ iterations the certified defect depth is\n\\[\n m_j=2e+2^j(m-2e),\n\\]\nand the next change has depth at least $m_j-e$. In particular, the sequence converges uniformly on $G$ and all prescribed multiplication relations hold exactly in the limit.\n\\end{theorem}\n\\begin{proof}\nSet $m_0=m$ and $m_{j+1}=2m_j-2e$. Induction gives the displayed formula, so $m_j>2e$ at every stage and $m_j\\to\\infty$. The correction lemma applies repeatedly. Differences of consecutive matrices have entry valuations tending to infinity, hence their telescoping sums converge in the complete ring $M_d(\\mathbb Z_p)$. Every correction preserves reduction modulo $p$, so the limits remain invertible. Their reduction modulo $p^{m-e}$ is the original one.","ngrightarrow\\operatorname{GL}_d(\\mathbb Z_p)\n \\quad\\text{with}\\quad\n \\rho(g)\\equiv A_g\\pmod{p^{m-e}}.\n\\]\nAfter $j$ iterations the certified defect depth is\n\\[\n m_j=2e+2^j(m-2e),\n\\]\nand the next change has depth at least $m_j-e$. In particular, the sequence converges uniformly on $G$ and all prescribed multiplication relations hold exactly in the limit.\n\\end{theorem}\n\\begin{proof}\nSet $m_0=m$ and $m_{j+1}=2m_j-2e$. Induction gives the displayed formula, so $m_j>2e$ at every stage and $m_j\\to\\infty$. The correction lemma applies repeatedly. Differences of consecutive matrices have entry valuations tending to infinity, hence their telescoping sums converge in the complete ring $M_d(\\mathbb Z_p)$. Every correction preserves reduction modulo $p$, so the limits remain invertible. Their reduction modulo $p^{m-e}$ is the original one.","}\\quad\n \\rho(g)\\equiv A_g\\pmod{p^{m-e}}.\n\\]\nAfter $j$ iterations the certified defect depth is\n\\[\n m_j=2e+2^j(m-2e),\n\\]\nand the next change has depth at least $m_j-e$. In particular, the sequence converges uniformly on $G$ and all prescribed multiplication relations hold exactly in the limit.\n\\end{theorem}\n\\begin{proof}\nSet $m_0=m$ and $m_{j+1}=2m_j-2e$. Induction gives the displayed formula, so $m_j>2e$ at every stage and $m_j\\to\\infty$. The correction lemma applies repeatedly. Differences of consecutive matrices have entry valuations tending to infinity, hence their telescoping sums converge in the complete ring $M_d(\\mathbb Z_p)$. Every correction preserves reduction modulo $p$, so the limits remain invertible. Their reduction modulo $p^{m-e}$ is the original one.","od{p^{m-e}}.\n\\]\nAfter $j$ iterations the certified defect depth is\n\\[\n m_j=2e+2^j(m-2e),\n\\]\nand the next change has depth at least $m_j-e$. In particular, the sequence converges uniformly on $G$ and all prescribed multiplication relations hold exactly in the limit.\n\\end{theorem}\n\\begin{proof}\nSet $m_0=m$ and $m_{j+1}=2m_j-2e$. Induction gives the displayed formula, so $m_j>2e$ at every stage and $m_j\\to\\infty$. The correction lemma applies repeatedly. Differences of consecutive matrices have entry valuations tending to infinity, hence their telescoping sums converge in the complete ring $M_d(\\mathbb Z_p)$. Every correction preserves reduction modulo $p$, so the limits remain invertible. Their reduction modulo $p^{m-e}$ is the original one.","and all prescribed multiplication relations hold exactly in the limit.\n\\end{theorem}\n\\begin{proof}\nSet $m_0=m$ and $m_{j+1}=2m_j-2e$. Induction gives the displayed formula, so $m_j>2e$ at every stage and $m_j\\to\\infty$. The correction lemma applies repeatedly. Differences of consecutive matrices have entry valuations tending to infinity, hence their telescoping sums converge in the complete ring $M_d(\\mathbb Z_p)$. Every correction preserves reduction modulo $p$, so the limits remain invertible. Their reduction modulo $p^{m-e}$ is the original one.","limit.\n\\end{theorem}\n\\begin{proof}\nSet $m_0=m$ and $m_{j+1}=2m_j-2e$. Induction gives the displayed formula, so $m_j>2e$ at every stage and $m_j\\to\\infty$. The correction lemma applies repeatedly. Differences of consecutive matrices have entry valuations tending to infinity, hence their telescoping sums converge in the complete ring $M_d(\\mathbb Z_p)$. Every correction preserves reduction modulo $p$, so the limits remain invertible. Their reduction modulo $p^{m-e}$ is the original one.","m_j-2e$. Induction gives the displayed formula, so $m_j>2e$ at every stage and $m_j\\to\\infty$. The correction lemma applies repeatedly. Differences of consecutive matrices have entry valuations tending to infinity, hence their telescoping sums converge in the complete ring $M_d(\\mathbb Z_p)$. Every correction preserves reduction modulo $p$, so the limits remain invertible. Their reduction modulo $p^{m-e}$ is the original one.","Matrix multiplication and inversion on $\\operatorname{GL}_d(\\mathbb Z_p)$ are continuous. The defects converge to zero because their depths are at least $m_j$. Passing to the limit gives $\\rho(g)\\rho(h)=\\rho(gh)$ and $\\rho(1)=I$.\n\\end{proof}","The recurrence is a quantitative certificate of convergence, not a claim that arbitrary small residuals can always be removed. At $m=2e$ this estimate does not improve the defect depth. We do not assert that the threshold is optimal for every group, rank, or representation.","\\begin{theorem}[Finite horizon for fixed finite-group representations]\\label{thm:finite-action-horizon}\nLet $G$ be a finite group, $d,n\\geq1$, and $e=v_p(|G|)$. Set\n\\[\n \\boxed{h_G(n)=\\max\\{n+e,\\,2e+1\\}.}\n\\]\nFor every $\\rho_n:G\\to\\operatorname{GL}_d(\\mathbb Z/p^n)$, the following are equivalent:\n\\begin{enumerate}\n\\item There is a representation $\\rho:G\\to\\operatorname{GL}_d(\\mathbb Z_p)$ with reduction $\\rho_n$.\n\\item There is a representation $\\rho_h:G\\to\\operatorname{GL}_d(\\mathbb Z/p^{h_G(n)})$ with reduction $\\rho_n$.\n\\end{enumerate}\nThe implication from (2) to (1) is constructive to arbitrary requested precision by \\cref{thm:convergent-action-repair}. It preserves the original level $n$, not necessarily every digit of the chosen level-$h$ witness.\n\\end{theorem}\n\\begin{proof}\nReduction proves (1)","\\begin{theorem}[Finite horizon for fixed finite-group representations]\\label{thm:finite-action-horizon}\nLet $G$ be a finite group, $d,n\\geq1$, and $e=v_p(|G|)$. Set\n\\[\n \\boxed{h_G(n)=\\max\\{n+e,\\,2e+1\\}.}\n\\]\nFor every $\\rho_n:G\\to\\operatorname{GL}_d(\\mathbb Z/p^n)$, the following are equivalent:\n\\begin{enumerate}\n\\item There is a representation $\\rho:G\\to\\operatorname{GL}_d(\\mathbb Z_p)$ with reduction $\\rho_n$.\n\\item There is a representation $\\rho_h:G\\to\\operatorname{GL}_d(\\mathbb Z/p^{h_G(n)})$ with reduction $\\rho_n$.\n\\end{enumerate}\nThe implication from (2) to (1) is constructive to arbitrary requested precision by \\cref{thm:convergent-action-repair}. It preserves the original level $n$, not necessarily every digit of the chosen level-$h$ witness.\n\\end{theorem}\n\\begin{proof}\nReduction proves (1) implies (2). Conversely, lift","\\begin{theorem}[Finite horizon for fixed finite-group representations]\\label{thm:finite-action-horizon}\nLet $G$ be a finite group, $d,n\\geq1$, and $e=v_p(|G|)$. Set\n\\[\n \\boxed{h_G(n)=\\max\\{n+e,\\,2e+1\\}.}\n\\]\nFor every $\\rho_n:G\\to\\operatorname{GL}_d(\\mathbb Z/p^n)$, the following are equivalent:\n\\begin{enumerate}\n\\item There is a representation $\\rho:G\\to\\operatorname{GL}_d(\\mathbb Z_p)$ with reduction $\\rho_n$.\n\\item There is a representation $\\rho_h:G\\to\\operatorname{GL}_d(\\mathbb Z/p^{h_G(n)})$ with reduction $\\rho_n$.\n\\end{enumerate}\nThe implication from (2) to (1) is constructive to arbitrary requested precision by \\cref{thm:convergent-action-repair}. It preserves the original level $n$, not necessarily every digit of the chosen level-$h$ witness.\n\\end{theorem}\n\\begin{proof}\nReduction proves (1) implies (2). Conversely, lift every matrix of $","\\begin{theorem}[Finite horizon for fixed finite-group representations]\\label{thm:finite-action-horizon}\nLet $G$ be a finite group, $d,n\\geq1$, and $e=v_p(|G|)$. Set\n\\[\n \\boxed{h_G(n)=\\max\\{n+e,\\,2e+1\\}.}\n\\]\nFor every $\\rho_n:G\\to\\operatorname{GL}_d(\\mathbb Z/p^n)$, the following are equivalent:\n\\begin{enumerate}\n\\item There is a representation $\\rho:G\\to\\operatorname{GL}_d(\\mathbb Z_p)$ with reduction $\\rho_n$.\n\\item There is a representation $\\rho_h:G\\to\\operatorname{GL}_d(\\mathbb Z/p^{h_G(n)})$ with reduction $\\rho_n$.\n\\end{enumerate}\nThe implication from (2) to (1) is constructive to arbitrary requested precision by \\cref{thm:convergent-action-repair}. It preserves the original level $n$, not necessarily every digit of the chosen level-$h$ witness.\n\\end{theorem}\n\\begin{proof}\nReduction proves (1) implies (2). Conversely, lift every matrix of $\\rho_h$ arbitrarily to an invertible integral","\\begin{theorem}[Finite horizon for fixed finite-group representations]\\label{thm:finite-action-horizon}\nLet $G$ be a finite group, $d,n\\geq1$, and $e=v_p(|G|)$. Set\n\\[\n \\boxed{h_G(n)=\\max\\{n+e,\\,2e+1\\}.}\n\\]\nFor every $\\rho_n:G\\to\\operatorname{GL}_d(\\mathbb Z/p^n)$, the following are equivalent:\n\\begin{enumerate}\n\\item There is a representation $\\rho:G\\to\\operatorname{GL}_d(\\mathbb Z_p)$ with reduction $\\rho_n$.\n\\item There is a representation $\\rho_h:G\\to\\operatorname{GL}_d(\\mathbb Z/p^{h_G(n)})$ with reduction $\\rho_n$.\n\\end{enumerate}\nThe implication from (2) to (1) is constructive to arbitrary requested precision by \\cref{thm:convergent-action-repair}. It preserves the original level $n$, not necessarily every digit of the chosen level-$h$ witness.\n\\end{theorem}\n\\begin{proof}\nReduction proves (1) implies (2). Conversely, lift every matrix of $\\rho_h$ arbitrarily to an invertible integral matrix, keeping the identity equal to $I$. Its multiplicati","\\begin{theorem}[Finite horizon for fixed finite-group representations]\\label{thm:finite-action-horizon}\nLet $G$ be a finite group, $d,n\\geq1$, and $e=v_p(|G|)$. Set\n\\[\n \\boxed{h_G(n)=\\max\\{n+e,\\,2e+1\\}.}\n\\]\nFor every $\\rho_n:G\\to\\operatorname{GL}_d(\\mathbb Z/p^n)$, the following are equivalent:\n\\begin{enumerate}\n\\item There is a representation $\\rho:G\\to\\operatorname{GL}_d(\\mathbb Z_p)$ with reduction $\\rho_n$.\n\\item There is a representation $\\rho_h:G\\to\\operatorname{GL}_d(\\mathbb Z/p^{h_G(n)})$ with reduction $\\rho_n$.\n\\end{enumerate}\nThe implication from (2) to (1) is constructive to arbitrary requested precision by \\cref{thm:convergent-action-repair}. It preserves the original level $n$, not necessarily every digit of the chosen level-$h$ witness.\n\\end{theorem}\n\\begin{proof}\nReduction proves (1) implies (2). Conversely, lift every matrix of $\\rho_h$ arbitrarily to an invertible integral matrix, keeping the identity equal to $I$. Its multiplicativity defects are divisible by $p^h$. Since $h>2e$, convergent repair applies and returns an action agreeing with these matric","{theorem}[Finite horizon for fixed finite-group representations]\\label{thm:finite-action-horizon}\nLet $G$ be a finite group, $d,n\\geq1$, and $e=v_p(|G|)$. Set\n\\[\n \\boxed{h_G(n)=\\max\\{n+e,\\,2e+1\\}.}\n\\]\nFor every $\\rho_n:G\\to\\operatorname{GL}_d(\\mathbb Z/p^n)$, the following are equivalent:\n\\begin{enumerate}\n\\item There is a representation $\\rho:G\\to\\operatorname{GL}_d(\\mathbb Z_p)$ with reduction $\\rho_n$.\n\\item There is a representation $\\rho_h:G\\to\\operatorname{GL}_d(\\mathbb Z/p^{h_G(n)})$ with reduction $\\rho_n$.\n\\end{enumerate}\nThe implication from (2) to (1) is constructive to arbitrary requested precision by \\cref{thm:convergent-action-repair}. It preserves the original level $n$, not necessarily every digit of the chosen level-$h$ witness.\n\\end{theorem}\n\\begin{proof}\nReduction proves (1) implies (2). Conversely, lift every matrix of $\\rho_h$ arbitrarily to an invertible integral matrix, keeping the identity equal to $I$. Its multiplicativity defects are divisible by $p^h$. Since $h>2e$, convergent repair applies and returns an action agreeing with these matrices modulo $p^{h-e}…","group representations]\\label{thm:finite-action-horizon}\nLet $G$ be a finite group, $d,n\\geq1$, and $e=v_p(|G|)$. Set\n\\[\n \\boxed{h_G(n)=\\max\\{n+e,\\,2e+1\\}.}\n\\]\nFor every $\\rho_n:G\\to\\operatorname{GL}_d(\\mathbb Z/p^n)$, the following are equivalent:\n\\begin{enumerate}\n\\item There is a representation $\\rho:G\\to\\operatorname{GL}_d(\\mathbb Z_p)$ with reduction $\\rho_n$.\n\\item There is a representation $\\rho_h:G\\to\\operatorname{GL}_d(\\mathbb Z/p^{h_G(n)})$ with reduction $\\rho_n$.\n\\end{enumerate}\nThe implication from (2) to (1) is constructive to arbitrary requested precision by \\cref{thm:convergent-action-repair}. It preserves the original level $n$, not necessarily every digit of the chosen level-$h$ witness.\n\\end{theorem}\n\\begin{proof}\nReduction proves (1) implies (2). Conversely, lift every matrix of $\\rho_h$ arbitrarily to an invertible integral matrix, keeping the identity equal to $I$. Its multiplicativity defects are divisible by $p^h$. Since $h>2e$, convergent repair applies and returns an action agreeing with these matrices modulo $p^{h-e}$. Because $h-e\\geq n$, it returns precise…","inite group, $d,n\\geq1$, and $e=v_p(|G|)$. Set\n\\[\n \\boxed{h_G(n)=\\max\\{n+e,\\,2e+1\\}.}\n\\]\nFor every $\\rho_n:G\\to\\operatorname{GL}_d(\\mathbb Z/p^n)$, the following are equivalent:\n\\begin{enumerate}\n\\item There is a representation $\\rho:G\\to\\operatorname{GL}_d(\\mathbb Z_p)$ with reduction $\\rho_n$.\n\\item There is a representation $\\rho_h:G\\to\\operatorname{GL}_d(\\mathbb Z/p^{h_G(n)})$ with reduction $\\rho_n$.\n\\end{enumerate}\nThe implication from (2) to (1) is constructive to arbitrary requested precision by \\cref{thm:convergent-action-repair}. It preserves the original level $n$, not necessarily every digit of the chosen level-$h$ witness.\n\\end{theorem}\n\\begin{proof}\nReduction proves (1) implies (2). Conversely, lift every matrix of $\\rho_h$ arbitrarily to an invertible integral matrix, keeping the identity equal to $I$. Its multiplicativity defects are divisible by $p^h$. Since $h>2e$, convergent repair applies and returns an action agreeing with these matrices modulo $p^{h-e}$. Because $h-e\\geq n$, it returns precisely the specified $\\rho_n$ on reduction.\n\\end{proof}","begin{enumerate}\n\\item There is a representation $\\rho:G\\to\\operatorname{GL}_d(\\mathbb Z_p)$ with reduction $\\rho_n$.\n\\item There is a representation $\\rho_h:G\\to\\operatorname{GL}_d(\\mathbb Z/p^{h_G(n)})$ with reduction $\\rho_n$.\n\\end{enumerate}\nThe implication from (2) to (1) is constructive to arbitrary requested precision by \\cref{thm:convergent-action-repair}. It preserves the original level $n$, not necessarily every digit of the chosen level-$h$ witness.\n\\end{theorem}\n\\begin{proof}\nReduction proves (1) implies (2). Conversely, lift every matrix of $\\rho_h$ arbitrarily to an invertible integral matrix, keeping the identity equal to $I$. Its multiplicativity defects are divisible by $p^h$. Since $h>2e$, convergent repair applies and returns an action agreeing with these matrices modulo $p^{h-e}$. Because $h-e\\geq n$, it returns precisely the specified $\\rho_n$ on reduction.\n\\end{proof}","o:G\\to\\operatorname{GL}_d(\\mathbb Z_p)$ with reduction $\\rho_n$.\n\\item There is a representation $\\rho_h:G\\to\\operatorname{GL}_d(\\mathbb Z/p^{h_G(n)})$ with reduction $\\rho_n$.\n\\end{enumerate}\nThe implication from (2) to (1) is constructive to arbitrary requested precision by \\cref{thm:convergent-action-repair}. It preserves the original level $n$, not necessarily every digit of the chosen level-$h$ witness.\n\\end{theorem}\n\\begin{proof}\nReduction proves (1) implies (2). Conversely, lift every matrix of $\\rho_h$ arbitrarily to an invertible integral matrix, keeping the identity equal to $I$. Its multiplicativity defects are divisible by $p^h$. Since $h>2e$, convergent repair applies and returns an action agreeing with these matrices modulo $p^{h-e}$. Because $h-e\\geq n$, it returns precisely the specified $\\rho_n$ on reduction.\n\\end{proof}","o\\operatorname{GL}_d(\\mathbb Z/p^{h_G(n)})$ with reduction $\\rho_n$.\n\\end{enumerate}\nThe implication from (2) to (1) is constructive to arbitrary requested precision by \\cref{thm:convergent-action-repair}. It preserves the original level $n$, not necessarily every digit of the chosen level-$h$ witness.\n\\end{theorem}\n\\begin{proof}\nReduction proves (1) implies (2). Conversely, lift every matrix of $\\rho_h$ arbitrarily to an invertible integral matrix, keeping the identity equal to $I$. Its multiplicativity defects are divisible by $p^h$. Since $h>2e$, convergent repair applies and returns an action agreeing with these matrices modulo $p^{h-e}$. Because $h-e\\geq n$, it returns precisely the specified $\\rho_n$ on reduction.\n\\end{proof}","The implication from (2) to (1) is constructive to arbitrary requested precision by \\cref{thm:convergent-action-repair}. It preserves the original level $n$, not necessarily every digit of the chosen level-$h$ witness.\n\\end{theorem}\n\\begin{proof}\nReduction proves (1) implies (2). Conversely, lift every matrix of $\\rho_h$ arbitrarily to an invertible integral matrix, keeping the identity equal to $I$. Its multiplicativity defects are divisible by $p^h$. Since $h>2e$, convergent repair applies and returns an action agreeing with these matrices modulo $p^{h-e}$. Because $h-e\\geq n$, it returns precisely the specified $\\rho_n$ on reduction.\n\\end{proof}","arbitrary requested precision by \\cref{thm:convergent-action-repair}. It preserves the original level $n$, not necessarily every digit of the chosen level-$h$ witness.\n\\end{theorem}\n\\begin{proof}\nReduction proves (1) implies (2). Conversely, lift every matrix of $\\rho_h$ arbitrarily to an invertible integral matrix, keeping the identity equal to $I$. Its multiplicativity defects are divisible by $p^h$. Since $h>2e$, convergent repair applies and returns an action agreeing with these matrices modulo $p^{h-e}$. Because $h-e\\geq n$, it returns precisely the specified $\\rho_n$ on reduction.\n\\end{proof}","uested precision by \\cref{thm:convergent-action-repair}. It preserves the original level $n$, not necessarily every digit of the chosen level-$h$ witness.\n\\end{theorem}\n\\begin{proof}\nReduction proves (1) implies (2). Conversely, lift every matrix of $\\rho_h$ arbitrarily to an invertible integral matrix, keeping the identity equal to $I$. Its multiplicativity defects are divisible by $p^h$. Since $h>2e$, convergent repair applies and returns an action agreeing with these matrices modulo $p^{h-e}$. Because $h-e\\geq n$, it returns precisely the specified $\\rho_n$ on reduction.\n\\end{proof}",", not necessarily every digit of the chosen level-$h$ witness.\n\\end{theorem}\n\\begin{proof}\nReduction proves (1) implies (2). Conversely, lift every matrix of $\\rho_h$ arbitrarily to an invertible integral matrix, keeping the identity equal to $I$. Its multiplicativity defects are divisible by $p^h$. Since $h>2e$, convergent repair applies and returns an action agreeing with these matrices modulo $p^{h-e}$. Because $h-e\\geq n$, it returns precisely the specified $\\rho_n$ on reduction.\n\\end{proof}","very digit of the chosen level-$h$ witness.\n\\end{theorem}\n\\begin{proof}\nReduction proves (1) implies (2). Conversely, lift every matrix of $\\rho_h$ arbitrarily to an invertible integral matrix, keeping the identity equal to $I$. Its multiplicativity defects are divisible by $p^h$. Since $h>2e$, convergent repair applies and returns an action agreeing with these matrices modulo $p^{h-e}$. Because $h-e\\geq n$, it returns precisely the specified $\\rho_n$ on reduction.\n\\end{proof}","{theorem}\n\\begin{proof}\nReduction proves (1) implies (2). Conversely, lift every matrix of $\\rho_h$ arbitrarily to an invertible integral matrix, keeping the identity equal to $I$. Its multiplicativity defects are divisible by $p^h$. Since $h>2e$, convergent repair applies and returns an action agreeing with these matrices modulo $p^{h-e}$. Because $h-e\\geq n$, it returns precisely the specified $\\rho_n$ on reduction.\n\\end{proof}","Thus a supplied high-precision action is a finite certificate for existence of an entire integral action. A level-$h$ witness may itself have an obstruction at its next level: the repair is allowed to alter the top $e$ digits while keeping the requested lower data. There is no claim that a positive certificate yields an exact lift of that witness at every digit.","A terminating decision method follows. Begin with $\\rho_n$ and use \\cref{thm:relation-H2,prop:repair-H1} to enumerate all one-step solutions, retaining their actual framed matrices, until level $h_G(n)$. A surviving branch is a positive witness. If all branches die before that finite level, no integral lift exists. Every layer is finite, and Gaussian elimination provides checkable obstruction certificates at dead nodes. Choosing a single convenient branch is not sufficient for a negative verdict. Quotienting branches by conjugacy is optional and requires preserving their lifting data.","For $p\\nmid|G|$, one has $e=0$ and $h_G(n)=n$. Every finite-level representation in this fixed finite-group problem then lifts. When $e>0$, the test is still finite, but its number of branches may be enormous. No polynomial-time or uniformly practical algorithm is claimed.","\\begin{example}[A positive witness need not lift itself]\\label{ex:horizon-witness}\nTake $p=2$, $G=C_2$, and rank one. The level-two representation with generator $3\\pmod4$ has the level-three witness $3\\pmod8$, since $3^2\\equiv1\\pmod8$. Here $e=1$ and $h_G(2)=3$. Nevertheless neither lift of $3\\pmod8$ to $\\mathbb Z/16$ has square one: both $3^2$ and $11^2$ are $9\\pmod{16}$.","Starting with the integer matrix $a_0=3$, the averaged repair becomes\n\\[\n a_{j+1}=\\frac{3-a_j^2}{2}\\,a_j.\n\\]\nThe first values are $3,-9,351$, whose relation defects have valuations $3,4,6$. The convergence theorem gives an exact $2$-adic involution congruent to $3\\pmod4$. In the integral domain $\\mathbb Z_2$, the equation $a^2=1$ has only the solutions $1,-1$. Thus the repaired limit is $-1$, which is $7\\pmod8$, not the selected witness $3\\pmod8$. This is exactly the distinction between preserving the requested seed and preserving every exploratory refinement.\n\\end{example}","ng with the integer matrix $a_0=3$, the averaged repair becomes\n\\[\n a_{j+1}=\\frac{3-a_j^2}{2}\\,a_j.\n\\]\nThe first values are $3,-9,351$, whose relation defects have valuations $3,4,6$. The convergence theorem gives an exact $2$-adic involution congruent to $3\\pmod4$. In the integral domain $\\mathbb Z_2$, the equation $a^2=1$ has only the solutions $1,-1$. Thus the repaired limit is $-1$, which is $7\\pmod8$, not the selected witness $3\\pmod8$. This is exactly the distinction between preserving the requested seed and preserving every exploratory refinement.\n\\end{example}","the averaged repair becomes\n\\[\n a_{j+1}=\\frac{3-a_j^2}{2}\\,a_j.\n\\]\nThe first values are $3,-9,351$, whose relation defects have valuations $3,4,6$. The convergence theorem gives an exact $2$-adic involution congruent to $3\\pmod4$. In the integral domain $\\mathbb Z_2$, the equation $a^2=1$ has only the solutions $1,-1$. Thus the repaired limit is $-1$, which is $7\\pmod8$, not the selected witness $3\\pmod8$. This is exactly the distinction between preserving the requested seed and preserving every exploratory refinement.\n\\end{example}","\\begin{theorem}[Close integral actions are conjugate]\\label{thm:close-action-conjugacy}\nLet $\\rho,\\sigma:G\\to\\operatorname{GL}_d(\\mathbb Z_p)$ be representations satisfying\n\\[\n \\sigma(g)\\equiv\\rho(g)\\pmod{p^s}\\quad(g\\in G),\\qquad s>e.\n\\]\nThen\n\\[\n T=\\frac1{|G|}\\sum_{g\\in G}\\sigma(g)\\rho(g)^{-1}\n \\in I+p^{s-e}M_d(\\mathbb Z_p)\n\\]\nis invertible and satisfies\n\\[\n \\sigma(g)T=T\\rho(g),\\qquad\n \\sigma(g)=T\\rho(g)T^{-1}.\n\\]\n\\end{theorem}\n\\begin{proof}\nEvery summand is congruent to $I$ modulo $p^s$. Its average is therefore $I$ modulo $p^{s-e}$, so it is integral and invertible. For $h\\in G$,\n\\[\n \\sigma(h)T\\rho(h)^{-1}\n =\\frac1{|G|}\\sum_g\\sigma(hg)\\rho(hg)^{-1}=T,\n\\]\nby reindexing the finite sum. This is the required intertwining identity.\n\\end{proof}","conjugacy}\nLet $\\rho,\\sigma:G\\to\\operatorname{GL}_d(\\mathbb Z_p)$ be representations satisfying\n\\[\n \\sigma(g)\\equiv\\rho(g)\\pmod{p^s}\\quad(g\\in G),\\qquad s>e.\n\\]\nThen\n\\[\n T=\\frac1{|G|}\\sum_{g\\in G}\\sigma(g)\\rho(g)^{-1}\n \\in I+p^{s-e}M_d(\\mathbb Z_p)\n\\]\nis invertible and satisfies\n\\[\n \\sigma(g)T=T\\rho(g),\\qquad\n \\sigma(g)=T\\rho(g)T^{-1}.\n\\]\n\\end{theorem}\n\\begin{proof}\nEvery summand is congruent to $I$ modulo $p^s$. Its average is therefore $I$ modulo $p^{s-e}$, so it is integral and invertible. For $h\\in G$,\n\\[\n \\sigma(h)T\\rho(h)^{-1}\n =\\frac1{|G|}\\sum_g\\sigma(hg)\\rho(hg)^{-1}=T,\n\\]\nby reindexing the finite sum. This is the required intertwining identity.\n\\end{proof}","Let $\\rho,\\sigma:G\\to\\operatorname{GL}_d(\\mathbb Z_p)$ be representations satisfying\n\\[\n \\sigma(g)\\equiv\\rho(g)\\pmod{p^s}\\quad(g\\in G),\\qquad s>e.\n\\]\nThen\n\\[\n T=\\frac1{|G|}\\sum_{g\\in G}\\sigma(g)\\rho(g)^{-1}\n \\in I+p^{s-e}M_d(\\mathbb Z_p)\n\\]\nis invertible and satisfies\n\\[\n \\sigma(g)T=T\\rho(g),\\qquad\n \\sigma(g)=T\\rho(g)T^{-1}.\n\\]\n\\end{theorem}\n\\begin{proof}\nEvery summand is congruent to $I$ modulo $p^s$. Its average is therefore $I$ modulo $p^{s-e}$, so it is integral and invertible. For $h\\in G$,\n\\[\n \\sigma(h)T\\rho(h)^{-1}\n =\\frac1{|G|}\\sum_g\\sigma(hg)\\rho(hg)^{-1}=T,\n\\]\nby reindexing the finite sum. This is the required intertwining identity.\n\\end{proof}","me{GL}_d(\\mathbb Z_p)$ be representations satisfying\n\\[\n \\sigma(g)\\equiv\\rho(g)\\pmod{p^s}\\quad(g\\in G),\\qquad s>e.\n\\]\nThen\n\\[\n T=\\frac1{|G|}\\sum_{g\\in G}\\sigma(g)\\rho(g)^{-1}\n \\in I+p^{s-e}M_d(\\mathbb Z_p)\n\\]\nis invertible and satisfies\n\\[\n \\sigma(g)T=T\\rho(g),\\qquad\n \\sigma(g)=T\\rho(g)T^{-1}.\n\\]\n\\end{theorem}\n\\begin{proof}\nEvery summand is congruent to $I$ modulo $p^s$. Its average is therefore $I$ modulo $p^{s-e}$, so it is integral and invertible. For $h\\in G$,\n\\[\n \\sigma(h)T\\rho(h)^{-1}\n =\\frac1{|G|}\\sum_g\\sigma(hg)\\rho(hg)^{-1}=T,\n\\]\nby reindexing the finite sum. This is the required intertwining identity.\n\\end{proof}","athbb Z_p)$ be representations satisfying\n\\[\n \\sigma(g)\\equiv\\rho(g)\\pmod{p^s}\\quad(g\\in G),\\qquad s>e.\n\\]\nThen\n\\[\n T=\\frac1{|G|}\\sum_{g\\in G}\\sigma(g)\\rho(g)^{-1}\n \\in I+p^{s-e}M_d(\\mathbb Z_p)\n\\]\nis invertible and satisfies\n\\[\n \\sigma(g)T=T\\rho(g),\\qquad\n \\sigma(g)=T\\rho(g)T^{-1}.\n\\]\n\\end{theorem}\n\\begin{proof}\nEvery summand is congruent to $I$ modulo $p^s$. Its average is therefore $I$ modulo $p^{s-e}$, so it is integral and invertible. For $h\\in G$,\n\\[\n \\sigma(h)T\\rho(h)^{-1}\n =\\frac1{|G|}\\sum_g\\sigma(hg)\\rho(hg)^{-1}=T,\n\\]\nby reindexing the finite sum. This is the required intertwining identity.\n\\end{proof}","gma(g)\\equiv\\rho(g)\\pmod{p^s}\\quad(g\\in G),\\qquad s>e.\n\\]\nThen\n\\[\n T=\\frac1{|G|}\\sum_{g\\in G}\\sigma(g)\\rho(g)^{-1}\n \\in I+p^{s-e}M_d(\\mathbb Z_p)\n\\]\nis invertible and satisfies\n\\[\n \\sigma(g)T=T\\rho(g),\\qquad\n \\sigma(g)=T\\rho(g)T^{-1}.\n\\]\n\\end{theorem}\n\\begin{proof}\nEvery summand is congruent to $I$ modulo $p^s$. Its average is therefore $I$ modulo $p^{s-e}$, so it is integral and invertible. For $h\\in G$,\n\\[\n \\sigma(h)T\\rho(h)^{-1}\n =\\frac1{|G|}\\sum_g\\sigma(hg)\\rho(hg)^{-1}=T,\n\\]\nby reindexing the finite sum. This is the required intertwining identity.\n\\end{proof}","v\\rho(g)\\pmod{p^s}\\quad(g\\in G),\\qquad s>e.\n\\]\nThen\n\\[\n T=\\frac1{|G|}\\sum_{g\\in G}\\sigma(g)\\rho(g)^{-1}\n \\in I+p^{s-e}M_d(\\mathbb Z_p)\n\\]\nis invertible and satisfies\n\\[\n \\sigma(g)T=T\\rho(g),\\qquad\n \\sigma(g)=T\\rho(g)T^{-1}.\n\\]\n\\end{theorem}\n\\begin{proof}\nEvery summand is congruent to $I$ modulo $p^s$. Its average is therefore $I$ modulo $p^{s-e}$, so it is integral and invertible. For $h\\in G$,\n\\[\n \\sigma(h)T\\rho(h)^{-1}\n =\\frac1{|G|}\\sum_g\\sigma(hg)\\rho(hg)^{-1}=T,\n\\]\nby reindexing the finite sum. This is the required intertwining identity.\n\\end{proof}","This is the averaging mechanism underlying finite-precision rigidity of integral finite-group representations, closely related to the classical rigidity of integral finite-group lattices. The proof above is independent of any such external rigidity theorem. The strict inequality matters: for $p=2$, $G=C_2$, and rank one, the trivial and sign representations agree modulo $2$ but are not conjugate. Here $s=e=1$.","s the averaging mechanism underlying finite-precision rigidity of integral finite-group representations, closely related to the classical rigidity of integral finite-group lattices. The proof above is independent of any such external rigidity theorem. The strict inequality matters: for $p=2$, $G=C_2$, and rank one, the trivial and sign representations agree modulo $2$ but are not conjugate. Here $s=e=1$.","\\begin{corollary}[A finite catalogue of integral action classes]\\label{cor:finite-action-catalogue}\nLet $s=e+1$ and $h=2e+1$. Denote by $\\mathscr L_s$ the image of\n\\[\n \\operatorname{Hom}(G,\\operatorname{GL}_d(\\mathbb Z/p^h))\n \\longrightarrow\n \\operatorname{Hom}(G,\\operatorname{GL}_d(\\mathbb Z/p^s)).\n\\]\nReduction gives a bijection\n\\[\n \\boxed{\n \\frac{\\operatorname{Hom}(G,\\operatorname{GL}_d(\\mathbb Z_p))}\n {\\operatorname{GL}_d(\\mathbb Z_p)}\n \\ \\longleftrightarrow\\\n \\frac{\\mathscr L_s}{\\operatorname{GL}_d(\\mathbb Z/p^s)}.}\n\\]\nConsequently this class set is finite and can be determined from a finite multiplication table and finite matrix data.\n\\end{corollary}\n\\begin{proof}\nThe finite-horizon theorem gives exactly $h_G(s)=2e+1$, so every element of $\\mathscr L_s$ lifts and every integral action reduces i","\\begin{corollary}[A finite catalogue of integral action classes]\\label{cor:finite-action-catalogue}\nLet $s=e+1$ and $h=2e+1$. Denote by $\\mathscr L_s$ the image of\n\\[\n \\operatorname{Hom}(G,\\operatorname{GL}_d(\\mathbb Z/p^h))\n \\longrightarrow\n \\operatorname{Hom}(G,\\operatorname{GL}_d(\\mathbb Z/p^s)).\n\\]\nReduction gives a bijection\n\\[\n \\boxed{\n \\frac{\\operatorname{Hom}(G,\\operatorname{GL}_d(\\mathbb Z_p))}\n {\\operatorname{GL}_d(\\mathbb Z_p)}\n \\ \\longleftrightarrow\\\n \\frac{\\mathscr L_s}{\\operatorname{GL}_d(\\mathbb Z/p^s)}.}\n\\]\nConsequently this class set is finite and can be determined from a finite multiplication table and finite matrix data.\n\\end{corollary}\n\\begin{proof}\nThe finite-horizon theorem gives exactly $h_G(s)=2e+1$, so every element of $\\mathscr L_s$ lifts and every integral action reduces into it. If tw","\\begin{corollary}[A finite catalogue of integral action classes]\\label{cor:finite-action-catalogue}\nLet $s=e+1$ and $h=2e+1$. Denote by $\\mathscr L_s$ the image of\n\\[\n \\operatorname{Hom}(G,\\operatorname{GL}_d(\\mathbb Z/p^h))\n \\longrightarrow\n \\operatorname{Hom}(G,\\operatorname{GL}_d(\\mathbb Z/p^s)).\n\\]\nReduction gives a bijection\n\\[\n \\boxed{\n \\frac{\\operatorname{Hom}(G,\\operatorname{GL}_d(\\mathbb Z_p))}\n {\\operatorname{GL}_d(\\mathbb Z_p)}\n \\ \\longleftrightarrow\\\n \\frac{\\mathscr L_s}{\\operatorname{GL}_d(\\mathbb Z/p^s)}.}\n\\]\nConsequently this class set is finite and can be determined from a finite multiplication table and finite matrix data.\n\\end{corollary}\n\\begin{proof}\nThe finite-horizon theorem gives exactly $h_G(s)=2e+1$, so every element of $\\mathscr L_s$ lifts and every integral action reduces into it. If two reductions are conjugate","\\begin{corollary}[A finite catalogue of integral action classes]\\label{cor:finite-action-catalogue}\nLet $s=e+1$ and $h=2e+1$. Denote by $\\mathscr L_s$ the image of\n\\[\n \\operatorname{Hom}(G,\\operatorname{GL}_d(\\mathbb Z/p^h))\n \\longrightarrow\n \\operatorname{Hom}(G,\\operatorname{GL}_d(\\mathbb Z/p^s)).\n\\]\nReduction gives a bijection\n\\[\n \\boxed{\n \\frac{\\operatorname{Hom}(G,\\operatorname{GL}_d(\\mathbb Z_p))}\n {\\operatorname{GL}_d(\\mathbb Z_p)}\n \\ \\longleftrightarrow\\\n \\frac{\\mathscr L_s}{\\operatorname{GL}_d(\\mathbb Z/p^s)}.}\n\\]\nConsequently this class set is finite and can be determined from a finite multiplication table and finite matrix data.\n\\end{corollary}\n\\begin{proof}\nThe finite-horizon theorem gives exactly $h_G(s)=2e+1$, so every element of $\\mathscr L_s$ lifts and every integral action reduces into it. If two reductions are conjugate, lift their conjugating matrix to $\\operatorname{GL}_d(\\mathbb Z_p)$ and first conjugate one action by that lift. Their reductions now agree modulo $p^","\\begin{corollary}[A finite catalogue of integral action classes]\\label{cor:finite-action-catalogue}\nLet $s=e+1$ and $h=2e+1$. Denote by $\\mathscr L_s$ the image of\n\\[\n \\operatorname{Hom}(G,\\operatorname{GL}_d(\\mathbb Z/p^h))\n \\longrightarrow\n \\operatorname{Hom}(G,\\operatorname{GL}_d(\\mathbb Z/p^s)).\n\\]\nReduction gives a bijection\n\\[\n \\boxed{\n \\frac{\\operatorname{Hom}(G,\\operatorname{GL}_d(\\mathbb Z_p))}\n {\\operatorname{GL}_d(\\mathbb Z_p)}\n \\ \\longleftrightarrow\\\n \\frac{\\mathscr L_s}{\\operatorname{GL}_d(\\mathbb Z/p^s)}.}\n\\]\nConsequently this class set is finite and can be determined from a finite multiplication table and finite matrix data.\n\\end{corollary}\n\\begin{proof}\nThe finite-horizon theorem gives exactly $h_G(s)=2e+1$, so every element of $\\mathscr L_s$ lifts and every integral action reduces into it. If two reductions are conjugate, lift their conjugating matrix to $\\operatorname{GL}_d(\\mathbb Z_p)$ and first conjugate one action by that lift. Their reductions now agree modulo $p^s$. Since $s>e$, \\cref{thm:close-action-conjugacy} makes the integral actions conjugate. The target…","a bijection\n\\[\n \\boxed{\n \\frac{\\operatorname{Hom}(G,\\operatorname{GL}_d(\\mathbb Z_p))}\n {\\operatorname{GL}_d(\\mathbb Z_p)}\n \\ \\longleftrightarrow\\\n \\frac{\\mathscr L_s}{\\operatorname{GL}_d(\\mathbb Z/p^s)}.}\n\\]\nConsequently this class set is finite and can be determined from a finite multiplication table and finite matrix data.\n\\end{corollary}\n\\begin{proof}\nThe finite-horizon theorem gives exactly $h_G(s)=2e+1$, so every element of $\\mathscr L_s$ lifts and every integral action reduces into it. If two reductions are conjugate, lift their conjugating matrix to $\\operatorname{GL}_d(\\mathbb Z_p)$ and first conjugate one action by that lift. Their reductions now agree modulo $p^s$. Since $s>e$, \\cref{thm:close-action-conjugacy} makes the integral actions conjugate. The target is finite because it is a quotient of a finite set.\n\\end{proof}","ratorname{Hom}(G,\\operatorname{GL}_d(\\mathbb Z_p))}\n {\\operatorname{GL}_d(\\mathbb Z_p)}\n \\ \\longleftrightarrow\\\n \\frac{\\mathscr L_s}{\\operatorname{GL}_d(\\mathbb Z/p^s)}.}\n\\]\nConsequently this class set is finite and can be determined from a finite multiplication table and finite matrix data.\n\\end{corollary}\n\\begin{proof}\nThe finite-horizon theorem gives exactly $h_G(s)=2e+1$, so every element of $\\mathscr L_s$ lifts and every integral action reduces into it. If two reductions are conjugate, lift their conjugating matrix to $\\operatorname{GL}_d(\\mathbb Z_p)$ and first conjugate one action by that lift. Their reductions now agree modulo $p^s$. Since $s>e$, \\cref{thm:close-action-conjugacy} makes the integral actions conjugate. The target is finite because it is a quotient of a finite set.\n\\end{proof}","s}{\\operatorname{GL}_d(\\mathbb Z/p^s)}.}\n\\]\nConsequently this class set is finite and can be determined from a finite multiplication table and finite matrix data.\n\\end{corollary}\n\\begin{proof}\nThe finite-horizon theorem gives exactly $h_G(s)=2e+1$, so every element of $\\mathscr L_s$ lifts and every integral action reduces into it. If two reductions are conjugate, lift their conjugating matrix to $\\operatorname{GL}_d(\\mathbb Z_p)$ and first conjugate one action by that lift. Their reductions now agree modulo $p^s$. Since $s>e$, \\cref{thm:close-action-conjugacy} makes the integral actions conjugate. The target is finite because it is a quotient of a finite set.\n\\end{proof}","ite multiplication table and finite matrix data.\n\\end{corollary}\n\\begin{proof}\nThe finite-horizon theorem gives exactly $h_G(s)=2e+1$, so every element of $\\mathscr L_s$ lifts and every integral action reduces into it. If two reductions are conjugate, lift their conjugating matrix to $\\operatorname{GL}_d(\\mathbb Z_p)$ and first conjugate one action by that lift. Their reductions now agree modulo $p^s$. Since $s>e$, \\cref{thm:close-action-conjugacy} makes the integral actions conjugate. The target is finite because it is a quotient of a finite set.\n\\end{proof}","ation table and finite matrix data.\n\\end{corollary}\n\\begin{proof}\nThe finite-horizon theorem gives exactly $h_G(s)=2e+1$, so every element of $\\mathscr L_s$ lifts and every integral action reduces into it. If two reductions are conjugate, lift their conjugating matrix to $\\operatorname{GL}_d(\\mathbb Z_p)$ and first conjugate one action by that lift. Their reductions now agree modulo $p^s$. Since $s>e$, \\cref{thm:close-action-conjugacy} makes the integral actions conjugate. The target is finite because it is a quotient of a finite set.\n\\end{proof}","This is a classification of isomorphism classes, not an equivalence with a finite groupoid. Integral automorphism groups and noninvertible morphisms need not be determined by finite reduction. Likewise the averaging construction is not asserted to be a canonical framed lift. Two outputs agreeing modulo $p^{m-e}$ are conjugate by a matrix congruent to $I$ modulo $p^{m-2e}$ when $m>2e$, but need not be literally the same matrices.","\\begin{proposition}[Cyclic relation Jacobian]\\label{prop:cyclic-jacobian}\nLet $G=C_q$ for an integer $q\\geq1$, and let $A\\in\\operatorname{GL}_d(\\mathbb Z/p^n)$ satisfy $A^q=I$. Choose a lift $\\widetilde A$ modulo $p^{n+1}$ and set\n\\[\n c=\\frac{\\widetilde A^q-I}{p^n}\\pmod p,\n \\qquad L_{\\bar A}(B)=\\sum_{i=0}^{q-1}\\bar A^iB\\bar A^{q-1-i}.\n\\]\nThen the lifts of the cyclic action to level $n+1$ are exactly the matrices\n\\[\n \\widetilde A+p^nB\\quad\\text{such that}\\quad L_{\\bar A}(B)=-c.\n\\]\nIf the affine solution set is nonempty, it has $p^{\\dim_k\\ker L_{\\bar A}}$ elements. This criterion uses $d^2$ variables instead of all group-table cochains.\n\\end{proposition}\n\\begin{proof}\nExpand the noncommutative product $(\\widetilde A+p^nB)^q$. Terms with two or more factors $p^nB$ vanish modul","\\begin{proposition}[Cyclic relation Jacobian]\\label{prop:cyclic-jacobian}\nLet $G=C_q$ for an integer $q\\geq1$, and let $A\\in\\operatorname{GL}_d(\\mathbb Z/p^n)$ satisfy $A^q=I$. Choose a lift $\\widetilde A$ modulo $p^{n+1}$ and set\n\\[\n c=\\frac{\\widetilde A^q-I}{p^n}\\pmod p,\n \\qquad L_{\\bar A}(B)=\\sum_{i=0}^{q-1}\\bar A^iB\\bar A^{q-1-i}.\n\\]\nThen the lifts of the cyclic action to level $n+1$ are exactly the matrices\n\\[\n \\widetilde A+p^nB\\quad\\text{such that}\\quad L_{\\bar A}(B)=-c.\n\\]\nIf the affine solution set is nonempty, it has $p^{\\dim_k\\ker L_{\\bar A}}$ elements. This criterion uses $d^2$ variables instead of all group-table cochains.\n\\end{proposition}\n\\begin{proof}\nExpand the noncommutative product $(\\widetilde A+p^nB)^q$. Terms with two or more factors $p^nB$ vanish modulo $p^{n+1}$. The terms w","\\begin{proposition}[Cyclic relation Jacobian]\\label{prop:cyclic-jacobian}\nLet $G=C_q$ for an integer $q\\geq1$, and let $A\\in\\operatorname{GL}_d(\\mathbb Z/p^n)$ satisfy $A^q=I$. Choose a lift $\\widetilde A$ modulo $p^{n+1}$ and set\n\\[\n c=\\frac{\\widetilde A^q-I}{p^n}\\pmod p,\n \\qquad L_{\\bar A}(B)=\\sum_{i=0}^{q-1}\\bar A^iB\\bar A^{q-1-i}.\n\\]\nThen the lifts of the cyclic action to level $n+1$ are exactly the matrices\n\\[\n \\widetilde A+p^nB\\quad\\text{such that}\\quad L_{\\bar A}(B)=-c.\n\\]\nIf the affine solution set is nonempty, it has $p^{\\dim_k\\ker L_{\\bar A}}$ elements. This criterion uses $d^2$ variables instead of all group-table cochains.\n\\end{proposition}\n\\begin{proof}\nExpand the noncommutative product $(\\widetilde A+p^nB)^q$. Terms with two or more factors $p^nB$ vanish modulo $p^{n+1}$. The terms with one such factor give exactly $p^nL_{\\bar A}(B)","\\begin{proposition}[Cyclic relation Jacobian]\\label{prop:cyclic-jacobian}\nLet $G=C_q$ for an integer $q\\geq1$, and let $A\\in\\operatorname{GL}_d(\\mathbb Z/p^n)$ satisfy $A^q=I$. Choose a lift $\\widetilde A$ modulo $p^{n+1}$ and set\n\\[\n c=\\frac{\\widetilde A^q-I}{p^n}\\pmod p,\n \\qquad L_{\\bar A}(B)=\\sum_{i=0}^{q-1}\\bar A^iB\\bar A^{q-1-i}.\n\\]\nThen the lifts of the cyclic action to level $n+1$ are exactly the matrices\n\\[\n \\widetilde A+p^nB\\quad\\text{such that}\\quad L_{\\bar A}(B)=-c.\n\\]\nIf the affine solution set is nonempty, it has $p^{\\dim_k\\ker L_{\\bar A}}$ elements. This criterion uses $d^2$ variables instead of all group-table cochains.\n\\end{proposition}\n\\begin{proof}\nExpand the noncommutative product $(\\widetilde A+p^nB)^q$. Terms with two or more factors $p^nB$ vanish modulo $p^{n+1}$. The terms with one such factor give exactly $p^nL_{\\bar A}(B)$. Therefore the","\\begin{proposition}[Cyclic relation Jacobian]\\label{prop:cyclic-jacobian}\nLet $G=C_q$ for an integer $q\\geq1$, and let $A\\in\\operatorname{GL}_d(\\mathbb Z/p^n)$ satisfy $A^q=I$. Choose a lift $\\widetilde A$ modulo $p^{n+1}$ and set\n\\[\n c=\\frac{\\widetilde A^q-I}{p^n}\\pmod p,\n \\qquad L_{\\bar A}(B)=\\sum_{i=0}^{q-1}\\bar A^iB\\bar A^{q-1-i}.\n\\]\nThen the lifts of the cyclic action to level $n+1$ are exactly the matrices\n\\[\n \\widetilde A+p^nB\\quad\\text{such that}\\quad L_{\\bar A}(B)=-c.\n\\]\nIf the affine solution set is nonempty, it has $p^{\\dim_k\\ker L_{\\bar A}}$ elements. This criterion uses $d^2$ variables instead of all group-table cochains.\n\\end{proposition}\n\\begin{proof}\nExpand the noncommutative product $(\\widetilde A+p^nB)^q$. Terms with two or more factors $p^nB$ vanish modulo $p^{n+1}$. The terms with one such factor give exactly $p^nL_{\\bar A}(B)$. Therefore the cyclic relation is equivalent","\\begin{proposition}[Cyclic relation Jacobian]\\label{prop:cyclic-jacobian}\nLet $G=C_q$ for an integer $q\\geq1$, and let $A\\in\\operatorname{GL}_d(\\mathbb Z/p^n)$ satisfy $A^q=I$. Choose a lift $\\widetilde A$ modulo $p^{n+1}$ and set\n\\[\n c=\\frac{\\widetilde A^q-I}{p^n}\\pmod p,\n \\qquad L_{\\bar A}(B)=\\sum_{i=0}^{q-1}\\bar A^iB\\bar A^{q-1-i}.\n\\]\nThen the lifts of the cyclic action to level $n+1$ are exactly the matrices\n\\[\n \\widetilde A+p^nB\\quad\\text{such that}\\quad L_{\\bar A}(B)=-c.\n\\]\nIf the affine solution set is nonempty, it has $p^{\\dim_k\\ker L_{\\bar A}}$ elements. This criterion uses $d^2$ variables instead of all group-table cochains.\n\\end{proposition}\n\\begin{proof}\nExpand the noncommutative product $(\\widetilde A+p^nB)^q$. Terms with two or more factors $p^nB$ vanish modulo $p^{n+1}$. The terms with one such factor give exactly $p^nL_{\\bar A}(B)$. Therefore the cyclic relation is equivalent to the stated li","\\begin{proposition}[Cyclic relation Jacobian]\\label{prop:cyclic-jacobian}\nLet $G=C_q$ for an integer $q\\geq1$, and let $A\\in\\operatorname{GL}_d(\\mathbb Z/p^n)$ satisfy $A^q=I$. Choose a lift $\\widetilde A$ modulo $p^{n+1}$ and set\n\\[\n c=\\frac{\\widetilde A^q-I}{p^n}\\pmod p,\n \\qquad L_{\\bar A}(B)=\\sum_{i=0}^{q-1}\\bar A^iB\\bar A^{q-1-i}.\n\\]\nThen the lifts of the cyclic action to level $n+1$ are exactly the matrices\n\\[\n \\widetilde A+p^nB\\quad\\text{such that}\\quad L_{\\bar A}(B)=-c.\n\\]\nIf the affine solution set is nonempty, it has $p^{\\dim_k\\ker L_{\\bar A}}$ elements. This criterion uses $d^2$ variables instead of all group-table cochains.\n\\end{proposition}\n\\begin{proof}\nExpand the noncommutative product $(\\widetilde A+p^nB)^q$. Terms with two or more factors $p^nB$ vanish modulo $p^{n+1}$. The terms with one such factor give exactly $p^nL_{\\bar A}(B)$. Therefore the cyclic relation is equivalent to the stated linear equation. All candidate matrices remain invertible because their mod-$p$ reductions are fixed. A nonempty fiber","\\begin{proposition}[Cyclic relation Jacobian]\\label{prop:cyclic-jacobian}\nLet $G=C_q$ for an integer $q\\geq1$, and let $A\\in\\operatorname{GL}_d(\\mathbb Z/p^n)$ satisfy $A^q=I$. Choose a lift $\\widetilde A$ modulo $p^{n+1}$ and set\n\\[\n c=\\frac{\\widetilde A^q-I}{p^n}\\pmod p,\n \\qquad L_{\\bar A}(B)=\\sum_{i=0}^{q-1}\\bar A^iB\\bar A^{q-1-i}.\n\\]\nThen the lifts of the cyclic action to level $n+1$ are exactly the matrices\n\\[\n \\widetilde A+p^nB\\quad\\text{such that}\\quad L_{\\bar A}(B)=-c.\n\\]\nIf the affine solution set is nonempty, it has $p^{\\dim_k\\ker L_{\\bar A}}$ elements. This criterion uses $d^2$ variables instead of all group-table cochains.\n\\end{proposition}\n\\begin{proof}\nExpand the noncommutative product $(\\widetilde A+p^nB)^q$. Terms with two or more factors $p^nB$ vanish modulo $p^{n+1}$. The terms with one such factor give exactly $p^nL_{\\bar A}(B)$. Therefore the cyclic relation is equivalent to the stated linear equation. All candidate matrices remain invertible because their mod-$p$ reductions are fixed. A nonempty fiber of a linear map over $k$ is a translate of its kern","}[Cyclic relation Jacobian]\\label{prop:cyclic-jacobian}\nLet $G=C_q$ for an integer $q\\geq1$, and let $A\\in\\operatorname{GL}_d(\\mathbb Z/p^n)$ satisfy $A^q=I$. Choose a lift $\\widetilde A$ modulo $p^{n+1}$ and set\n\\[\n c=\\frac{\\widetilde A^q-I}{p^n}\\pmod p,\n \\qquad L_{\\bar A}(B)=\\sum_{i=0}^{q-1}\\bar A^iB\\bar A^{q-1-i}.\n\\]\nThen the lifts of the cyclic action to level $n+1$ are exactly the matrices\n\\[\n \\widetilde A+p^nB\\quad\\text{such that}\\quad L_{\\bar A}(B)=-c.\n\\]\nIf the affine solution set is nonempty, it has $p^{\\dim_k\\ker L_{\\bar A}}$ elements. This criterion uses $d^2$ variables instead of all group-table cochains.\n\\end{proposition}\n\\begin{proof}\nExpand the noncommutative product $(\\widetilde A+p^nB)^q$. Terms with two or more factors $p^nB$ vanish modulo $p^{n+1}$. The terms with one such factor give exactly $p^nL_{\\bar A}(B)$. Therefore the cyclic relation is equivalent to the stated linear equation. All candidate matrices remain invertible because their mod-$p$ reductions are fixed. A nonempty fiber of a linear map over $k$ is a translate of its kernel and has the claimed cardin…","name{GL}_d(\\mathbb Z/p^n)$ satisfy $A^q=I$. Choose a lift $\\widetilde A$ modulo $p^{n+1}$ and set\n\\[\n c=\\frac{\\widetilde A^q-I}{p^n}\\pmod p,\n \\qquad L_{\\bar A}(B)=\\sum_{i=0}^{q-1}\\bar A^iB\\bar A^{q-1-i}.\n\\]\nThen the lifts of the cyclic action to level $n+1$ are exactly the matrices\n\\[\n \\widetilde A+p^nB\\quad\\text{such that}\\quad L_{\\bar A}(B)=-c.\n\\]\nIf the affine solution set is nonempty, it has $p^{\\dim_k\\ker L_{\\bar A}}$ elements. This criterion uses $d^2$ variables instead of all group-table cochains.\n\\end{proposition}\n\\begin{proof}\nExpand the noncommutative product $(\\widetilde A+p^nB)^q$. Terms with two or more factors $p^nB$ vanish modulo $p^{n+1}$. The terms with one such factor give exactly $p^nL_{\\bar A}(B)$. Therefore the cyclic relation is equivalent to the stated linear equation. All candidate matrices remain invertible because their mod-$p$ reductions are fixed. A nonempty fiber of a linear map over $k$ is a translate of its kernel and has the claimed cardinality.\n\\end{proof}","$\\widetilde A$ modulo $p^{n+1}$ and set\n\\[\n c=\\frac{\\widetilde A^q-I}{p^n}\\pmod p,\n \\qquad L_{\\bar A}(B)=\\sum_{i=0}^{q-1}\\bar A^iB\\bar A^{q-1-i}.\n\\]\nThen the lifts of the cyclic action to level $n+1$ are exactly the matrices\n\\[\n \\widetilde A+p^nB\\quad\\text{such that}\\quad L_{\\bar A}(B)=-c.\n\\]\nIf the affine solution set is nonempty, it has $p^{\\dim_k\\ker L_{\\bar A}}$ elements. This criterion uses $d^2$ variables instead of all group-table cochains.\n\\end{proposition}\n\\begin{proof}\nExpand the noncommutative product $(\\widetilde A+p^nB)^q$. Terms with two or more factors $p^nB$ vanish modulo $p^{n+1}$. The terms with one such factor give exactly $p^nL_{\\bar A}(B)$. Therefore the cyclic relation is equivalent to the stated linear equation. All candidate matrices remain invertible because their mod-$p$ reductions are fixed. A nonempty fiber of a linear map over $k$ is a translate of its kernel and has the claimed cardinality.\n\\end{proof}","1}\\bar A^iB\\bar A^{q-1-i}.\n\\]\nThen the lifts of the cyclic action to level $n+1$ are exactly the matrices\n\\[\n \\widetilde A+p^nB\\quad\\text{such that}\\quad L_{\\bar A}(B)=-c.\n\\]\nIf the affine solution set is nonempty, it has $p^{\\dim_k\\ker L_{\\bar A}}$ elements. This criterion uses $d^2$ variables instead of all group-table cochains.\n\\end{proposition}\n\\begin{proof}\nExpand the noncommutative product $(\\widetilde A+p^nB)^q$. Terms with two or more factors $p^nB$ vanish modulo $p^{n+1}$. The terms with one such factor give exactly $p^nL_{\\bar A}(B)$. Therefore the cyclic relation is equivalent to the stated linear equation. All candidate matrices remain invertible because their mod-$p$ reductions are fixed. A nonempty fiber of a linear map over $k$ is a translate of its kernel and has the claimed cardinality.\n\\end{proof}","ic action to level $n+1$ are exactly the matrices\n\\[\n \\widetilde A+p^nB\\quad\\text{such that}\\quad L_{\\bar A}(B)=-c.\n\\]\nIf the affine solution set is nonempty, it has $p^{\\dim_k\\ker L_{\\bar A}}$ elements. This criterion uses $d^2$ variables instead of all group-table cochains.\n\\end{proposition}\n\\begin{proof}\nExpand the noncommutative product $(\\widetilde A+p^nB)^q$. Terms with two or more factors $p^nB$ vanish modulo $p^{n+1}$. The terms with one such factor give exactly $p^nL_{\\bar A}(B)$. Therefore the cyclic relation is equivalent to the stated linear equation. All candidate matrices remain invertible because their mod-$p$ reductions are fixed. A nonempty fiber of a linear map over $k$ is a translate of its kernel and has the claimed cardinality.\n\\end{proof}","+1$ are exactly the matrices\n\\[\n \\widetilde A+p^nB\\quad\\text{such that}\\quad L_{\\bar A}(B)=-c.\n\\]\nIf the affine solution set is nonempty, it has $p^{\\dim_k\\ker L_{\\bar A}}$ elements. This criterion uses $d^2$ variables instead of all group-table cochains.\n\\end{proposition}\n\\begin{proof}\nExpand the noncommutative product $(\\widetilde A+p^nB)^q$. Terms with two or more factors $p^nB$ vanish modulo $p^{n+1}$. The terms with one such factor give exactly $p^nL_{\\bar A}(B)$. Therefore the cyclic relation is equivalent to the stated linear equation. All candidate matrices remain invertible because their mod-$p$ reductions are fixed. A nonempty fiber of a linear map over $k$ is a translate of its kernel and has the claimed cardinality.\n\\end{proof}","\\text{such that}\\quad L_{\\bar A}(B)=-c.\n\\]\nIf the affine solution set is nonempty, it has $p^{\\dim_k\\ker L_{\\bar A}}$ elements. This criterion uses $d^2$ variables instead of all group-table cochains.\n\\end{proposition}\n\\begin{proof}\nExpand the noncommutative product $(\\widetilde A+p^nB)^q$. Terms with two or more factors $p^nB$ vanish modulo $p^{n+1}$. The terms with one such factor give exactly $p^nL_{\\bar A}(B)$. Therefore the cyclic relation is equivalent to the stated linear equation. All candidate matrices remain invertible because their mod-$p$ reductions are fixed. A nonempty fiber of a linear map over $k$ is a translate of its kernel and has the claimed cardinality.\n\\end{proof}","variables instead of all group-table cochains.\n\\end{proposition}\n\\begin{proof}\nExpand the noncommutative product $(\\widetilde A+p^nB)^q$. Terms with two or more factors $p^nB$ vanish modulo $p^{n+1}$. The terms with one such factor give exactly $p^nL_{\\bar A}(B)$. Therefore the cyclic relation is equivalent to the stated linear equation. All candidate matrices remain invertible because their mod-$p$ reductions are fixed. A nonempty fiber of a linear map over $k$ is a translate of its kernel and has the claimed cardinality.\n\\end{proof}","\\begin{proof}\nExpand the noncommutative product $(\\widetilde A+p^nB)^q$. Terms with two or more factors $p^nB$ vanish modulo $p^{n+1}$. The terms with one such factor give exactly $p^nL_{\\bar A}(B)$. Therefore the cyclic relation is equivalent to the stated linear equation. All candidate matrices remain invertible because their mod-$p$ reductions are fixed. A nonempty fiber of a linear map over $k$ is a translate of its kernel and has the claimed cardinality.\n\\end{proof}","For $A=I+pJ$ modulo $p^2$ and $q=p$, the residual is $c=J$ while $L_I(B)=pB=0$ in $k$. This recovers the one-step obstruction without exhaustive testing. At other inputs the Jacobian may have nonzero rank, so some nonzero initial relation residuals are removable rather than persistent.","\\begin{proposition}[A profinite lift need not preserve the finite quotient]\\label{prop:profinite-quotient-boundary}\nLet $p$ be odd and $J\\in M_d(\\mathbb Z_p)$ be invertible modulo $p$. The action $C_p\\to\\operatorname{GL}_d(\\mathbb Z/p^2)$ given by $I+pJ$ has no integral lift as an action of $C_p$. Its pullback along $\\mathbb Z_p\\twoheadrightarrow C_p$, however, has a continuous integral lift\n\\[\n \\rho(x)=(I+pJ)^x\n =\\sum_{j\\geq0}\\binom{x}{j}p^jJ^j,\n \\qquad x\\in\\mathbb Z_p.\n\\]\n\\end{proposition}\n\\begin{proof}\nFor each $j$, the polynomial $\\binom{x}{j}$ is continuous on $\\mathbb Z_p$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible b","\\begin{proposition}[A profinite lift need not preserve the finite quotient]\\label{prop:profinite-quotient-boundary}\nLet $p$ be odd and $J\\in M_d(\\mathbb Z_p)$ be invertible modulo $p$. The action $C_p\\to\\operatorname{GL}_d(\\mathbb Z/p^2)$ given by $I+pJ$ has no integral lift as an action of $C_p$. Its pullback along $\\mathbb Z_p\\twoheadrightarrow C_p$, however, has a continuous integral lift\n\\[\n \\rho(x)=(I+pJ)^x\n =\\sum_{j\\geq0}\\binom{x}{j}p^jJ^j,\n \\qquad x\\in\\mathbb Z_p.\n\\]\n\\end{proposition}\n\\begin{proof}\nFor each $j$, the polynomial $\\binom{x}{j}$ is continuous on $\\mathbb Z_p$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer ar","\\begin{proposition}[A profinite lift need not preserve the finite quotient]\\label{prop:profinite-quotient-boundary}\nLet $p$ be odd and $J\\in M_d(\\mathbb Z_p)$ be invertible modulo $p$. The action $C_p\\to\\operatorname{GL}_d(\\mathbb Z/p^2)$ given by $I+pJ$ has no integral lift as an action of $C_p$. Its pullback along $\\mathbb Z_p\\twoheadrightarrow C_p$, however, has a continuous integral lift\n\\[\n \\rho(x)=(I+pJ)^x\n =\\sum_{j\\geq0}\\binom{x}{j}p^jJ^j,\n \\qquad x\\in\\mathbb Z_p.\n\\]\n\\end{proposition}\n\\begin{proof}\nFor each $j$, the polynomial $\\binom{x}{j}$ is continuous on $\\mathbb Z_p$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual m","\\begin{proposition}[A profinite lift need not preserve the finite quotient]\\label{prop:profinite-quotient-boundary}\nLet $p$ be odd and $J\\in M_d(\\mathbb Z_p)$ be invertible modulo $p$. The action $C_p\\to\\operatorname{GL}_d(\\mathbb Z/p^2)$ given by $I+pJ$ has no integral lift as an action of $C_p$. Its pullback along $\\mathbb Z_p\\twoheadrightarrow C_p$, however, has a continuous integral lift\n\\[\n \\rho(x)=(I+pJ)^x\n =\\sum_{j\\geq0}\\binom{x}{j}p^jJ^j,\n \\qquad x\\in\\mathbb Z_p.\n\\]\n\\end{proposition}\n\\begin{proof}\nFor each $j$, the polynomial $\\binom{x}{j}$ is continuous on $\\mathbb Z_p$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual matrix power. The nonnegative integers are dense in $\\ma","\\begin{proposition}[A profinite lift need not preserve the finite quotient]\\label{prop:profinite-quotient-boundary}\nLet $p$ be odd and $J\\in M_d(\\mathbb Z_p)$ be invertible modulo $p$. The action $C_p\\to\\operatorname{GL}_d(\\mathbb Z/p^2)$ given by $I+pJ$ has no integral lift as an action of $C_p$. Its pullback along $\\mathbb Z_p\\twoheadrightarrow C_p$, however, has a continuous integral lift\n\\[\n \\rho(x)=(I+pJ)^x\n =\\sum_{j\\geq0}\\binom{x}{j}p^jJ^j,\n \\qquad x\\in\\mathbb Z_p.\n\\]\n\\end{proposition}\n\\begin{proof}\nFor each $j$, the polynomial $\\binom{x}{j}$ is continuous on $\\mathbb Z_p$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual matrix power. The nonnegative integers are dense in $\\mathbb Z_p$, so co","\\begin{proposition}[A profinite lift need not preserve the finite quotient]\\label{prop:profinite-quotient-boundary}\nLet $p$ be odd and $J\\in M_d(\\mathbb Z_p)$ be invertible modulo $p$. The action $C_p\\to\\operatorname{GL}_d(\\mathbb Z/p^2)$ given by $I+pJ$ has no integral lift as an action of $C_p$. Its pullback along $\\mathbb Z_p\\twoheadrightarrow C_p$, however, has a continuous integral lift\n\\[\n \\rho(x)=(I+pJ)^x\n =\\sum_{j\\geq0}\\binom{x}{j}p^jJ^j,\n \\qquad x\\in\\mathbb Z_p.\n\\]\n\\end{proposition}\n\\begin{proof}\nFor each $j$, the polynomial $\\binom{x}{j}$ is continuous on $\\mathbb Z_p$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual matrix power. The nonnegative integers are dense in $\\mathbb Z_p$, so continuity extends the identity $\\rho(x+y)=\\r","\\begin{proposition}[A profinite lift need not preserve the finite quotient]\\label{prop:profinite-quotient-boundary}\nLet $p$ be odd and $J\\in M_d(\\mathbb Z_p)$ be invertible modulo $p$. The action $C_p\\to\\operatorname{GL}_d(\\mathbb Z/p^2)$ given by $I+pJ$ has no integral lift as an action of $C_p$. Its pullback along $\\mathbb Z_p\\twoheadrightarrow C_p$, however, has a continuous integral lift\n\\[\n \\rho(x)=(I+pJ)^x\n =\\sum_{j\\geq0}\\binom{x}{j}p^jJ^j,\n \\qquad x\\in\\mathbb Z_p.\n\\]\n\\end{proposition}\n\\begin{proof}\nFor each $j$, the polynomial $\\binom{x}{j}$ is continuous on $\\mathbb Z_p$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual matrix power. The nonnegative integers are dense in $\\mathbb Z_p$, so continuity extends the identity $\\rho(x+y)=\\rho(x)\\rho(y)$ to all pairs. Each value is $I$ modulo $p$","\\begin{proposition}[A profinite lift need not preserve the finite quotient]\\label{prop:profinite-quotient-boundary}\nLet $p$ be odd and $J\\in M_d(\\mathbb Z_p)$ be invertible modulo $p$. The action $C_p\\to\\operatorname{GL}_d(\\mathbb Z/p^2)$ given by $I+pJ$ has no integral lift as an action of $C_p$. Its pullback along $\\mathbb Z_p\\twoheadrightarrow C_p$, however, has a continuous integral lift\n\\[\n \\rho(x)=(I+pJ)^x\n =\\sum_{j\\geq0}\\binom{x}{j}p^jJ^j,\n \\qquad x\\in\\mathbb Z_p.\n\\]\n\\end{proposition}\n\\begin{proof}\nFor each $j$, the polynomial $\\binom{x}{j}$ is continuous on $\\mathbb Z_p$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual matrix power. The nonnegative integers are dense in $\\mathbb Z_p$, so continuity extends the identity $\\rho(x+y)=\\rho(x)\\rho(y)$ to all pairs. Each value is $I$ modulo $p$ and hence invertible.","p$ be odd and $J\\in M_d(\\mathbb Z_p)$ be invertible modulo $p$. The action $C_p\\to\\operatorname{GL}_d(\\mathbb Z/p^2)$ given by $I+pJ$ has no integral lift as an action of $C_p$. Its pullback along $\\mathbb Z_p\\twoheadrightarrow C_p$, however, has a continuous integral lift\n\\[\n \\rho(x)=(I+pJ)^x\n =\\sum_{j\\geq0}\\binom{x}{j}p^jJ^j,\n \\qquad x\\in\\mathbb Z_p.\n\\]\n\\end{proposition}\n\\begin{proof}\nFor each $j$, the polynomial $\\binom{x}{j}$ is continuous on $\\mathbb Z_p$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual matrix power. The nonnegative integers are dense in $\\mathbb Z_p$, so continuity extends the identity $\\rho(x+y)=\\rho(x)\\rho(y)$ to all pairs. Each value is $I$ modulo $p$ and hence invertible.","M_d(\\mathbb Z_p)$ be invertible modulo $p$. The action $C_p\\to\\operatorname{GL}_d(\\mathbb Z/p^2)$ given by $I+pJ$ has no integral lift as an action of $C_p$. Its pullback along $\\mathbb Z_p\\twoheadrightarrow C_p$, however, has a continuous integral lift\n\\[\n \\rho(x)=(I+pJ)^x\n =\\sum_{j\\geq0}\\binom{x}{j}p^jJ^j,\n \\qquad x\\in\\mathbb Z_p.\n\\]\n\\end{proposition}\n\\begin{proof}\nFor each $j$, the polynomial $\\binom{x}{j}$ is continuous on $\\mathbb Z_p$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual matrix power. The nonnegative integers are dense in $\\mathbb Z_p$, so continuity extends the identity $\\rho(x+y)=\\rho(x)\\rho(y)$ to all pairs. Each value is $I$ modulo $p$ and hence invertible.","modulo $p$. The action $C_p\\to\\operatorname{GL}_d(\\mathbb Z/p^2)$ given by $I+pJ$ has no integral lift as an action of $C_p$. Its pullback along $\\mathbb Z_p\\twoheadrightarrow C_p$, however, has a continuous integral lift\n\\[\n \\rho(x)=(I+pJ)^x\n =\\sum_{j\\geq0}\\binom{x}{j}p^jJ^j,\n \\qquad x\\in\\mathbb Z_p.\n\\]\n\\end{proposition}\n\\begin{proof}\nFor each $j$, the polynomial $\\binom{x}{j}$ is continuous on $\\mathbb Z_p$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual matrix power. The nonnegative integers are dense in $\\mathbb Z_p$, so continuity extends the identity $\\rho(x+y)=\\rho(x)\\rho(y)$ to all pairs. Each value is $I$ modulo $p$ and hence invertible.","\\mathbb Z/p^2)$ given by $I+pJ$ has no integral lift as an action of $C_p$. Its pullback along $\\mathbb Z_p\\twoheadrightarrow C_p$, however, has a continuous integral lift\n\\[\n \\rho(x)=(I+pJ)^x\n =\\sum_{j\\geq0}\\binom{x}{j}p^jJ^j,\n \\qquad x\\in\\mathbb Z_p.\n\\]\n\\end{proposition}\n\\begin{proof}\nFor each $j$, the polynomial $\\binom{x}{j}$ is continuous on $\\mathbb Z_p$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual matrix power. The nonnegative integers are dense in $\\mathbb Z_p$, so continuity extends the identity $\\rho(x+y)=\\rho(x)\\rho(y)$ to all pairs. Each value is $I$ modulo $p$ and hence invertible.","$\\mathbb Z_p\\twoheadrightarrow C_p$, however, has a continuous integral lift\n\\[\n \\rho(x)=(I+pJ)^x\n =\\sum_{j\\geq0}\\binom{x}{j}p^jJ^j,\n \\qquad x\\in\\mathbb Z_p.\n\\]\n\\end{proposition}\n\\begin{proof}\nFor each $j$, the polynomial $\\binom{x}{j}$ is continuous on $\\mathbb Z_p$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual matrix power. The nonnegative integers are dense in $\\mathbb Z_p$, so continuity extends the identity $\\rho(x+y)=\\rho(x)\\rho(y)$ to all pairs. Each value is $I$ modulo $p$ and hence invertible.","eadrightarrow C_p$, however, has a continuous integral lift\n\\[\n \\rho(x)=(I+pJ)^x\n =\\sum_{j\\geq0}\\binom{x}{j}p^jJ^j,\n \\qquad x\\in\\mathbb Z_p.\n\\]\n\\end{proposition}\n\\begin{proof}\nFor each $j$, the polynomial $\\binom{x}{j}$ is continuous on $\\mathbb Z_p$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual matrix power. The nonnegative integers are dense in $\\mathbb Z_p$, so continuity extends the identity $\\rho(x+y)=\\rho(x)\\rho(y)$ to all pairs. Each value is $I$ modulo $p$ and hence invertible.","\\rho(x)=(I+pJ)^x\n =\\sum_{j\\geq0}\\binom{x}{j}p^jJ^j,\n \\qquad x\\in\\mathbb Z_p.\n\\]\n\\end{proposition}\n\\begin{proof}\nFor each $j$, the polynomial $\\binom{x}{j}$ is continuous on $\\mathbb Z_p$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual matrix power. The nonnegative integers are dense in $\\mathbb Z_p$, so continuity extends the identity $\\rho(x+y)=\\rho(x)\\rho(y)$ to all pairs. Each value is $I$ modulo $p$ and hence invertible.","\\geq0}\\binom{x}{j}p^jJ^j,\n \\qquad x\\in\\mathbb Z_p.\n\\]\n\\end{proposition}\n\\begin{proof}\nFor each $j$, the polynomial $\\binom{x}{j}$ is continuous on $\\mathbb Z_p$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual matrix power. The nonnegative integers are dense in $\\mathbb Z_p$, so continuity extends the identity $\\rho(x+y)=\\rho(x)\\rho(y)$ to all pairs. Each value is $I$ modulo $p$ and hence invertible.","omial $\\binom{x}{j}$ is continuous on $\\mathbb Z_p$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual matrix power. The nonnegative integers are dense in $\\mathbb Z_p$, so continuity extends the identity $\\rho(x+y)=\\rho(x)\\rho(y)$ to all pairs. Each value is $I$ modulo $p$ and hence invertible.","$ and takes values there: approximate $x$ by nonnegative integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual matrix power. The nonnegative integers are dense in $\\mathbb Z_p$, so continuity extends the identity $\\rho(x+y)=\\rho(x)\\rho(y)$ to all pairs. Each value is $I$ modulo $p$ and hence invertible.","e integers, where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual matrix power. The nonnegative integers are dense in $\\mathbb Z_p$, so continuity extends the identity $\\rho(x+y)=\\rho(x)\\rho(y)$ to all pairs. Each value is $I$ modulo $p$ and hence invertible.","where the binomial coefficient is integral, and use closedness of $\\mathbb Z_p$ in $\\mathbb Q_p$. Thus the series converges uniformly because its $j$th term is divisible by $p^j$. For nonnegative integer arguments it is the usual matrix power. The nonnegative integers are dense in $\\mathbb Z_p$, so continuity extends the identity $\\rho(x+y)=\\rho(x)\\rho(y)$ to all pairs. Each value is $I$ modulo $p$ and hence invertible.","Modulo $p^2$ the series is $I+pxJ$, which depends only on $x\\bmod p$ and gives the specified finite-level action. But $\\rho(p)\\equiv I+p^2J\\ne I\\pmod{p^3}$, so the integral lift does not factor through $C_p$. Nonexistence of an integral $C_p$-lift is \\cref{thm:one-step-group-lift}.\n\\end{proof}","The same warning applies to finite quotients used in a Galois problem. A relation imposed by fixing one finite quotient may be absent from the full profinite source. Failure to lift through that fixed quotient therefore does not by itself obstruct a continuous lift with growing image. Conversely, the finite-group horizon depends on $v_p(|G|)$ and gives no uniform bound for arbitrary growing finite quotients.","There are now three distinct tests. A chosen functor's exchange cone is a necessary diagnostic on its detectable inputs, but the central-sign example shows it is not a complete lifting invariant. The class $o_n(\\rho_n)$ is a complete one-step obstruction because its vanishing is proved equivalent to solving the actual multiplicativity equations. Finally, the finite horizon plus convergent correction turns a finite witness into a full $\\mathbb Z_p$-action for a fixed finite group. This gives a positive reconstruction theorem without killing the residual in a Verdier quotient.","hosen functor's exchange cone is a necessary diagnostic on its detectable inputs, but the central-sign example shows it is not a complete lifting invariant. The class $o_n(\\rho_n)$ is a complete one-step obstruction because its vanishing is proved equivalent to solving the actual multiplicativity equations. Finally, the finite horizon plus convergent correction turns a finite witness into a full $\\mathbb Z_p$-action for a fixed finite group. This gives a positive reconstruction theorem without killing the residual in a Verdier quotient.","In the two-axis vocabulary, the resolution fiber is the actual lift groupoid, the transformation direction includes changes of basis, the relation residual is a normalized cocycle, admissible repairs are cochains, and its class in $H^2$ is the obstruction. The averaging theorem then supplies a quantitative regime in which repeated repairs converge. These identifications recover and connect classical deformation and ultrametric-stability mathematics. They do not make the abstract obstruction formalism a proof of automorphic existence or Langlands reciprocity.","This section keeps the profinite source fixed while allowing its finite images to grow. It separates two independent resolution coordinates: coefficient precision and the finite quotient through which a particular observation factors. Its strongest constructive conclusion is a complete finite-matrix lifting criterion for continuous rank-two representations of $\\mathbb Z_p^r$. In arbitrary rank the same criterion holds when a specified normalized generator is cyclic modulo $p$. No prescribed determinant, ramification type, Hodge condition, or automorphic realization is included in these statements.","to grow. It separates two independent resolution coordinates: coefficient precision and the finite quotient through which a particular observation factors. Its strongest constructive conclusion is a complete finite-matrix lifting criterion for continuous rank-two representations of $\\mathbb Z_p^r$. In arbitrary rank the same criterion holds when a specified normalized generator is cyclic modulo $p$. No prescribed determinant, ramification type, Hodge condition, or automorphic realization is included in these statements.","The continuous cohomology used below is the cohomology of continuous inhomogeneous cochains with a \\emph{finite discrete} coefficient module. The finite-quotient description is classical \\cite[Chapter II, Section 4, Proposition 4.2]{MilneCFT2020}; see also \\cite{StacksContinuous}. The free pro-$p$ construction and cyclic-centralizer argument are given directly. The claims are written deductions under explicit hypotheses, not assertions of historical novelty.","Fix a profinite group $\\Gamma$, a prime $p$, $n,d\\geq1$, and a continuous representation\n\\[\n \\rho_n:\\Gamma\\longrightarrow\\operatorname{GL}_d(R_n),\n \\qquad R_n=\\mathbb Z/p^n.\n\\]\nThe target is finite and discrete. Give $V=M_d(\\mathbb F_p)$ the adjoint action of $\\bar\\rho$. Choose a section of the reduction map on the finite set $\\rho_n(\\Gamma)$, taking the identity to the identity. Composing this section with $\\rho_n$ gives a continuous matrix-valued lift $A_g$ to $R_{n+1}$, not necessarily multiplicative. Define\n\\[\n c_A(g,h)=\\frac{A_gA_hA_{gh}^{-1}-I}{p^n}\\pmod p.\n\\]\nHere division is the identification of $p^nR_{n+1}$ with $\\mathbb F_p$, as in \\cref{thm:relation-H2}.","ite group $\\Gamma$, a prime $p$, $n,d\\geq1$, and a continuous representation\n\\[\n \\rho_n:\\Gamma\\longrightarrow\\operatorname{GL}_d(R_n),\n \\qquad R_n=\\mathbb Z/p^n.\n\\]\nThe target is finite and discrete. Give $V=M_d(\\mathbb F_p)$ the adjoint action of $\\bar\\rho$. Choose a section of the reduction map on the finite set $\\rho_n(\\Gamma)$, taking the identity to the identity. Composing this section with $\\rho_n$ gives a continuous matrix-valued lift $A_g$ to $R_{n+1}$, not necessarily multiplicative. Define\n\\[\n c_A(g,h)=\\frac{A_gA_hA_{gh}^{-1}-I}{p^n}\\pmod p.\n\\]\nHere division is the identification of $p^nR_{n+1}$ with $\\mathbb F_p$, as in \\cref{thm:relation-H2}.","d a continuous representation\n\\[\n \\rho_n:\\Gamma\\longrightarrow\\operatorname{GL}_d(R_n),\n \\qquad R_n=\\mathbb Z/p^n.\n\\]\nThe target is finite and discrete. Give $V=M_d(\\mathbb F_p)$ the adjoint action of $\\bar\\rho$. Choose a section of the reduction map on the finite set $\\rho_n(\\Gamma)$, taking the identity to the identity. Composing this section with $\\rho_n$ gives a continuous matrix-valued lift $A_g$ to $R_{n+1}$, not necessarily multiplicative. Define\n\\[\n c_A(g,h)=\\frac{A_gA_hA_{gh}^{-1}-I}{p^n}\\pmod p.\n\\]\nHere division is the identification of $p^nR_{n+1}$ with $\\mathbb F_p$, as in \\cref{thm:relation-H2}.","nuous representation\n\\[\n \\rho_n:\\Gamma\\longrightarrow\\operatorname{GL}_d(R_n),\n \\qquad R_n=\\mathbb Z/p^n.\n\\]\nThe target is finite and discrete. Give $V=M_d(\\mathbb F_p)$ the adjoint action of $\\bar\\rho$. Choose a section of the reduction map on the finite set $\\rho_n(\\Gamma)$, taking the identity to the identity. Composing this section with $\\rho_n$ gives a continuous matrix-valued lift $A_g$ to $R_{n+1}$, not necessarily multiplicative. Define\n\\[\n c_A(g,h)=\\frac{A_gA_hA_{gh}^{-1}-I}{p^n}\\pmod p.\n\\]\nHere division is the identification of $p^nR_{n+1}$ with $\\mathbb F_p$, as in \\cref{thm:relation-H2}.","ow\\operatorname{GL}_d(R_n),\n \\qquad R_n=\\mathbb Z/p^n.\n\\]\nThe target is finite and discrete. Give $V=M_d(\\mathbb F_p)$ the adjoint action of $\\bar\\rho$. Choose a section of the reduction map on the finite set $\\rho_n(\\Gamma)$, taking the identity to the identity. Composing this section with $\\rho_n$ gives a continuous matrix-valued lift $A_g$ to $R_{n+1}$, not necessarily multiplicative. Define\n\\[\n c_A(g,h)=\\frac{A_gA_hA_{gh}^{-1}-I}{p^n}\\pmod p.\n\\]\nHere division is the identification of $p^nR_{n+1}$ with $\\mathbb F_p$, as in \\cref{thm:relation-H2}.","ive $V=M_d(\\mathbb F_p)$ the adjoint action of $\\bar\\rho$. Choose a section of the reduction map on the finite set $\\rho_n(\\Gamma)$, taking the identity to the identity. Composing this section with $\\rho_n$ gives a continuous matrix-valued lift $A_g$ to $R_{n+1}$, not necessarily multiplicative. Define\n\\[\n c_A(g,h)=\\frac{A_gA_hA_{gh}^{-1}-I}{p^n}\\pmod p.\n\\]\nHere division is the identification of $p^nR_{n+1}$ with $\\mathbb F_p$, as in \\cref{thm:relation-H2}.","F_p)$ the adjoint action of $\\bar\\rho$. Choose a section of the reduction map on the finite set $\\rho_n(\\Gamma)$, taking the identity to the identity. Composing this section with $\\rho_n$ gives a continuous matrix-valued lift $A_g$ to $R_{n+1}$, not necessarily multiplicative. Define\n\\[\n c_A(g,h)=\\frac{A_gA_hA_{gh}^{-1}-I}{p^n}\\pmod p.\n\\]\nHere division is the identification of $p^nR_{n+1}$ with $\\mathbb F_p$, as in \\cref{thm:relation-H2}.","\\begin{theorem}[Complete continuous one-step obstruction]\\label{thm:continuous-H2}\nThe class\n\\[\n o_n^{\\mathrm{cts}}(\\rho_n)=[c_A]\\in H^2_{\\mathrm{cts}}(\\Gamma,V)\n\\]\nis independent of the continuous choices. It vanishes if and only if $\\rho_n$ has a \\emph{continuous} lift to $\\operatorname{GL}_d(R_{n+1})$ with its specified reduction. More precisely, the repair equation is\n\\[\n \\delta b=-c_A,\\qquad b:\\Gamma\\to V\\text{ continuous and normalized},\n\\]\nand the repaired matrices are $(I+p^nb(g))A_g$. When lifts exist, their framed set is a torsor for $Z^1_{\\mathrm{cts}}(\\Gamma,V)$. Strict-conjugacy classes and stabilizers are governed by $H^1_{\\mathrm{cts}}$ and $H^0_{\\mathrm{cts}}$, respectively.\n\\end{theorem}\n\\begin{proof}\nThe associativity and square-zero computations in \\cref{thm:relation-H2} hold pointwise without a finiteness assumption on the source.","\\begin{theorem}[Complete continuous one-step obstruction]\\label{thm:continuous-H2}\nThe class\n\\[\n o_n^{\\mathrm{cts}}(\\rho_n)=[c_A]\\in H^2_{\\mathrm{cts}}(\\Gamma,V)\n\\]\nis independent of the continuous choices. It vanishes if and only if $\\rho_n$ has a \\emph{continuous} lift to $\\operatorname{GL}_d(R_{n+1})$ with its specified reduction. More precisely, the repair equation is\n\\[\n \\delta b=-c_A,\\qquad b:\\Gamma\\to V\\text{ continuous and normalized},\n\\]\nand the repaired matrices are $(I+p^nb(g))A_g$. When lifts exist, their framed set is a torsor for $Z^1_{\\mathrm{cts}}(\\Gamma,V)$. Strict-conjugacy classes and stabilizers are governed by $H^1_{\\mathrm{cts}}$ and $H^0_{\\mathrm{cts}}$, respectively.\n\\end{theorem}\n\\begin{proof}\nThe associativity and square-zero computations in \\cref{thm:relation-H2} hold pointwise without a finiteness assumption on the source. All the resulting cochains are continuous because the targets are finite and t","\\begin{theorem}[Complete continuous one-step obstruction]\\label{thm:continuous-H2}\nThe class\n\\[\n o_n^{\\mathrm{cts}}(\\rho_n)=[c_A]\\in H^2_{\\mathrm{cts}}(\\Gamma,V)\n\\]\nis independent of the continuous choices. It vanishes if and only if $\\rho_n$ has a \\emph{continuous} lift to $\\operatorname{GL}_d(R_{n+1})$ with its specified reduction. More precisely, the repair equation is\n\\[\n \\delta b=-c_A,\\qquad b:\\Gamma\\to V\\text{ continuous and normalized},\n\\]\nand the repaired matrices are $(I+p^nb(g))A_g$. When lifts exist, their framed set is a torsor for $Z^1_{\\mathrm{cts}}(\\Gamma,V)$. Strict-conjugacy classes and stabilizers are governed by $H^1_{\\mathrm{cts}}$ and $H^0_{\\mathrm{cts}}$, respectively.\n\\end{theorem}\n\\begin{proof}\nThe associativity and square-zero computations in \\cref{thm:relation-H2} hold pointwise without a finiteness assumption on the source. All the resulting cochains are continuous because the targets are finite and the chosen lifts are continuous. Any other continuous lift diffe","\\begin{theorem}[Complete continuous one-step obstruction]\\label{thm:continuous-H2}\nThe class\n\\[\n o_n^{\\mathrm{cts}}(\\rho_n)=[c_A]\\in H^2_{\\mathrm{cts}}(\\Gamma,V)\n\\]\nis independent of the continuous choices. It vanishes if and only if $\\rho_n$ has a \\emph{continuous} lift to $\\operatorname{GL}_d(R_{n+1})$ with its specified reduction. More precisely, the repair equation is\n\\[\n \\delta b=-c_A,\\qquad b:\\Gamma\\to V\\text{ continuous and normalized},\n\\]\nand the repaired matrices are $(I+p^nb(g))A_g$. When lifts exist, their framed set is a torsor for $Z^1_{\\mathrm{cts}}(\\Gamma,V)$. Strict-conjugacy classes and stabilizers are governed by $H^1_{\\mathrm{cts}}$ and $H^0_{\\mathrm{cts}}$, respectively.\n\\end{theorem}\n\\begin{proof}\nThe associativity and square-zero computations in \\cref{thm:relation-H2} hold pointwise without a finiteness assumption on the source. All the resulting cochains are continuous because the targets are finite and the chosen lifts are continuous. Any other continuous lift differs uniquely by $(I+p^nb(g))$ with continuous $b$, and its residual is $c_A+\\delta b$. Hence a ze…","The class\n\\[\n o_n^{\\mathrm{cts}}(\\rho_n)=[c_A]\\in H^2_{\\mathrm{cts}}(\\Gamma,V)\n\\]\nis independent of the continuous choices. It vanishes if and only if $\\rho_n$ has a \\emph{continuous} lift to $\\operatorname{GL}_d(R_{n+1})$ with its specified reduction. More precisely, the repair equation is\n\\[\n \\delta b=-c_A,\\qquad b:\\Gamma\\to V\\text{ continuous and normalized},\n\\]\nand the repaired matrices are $(I+p^nb(g))A_g$. When lifts exist, their framed set is a torsor for $Z^1_{\\mathrm{cts}}(\\Gamma,V)$. Strict-conjugacy classes and stabilizers are governed by $H^1_{\\mathrm{cts}}$ and $H^0_{\\mathrm{cts}}$, respectively.\n\\end{theorem}\n\\begin{proof}\nThe associativity and square-zero computations in \\cref{thm:relation-H2} hold pointwise without a finiteness assumption on the source. All the resulting cochains are continuous because the targets are finite and the chosen lifts are continuous. Any other continuous lift differs uniquely by $(I+p^nb(g))$ with continuous $b$, and its residual is $c_A+\\delta b$. Hence a zero class is equivalent to an actual continuous repair. Conversely an actual continu…","(\\Gamma,V)\n\\]\nis independent of the continuous choices. It vanishes if and only if $\\rho_n$ has a \\emph{continuous} lift to $\\operatorname{GL}_d(R_{n+1})$ with its specified reduction. More precisely, the repair equation is\n\\[\n \\delta b=-c_A,\\qquad b:\\Gamma\\to V\\text{ continuous and normalized},\n\\]\nand the repaired matrices are $(I+p^nb(g))A_g$. When lifts exist, their framed set is a torsor for $Z^1_{\\mathrm{cts}}(\\Gamma,V)$. Strict-conjugacy classes and stabilizers are governed by $H^1_{\\mathrm{cts}}$ and $H^0_{\\mathrm{cts}}$, respectively.\n\\end{theorem}\n\\begin{proof}\nThe associativity and square-zero computations in \\cref{thm:relation-H2} hold pointwise without a finiteness assumption on the source. All the resulting cochains are continuous because the targets are finite and the chosen lifts are continuous. Any other continuous lift differs uniquely by $(I+p^nb(g))$ with continuous $b$, and its residual is $c_A+\\delta b$. Hence a zero class is equivalent to an actual continuous repair. Conversely an actual continuous homomorphism is such a repair. The difference of two repairs is …","n$ has a \\emph{continuous} lift to $\\operatorname{GL}_d(R_{n+1})$ with its specified reduction. More precisely, the repair equation is\n\\[\n \\delta b=-c_A,\\qquad b:\\Gamma\\to V\\text{ continuous and normalized},\n\\]\nand the repaired matrices are $(I+p^nb(g))A_g$. When lifts exist, their framed set is a torsor for $Z^1_{\\mathrm{cts}}(\\Gamma,V)$. Strict-conjugacy classes and stabilizers are governed by $H^1_{\\mathrm{cts}}$ and $H^0_{\\mathrm{cts}}$, respectively.\n\\end{theorem}\n\\begin{proof}\nThe associativity and square-zero computations in \\cref{thm:relation-H2} hold pointwise without a finiteness assumption on the source. All the resulting cochains are continuous because the targets are finite and the chosen lifts are continuous. Any other continuous lift differs uniquely by $(I+p^nb(g))$ with continuous $b$, and its residual is $c_A+\\delta b$. Hence a zero class is equivalent to an actual continuous repair. Conversely an actual continuous homomorphism is such a repair. The difference of two repairs is a continuous $1$-cocycle. Conjugation by $I+p^nX$ changes it by a coboundary, with kernel…","} lift to $\\operatorname{GL}_d(R_{n+1})$ with its specified reduction. More precisely, the repair equation is\n\\[\n \\delta b=-c_A,\\qquad b:\\Gamma\\to V\\text{ continuous and normalized},\n\\]\nand the repaired matrices are $(I+p^nb(g))A_g$. When lifts exist, their framed set is a torsor for $Z^1_{\\mathrm{cts}}(\\Gamma,V)$. Strict-conjugacy classes and stabilizers are governed by $H^1_{\\mathrm{cts}}$ and $H^0_{\\mathrm{cts}}$, respectively.\n\\end{theorem}\n\\begin{proof}\nThe associativity and square-zero computations in \\cref{thm:relation-H2} hold pointwise without a finiteness assumption on the source. All the resulting cochains are continuous because the targets are finite and the chosen lifts are continuous. Any other continuous lift differs uniquely by $(I+p^nb(g))$ with continuous $b$, and its residual is $c_A+\\delta b$. Hence a zero class is equivalent to an actual continuous repair. Conversely an actual continuous homomorphism is such a repair. The difference of two repairs is a continuous $1$-cocycle. Conjugation by $I+p^nX$ changes it by a coboundary, with kernel the invariant elements o…","rs are governed by $H^1_{\\mathrm{cts}}$ and $H^0_{\\mathrm{cts}}$, respectively.\n\\end{theorem}\n\\begin{proof}\nThe associativity and square-zero computations in \\cref{thm:relation-H2} hold pointwise without a finiteness assumption on the source. All the resulting cochains are continuous because the targets are finite and the chosen lifts are continuous. Any other continuous lift differs uniquely by $(I+p^nb(g))$ with continuous $b$, and its residual is $c_A+\\delta b$. Hence a zero class is equivalent to an actual continuous repair. Conversely an actual continuous homomorphism is such a repair. The difference of two repairs is a continuous $1$-cocycle. Conjugation by $I+p^nX$ changes it by a coboundary, with kernel the invariant elements of $V$, exactly as in the finite case.\n\\end{proof}","rm{cts}}$ and $H^0_{\\mathrm{cts}}$, respectively.\n\\end{theorem}\n\\begin{proof}\nThe associativity and square-zero computations in \\cref{thm:relation-H2} hold pointwise without a finiteness assumption on the source. All the resulting cochains are continuous because the targets are finite and the chosen lifts are continuous. Any other continuous lift differs uniquely by $(I+p^nb(g))$ with continuous $b$, and its residual is $c_A+\\delta b$. Hence a zero class is equivalent to an actual continuous repair. Conversely an actual continuous homomorphism is such a repair. The difference of two repairs is a continuous $1$-cocycle. Conjugation by $I+p^nX$ changes it by a coboundary, with kernel the invariant elements of $V$, exactly as in the finite case.\n\\end{proof}","rm{cts}}$, respectively.\n\\end{theorem}\n\\begin{proof}\nThe associativity and square-zero computations in \\cref{thm:relation-H2} hold pointwise without a finiteness assumption on the source. All the resulting cochains are continuous because the targets are finite and the chosen lifts are continuous. Any other continuous lift differs uniquely by $(I+p^nb(g))$ with continuous $b$, and its residual is $c_A+\\delta b$. Hence a zero class is equivalent to an actual continuous repair. Conversely an actual continuous homomorphism is such a repair. The difference of two repairs is a continuous $1$-cocycle. Conjugation by $I+p^nX$ changes it by a coboundary, with kernel the invariant elements of $V$, exactly as in the finite case.\n\\end{proof}","l the resulting cochains are continuous because the targets are finite and the chosen lifts are continuous. Any other continuous lift differs uniquely by $(I+p^nb(g))$ with continuous $b$, and its residual is $c_A+\\delta b$. Hence a zero class is equivalent to an actual continuous repair. Conversely an actual continuous homomorphism is such a repair. The difference of two repairs is a continuous $1$-cocycle. Conjugation by $I+p^nX$ changes it by a coboundary, with kernel the invariant elements of $V$, exactly as in the finite case.\n\\end{proof}","continuous because the targets are finite and the chosen lifts are continuous. Any other continuous lift differs uniquely by $(I+p^nb(g))$ with continuous $b$, and its residual is $c_A+\\delta b$. Hence a zero class is equivalent to an actual continuous repair. Conversely an actual continuous homomorphism is such a repair. The difference of two repairs is a continuous $1$-cocycle. Conjugation by $I+p^nX$ changes it by a coboundary, with kernel the invariant elements of $V$, exactly as in the finite case.\n\\end{proof}","ous. Any other continuous lift differs uniquely by $(I+p^nb(g))$ with continuous $b$, and its residual is $c_A+\\delta b$. Hence a zero class is equivalent to an actual continuous repair. Conversely an actual continuous homomorphism is such a repair. The difference of two repairs is a continuous $1$-cocycle. Conjugation by $I+p^nX$ changes it by a coboundary, with kernel the invariant elements of $V$, exactly as in the finite case.\n\\end{proof}","\\begin{theorem}[Source refinement and inflation]\\label{thm:source-inflation}\nChoose an open normal $U\\subseteq\\ker\\rho_n$ and compute $c_A$ on $Q=\\Gamma/U$. Then\n\\[\n o_n^{\\mathrm{cts}}(\\rho_n)=0\n \\quad\\Longleftrightarrow\\quad\n \\exists\\, U'\\triangleleft_{\\mathrm{open}}\\Gamma,\\ U'\\subseteq U:\n \\operatorname{Inf}_{\\Gamma/U}^{\\Gamma/U'}[c_A]=0.\n\\]\nThus a continuous next-level lift exists exactly when a repair exists at \\emph{some finer source quotient}, not necessarily the quotient used initially. For every $j\\geq0$,\n\\[\n H^j_{\\mathrm{cts}}(\\Gamma,V)\n \\cong\\underset{U'\\subseteq U}{\\operatorname{colim}}\n H^j(\\Gamma/U',V).\n\\]\n\\end{theorem}\n\\begin{proof}\nEvery continuous map $\\Gamma^j\\to V$ is locally constant. Cover its compact domain by finitely many products of cosets on which the map is constant. Intersect the fi","\\begin{theorem}[Source refinement and inflation]\\label{thm:source-inflation}\nChoose an open normal $U\\subseteq\\ker\\rho_n$ and compute $c_A$ on $Q=\\Gamma/U$. Then\n\\[\n o_n^{\\mathrm{cts}}(\\rho_n)=0\n \\quad\\Longleftrightarrow\\quad\n \\exists\\, U'\\triangleleft_{\\mathrm{open}}\\Gamma,\\ U'\\subseteq U:\n \\operatorname{Inf}_{\\Gamma/U}^{\\Gamma/U'}[c_A]=0.\n\\]\nThus a continuous next-level lift exists exactly when a repair exists at \\emph{some finer source quotient}, not necessarily the quotient used initially. For every $j\\geq0$,\n\\[\n H^j_{\\mathrm{cts}}(\\Gamma,V)\n \\cong\\underset{U'\\subseteq U}{\\operatorname{colim}}\n H^j(\\Gamma/U',V).\n\\]\n\\end{theorem}\n\\begin{proof}\nEvery continuous map $\\Gamma^j\\to V$ is locally constant. Cover its compact domain by finitely many products of cosets on which the map is constant. Intersect the finitely many open n","\\begin{theorem}[Source refinement and inflation]\\label{thm:source-inflation}\nChoose an open normal $U\\subseteq\\ker\\rho_n$ and compute $c_A$ on $Q=\\Gamma/U$. Then\n\\[\n o_n^{\\mathrm{cts}}(\\rho_n)=0\n \\quad\\Longleftrightarrow\\quad\n \\exists\\, U'\\triangleleft_{\\mathrm{open}}\\Gamma,\\ U'\\subseteq U:\n \\operatorname{Inf}_{\\Gamma/U}^{\\Gamma/U'}[c_A]=0.\n\\]\nThus a continuous next-level lift exists exactly when a repair exists at \\emph{some finer source quotient}, not necessarily the quotient used initially. For every $j\\geq0$,\n\\[\n H^j_{\\mathrm{cts}}(\\Gamma,V)\n \\cong\\underset{U'\\subseteq U}{\\operatorname{colim}}\n H^j(\\Gamma/U',V).\n\\]\n\\end{theorem}\n\\begin{proof}\nEvery continuous map $\\Gamma^j\\to V$ is locally constant. Cover its compact domain by finitely many products of cosets on which the map is constant. Intersect the finitely many open normal subgroups","\\begin{theorem}[Source refinement and inflation]\\label{thm:source-inflation}\nChoose an open normal $U\\subseteq\\ker\\rho_n$ and compute $c_A$ on $Q=\\Gamma/U$. Then\n\\[\n o_n^{\\mathrm{cts}}(\\rho_n)=0\n \\quad\\Longleftrightarrow\\quad\n \\exists\\, U'\\triangleleft_{\\mathrm{open}}\\Gamma,\\ U'\\subseteq U:\n \\operatorname{Inf}_{\\Gamma/U}^{\\Gamma/U'}[c_A]=0.\n\\]\nThus a continuous next-level lift exists exactly when a repair exists at \\emph{some finer source quotient}, not necessarily the quotient used initially. For every $j\\geq0$,\n\\[\n H^j_{\\mathrm{cts}}(\\Gamma,V)\n \\cong\\underset{U'\\subseteq U}{\\operatorname{colim}}\n H^j(\\Gamma/U',V).\n\\]\n\\end{theorem}\n\\begin{proof}\nEvery continuous map $\\Gamma^j\\to V$ is locally constant. Cover its compact domain by finitely many products of cosets on which the map is constant. Intersect the finitely many open normal subgroups defining these cosets and further intersect with $U$. The map then factors through $(\\Gamma/U')^j$. This applies to cocycles, coboundaries, and cochains witnessing an equality. The finite-q","\\ker\\rho_n$ and compute $c_A$ on $Q=\\Gamma/U$. Then\n\\[\n o_n^{\\mathrm{cts}}(\\rho_n)=0\n \\quad\\Longleftrightarrow\\quad\n \\exists\\, U'\\triangleleft_{\\mathrm{open}}\\Gamma,\\ U'\\subseteq U:\n \\operatorname{Inf}_{\\Gamma/U}^{\\Gamma/U'}[c_A]=0.\n\\]\nThus a continuous next-level lift exists exactly when a repair exists at \\emph{some finer source quotient}, not necessarily the quotient used initially. For every $j\\geq0$,\n\\[\n H^j_{\\mathrm{cts}}(\\Gamma,V)\n \\cong\\underset{U'\\subseteq U}{\\operatorname{colim}}\n H^j(\\Gamma/U',V).\n\\]\n\\end{theorem}\n\\begin{proof}\nEvery continuous map $\\Gamma^j\\to V$ is locally constant. Cover its compact domain by finitely many products of cosets on which the map is constant. Intersect the finitely many open normal subgroups defining these cosets and further intersect with $U$. The map then factors through $(\\Gamma/U')^j$. This applies to cocycles, coboundaries, and cochains witnessing an equality. The finite-quotient cochain complexes therefore have the continuous complex as their filtered union, and the same observation about witnesses proves the asserted cohomology colimi…","and compute $c_A$ on $Q=\\Gamma/U$. Then\n\\[\n o_n^{\\mathrm{cts}}(\\rho_n)=0\n \\quad\\Longleftrightarrow\\quad\n \\exists\\, U'\\triangleleft_{\\mathrm{open}}\\Gamma,\\ U'\\subseteq U:\n \\operatorname{Inf}_{\\Gamma/U}^{\\Gamma/U'}[c_A]=0.\n\\]\nThus a continuous next-level lift exists exactly when a repair exists at \\emph{some finer source quotient}, not necessarily the quotient used initially. For every $j\\geq0$,\n\\[\n H^j_{\\mathrm{cts}}(\\Gamma,V)\n \\cong\\underset{U'\\subseteq U}{\\operatorname{colim}}\n H^j(\\Gamma/U',V).\n\\]\n\\end{theorem}\n\\begin{proof}\nEvery continuous map $\\Gamma^j\\to V$ is locally constant. Cover its compact domain by finitely many products of cosets on which the map is constant. Intersect the finitely many open normal subgroups defining these cosets and further intersect with $U$. The map then factors through $(\\Gamma/U')^j$. This applies to cocycles, coboundaries, and cochains witnessing an equality. The finite-quotient cochain complexes therefore have the continuous complex as their filtered union, and the same observation about witnesses proves the asserted cohomology colimit. In partic…","U'\\subseteq U:\n \\operatorname{Inf}_{\\Gamma/U}^{\\Gamma/U'}[c_A]=0.\n\\]\nThus a continuous next-level lift exists exactly when a repair exists at \\emph{some finer source quotient}, not necessarily the quotient used initially. For every $j\\geq0$,\n\\[\n H^j_{\\mathrm{cts}}(\\Gamma,V)\n \\cong\\underset{U'\\subseteq U}{\\operatorname{colim}}\n H^j(\\Gamma/U',V).\n\\]\n\\end{theorem}\n\\begin{proof}\nEvery continuous map $\\Gamma^j\\to V$ is locally constant. Cover its compact domain by finitely many products of cosets on which the map is constant. Intersect the finitely many open normal subgroups defining these cosets and further intersect with $U$. The map then factors through $(\\Gamma/U')^j$. This applies to cocycles, coboundaries, and cochains witnessing an equality. The finite-quotient cochain complexes therefore have the continuous complex as their filtered union, and the same observation about witnesses proves the asserted cohomology colimit. In particular a class vanishes in that colimit exactly when it vanishes at some subsequent stage. Apply \\cref{thm:continuous-H2}. This is the finite-module speciali…","very $j\\geq0$,\n\\[\n H^j_{\\mathrm{cts}}(\\Gamma,V)\n \\cong\\underset{U'\\subseteq U}{\\operatorname{colim}}\n H^j(\\Gamma/U',V).\n\\]\n\\end{theorem}\n\\begin{proof}\nEvery continuous map $\\Gamma^j\\to V$ is locally constant. Cover its compact domain by finitely many products of cosets on which the map is constant. Intersect the finitely many open normal subgroups defining these cosets and further intersect with $U$. The map then factors through $(\\Gamma/U')^j$. This applies to cocycles, coboundaries, and cochains witnessing an equality. The finite-quotient cochain complexes therefore have the continuous complex as their filtered union, and the same observation about witnesses proves the asserted cohomology colimit. In particular a class vanishes in that colimit exactly when it vanishes at some subsequent stage. Apply \\cref{thm:continuous-H2}. This is the finite-module specialization of the cited classical inflation theorem.\n\\end{proof}","s}}(\\Gamma,V)\n \\cong\\underset{U'\\subseteq U}{\\operatorname{colim}}\n H^j(\\Gamma/U',V).\n\\]\n\\end{theorem}\n\\begin{proof}\nEvery continuous map $\\Gamma^j\\to V$ is locally constant. Cover its compact domain by finitely many products of cosets on which the map is constant. Intersect the finitely many open normal subgroups defining these cosets and further intersect with $U$. The map then factors through $(\\Gamma/U')^j$. This applies to cocycles, coboundaries, and cochains witnessing an equality. The finite-quotient cochain complexes therefore have the continuous complex as their filtered union, and the same observation about witnesses proves the asserted cohomology colimit. In particular a class vanishes in that colimit exactly when it vanishes at some subsequent stage. Apply \\cref{thm:continuous-H2}. This is the finite-module specialization of the cited classical inflation theorem.\n\\end{proof}","\\begin{example}[An explicit obstruction killed by refining the source]\\label{ex:inflation-carry}\nLet $p$ be odd, $J\\in M_d(\\mathbb Z)$ nonzero modulo $p$, and $A=I+pJ$ modulo $p^3$. Then $A^p=I+p^2J$ and $A^{p^2}=I$ modulo $p^3$. The representation of $C_p$ modulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit sh","\\begin{example}[An explicit obstruction killed by refining the source]\\label{ex:inflation-carry}\nLet $p$ be odd, $J\\in M_d(\\mathbb Z)$ nonzero modulo $p$, and $A=I+pJ$ modulo $p^3$. Then $A^p=I+p^2J$ and $A^{p^2}=I$ modulo $p^3$. The representation of $C_p$ modulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{I","\\begin{example}[An explicit obstruction killed by refining the source]\\label{ex:inflation-carry}\nLet $p$ be odd, $J\\in M_d(\\mathbb Z)$ nonzero modulo $p$, and $A=I+pJ$ modulo $p^3$. Then $A^p=I+p^2J$ and $A^{p^2}=I$ modulo $p^3$. The representation of $C_p$ modulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-d","\\begin{example}[An explicit obstruction killed by refining the source]\\label{ex:inflation-carry}\nLet $p$ be odd, $J\\in M_d(\\mathbb Z)$ nonzero modulo $p$, and $A=I+pJ$ modulo $p^3$. Then $A^p=I+p^2J$ and $A^{p^2}=I$ modulo $p^3$. The representation of $C_p$ modulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is","\\begin{example}[An explicit obstruction killed by refining the source]\\label{ex:inflation-carry}\nLet $p$ be odd, $J\\in M_d(\\mathbb Z)$ nonzero modulo $p$, and $A=I+pJ$ modulo $p^3$. Then $A^p=I+p^2J$ and $A^{p^2}=I$ modulo $p^3$. The representation of $C_p$ modulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative","\\begin{example}[An explicit obstruction killed by refining the source]\\label{ex:inflation-carry}\nLet $p$ be odd, $J\\in M_d(\\mathbb Z)$ nonzero modulo $p$, and $A=I+pJ$ modulo $p^3$. Then $A^p=I+p^2J$ and $A^{p^2}=I$ modulo $p^3$. The representation of $C_p$ modulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit ca","\\begin{example}[An explicit obstruction killed by refining the source]\\label{ex:inflation-carry}\nLet $p$ be odd, $J\\in M_d(\\mathbb Z)$ nonzero modulo $p$, and $A=I+pJ$ modulo $p^3$. Then $A^p=I+p^2J$ and $A^{p^2}=I$ modulo $p^3$. The representation of $C_p$ modulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$.","\\begin{example}[An explicit obstruction killed by refining the source]\\label{ex:inflation-carry}\nLet $p$ be odd, $J\\in M_d(\\mathbb Z)$ nonzero modulo $p$, and $A=I+pJ$ modulo $p^3$. Then $A^p=I+p^2J$ and $A^{p^2}=I$ modulo $p^3$. The representation of $C_p$ modulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$. The repaired","\\begin{example}[An explicit obstruction killed by refining the source]\\label{ex:inflation-carry}\nLet $p$ be odd, $J\\in M_d(\\mathbb Z)$ nonzero modulo $p$, and $A=I+pJ$ modulo $p^3$. Then $A^p=I+p^2J$ and $A^{p^2}=I$ modulo $p^3$. The representation of $C_p$ modulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$. The repaired map is exactly $x\\mapsto A^x$","\\begin{example}[An explicit obstruction killed by refining the source]\\label{ex:inflation-carry}\nLet $p$ be odd, $J\\in M_d(\\mathbb Z)$ nonzero modulo $p$, and $A=I+pJ$ modulo $p^3$. Then $A^p=I+p^2J$ and $A^{p^2}=I$ modulo $p^3$. The representation of $C_p$ modulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$. The repaired map is exactly $x\\mapsto A^x$. Thus inflat","\\begin{example}[An explicit obstruction killed by refining the source]\\label{ex:inflation-carry}\nLet $p$ be odd, $J\\in M_d(\\mathbb Z)$ nonzero modulo $p$, and $A=I+pJ$ modulo $p^3$. Then $A^p=I+p^2J$ and $A^{p^2}=I$ modulo $p^3$. The representation of $C_p$ modulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$. The repaired map is exactly $x\\mapsto A^x$. Thus inflation kills a genuine","\\begin{example}[An explicit obstruction killed by refining the source]\\label{ex:inflation-carry}\nLet $p$ be odd, $J\\in M_d(\\mathbb Z)$ nonzero modulo $p$, and $A=I+pJ$ modulo $p^3$. Then $A^p=I+p^2J$ and $A^{p^2}=I$ modulo $p^3$. The representation of $C_p$ modulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$. The repaired map is exactly $x\\mapsto A^x$. Thus inflation kills a genuine finite-quotient obstruction. This","\\begin{example}[An explicit obstruction killed by refining the source]\\label{ex:inflation-carry}\nLet $p$ be odd, $J\\in M_d(\\mathbb Z)$ nonzero modulo $p$, and $A=I+pJ$ modulo $p^3$. Then $A^p=I+p^2J$ and $A^{p^2}=I$ modulo $p^3$. The representation of $C_p$ modulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$. The repaired map is exactly $x\\mapsto A^x$. Thus inflation kills a genuine finite-quotient obstruction. This calculation does n","\\begin{example}[An explicit obstruction killed by refining the source]\\label{ex:inflation-carry}\nLet $p$ be odd, $J\\in M_d(\\mathbb Z)$ nonzero modulo $p$, and $A=I+pJ$ modulo $p^3$. Then $A^p=I+p^2J$ and $A^{p^2}=I$ modulo $p^3$. The representation of $C_p$ modulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$. The repaired map is exactly $x\\mapsto A^x$. Thus inflation kills a genuine finite-quotient obstruction. This calculation does not declare every continuous obstruction removable.\n\\end{exa…","odd, $J\\in M_d(\\mathbb Z)$ nonzero modulo $p$, and $A=I+pJ$ modulo $p^3$. Then $A^p=I+p^2J$ and $A^{p^2}=I$ modulo $p^3$. The representation of $C_p$ modulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$. The repaired map is exactly $x\\mapsto A^x$. Thus inflation kills a genuine finite-quotient obstruction. This calculation does not declare every continuous obstruction removable.\n\\end{example}","\\in M_d(\\mathbb Z)$ nonzero modulo $p$, and $A=I+pJ$ modulo $p^3$. Then $A^p=I+p^2J$ and $A^{p^2}=I$ modulo $p^3$. The representation of $C_p$ modulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$. The repaired map is exactly $x\\mapsto A^x$. Thus inflation kills a genuine finite-quotient obstruction. This calculation does not declare every continuous obstruction removable.\n\\end{example}","\\mathbb Z)$ nonzero modulo $p$, and $A=I+pJ$ modulo $p^3$. Then $A^p=I+p^2J$ and $A^{p^2}=I$ modulo $p^3$. The representation of $C_p$ modulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$. The repaired map is exactly $x\\mapsto A^x$. Thus inflation kills a genuine finite-quotient obstruction. This calculation does not declare every continuous obstruction removable.\n\\end{example}","ion of $C_p$ modulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$. The repaired map is exactly $x\\mapsto A^x$. Thus inflation kills a genuine finite-quotient obstruction. This calculation does not declare every continuous obstruction removable.\n\\end{example}","dulo $p^2$ uses powers of $A$. For representatives $0\\leq a,b<p$, choose $A_a=A^a$ at the next level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$. The repaired map is exactly $x\\mapsto A^x$. Thus inflation kills a genuine finite-quotient obstruction. This calculation does not declare every continuous obstruction removable.\n\\end{example}","ext level. Its cocycle is\n\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$. The repaired map is exactly $x\\mapsto A^x$. Thus inflation kills a genuine finite-quotient obstruction. This calculation does not declare every continuous obstruction removable.\n\\end{example}","\\[\n c(a,b)=\\left\\lfloor\\frac{a+b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$. The repaired map is exactly $x\\mapsto A^x$. Thus inflation kills a genuine finite-quotient obstruction. This calculation does not declare every continuous obstruction removable.\n\\end{example}","b}{p}\\right\\rfloor J\\pmod p.\n\\]\nThe adjoint residual action is trivial. For any normalized cochain $u$ on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$. The repaired map is exactly $x\\mapsto A^x$. Thus inflation kills a genuine finite-quotient obstruction. This calculation does not declare every continuous obstruction removable.\n\\end{example}","on $C_p$,\n\\[\n \\sum_{t=0}^{p-1}(\\delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$. The repaired map is exactly $x\\mapsto A^x$. Thus inflation kills a genuine finite-quotient obstruction. This calculation does not declare every continuous obstruction removable.\n\\end{example}","delta u)(1,t)=0,\n \\qquad\\sum_{t=0}^{p-1}c(1,t)=J\\ne0.\n\\]\nSo the finite-quotient class is nonzero. On $C_{p^2}$ put\n\\[\n b(x)=\\left\\lfloor\\frac{x}{p}\\right\\rfloor J\\pmod p\n \\quad(0\\leq x<p^2).\n\\]\nWriting each $x$ as its low digit plus $p$ times its high digit shows\n$\\delta b=-\\operatorname{Inf}(c)$: the high-digit carry is the negative of the low-digit carry modulo $p$. The repaired map is exactly $x\\mapsto A^x$. Thus inflation kills a genuine finite-quotient obstruction. This calculation does not declare every continuous obstruction removable.\n\\end{example}","\\begin{theorem}[Continuous reconstruction from unobstructed stages]\\label{thm:continuous-tower}\nIf $H^2_{\\mathrm{cts}}(\\Gamma,\\operatorname{Ad}\\bar\\rho)=0$, every given $\\rho_n$ with that residual action extends to a continuous representation\n\\[\n \\rho:\\Gamma\\longrightarrow\\operatorname{GL}_d(\\mathbb Z_p).\n\\]\nWithout the cohomology-vanishing hypothesis, a compatible sequence of continuous lifts still reconstructs uniquely as a framed representation. If $\\Gamma$ is topologically finitely generated, existence of \\emph{some} extension of the fixed $\\rho_n$ to every coefficient level is enough to obtain a compatible sequence and hence an integral lift.\n\\end{theorem}\n\\begin{proof}\nThe residual adjoint module is unchanged by each coefficient lift. Under the vanishing hypothesis, \\cref{thm:continuous-H2} permits recursive choice of compatible lifts $","\\begin{theorem}[Continuous reconstruction from unobstructed stages]\\label{thm:continuous-tower}\nIf $H^2_{\\mathrm{cts}}(\\Gamma,\\operatorname{Ad}\\bar\\rho)=0$, every given $\\rho_n$ with that residual action extends to a continuous representation\n\\[\n \\rho:\\Gamma\\longrightarrow\\operatorname{GL}_d(\\mathbb Z_p).\n\\]\nWithout the cohomology-vanishing hypothesis, a compatible sequence of continuous lifts still reconstructs uniquely as a framed representation. If $\\Gamma$ is topologically finitely generated, existence of \\emph{some} extension of the fixed $\\rho_n$ to every coefficient level is enough to obtain a compatible sequence and hence an integral lift.\n\\end{theorem}\n\\begin{proof}\nThe residual adjoint module is unchanged by each coefficient lift. Under the vanishing hypothesis, \\cref{thm:continuous-H2} permits recursive choice of compatible lifts $\\rho_m$ for all $m\\geq","\\begin{theorem}[Continuous reconstruction from unobstructed stages]\\label{thm:continuous-tower}\nIf $H^2_{\\mathrm{cts}}(\\Gamma,\\operatorname{Ad}\\bar\\rho)=0$, every given $\\rho_n$ with that residual action extends to a continuous representation\n\\[\n \\rho:\\Gamma\\longrightarrow\\operatorname{GL}_d(\\mathbb Z_p).\n\\]\nWithout the cohomology-vanishing hypothesis, a compatible sequence of continuous lifts still reconstructs uniquely as a framed representation. If $\\Gamma$ is topologically finitely generated, existence of \\emph{some} extension of the fixed $\\rho_n$ to every coefficient level is enough to obtain a compatible sequence and hence an integral lift.\n\\end{theorem}\n\\begin{proof}\nThe residual adjoint module is unchanged by each coefficient lift. Under the vanishing hypothesis, \\cref{thm:continuous-H2} permits recursive choice of compatible lifts $\\rho_m$ for all $m\\geq n$. Take the inverse limit of their matrix entries. Multiplication is compatible with this limit, and determinants remain units be","ed stages]\\label{thm:continuous-tower}\nIf $H^2_{\\mathrm{cts}}(\\Gamma,\\operatorname{Ad}\\bar\\rho)=0$, every given $\\rho_n$ with that residual action extends to a continuous representation\n\\[\n \\rho:\\Gamma\\longrightarrow\\operatorname{GL}_d(\\mathbb Z_p).\n\\]\nWithout the cohomology-vanishing hypothesis, a compatible sequence of continuous lifts still reconstructs uniquely as a framed representation. If $\\Gamma$ is topologically finitely generated, existence of \\emph{some} extension of the fixed $\\rho_n$ to every coefficient level is enough to obtain a compatible sequence and hence an integral lift.\n\\end{theorem}\n\\begin{proof}\nThe residual adjoint module is unchanged by each coefficient lift. Under the vanishing hypothesis, \\cref{thm:continuous-H2} permits recursive choice of compatible lifts $\\rho_m$ for all $m\\geq n$. Take the inverse limit of their matrix entries. Multiplication is compatible with this limit, and determinants remain units because their reductions modulo $p$ are units. Each finite projection is continuous, so the map into the inverse-limit topology is continuous. The proje…",")=0$, every given $\\rho_n$ with that residual action extends to a continuous representation\n\\[\n \\rho:\\Gamma\\longrightarrow\\operatorname{GL}_d(\\mathbb Z_p).\n\\]\nWithout the cohomology-vanishing hypothesis, a compatible sequence of continuous lifts still reconstructs uniquely as a framed representation. If $\\Gamma$ is topologically finitely generated, existence of \\emph{some} extension of the fixed $\\rho_n$ to every coefficient level is enough to obtain a compatible sequence and hence an integral lift.\n\\end{theorem}\n\\begin{proof}\nThe residual adjoint module is unchanged by each coefficient lift. Under the vanishing hypothesis, \\cref{thm:continuous-H2} permits recursive choice of compatible lifts $\\rho_m$ for all $m\\geq n$. Take the inverse limit of their matrix entries. Multiplication is compatible with this limit, and determinants remain units because their reductions modulo $p$ are units. Each finite projection is continuous, so the map into the inverse-limit topology is continuous. The projections determine it uniquely.","f $\\Gamma$ is topologically finitely generated, existence of \\emph{some} extension of the fixed $\\rho_n$ to every coefficient level is enough to obtain a compatible sequence and hence an integral lift.\n\\end{theorem}\n\\begin{proof}\nThe residual adjoint module is unchanged by each coefficient lift. Under the vanishing hypothesis, \\cref{thm:continuous-H2} permits recursive choice of compatible lifts $\\rho_m$ for all $m\\geq n$. Take the inverse limit of their matrix entries. Multiplication is compatible with this limit, and determinants remain units because their reductions modulo $p$ are units. Each finite projection is continuous, so the map into the inverse-limit topology is continuous. The projections determine it uniquely.","ologically finitely generated, existence of \\emph{some} extension of the fixed $\\rho_n$ to every coefficient level is enough to obtain a compatible sequence and hence an integral lift.\n\\end{theorem}\n\\begin{proof}\nThe residual adjoint module is unchanged by each coefficient lift. Under the vanishing hypothesis, \\cref{thm:continuous-H2} permits recursive choice of compatible lifts $\\rho_m$ for all $m\\geq n$. Take the inverse limit of their matrix entries. Multiplication is compatible with this limit, and determinants remain units because their reductions modulo $p$ are units. Each finite projection is continuous, so the map into the inverse-limit topology is continuous. The projections determine it uniquely.","an integral lift.\n\\end{theorem}\n\\begin{proof}\nThe residual adjoint module is unchanged by each coefficient lift. Under the vanishing hypothesis, \\cref{thm:continuous-H2} permits recursive choice of compatible lifts $\\rho_m$ for all $m\\geq n$. Take the inverse limit of their matrix entries. Multiplication is compatible with this limit, and determinants remain units because their reductions modulo $p$ are units. Each finite projection is continuous, so the map into the inverse-limit topology is continuous. The projections determine it uniquely.","For the last assertion, choose $r$ topological generators. A continuous map into a finite group is determined by their images, since the equalizer of two such maps is closed and contains a dense subgroup when the images agree. Thus there are finitely many continuous representations at each level. Form the rooted tree of extensions of $\\rho_n$, with edges given by one-step reduction. It is finitely branching. If every level is nonempty, the root has descendants at arbitrarily large depth. At least one child has the same property, since there are only finitely many children. Repeating gives an infinite compatible path. Apply the first limit argument. This compactness argument supplies no numerical stopping bound and no algori","For the last assertion, choose $r$ topological generators. A continuous map into a finite group is determined by their images, since the equalizer of two such maps is closed and contains a dense subgroup when the images agree. Thus there are finitely many continuous representations at each level. Form the rooted tree of extensions of $\\rho_n$, with edges given by one-step reduction. It is finitely branching. If every level is nonempty, the root has descendants at arbitrarily large depth. At least one child has the same property, since there are only finitely many children. Repeating gives an infinite compatible path. Apply the first limit argument. This compactness argument supplies no numerical stopping bound and no algorithm for computing $H^2_{\\mathrm{cts}}$ of an arbitrary source.\n\\end{proof}","dges given by one-step reduction. It is finitely branching. If every level is nonempty, the root has descendants at arbitrarily large depth. At least one child has the same property, since there are only finitely many children. Repeating gives an infinite compatible path. Apply the first limit argument. This compactness argument supplies no numerical stopping bound and no algorithm for computing $H^2_{\\mathrm{cts}}$ of an arbitrary source.\n\\end{proof}","\\begin{theorem}[Unrestricted coefficient lifting for a free pro-$p$ source]\\label{thm:free-pro-p-lift}\nLet $F_r^{(p)}$ be the free pro-$p$ group on finitely many generators. Every continuous\n\\[\n F_r^{(p)}\\longrightarrow\\operatorname{GL}_d(R_n)\n\\]\nhas a continuous lift to $\\operatorname{GL}_d(\\mathbb Z_p)$. One may choose arbitrary integral lifts of its generator matrices.\n\\end{theorem}\n\\begin{proof}\nConstruct $F_r^{(p)}$ as the inverse limit of the finite $p$-group quotients of the abstract free group on these generators. A map from the generators to a pro-$p$ group induces compatible maps to each of its finite $p$-group quotients, and hence a unique continuous map from this inverse limit. This proves the universal property needed here.","}[Unrestricted coefficient lifting for a free pro-$p$ source]\\label{thm:free-pro-p-lift}\nLet $F_r^{(p)}$ be the free pro-$p$ group on finitely many generators. Every continuous\n\\[\n F_r^{(p)}\\longrightarrow\\operatorname{GL}_d(R_n)\n\\]\nhas a continuous lift to $\\operatorname{GL}_d(\\mathbb Z_p)$. One may choose arbitrary integral lifts of its generator matrices.\n\\end{theorem}\n\\begin{proof}\nConstruct $F_r^{(p)}$ as the inverse limit of the finite $p$-group quotients of the abstract free group on these generators. A map from the generators to a pro-$p$ group induces compatible maps to each of its finite $p$-group quotients, and hence a unique continuous map from this inverse limit. This proves the universal property needed here.","ro-$p$ source]\\label{thm:free-pro-p-lift}\nLet $F_r^{(p)}$ be the free pro-$p$ group on finitely many generators. Every continuous\n\\[\n F_r^{(p)}\\longrightarrow\\operatorname{GL}_d(R_n)\n\\]\nhas a continuous lift to $\\operatorname{GL}_d(\\mathbb Z_p)$. One may choose arbitrary integral lifts of its generator matrices.\n\\end{theorem}\n\\begin{proof}\nConstruct $F_r^{(p)}$ as the inverse limit of the finite $p$-group quotients of the abstract free group on these generators. A map from the generators to a pro-$p$ group induces compatible maps to each of its finite $p$-group quotients, and hence a unique continuous map from this inverse limit. This proves the universal property needed here.","nerators. Every continuous\n\\[\n F_r^{(p)}\\longrightarrow\\operatorname{GL}_d(R_n)\n\\]\nhas a continuous lift to $\\operatorname{GL}_d(\\mathbb Z_p)$. One may choose arbitrary integral lifts of its generator matrices.\n\\end{theorem}\n\\begin{proof}\nConstruct $F_r^{(p)}$ as the inverse limit of the finite $p$-group quotients of the abstract free group on these generators. A map from the generators to a pro-$p$ group induces compatible maps to each of its finite $p$-group quotients, and hence a unique continuous map from this inverse limit. This proves the universal property needed here.","operatorname{GL}_d(R_n)\n\\]\nhas a continuous lift to $\\operatorname{GL}_d(\\mathbb Z_p)$. One may choose arbitrary integral lifts of its generator matrices.\n\\end{theorem}\n\\begin{proof}\nConstruct $F_r^{(p)}$ as the inverse limit of the finite $p$-group quotients of the abstract free group on these generators. A map from the generators to a pro-$p$ group induces compatible maps to each of its finite $p$-group quotients, and hence a unique continuous map from this inverse limit. This proves the universal property needed here.","The residual image $P\\subseteq\\operatorname{GL}_d(\\mathbb F_p)$ is a $p$-group, being a finite quotient of a pro-$p$ group. Let $H$ be its inverse image in $\\operatorname{GL}_d(\\mathbb Z_p)$. Its quotient modulo $p^m$ is a finite $p$-group: its kernel over $P$ has the filtration by congruence subgroups with additive $M_d(\\mathbb F_p)$ subquotients. Therefore $H$ is pro-$p$, including at $p=2$. Arbitrary integral lifts of the generator matrices lie in $H$. The just-proved universal property extends them to a continuous map. Reduction agrees with the input on the generators and therefore everywhere. This does not assert that the lift is faithful or has finite image.\n\\end{proof}","b F_p)$ is a $p$-group, being a finite quotient of a pro-$p$ group. Let $H$ be its inverse image in $\\operatorname{GL}_d(\\mathbb Z_p)$. Its quotient modulo $p^m$ is a finite $p$-group: its kernel over $P$ has the filtration by congruence subgroups with additive $M_d(\\mathbb F_p)$ subquotients. Therefore $H$ is pro-$p$, including at $p=2$. Arbitrary integral lifts of the generator matrices lie in $H$. The just-proved universal property extends them to a continuous map. Reduction agrees with the input on the generators and therefore everywhere. This does not assert that the lift is faithful or has finite image.\n\\end{proof}","\\begin{lemma}[When an integral matrix has continuous procyclic powers]\\label{lem:procyclic-powers}\nLet $A\\in\\operatorname{GL}_d(\\mathbb Z_p)$. The map $m\\mapsto A^m$ on $\\mathbb Z$ extends uniquely to a continuous homomorphism $\\mathbb Z_p\\to\\operatorname{GL}_d(\\mathbb Z_p)$ if and only if the reduction of $A$ modulo $p$ has $p$-power order. In particular a commuting tuple of such matrices determines a continuous representation of $\\mathbb Z_p^r$.\n\\end{lemma}\n\\begin{proof}\nNecessity follows because a continuous image of $\\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows t","\\begin{lemma}[When an integral matrix has continuous procyclic powers]\\label{lem:procyclic-powers}\nLet $A\\in\\operatorname{GL}_d(\\mathbb Z_p)$. The map $m\\mapsto A^m$ on $\\mathbb Z$ extends uniquely to a continuous homomorphism $\\mathbb Z_p\\to\\operatorname{GL}_d(\\mathbb Z_p)$ if and only if the reduction of $A$ modulo $p$ has $p$-power order. In particular a commuting tuple of such matrices determines a continuous representation of $\\mathbb Z_p^r$.\n\\end{lemma}\n\\begin{proof}\nNecessity follows because a continuous image of $\\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I","\\begin{lemma}[When an integral matrix has continuous procyclic powers]\\label{lem:procyclic-powers}\nLet $A\\in\\operatorname{GL}_d(\\mathbb Z_p)$. The map $m\\mapsto A^m$ on $\\mathbb Z$ extends uniquely to a continuous homomorphism $\\mathbb Z_p\\to\\operatorname{GL}_d(\\mathbb Z_p)$ if and only if the reduction of $A$ modulo $p$ has $p$-power order. In particular a commuting tuple of such matrices determines a continuous representation of $\\mathbb Z_p^r$.\n\\end{lemma}\n\\begin{proof}\nNecessity follows because a continuous image of $\\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\ma","\\begin{lemma}[When an integral matrix has continuous procyclic powers]\\label{lem:procyclic-powers}\nLet $A\\in\\operatorname{GL}_d(\\mathbb Z_p)$. The map $m\\mapsto A^m$ on $\\mathbb Z$ extends uniquely to a continuous homomorphism $\\mathbb Z_p\\to\\operatorname{GL}_d(\\mathbb Z_p)$ if and only if the reduction of $A$ modulo $p$ has $p$-power order. In particular a commuting tuple of such matrices determines a continuous representation of $\\mathbb Z_p^r$.\n\\end{lemma}\n\\begin{proof}\nNecessity follows because a continuous image of $\\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\mathbb Z_p).\n\\]\nHence exponents congruent modulo $p^{t+j}$ give powers congruent modulo $p^{j+1}$","\\begin{lemma}[When an integral matrix has continuous procyclic powers]\\label{lem:procyclic-powers}\nLet $A\\in\\operatorname{GL}_d(\\mathbb Z_p)$. The map $m\\mapsto A^m$ on $\\mathbb Z$ extends uniquely to a continuous homomorphism $\\mathbb Z_p\\to\\operatorname{GL}_d(\\mathbb Z_p)$ if and only if the reduction of $A$ modulo $p$ has $p$-power order. In particular a commuting tuple of such matrices determines a continuous representation of $\\mathbb Z_p^r$.\n\\end{lemma}\n\\begin{proof}\nNecessity follows because a continuous image of $\\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\mathbb Z_p).\n\\]\nHence exponents congruent modulo $p^{t+j}$ give powers congruent modulo $p^{j+1}$, also for negative integer exponent","\\begin{lemma}[When an integral matrix has continuous procyclic powers]\\label{lem:procyclic-powers}\nLet $A\\in\\operatorname{GL}_d(\\mathbb Z_p)$. The map $m\\mapsto A^m$ on $\\mathbb Z$ extends uniquely to a continuous homomorphism $\\mathbb Z_p\\to\\operatorname{GL}_d(\\mathbb Z_p)$ if and only if the reduction of $A$ modulo $p$ has $p$-power order. In particular a commuting tuple of such matrices determines a continuous representation of $\\mathbb Z_p^r$.\n\\end{lemma}\n\\begin{proof}\nNecessity follows because a continuous image of $\\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\mathbb Z_p).\n\\]\nHence exponents congruent modulo $p^{t+j}$ give powers congruent modulo $p^{j+1}$, also for negative integer exponents by invert","\\begin{lemma}[When an integral matrix has continuous procyclic powers]\\label{lem:procyclic-powers}\nLet $A\\in\\operatorname{GL}_d(\\mathbb Z_p)$. The map $m\\mapsto A^m$ on $\\mathbb Z$ extends uniquely to a continuous homomorphism $\\mathbb Z_p\\to\\operatorname{GL}_d(\\mathbb Z_p)$ if and only if the reduction of $A$ modulo $p$ has $p$-power order. In particular a commuting tuple of such matrices determines a continuous representation of $\\mathbb Z_p^r$.\n\\end{lemma}\n\\begin{proof}\nNecessity follows because a continuous image of $\\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\mathbb Z_p).\n\\]\nHence exponents congruent modulo $p^{t+j}$ give powers congruent modulo $p^{j+1}$, also for negative integer exponents by invertibility.","x has continuous procyclic powers]\\label{lem:procyclic-powers}\nLet $A\\in\\operatorname{GL}_d(\\mathbb Z_p)$. The map $m\\mapsto A^m$ on $\\mathbb Z$ extends uniquely to a continuous homomorphism $\\mathbb Z_p\\to\\operatorname{GL}_d(\\mathbb Z_p)$ if and only if the reduction of $A$ modulo $p$ has $p$-power order. In particular a commuting tuple of such matrices determines a continuous representation of $\\mathbb Z_p^r$.\n\\end{lemma}\n\\begin{proof}\nNecessity follows because a continuous image of $\\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\mathbb Z_p).\n\\]\nHence exponents congruent modulo $p^{t+j}$ give powers congruent modulo $p^{j+1}$, also for negative integer exponents by invertibility. Completeness extends the map continuously from the dense subgroup $\\mathbb Z$ to $\\mathbb Z_p$. The group …","(\\mathbb Z_p)$. The map $m\\mapsto A^m$ on $\\mathbb Z$ extends uniquely to a continuous homomorphism $\\mathbb Z_p\\to\\operatorname{GL}_d(\\mathbb Z_p)$ if and only if the reduction of $A$ modulo $p$ has $p$-power order. In particular a commuting tuple of such matrices determines a continuous representation of $\\mathbb Z_p^r$.\n\\end{lemma}\n\\begin{proof}\nNecessity follows because a continuous image of $\\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\mathbb Z_p).\n\\]\nHence exponents congruent modulo $p^{t+j}$ give powers congruent modulo $p^{j+1}$, also for negative integer exponents by invertibility. Completeness extends the map continuously from the dense subgroup $\\mathbb Z$ to $\\mathbb Z_p$. The group law and uniqueness follow by continuity. For commuting matrices the product of these powers…","mathbb Z$ extends uniquely to a continuous homomorphism $\\mathbb Z_p\\to\\operatorname{GL}_d(\\mathbb Z_p)$ if and only if the reduction of $A$ modulo $p$ has $p$-power order. In particular a commuting tuple of such matrices determines a continuous representation of $\\mathbb Z_p^r$.\n\\end{lemma}\n\\begin{proof}\nNecessity follows because a continuous image of $\\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\mathbb Z_p).\n\\]\nHence exponents congruent modulo $p^{t+j}$ give powers congruent modulo $p^{j+1}$, also for negative integer exponents by invertibility. Completeness extends the map continuously from the dense subgroup $\\mathbb Z$ to $\\mathbb Z_p$. The group law and uniqueness follow by continuity. For commuting matrices the product of these powers defines the final assertion.\n\\end{proof}","a continuous homomorphism $\\mathbb Z_p\\to\\operatorname{GL}_d(\\mathbb Z_p)$ if and only if the reduction of $A$ modulo $p$ has $p$-power order. In particular a commuting tuple of such matrices determines a continuous representation of $\\mathbb Z_p^r$.\n\\end{lemma}\n\\begin{proof}\nNecessity follows because a continuous image of $\\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\mathbb Z_p).\n\\]\nHence exponents congruent modulo $p^{t+j}$ give powers congruent modulo $p^{j+1}$, also for negative integer exponents by invertibility. Completeness extends the map continuously from the dense subgroup $\\mathbb Z$ to $\\mathbb Z_p$. The group law and uniqueness follow by continuity. For commuting matrices the product of these powers defines the final assertion.\n\\end{proof}","\\to\\operatorname{GL}_d(\\mathbb Z_p)$ if and only if the reduction of $A$ modulo $p$ has $p$-power order. In particular a commuting tuple of such matrices determines a continuous representation of $\\mathbb Z_p^r$.\n\\end{lemma}\n\\begin{proof}\nNecessity follows because a continuous image of $\\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\mathbb Z_p).\n\\]\nHence exponents congruent modulo $p^{t+j}$ give powers congruent modulo $p^{j+1}$, also for negative integer exponents by invertibility. Completeness extends the map continuously from the dense subgroup $\\mathbb Z$ to $\\mathbb Z_p$. The group law and uniqueness follow by continuity. For commuting matrices the product of these powers defines the final assertion.\n\\end{proof}",")$ if and only if the reduction of $A$ modulo $p$ has $p$-power order. In particular a commuting tuple of such matrices determines a continuous representation of $\\mathbb Z_p^r$.\n\\end{lemma}\n\\begin{proof}\nNecessity follows because a continuous image of $\\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\mathbb Z_p).\n\\]\nHence exponents congruent modulo $p^{t+j}$ give powers congruent modulo $p^{j+1}$, also for negative integer exponents by invertibility. Completeness extends the map continuously from the dense subgroup $\\mathbb Z$ to $\\mathbb Z_p$. The group law and uniqueness follow by continuity. For commuting matrices the product of these powers defines the final assertion.\n\\end{proof}","$A$ modulo $p$ has $p$-power order. In particular a commuting tuple of such matrices determines a continuous representation of $\\mathbb Z_p^r$.\n\\end{lemma}\n\\begin{proof}\nNecessity follows because a continuous image of $\\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\mathbb Z_p).\n\\]\nHence exponents congruent modulo $p^{t+j}$ give powers congruent modulo $p^{j+1}$, also for negative integer exponents by invertibility. Completeness extends the map continuously from the dense subgroup $\\mathbb Z$ to $\\mathbb Z_p$. The group law and uniqueness follow by continuity. For commuting matrices the product of these powers defines the final assertion.\n\\end{proof}","articular a commuting tuple of such matrices determines a continuous representation of $\\mathbb Z_p^r$.\n\\end{lemma}\n\\begin{proof}\nNecessity follows because a continuous image of $\\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\mathbb Z_p).\n\\]\nHence exponents congruent modulo $p^{t+j}$ give powers congruent modulo $p^{j+1}$, also for negative integer exponents by invertibility. Completeness extends the map continuously from the dense subgroup $\\mathbb Z$ to $\\mathbb Z_p$. The group law and uniqueness follow by continuity. For commuting matrices the product of these powers defines the final assertion.\n\\end{proof}","matrices determines a continuous representation of $\\mathbb Z_p^r$.\n\\end{lemma}\n\\begin{proof}\nNecessity follows because a continuous image of $\\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\mathbb Z_p).\n\\]\nHence exponents congruent modulo $p^{t+j}$ give powers congruent modulo $p^{j+1}$, also for negative integer exponents by invertibility. Completeness extends the map continuously from the dense subgroup $\\mathbb Z$ to $\\mathbb Z_p$. The group law and uniqueness follow by continuity. For commuting matrices the product of these powers defines the final assertion.\n\\end{proof}","p^r$.\n\\end{lemma}\n\\begin{proof}\nNecessity follows because a continuous image of $\\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\mathbb Z_p).\n\\]\nHence exponents congruent modulo $p^{t+j}$ give powers congruent modulo $p^{j+1}$, also for negative integer exponents by invertibility. Completeness extends the map continuously from the dense subgroup $\\mathbb Z$ to $\\mathbb Z_p$. The group law and uniqueness follow by continuity. For commuting matrices the product of these powers defines the final assertion.\n\\end{proof}","\\mathbb Z_p$ in a finite group is a cyclic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\mathbb Z_p).\n\\]\nHence exponents congruent modulo $p^{t+j}$ give powers congruent modulo $p^{j+1}$, also for negative integer exponents by invertibility. Completeness extends the map continuously from the dense subgroup $\\mathbb Z$ to $\\mathbb Z_p$. The group law and uniqueness follow by continuity. For commuting matrices the product of these powers defines the final assertion.\n\\end{proof}","lic $p$-group. Conversely suppose $A^{p^t}\\in I+pM_d(\\mathbb Z_p)$. For $B\\in I+p^sM_d(\\mathbb Z_p)$ with $s\\geq1$, binomial expansion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\mathbb Z_p).\n\\]\nHence exponents congruent modulo $p^{t+j}$ give powers congruent modulo $p^{j+1}$, also for negative integer exponents by invertibility. Completeness extends the map continuously from the dense subgroup $\\mathbb Z$ to $\\mathbb Z_p$. The group law and uniqueness follow by continuity. For commuting matrices the product of these powers defines the final assertion.\n\\end{proof}","ion shows $B^p\\in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\mathbb Z_p).\n\\]\nHence exponents congruent modulo $p^{t+j}$ give powers congruent modulo $p^{j+1}$, also for negative integer exponents by invertibility. Completeness extends the map continuously from the dense subgroup $\\mathbb Z$ to $\\mathbb Z_p$. The group law and uniqueness follow by continuity. For commuting matrices the product of these powers defines the final assertion.\n\\end{proof}","in I+p^{s+1}M_d(\\mathbb Z_p)$. For $p=2$ the quadratic term has depth $2s\\geq s+1$, so this estimate remains valid. It follows that\n\\[\n A^{p^{t+j}}\\in I+p^{j+1}M_d(\\mathbb Z_p).\n\\]\nHence exponents congruent modulo $p^{t+j}$ give powers congruent modulo $p^{j+1}$, also for negative integer exponents by invertibility. Completeness extends the map continuously from the dense subgroup $\\mathbb Z$ to $\\mathbb Z_p$. The group law and uniqueness follow by continuity. For commuting matrices the product of these powers defines the final assertion.\n\\end{proof}","\\begin{lemma}[Cyclic centralizers over a coefficient ring]\\label{lem:cyclic-centralizer}\nLet $R$ be a commutative ring, $B\\in M_d(R)$, and suppose a vector $v$ makes\n\\[\n Q=(v,Bv,\\ldots,B^{d-1}v)\\in\\operatorname{GL}_d(R).\n\\]\nThen every matrix commuting with $B$ is uniquely $f(B)$ for a polynomial $f\\in R[t]$ of degree less than $d$. The coefficients of $f$ are the entries of $Q^{-1}Mv$. In particular, the centralizer is commutative.","Over $R=\\mathbb Z/p^s$, it is enough that $\\bar B$ have a cyclic vector over $\\mathbb F_p$. Every nonscalar $2\\times2$ matrix over a field has such a vector.\n\\end{lemma}\n\\begin{proof}\nThere is a unique polynomial of degree less than $d$ satisfying $f(B)v=Mv$, since the columns of $Q$ are a basis. If $MB=BM$, then $M-f(B)$ annihilates every $B^jv$, so it is zero. Conversely every polynomial in $B$ commutes with $B$. This also proves uniqueness and the coefficient formula. A lifted cyclic vector modulo $p$ makes $\\det Q$ a unit modulo $p^s$.","Z/p^s$, it is enough that $\\bar B$ have a cyclic vector over $\\mathbb F_p$. Every nonscalar $2\\times2$ matrix over a field has such a vector.\n\\end{lemma}\n\\begin{proof}\nThere is a unique polynomial of degree less than $d$ satisfying $f(B)v=Mv$, since the columns of $Q$ are a basis. If $MB=BM$, then $M-f(B)$ annihilates every $B^jv$, so it is zero. Conversely every polynomial in $B$ commutes with $B$. This also proves uniqueness and the coefficient formula. A lifted cyclic vector modulo $p$ makes $\\det Q$ a unit modulo $p^s$.","$2\\times2$ matrix over a field has such a vector.\n\\end{lemma}\n\\begin{proof}\nThere is a unique polynomial of degree less than $d$ satisfying $f(B)v=Mv$, since the columns of $Q$ are a basis. If $MB=BM$, then $M-f(B)$ annihilates every $B^jv$, so it is zero. Conversely every polynomial in $B$ commutes with $B$. This also proves uniqueness and the coefficient formula. A lifted cyclic vector modulo $p$ makes $\\det Q$ a unit modulo $p^s$.","$ matrix over a field has such a vector.\n\\end{lemma}\n\\begin{proof}\nThere is a unique polynomial of degree less than $d$ satisfying $f(B)v=Mv$, since the columns of $Q$ are a basis. If $MB=BM$, then $M-f(B)$ annihilates every $B^jv$, so it is zero. Conversely every polynomial in $B$ commutes with $B$. This also proves uniqueness and the coefficient formula. A lifted cyclic vector modulo $p$ makes $\\det Q$ a unit modulo $p^s$.","as such a vector.\n\\end{lemma}\n\\begin{proof}\nThere is a unique polynomial of degree less than $d$ satisfying $f(B)v=Mv$, since the columns of $Q$ are a basis. If $MB=BM$, then $M-f(B)$ annihilates every $B^jv$, so it is zero. Conversely every polynomial in $B$ commutes with $B$. This also proves uniqueness and the coefficient formula. A lifted cyclic vector modulo $p$ makes $\\det Q$ a unit modulo $p^s$.","Finally, if a $2\\times2$ matrix has no cyclic vector, it takes each vector into its own span. On a basis $e_1,e_2$ it is diagonal, and applying this condition to $e_1+e_2$ makes the two diagonal entries equal. Thus it is scalar. This argument includes the field $\\mathbb F_2$.\n\\end{proof}","Here the source is $\\Gamma=\\mathbb Z_p^r$, where $r\\geq1$ is finite. Write the generator matrices of the given continuous representation $\\rho_n$ as $A_1,\\ldots,A_r$. They commute in $M_d(R_n)$ and each has $p$-power order modulo $p$.","Choose $0\\leq a<n$ and integer scalars $\\lambda_i$ such that\n\\[\n A_i\\equiv\\lambda_i I\\pmod{p^a}\\quad\\text{for every }i.\n\\]\nThere is no restriction in this display when $a=0$. Define the \\emph{normalized matrices}\n\\[\n B_i=\\frac{A_i-\\lambda_i I}{p^a}\n \\quad\\in M_d(R_{n-a}).\n\\]\nThis is well-defined from the given finite data. It does not invert $p$ inside $R_n$. Assume that, for some distinguished index $*$, the matrix $B_*$ is cyclic modulo $p$. Define the finite obstruction record\n\\[\n \\boxed{\\Omega_{a,*}(\\rho_n)=\n \\bigl([B_*,B_i]\\bigr)_{i=1}^r\\in M_d(R_{n-a})^r,}\n \\qquad [B,C]=BC-CB.\n\\]","e $0\\leq a<n$ and integer scalars $\\lambda_i$ such that\n\\[\n A_i\\equiv\\lambda_i I\\pmod{p^a}\\quad\\text{for every }i.\n\\]\nThere is no restriction in this display when $a=0$. Define the \\emph{normalized matrices}\n\\[\n B_i=\\frac{A_i-\\lambda_i I}{p^a}\n \\quad\\in M_d(R_{n-a}).\n\\]\nThis is well-defined from the given finite data. It does not invert $p$ inside $R_n$. Assume that, for some distinguished index $*$, the matrix $B_*$ is cyclic modulo $p$. Define the finite obstruction record\n\\[\n \\boxed{\\Omega_{a,*}(\\rho_n)=\n \\bigl([B_*,B_i]\\bigr)_{i=1}^r\\in M_d(R_{n-a})^r,}\n \\qquad [B,C]=BC-CB.\n\\]","d integer scalars $\\lambda_i$ such that\n\\[\n A_i\\equiv\\lambda_i I\\pmod{p^a}\\quad\\text{for every }i.\n\\]\nThere is no restriction in this display when $a=0$. Define the \\emph{normalized matrices}\n\\[\n B_i=\\frac{A_i-\\lambda_i I}{p^a}\n \\quad\\in M_d(R_{n-a}).\n\\]\nThis is well-defined from the given finite data. It does not invert $p$ inside $R_n$. Assume that, for some distinguished index $*$, the matrix $B_*$ is cyclic modulo $p$. Define the finite obstruction record\n\\[\n \\boxed{\\Omega_{a,*}(\\rho_n)=\n \\bigl([B_*,B_i]\\bigr)_{i=1}^r\\in M_d(R_{n-a})^r,}\n \\qquad [B,C]=BC-CB.\n\\]","bda_i$ such that\n\\[\n A_i\\equiv\\lambda_i I\\pmod{p^a}\\quad\\text{for every }i.\n\\]\nThere is no restriction in this display when $a=0$. Define the \\emph{normalized matrices}\n\\[\n B_i=\\frac{A_i-\\lambda_i I}{p^a}\n \\quad\\in M_d(R_{n-a}).\n\\]\nThis is well-defined from the given finite data. It does not invert $p$ inside $R_n$. Assume that, for some distinguished index $*$, the matrix $B_*$ is cyclic modulo $p$. Define the finite obstruction record\n\\[\n \\boxed{\\Omega_{a,*}(\\rho_n)=\n \\bigl([B_*,B_i]\\bigr)_{i=1}^r\\in M_d(R_{n-a})^r,}\n \\qquad [B,C]=BC-CB.\n\\]","mod{p^a}\\quad\\text{for every }i.\n\\]\nThere is no restriction in this display when $a=0$. Define the \\emph{normalized matrices}\n\\[\n B_i=\\frac{A_i-\\lambda_i I}{p^a}\n \\quad\\in M_d(R_{n-a}).\n\\]\nThis is well-defined from the given finite data. It does not invert $p$ inside $R_n$. Assume that, for some distinguished index $*$, the matrix $B_*$ is cyclic modulo $p$. Define the finite obstruction record\n\\[\n \\boxed{\\Omega_{a,*}(\\rho_n)=\n \\bigl([B_*,B_i]\\bigr)_{i=1}^r\\in M_d(R_{n-a})^r,}\n \\qquad [B,C]=BC-CB.\n\\]","\\begin{theorem}[Normalized cyclic lifting criterion]\\label{thm:normalized-cyclic-lift}\nUnder the preceding hypotheses,\n\\[\n \\boxed{\\rho_n\\text{ has a continuous integral lift}\n \\quad\\Longleftrightarrow\\quad\\Omega_{a,*}(\\rho_n)=0.}\n\\]\nIf the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express $B_i=f_i(B_*)$ in $R_{n-a}$, with $\\deg f_i<d$ and $f_*(t)=t$ when $d>1$. Choose an integral lift $\\widehat B$ of $B_*$ and integer lifts $\\widehat f_i$ of these polynomials, taking the distinguished generator itself to $\\widehat B$. Then put\n\\[\n \\widehat A_i=\\lambda_i I+p^a\\widehat f_i(\\widehat B),\n \\qquad\n \\rho(x_1,\\ldots,x_r)=\\prod_i\\widehat A_i^{x_i}.\n\\]\nFor $d=1$, use the scalar matrices directly. The construction preserves the prescribed reduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any comm","\\begin{theorem}[Normalized cyclic lifting criterion]\\label{thm:normalized-cyclic-lift}\nUnder the preceding hypotheses,\n\\[\n \\boxed{\\rho_n\\text{ has a continuous integral lift}\n \\quad\\Longleftrightarrow\\quad\\Omega_{a,*}(\\rho_n)=0.}\n\\]\nIf the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express $B_i=f_i(B_*)$ in $R_{n-a}$, with $\\deg f_i<d$ and $f_*(t)=t$ when $d>1$. Choose an integral lift $\\widehat B$ of $B_*$ and integer lifts $\\widehat f_i$ of these polynomials, taking the distinguished generator itself to $\\widehat B$. Then put\n\\[\n \\widehat A_i=\\lambda_i I+p^a\\widehat f_i(\\widehat B),\n \\qquad\n \\rho(x_1,\\ldots,x_r)=\\prod_i\\widehat A_i^{x_i}.\n\\]\nFor $d=1$, use the scalar matrices directly. The construction preserves the prescribed reduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\","\\begin{theorem}[Normalized cyclic lifting criterion]\\label{thm:normalized-cyclic-lift}\nUnder the preceding hypotheses,\n\\[\n \\boxed{\\rho_n\\text{ has a continuous integral lift}\n \\quad\\Longleftrightarrow\\quad\\Omega_{a,*}(\\rho_n)=0.}\n\\]\nIf the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express $B_i=f_i(B_*)$ in $R_{n-a}$, with $\\deg f_i<d$ and $f_*(t)=t$ when $d>1$. Choose an integral lift $\\widehat B$ of $B_*$ and integer lifts $\\widehat f_i$ of these polynomials, taking the distinguished generator itself to $\\widehat B$. Then put\n\\[\n \\widehat A_i=\\lambda_i I+p^a\\widehat f_i(\\widehat B),\n \\qquad\n \\rho(x_1,\\ldots,x_r)=\\prod_i\\widehat A_i^{x_i}.\n\\]\nFor $d=1$, use the scalar matrices directly. The construction preserves the prescribed reduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\widehat A_i]","\\begin{theorem}[Normalized cyclic lifting criterion]\\label{thm:normalized-cyclic-lift}\nUnder the preceding hypotheses,\n\\[\n \\boxed{\\rho_n\\text{ has a continuous integral lift}\n \\quad\\Longleftrightarrow\\quad\\Omega_{a,*}(\\rho_n)=0.}\n\\]\nIf the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express $B_i=f_i(B_*)$ in $R_{n-a}$, with $\\deg f_i<d$ and $f_*(t)=t$ when $d>1$. Choose an integral lift $\\widehat B$ of $B_*$ and integer lifts $\\widehat f_i$ of these polynomials, taking the distinguished generator itself to $\\widehat B$. Then put\n\\[\n \\widehat A_i=\\lambda_i I+p^a\\widehat f_i(\\widehat B),\n \\qquad\n \\rho(x_1,\\ldots,x_r)=\\prod_i\\widehat A_i^{x_i}.\n\\]\nFor $d=1$, use the scalar matrices directly. The construction preserves the prescribed reduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\widehat A_i]\n   =p^{2a}[\\widehat","\\begin{theorem}[Normalized cyclic lifting criterion]\\label{thm:normalized-cyclic-lift}\nUnder the preceding hypotheses,\n\\[\n \\boxed{\\rho_n\\text{ has a continuous integral lift}\n \\quad\\Longleftrightarrow\\quad\\Omega_{a,*}(\\rho_n)=0.}\n\\]\nIf the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express $B_i=f_i(B_*)$ in $R_{n-a}$, with $\\deg f_i<d$ and $f_*(t)=t$ when $d>1$. Choose an integral lift $\\widehat B$ of $B_*$ and integer lifts $\\widehat f_i$ of these polynomials, taking the distinguished generator itself to $\\widehat B$. Then put\n\\[\n \\widehat A_i=\\lambda_i I+p^a\\widehat f_i(\\widehat B),\n \\qquad\n \\rho(x_1,\\ldots,x_r)=\\prod_i\\widehat A_i^{x_i}.\n\\]\nFor $d=1$, use the scalar matrices directly. The construction preserves the prescribed reduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\widehat A_i]\n   =p^{2a}[\\widehat B_*,\\widehat B","theorem}[Normalized cyclic lifting criterion]\\label{thm:normalized-cyclic-lift}\nUnder the preceding hypotheses,\n\\[\n \\boxed{\\rho_n\\text{ has a continuous integral lift}\n \\quad\\Longleftrightarrow\\quad\\Omega_{a,*}(\\rho_n)=0.}\n\\]\nIf the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express $B_i=f_i(B_*)$ in $R_{n-a}$, with $\\deg f_i<d$ and $f_*(t)=t$ when $d>1$. Choose an integral lift $\\widehat B$ of $B_*$ and integer lifts $\\widehat f_i$ of these polynomials, taking the distinguished generator itself to $\\widehat B$. Then put\n\\[\n \\widehat A_i=\\lambda_i I+p^a\\widehat f_i(\\widehat B),\n \\qquad\n \\rho(x_1,\\ldots,x_r)=\\prod_i\\widehat A_i^{x_i}.\n\\]\nFor $d=1$, use the scalar matrices directly. The construction preserves the prescribed reduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\widehat A_i]\n   =p^{2a}[\\widehat B_*,\\widehat B_i]\n\\]\ni…","ing criterion]\\label{thm:normalized-cyclic-lift}\nUnder the preceding hypotheses,\n\\[\n \\boxed{\\rho_n\\text{ has a continuous integral lift}\n \\quad\\Longleftrightarrow\\quad\\Omega_{a,*}(\\rho_n)=0.}\n\\]\nIf the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express $B_i=f_i(B_*)$ in $R_{n-a}$, with $\\deg f_i<d$ and $f_*(t)=t$ when $d>1$. Choose an integral lift $\\widehat B$ of $B_*$ and integer lifts $\\widehat f_i$ of these polynomials, taking the distinguished generator itself to $\\widehat B$. Then put\n\\[\n \\widehat A_i=\\lambda_i I+p^a\\widehat f_i(\\widehat B),\n \\qquad\n \\rho(x_1,\\ldots,x_r)=\\prod_i\\widehat A_i^{x_i}.\n\\]\nFor $d=1$, use the scalar matrices directly. The construction preserves the prescribed reduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\widehat A_i]\n   =p^{2a}[\\widehat B_*,\\widehat B_i]\n\\]\nimply $[\\widehat B_*,\\widehat B_…","abel{thm:normalized-cyclic-lift}\nUnder the preceding hypotheses,\n\\[\n \\boxed{\\rho_n\\text{ has a continuous integral lift}\n \\quad\\Longleftrightarrow\\quad\\Omega_{a,*}(\\rho_n)=0.}\n\\]\nIf the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express $B_i=f_i(B_*)$ in $R_{n-a}$, with $\\deg f_i<d$ and $f_*(t)=t$ when $d>1$. Choose an integral lift $\\widehat B$ of $B_*$ and integer lifts $\\widehat f_i$ of these polynomials, taking the distinguished generator itself to $\\widehat B$. Then put\n\\[\n \\widehat A_i=\\lambda_i I+p^a\\widehat f_i(\\widehat B),\n \\qquad\n \\rho(x_1,\\ldots,x_r)=\\prod_i\\widehat A_i^{x_i}.\n\\]\nFor $d=1$, use the scalar matrices directly. The construction preserves the prescribed reduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\widehat A_i]\n   =p^{2a}[\\widehat B_*,\\widehat B_i]\n\\]\nimply $[\\widehat B_*,\\widehat B_i]=0$, since mat…","ic-lift}\nUnder the preceding hypotheses,\n\\[\n \\boxed{\\rho_n\\text{ has a continuous integral lift}\n \\quad\\Longleftrightarrow\\quad\\Omega_{a,*}(\\rho_n)=0.}\n\\]\nIf the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express $B_i=f_i(B_*)$ in $R_{n-a}$, with $\\deg f_i<d$ and $f_*(t)=t$ when $d>1$. Choose an integral lift $\\widehat B$ of $B_*$ and integer lifts $\\widehat f_i$ of these polynomials, taking the distinguished generator itself to $\\widehat B$. Then put\n\\[\n \\widehat A_i=\\lambda_i I+p^a\\widehat f_i(\\widehat B),\n \\qquad\n \\rho(x_1,\\ldots,x_r)=\\prod_i\\widehat A_i^{x_i}.\n\\]\nFor $d=1$, use the scalar matrices directly. The construction preserves the prescribed reduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\widehat A_i]\n   =p^{2a}[\\widehat B_*,\\widehat B_i]\n\\]\nimply $[\\widehat B_*,\\widehat B_i]=0$, since matrices over $\\mathbb Z_p$…","integral lift}\n \\quad\\Longleftrightarrow\\quad\\Omega_{a,*}(\\rho_n)=0.}\n\\]\nIf the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express $B_i=f_i(B_*)$ in $R_{n-a}$, with $\\deg f_i<d$ and $f_*(t)=t$ when $d>1$. Choose an integral lift $\\widehat B$ of $B_*$ and integer lifts $\\widehat f_i$ of these polynomials, taking the distinguished generator itself to $\\widehat B$. Then put\n\\[\n \\widehat A_i=\\lambda_i I+p^a\\widehat f_i(\\widehat B),\n \\qquad\n \\rho(x_1,\\ldots,x_r)=\\prod_i\\widehat A_i^{x_i}.\n\\]\nFor $d=1$, use the scalar matrices directly. The construction preserves the prescribed reduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\widehat A_i]\n   =p^{2a}[\\widehat B_*,\\widehat B_i]\n\\]\nimply $[\\widehat B_*,\\widehat B_i]=0$, since matrices over $\\mathbb Z_p$ have no $p$-torsion. Reducing modulo $p^{n-a}$ proves necessity. This argument pe…","ngleftrightarrow\\quad\\Omega_{a,*}(\\rho_n)=0.}\n\\]\nIf the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express $B_i=f_i(B_*)$ in $R_{n-a}$, with $\\deg f_i<d$ and $f_*(t)=t$ when $d>1$. Choose an integral lift $\\widehat B$ of $B_*$ and integer lifts $\\widehat f_i$ of these polynomials, taking the distinguished generator itself to $\\widehat B$. Then put\n\\[\n \\widehat A_i=\\lambda_i I+p^a\\widehat f_i(\\widehat B),\n \\qquad\n \\rho(x_1,\\ldots,x_r)=\\prod_i\\widehat A_i^{x_i}.\n\\]\nFor $d=1$, use the scalar matrices directly. The construction preserves the prescribed reduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\widehat A_i]\n   =p^{2a}[\\widehat B_*,\\widehat B_i]\n\\]\nimply $[\\widehat B_*,\\widehat B_i]=0$, since matrices over $\\mathbb Z_p$ have no $p$-torsion. Reducing modulo $p^{n-a}$ proves necessity. This argument permits arbitrary higher d…","polynomial certificates. Express $B_i=f_i(B_*)$ in $R_{n-a}$, with $\\deg f_i<d$ and $f_*(t)=t$ when $d>1$. Choose an integral lift $\\widehat B$ of $B_*$ and integer lifts $\\widehat f_i$ of these polynomials, taking the distinguished generator itself to $\\widehat B$. Then put\n\\[\n \\widehat A_i=\\lambda_i I+p^a\\widehat f_i(\\widehat B),\n \\qquad\n \\rho(x_1,\\ldots,x_r)=\\prod_i\\widehat A_i^{x_i}.\n\\]\nFor $d=1$, use the scalar matrices directly. The construction preserves the prescribed reduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\widehat A_i]\n   =p^{2a}[\\widehat B_*,\\widehat B_i]\n\\]\nimply $[\\widehat B_*,\\widehat B_i]=0$, since matrices over $\\mathbb Z_p$ have no $p$-torsion. Reducing modulo $p^{n-a}$ proves necessity. This argument permits arbitrary higher digits in all the integral lifts.","$d>1$. Choose an integral lift $\\widehat B$ of $B_*$ and integer lifts $\\widehat f_i$ of these polynomials, taking the distinguished generator itself to $\\widehat B$. Then put\n\\[\n \\widehat A_i=\\lambda_i I+p^a\\widehat f_i(\\widehat B),\n \\qquad\n \\rho(x_1,\\ldots,x_r)=\\prod_i\\widehat A_i^{x_i}.\n\\]\nFor $d=1$, use the scalar matrices directly. The construction preserves the prescribed reduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\widehat A_i]\n   =p^{2a}[\\widehat B_*,\\widehat B_i]\n\\]\nimply $[\\widehat B_*,\\widehat B_i]=0$, since matrices over $\\mathbb Z_p$ have no $p$-torsion. Reducing modulo $p^{n-a}$ proves necessity. This argument permits arbitrary higher digits in all the integral lifts.","ynomials, taking the distinguished generator itself to $\\widehat B$. Then put\n\\[\n \\widehat A_i=\\lambda_i I+p^a\\widehat f_i(\\widehat B),\n \\qquad\n \\rho(x_1,\\ldots,x_r)=\\prod_i\\widehat A_i^{x_i}.\n\\]\nFor $d=1$, use the scalar matrices directly. The construction preserves the prescribed reduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\widehat A_i]\n   =p^{2a}[\\widehat B_*,\\widehat B_i]\n\\]\nimply $[\\widehat B_*,\\widehat B_i]=0$, since matrices over $\\mathbb Z_p$ have no $p$-torsion. Reducing modulo $p^{n-a}$ proves necessity. This argument permits arbitrary higher digits in all the integral lifts.","$\\widehat B$. Then put\n\\[\n \\widehat A_i=\\lambda_i I+p^a\\widehat f_i(\\widehat B),\n \\qquad\n \\rho(x_1,\\ldots,x_r)=\\prod_i\\widehat A_i^{x_i}.\n\\]\nFor $d=1$, use the scalar matrices directly. The construction preserves the prescribed reduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\widehat A_i]\n   =p^{2a}[\\widehat B_*,\\widehat B_i]\n\\]\nimply $[\\widehat B_*,\\widehat B_i]=0$, since matrices over $\\mathbb Z_p$ have no $p$-torsion. Reducing modulo $p^{n-a}$ proves necessity. This argument permits arbitrary higher digits in all the integral lifts.","at B),\n \\qquad\n \\rho(x_1,\\ldots,x_r)=\\prod_i\\widehat A_i^{x_i}.\n\\]\nFor $d=1$, use the scalar matrices directly. The construction preserves the prescribed reduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\widehat A_i]\n   =p^{2a}[\\widehat B_*,\\widehat B_i]\n\\]\nimply $[\\widehat B_*,\\widehat B_i]=0$, since matrices over $\\mathbb Z_p$ have no $p$-torsion. Reducing modulo $p^{n-a}$ proves necessity. This argument permits arbitrary higher digits in all the integral lifts.","e the scalar matrices directly. The construction preserves the prescribed reduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\widehat A_i]\n   =p^{2a}[\\widehat B_*,\\widehat B_i]\n\\]\nimply $[\\widehat B_*,\\widehat B_i]=0$, since matrices over $\\mathbb Z_p$ have no $p$-torsion. Reducing modulo $p^{n-a}$ proves necessity. This argument permits arbitrary higher digits in all the integral lifts.","erves the prescribed reduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\widehat A_i]\n   =p^{2a}[\\widehat B_*,\\widehat B_i]\n\\]\nimply $[\\widehat B_*,\\widehat B_i]=0$, since matrices over $\\mathbb Z_p$ have no $p$-torsion. Reducing modulo $p^{n-a}$ proves necessity. This argument permits arbitrary higher digits in all the integral lifts.","eduction modulo $p^n$ and does not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\widehat A_i]\n   =p^{2a}[\\widehat B_*,\\widehat B_i]\n\\]\nimply $[\\widehat B_*,\\widehat B_i]=0$, since matrices over $\\mathbb Z_p$ have no $p$-torsion. Reducing modulo $p^{n-a}$ proves necessity. This argument permits arbitrary higher digits in all the integral lifts.","s not pass to an obstruction quotient.\n\\end{theorem}\n\\begin{proof}\nSuppose first that $\\widehat A_i$ are any commuting integral lifts. Then\n$\\widehat B_i=(\\widehat A_i-\\lambda_iI)/p^a$ are integral. The identities\n\\[\n 0=[\\widehat A_*,\\widehat A_i]\n   =p^{2a}[\\widehat B_*,\\widehat B_i]\n\\]\nimply $[\\widehat B_*,\\widehat B_i]=0$, since matrices over $\\mathbb Z_p$ have no $p$-torsion. Reducing modulo $p^{n-a}$ proves necessity. This argument permits arbitrary higher digits in all the integral lifts.","Conversely, the cyclic-centralizer lemma supplies the unique polynomials $f_i$ from the vanishing commutators. Their integral polynomial evaluations commute exactly. Multiplication by $p^a$ shows that $\\widehat A_i$ reduces to $A_i$ modulo $p^n$. Each is invertible because its reduction modulo $p$ is invertible. That reduction has $p$-power order by the input's continuity. The procyclic-powers lemma now supplies the continuous representation, whose reduction agrees on the topological generators and hence on all of $\\mathbb Z_p^r$. In rank one the same argument is simply scalar lifting. Polynomial coefficients and a cyclic vector give a finite positive certificate; a nonzero entry of a displayed commutator gives a finite negative certificate under the verified cyclic","Conversely, the cyclic-centralizer lemma supplies the unique polynomials $f_i$ from the vanishing commutators. Their integral polynomial evaluations commute exactly. Multiplication by $p^a$ shows that $\\widehat A_i$ reduces to $A_i$ modulo $p^n$. Each is invertible because its reduction modulo $p$ is invertible. That reduction has $p$-power order by the input's continuity. The procyclic-powers lemma now supplies the continuous representation, whose reduction agrees on the topological generators and hence on all of $\\mathbb Z_p^r$. In rank one the same argument is simply scalar lifting. Polynomial coefficients and a cyclic vector give a finite positive certificate; a nonzero entry of a displayed commutator gives a finite negative certificate under the verified cyclic hypothesis.\n\\end{proof}","gral polynomial evaluations commute exactly. Multiplication by $p^a$ shows that $\\widehat A_i$ reduces to $A_i$ modulo $p^n$. Each is invertible because its reduction modulo $p$ is invertible. That reduction has $p$-power order by the input's continuity. The procyclic-powers lemma now supplies the continuous representation, whose reduction agrees on the topological generators and hence on all of $\\mathbb Z_p^r$. In rank one the same argument is simply scalar lifting. Polynomial coefficients and a cyclic vector give a finite positive certificate; a nonzero entry of a displayed commutator gives a finite negative certificate under the verified cyclic hypothesis.\n\\end{proof}","\\begin{remark}[What normalization reveals]\nThe input relations only force\n\\[\n p^{2a}[B_i,B_j]\\equiv0\\pmod{p^n}.\n\\]\nWhen $a>0$, this can conceal a nonzero commutator in $R_{n-a}$. The obstruction is therefore not just the original zero commutator re-labelled. It measures the additional commutativity required after removing a common scalar precision. Different permitted scalar choices change $B_i$ by scalar matrices and leave its commutators unchanged. Changing representatives changes them only by zero in $R_{n-a}$. No preferred normalized matrix is claimed independently of these data.\n\\end{remark}",".\n\\]\nWhen $a>0$, this can conceal a nonzero commutator in $R_{n-a}$. The obstruction is therefore not just the original zero commutator re-labelled. It measures the additional commutativity required after removing a common scalar precision. Different permitted scalar choices change $B_i$ by scalar matrices and leave its commutators unchanged. Changing representatives changes them only by zero in $R_{n-a}$. No preferred normalized matrix is claimed independently of these data.\n\\end{remark}","\\begin{corollary}[A source-fixed finite horizon in the cyclic stratum]\\label{cor:abelian-cyclic-horizon}\nUnder the hypotheses of \\cref{thm:normalized-cyclic-lift}, the following are equivalent:\n\\[\n \\rho_n\\text{ lifts over }\\mathbb Z_p,\n \\qquad \\rho_n\\text{ extends over }R_{n+a},\n \\qquad \\Omega_{a,*}(\\rho_n)=0.\n\\]\nThe extension remains a representation of the same $\\mathbb Z_p^r$. When $a=0$, the current representation is already a sufficient certificate.\n\\end{corollary}\n\\begin{proof}\nAn integral lift supplies the finite extension. Given a finite extension with commuting matrices $A_i'$ modulo $p^{n+a}$, subtract the same $\\lambda_iI$ and divide by $p^a$. The resulting matrices $B_i'$ are defined modulo $p^n$ and reduce to $B_i$ modulo $p^{n-a}$. The equality\n$p^{2a}[B_*',B_i']=0$ modulo $p^{n+a}$ implies\n$[B_*',B_i']=0$ modulo $p^{n-a}$, so $\\Omega_{a,*}=0$. The normalized criterion supplies an integral lift. This is a bound obtained from the source's actual commutativity relations, not from the o","\\begin{corollary}[A source-fixed finite horizon in the cyclic stratum]\\label{cor:abelian-cyclic-horizon}\nUnder the hypotheses of \\cref{thm:normalized-cyclic-lift}, the following are equivalent:\n\\[\n \\rho_n\\text{ lifts over }\\mathbb Z_p,\n \\qquad \\rho_n\\text{ extends over }R_{n+a},\n \\qquad \\Omega_{a,*}(\\rho_n)=0.\n\\]\nThe extension remains a representation of the same $\\mathbb Z_p^r$. When $a=0$, the current representation is already a sufficient certificate.\n\\end{corollary}\n\\begin{proof}\nAn integral lift supplies the finite extension. Given a finite extension with commuting matrices $A_i'$ modulo $p^{n+a}$, subtract the same $\\lambda_iI$ and divide by $p^a$. The resulting matrices $B_i'$ are defined modulo $p^n$ and reduce to $B_i$ modulo $p^{n-a}$. The equality\n$p^{2a}[B_*',B_i']=0$ modulo $p^{n+a}$ implies\n$[B_*',B_i']=0$ modulo $p^{n-a}$, so $\\Omega_{a,*}=0$. The normalized criterion supplies an integral lift. This is a bound obtained from the source's actual commutativity relations, not from the orders of its growing finite images.\n\\end{proof}","alent:\n\\[\n \\rho_n\\text{ lifts over }\\mathbb Z_p,\n \\qquad \\rho_n\\text{ extends over }R_{n+a},\n \\qquad \\Omega_{a,*}(\\rho_n)=0.\n\\]\nThe extension remains a representation of the same $\\mathbb Z_p^r$. When $a=0$, the current representation is already a sufficient certificate.\n\\end{corollary}\n\\begin{proof}\nAn integral lift supplies the finite extension. Given a finite extension with commuting matrices $A_i'$ modulo $p^{n+a}$, subtract the same $\\lambda_iI$ and divide by $p^a$. The resulting matrices $B_i'$ are defined modulo $p^n$ and reduce to $B_i$ modulo $p^{n-a}$. The equality\n$p^{2a}[B_*',B_i']=0$ modulo $p^{n+a}$ implies\n$[B_*',B_i']=0$ modulo $p^{n-a}$, so $\\Omega_{a,*}=0$. The normalized criterion supplies an integral lift. This is a bound obtained from the source's actual commutativity relations, not from the orders of its growing finite images.\n\\end{proof}","o_n\\text{ lifts over }\\mathbb Z_p,\n \\qquad \\rho_n\\text{ extends over }R_{n+a},\n \\qquad \\Omega_{a,*}(\\rho_n)=0.\n\\]\nThe extension remains a representation of the same $\\mathbb Z_p^r$. When $a=0$, the current representation is already a sufficient certificate.\n\\end{corollary}\n\\begin{proof}\nAn integral lift supplies the finite extension. Given a finite extension with commuting matrices $A_i'$ modulo $p^{n+a}$, subtract the same $\\lambda_iI$ and divide by $p^a$. The resulting matrices $B_i'$ are defined modulo $p^n$ and reduce to $B_i$ modulo $p^{n-a}$. The equality\n$p^{2a}[B_*',B_i']=0$ modulo $p^{n+a}$ implies\n$[B_*',B_i']=0$ modulo $p^{n-a}$, so $\\Omega_{a,*}=0$. The normalized criterion supplies an integral lift. This is a bound obtained from the source's actual commutativity relations, not from the orders of its growing finite images.\n\\end{proof}","Z_p,\n \\qquad \\rho_n\\text{ extends over }R_{n+a},\n \\qquad \\Omega_{a,*}(\\rho_n)=0.\n\\]\nThe extension remains a representation of the same $\\mathbb Z_p^r$. When $a=0$, the current representation is already a sufficient certificate.\n\\end{corollary}\n\\begin{proof}\nAn integral lift supplies the finite extension. Given a finite extension with commuting matrices $A_i'$ modulo $p^{n+a}$, subtract the same $\\lambda_iI$ and divide by $p^a$. The resulting matrices $B_i'$ are defined modulo $p^n$ and reduce to $B_i$ modulo $p^{n-a}$. The equality\n$p^{2a}[B_*',B_i']=0$ modulo $p^{n+a}$ implies\n$[B_*',B_i']=0$ modulo $p^{n-a}$, so $\\Omega_{a,*}=0$. The normalized criterion supplies an integral lift. This is a bound obtained from the source's actual commutativity relations, not from the orders of its growing finite images.\n\\end{proof}","extends over }R_{n+a},\n \\qquad \\Omega_{a,*}(\\rho_n)=0.\n\\]\nThe extension remains a representation of the same $\\mathbb Z_p^r$. When $a=0$, the current representation is already a sufficient certificate.\n\\end{corollary}\n\\begin{proof}\nAn integral lift supplies the finite extension. Given a finite extension with commuting matrices $A_i'$ modulo $p^{n+a}$, subtract the same $\\lambda_iI$ and divide by $p^a$. The resulting matrices $B_i'$ are defined modulo $p^n$ and reduce to $B_i$ modulo $p^{n-a}$. The equality\n$p^{2a}[B_*',B_i']=0$ modulo $p^{n+a}$ implies\n$[B_*',B_i']=0$ modulo $p^{n-a}$, so $\\Omega_{a,*}=0$. The normalized criterion supplies an integral lift. This is a bound obtained from the source's actual commutativity relations, not from the orders of its growing finite images.\n\\end{proof}","\\Omega_{a,*}(\\rho_n)=0.\n\\]\nThe extension remains a representation of the same $\\mathbb Z_p^r$. When $a=0$, the current representation is already a sufficient certificate.\n\\end{corollary}\n\\begin{proof}\nAn integral lift supplies the finite extension. Given a finite extension with commuting matrices $A_i'$ modulo $p^{n+a}$, subtract the same $\\lambda_iI$ and divide by $p^a$. The resulting matrices $B_i'$ are defined modulo $p^n$ and reduce to $B_i$ modulo $p^{n-a}$. The equality\n$p^{2a}[B_*',B_i']=0$ modulo $p^{n+a}$ implies\n$[B_*',B_i']=0$ modulo $p^{n-a}$, so $\\Omega_{a,*}=0$. The normalized criterion supplies an integral lift. This is a bound obtained from the source's actual commutativity relations, not from the orders of its growing finite images.\n\\end{proof}","]\nThe extension remains a representation of the same $\\mathbb Z_p^r$. When $a=0$, the current representation is already a sufficient certificate.\n\\end{corollary}\n\\begin{proof}\nAn integral lift supplies the finite extension. Given a finite extension with commuting matrices $A_i'$ modulo $p^{n+a}$, subtract the same $\\lambda_iI$ and divide by $p^a$. The resulting matrices $B_i'$ are defined modulo $p^n$ and reduce to $B_i$ modulo $p^{n-a}$. The equality\n$p^{2a}[B_*',B_i']=0$ modulo $p^{n+a}$ implies\n$[B_*',B_i']=0$ modulo $p^{n-a}$, so $\\Omega_{a,*}=0$. The normalized criterion supplies an integral lift. This is a bound obtained from the source's actual commutativity relations, not from the orders of its growing finite images.\n\\end{proof}","ins a representation of the same $\\mathbb Z_p^r$. When $a=0$, the current representation is already a sufficient certificate.\n\\end{corollary}\n\\begin{proof}\nAn integral lift supplies the finite extension. Given a finite extension with commuting matrices $A_i'$ modulo $p^{n+a}$, subtract the same $\\lambda_iI$ and divide by $p^a$. The resulting matrices $B_i'$ are defined modulo $p^n$ and reduce to $B_i$ modulo $p^{n-a}$. The equality\n$p^{2a}[B_*',B_i']=0$ modulo $p^{n+a}$ implies\n$[B_*',B_i']=0$ modulo $p^{n-a}$, so $\\Omega_{a,*}=0$. The normalized criterion supplies an integral lift. This is a bound obtained from the source's actual commutativity relations, not from the orders of its growing finite images.\n\\end{proof}","ntation of the same $\\mathbb Z_p^r$. When $a=0$, the current representation is already a sufficient certificate.\n\\end{corollary}\n\\begin{proof}\nAn integral lift supplies the finite extension. Given a finite extension with commuting matrices $A_i'$ modulo $p^{n+a}$, subtract the same $\\lambda_iI$ and divide by $p^a$. The resulting matrices $B_i'$ are defined modulo $p^n$ and reduce to $B_i$ modulo $p^{n-a}$. The equality\n$p^{2a}[B_*',B_i']=0$ modulo $p^{n+a}$ implies\n$[B_*',B_i']=0$ modulo $p^{n-a}$, so $\\Omega_{a,*}=0$. The normalized criterion supplies an integral lift. This is a bound obtained from the source's actual commutativity relations, not from the orders of its growing finite images.\n\\end{proof}","thbb Z_p^r$. When $a=0$, the current representation is already a sufficient certificate.\n\\end{corollary}\n\\begin{proof}\nAn integral lift supplies the finite extension. Given a finite extension with commuting matrices $A_i'$ modulo $p^{n+a}$, subtract the same $\\lambda_iI$ and divide by $p^a$. The resulting matrices $B_i'$ are defined modulo $p^n$ and reduce to $B_i$ modulo $p^{n-a}$. The equality\n$p^{2a}[B_*',B_i']=0$ modulo $p^{n+a}$ implies\n$[B_*',B_i']=0$ modulo $p^{n-a}$, so $\\Omega_{a,*}=0$. The normalized criterion supplies an integral lift. This is a bound obtained from the source's actual commutativity relations, not from the orders of its growing finite images.\n\\end{proof}","current representation is already a sufficient certificate.\n\\end{corollary}\n\\begin{proof}\nAn integral lift supplies the finite extension. Given a finite extension with commuting matrices $A_i'$ modulo $p^{n+a}$, subtract the same $\\lambda_iI$ and divide by $p^a$. The resulting matrices $B_i'$ are defined modulo $p^n$ and reduce to $B_i$ modulo $p^{n-a}$. The equality\n$p^{2a}[B_*',B_i']=0$ modulo $p^{n+a}$ implies\n$[B_*',B_i']=0$ modulo $p^{n-a}$, so $\\Omega_{a,*}=0$. The normalized criterion supplies an integral lift. This is a bound obtained from the source's actual commutativity relations, not from the orders of its growing finite images.\n\\end{proof}","tion is already a sufficient certificate.\n\\end{corollary}\n\\begin{proof}\nAn integral lift supplies the finite extension. Given a finite extension with commuting matrices $A_i'$ modulo $p^{n+a}$, subtract the same $\\lambda_iI$ and divide by $p^a$. The resulting matrices $B_i'$ are defined modulo $p^n$ and reduce to $B_i$ modulo $p^{n-a}$. The equality\n$p^{2a}[B_*',B_i']=0$ modulo $p^{n+a}$ implies\n$[B_*',B_i']=0$ modulo $p^{n-a}$, so $\\Omega_{a,*}=0$. The normalized criterion supplies an integral lift. This is a bound obtained from the source's actual commutativity relations, not from the orders of its growing finite images.\n\\end{proof}","cient certificate.\n\\end{corollary}\n\\begin{proof}\nAn integral lift supplies the finite extension. Given a finite extension with commuting matrices $A_i'$ modulo $p^{n+a}$, subtract the same $\\lambda_iI$ and divide by $p^a$. The resulting matrices $B_i'$ are defined modulo $p^n$ and reduce to $B_i$ modulo $p^{n-a}$. The equality\n$p^{2a}[B_*',B_i']=0$ modulo $p^{n+a}$ implies\n$[B_*',B_i']=0$ modulo $p^{n-a}$, so $\\Omega_{a,*}=0$. The normalized criterion supplies an integral lift. This is a bound obtained from the source's actual commutativity relations, not from the orders of its growing finite images.\n\\end{proof}","ate.\n\\end{corollary}\n\\begin{proof}\nAn integral lift supplies the finite extension. Given a finite extension with commuting matrices $A_i'$ modulo $p^{n+a}$, subtract the same $\\lambda_iI$ and divide by $p^a$. The resulting matrices $B_i'$ are defined modulo $p^n$ and reduce to $B_i$ modulo $p^{n-a}$. The equality\n$p^{2a}[B_*',B_i']=0$ modulo $p^{n+a}$ implies\n$[B_*',B_i']=0$ modulo $p^{n-a}$, so $\\Omega_{a,*}=0$. The normalized criterion supplies an integral lift. This is a bound obtained from the source's actual commutativity relations, not from the orders of its growing finite images.\n\\end{proof}","For $A\\in M_2(R_n)$ define its scalar depth by\n\\[\n a(A)=\\max\\{0\\leq t\\leq n:A\\text{ is scalar modulo }p^t\\}.\n\\]\nWriting entries in $R_n$, this is the minimum of the capped valuations of\n$A_{12},A_{21},A_{11}-A_{22}$. It is unchanged by invertible conjugation. For a tuple put $a=\\min_i a(A_i)$.","\\begin{theorem}[Complete rank-two profinite lifting test]\\label{thm:rank-two-profinite}\nLet $p$ be any prime and let $\\rho_n:\\mathbb Z_p^r\\to\\operatorname{GL}_2(R_n)$ be continuous, with finitely many topological generators as above.\n\\begin{enumerate}\n\\item If $a=n$, all the $A_i$ are scalar and $\\rho_n$ has a continuous integral lift.\n\\item If $a<n$, choose any generator $A_*$ with scalar depth $a$. Take $\\lambda_i$ to be an integer representative of $(A_i)_{11}$ and form $B_i$ modulo $p^{n-a}$. Then\n\\[\n \\boxed{\\rho_n\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [B_*,B_i]=0\\text{ in }M_2(R_{n-a})\\ (1\\leq i\\leq r).}\n\\]\nThe test is finite matrix arithmetic on the \\emph{existing input}. It does not require enumeration of deeper coefficient lifts. When it passes, the po","\\begin{theorem}[Complete rank-two profinite lifting test]\\label{thm:rank-two-profinite}\nLet $p$ be any prime and let $\\rho_n:\\mathbb Z_p^r\\to\\operatorname{GL}_2(R_n)$ be continuous, with finitely many topological generators as above.\n\\begin{enumerate}\n\\item If $a=n$, all the $A_i$ are scalar and $\\rho_n$ has a continuous integral lift.\n\\item If $a<n$, choose any generator $A_*$ with scalar depth $a$. Take $\\lambda_i$ to be an integer representative of $(A_i)_{11}$ and form $B_i$ modulo $p^{n-a}$. Then\n\\[\n \\boxed{\\rho_n\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [B_*,B_i]=0\\text{ in }M_2(R_{n-a})\\ (1\\leq i\\leq r).}\n\\]\nThe test is finite matrix arithmetic on the \\emph{existing input}. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in \\cref{thm:normalized-cyclic-lift} produces exa","\\begin{theorem}[Complete rank-two profinite lifting test]\\label{thm:rank-two-profinite}\nLet $p$ be any prime and let $\\rho_n:\\mathbb Z_p^r\\to\\operatorname{GL}_2(R_n)$ be continuous, with finitely many topological generators as above.\n\\begin{enumerate}\n\\item If $a=n$, all the $A_i$ are scalar and $\\rho_n$ has a continuous integral lift.\n\\item If $a<n$, choose any generator $A_*$ with scalar depth $a$. Take $\\lambda_i$ to be an integer representative of $(A_i)_{11}$ and form $B_i$ modulo $p^{n-a}$. Then\n\\[\n \\boxed{\\rho_n\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [B_*,B_i]=0\\text{ in }M_2(R_{n-a})\\ (1\\leq i\\leq r).}\n\\]\nThe test is finite matrix arithmetic on the \\emph{existing input}. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in \\cref{thm:normalized-cyclic-lift} produces exact commuting integral generator matrices.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf all generat","\\begin{theorem}[Complete rank-two profinite lifting test]\\label{thm:rank-two-profinite}\nLet $p$ be any prime and let $\\rho_n:\\mathbb Z_p^r\\to\\operatorname{GL}_2(R_n)$ be continuous, with finitely many topological generators as above.\n\\begin{enumerate}\n\\item If $a=n$, all the $A_i$ are scalar and $\\rho_n$ has a continuous integral lift.\n\\item If $a<n$, choose any generator $A_*$ with scalar depth $a$. Take $\\lambda_i$ to be an integer representative of $(A_i)_{11}$ and form $B_i$ modulo $p^{n-a}$. Then\n\\[\n \\boxed{\\rho_n\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [B_*,B_i]=0\\text{ in }M_2(R_{n-a})\\ (1\\leq i\\leq r).}\n\\]\nThe test is finite matrix arithmetic on the \\emph{existing input}. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in \\cref{thm:normalized-cyclic-lift} produces exact commuting integral generator matrices.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf all generators are scalar,","\\begin{theorem}[Complete rank-two profinite lifting test]\\label{thm:rank-two-profinite}\nLet $p$ be any prime and let $\\rho_n:\\mathbb Z_p^r\\to\\operatorname{GL}_2(R_n)$ be continuous, with finitely many topological generators as above.\n\\begin{enumerate}\n\\item If $a=n$, all the $A_i$ are scalar and $\\rho_n$ has a continuous integral lift.\n\\item If $a<n$, choose any generator $A_*$ with scalar depth $a$. Take $\\lambda_i$ to be an integer representative of $(A_i)_{11}$ and form $B_i$ modulo $p^{n-a}$. Then\n\\[\n \\boxed{\\rho_n\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [B_*,B_i]=0\\text{ in }M_2(R_{n-a})\\ (1\\leq i\\leq r).}\n\\]\nThe test is finite matrix arithmetic on the \\emph{existing input}. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in \\cref{thm:normalized-cyclic-lift} produces exact commuting integral generator matrices.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf all generators are scalar, lift their scalar entri","\\begin{theorem}[Complete rank-two profinite lifting test]\\label{thm:rank-two-profinite}\nLet $p$ be any prime and let $\\rho_n:\\mathbb Z_p^r\\to\\operatorname{GL}_2(R_n)$ be continuous, with finitely many topological generators as above.\n\\begin{enumerate}\n\\item If $a=n$, all the $A_i$ are scalar and $\\rho_n$ has a continuous integral lift.\n\\item If $a<n$, choose any generator $A_*$ with scalar depth $a$. Take $\\lambda_i$ to be an integer representative of $(A_i)_{11}$ and form $B_i$ modulo $p^{n-a}$. Then\n\\[\n \\boxed{\\rho_n\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [B_*,B_i]=0\\text{ in }M_2(R_{n-a})\\ (1\\leq i\\leq r).}\n\\]\nThe test is finite matrix arithmetic on the \\emph{existing input}. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in \\cref{thm:normalized-cyclic-lift} produces exact commuting integral generator matrices.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf all generators are scalar, lift their scalar entries integrally. Their residues modulo $p$ have $","\\begin{theorem}[Complete rank-two profinite lifting test]\\label{thm:rank-two-profinite}\nLet $p$ be any prime and let $\\rho_n:\\mathbb Z_p^r\\to\\operatorname{GL}_2(R_n)$ be continuous, with finitely many topological generators as above.\n\\begin{enumerate}\n\\item If $a=n$, all the $A_i$ are scalar and $\\rho_n$ has a continuous integral lift.\n\\item If $a<n$, choose any generator $A_*$ with scalar depth $a$. Take $\\lambda_i$ to be an integer representative of $(A_i)_{11}$ and form $B_i$ modulo $p^{n-a}$. Then\n\\[\n \\boxed{\\rho_n\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [B_*,B_i]=0\\text{ in }M_2(R_{n-a})\\ (1\\leq i\\leq r).}\n\\]\nThe test is finite matrix arithmetic on the \\emph{existing input}. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in \\cref{thm:normalized-cyclic-lift} produces exact commuting integral generator matrices.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf all generators are scalar, lift their scalar entries integrally. Their residues modulo $p$ have $p$-power order and therefore","\\begin{theorem}[Complete rank-two profinite lifting test]\\label{thm:rank-two-profinite}\nLet $p$ be any prime and let $\\rho_n:\\mathbb Z_p^r\\to\\operatorname{GL}_2(R_n)$ be continuous, with finitely many topological generators as above.\n\\begin{enumerate}\n\\item If $a=n$, all the $A_i$ are scalar and $\\rho_n$ has a continuous integral lift.\n\\item If $a<n$, choose any generator $A_*$ with scalar depth $a$. Take $\\lambda_i$ to be an integer representative of $(A_i)_{11}$ and form $B_i$ modulo $p^{n-a}$. Then\n\\[\n \\boxed{\\rho_n\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [B_*,B_i]=0\\text{ in }M_2(R_{n-a})\\ (1\\leq i\\leq r).}\n\\]\nThe test is finite matrix arithmetic on the \\emph{existing input}. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in \\cref{thm:normalized-cyclic-lift} produces exact commuting integral generator matrices.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf all generators are scalar, lift their scalar entries integrally. Their residues modulo $p$ have $p$-power order and therefore equal $1$, since $\\m…","orem}[Complete rank-two profinite lifting test]\\label{thm:rank-two-profinite}\nLet $p$ be any prime and let $\\rho_n:\\mathbb Z_p^r\\to\\operatorname{GL}_2(R_n)$ be continuous, with finitely many topological generators as above.\n\\begin{enumerate}\n\\item If $a=n$, all the $A_i$ are scalar and $\\rho_n$ has a continuous integral lift.\n\\item If $a<n$, choose any generator $A_*$ with scalar depth $a$. Take $\\lambda_i$ to be an integer representative of $(A_i)_{11}$ and form $B_i$ modulo $p^{n-a}$. Then\n\\[\n \\boxed{\\rho_n\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [B_*,B_i]=0\\text{ in }M_2(R_{n-a})\\ (1\\leq i\\leq r).}\n\\]\nThe test is finite matrix arithmetic on the \\emph{existing input}. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in \\cref{thm:normalized-cyclic-lift} produces exact commuting integral generator matrices.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf all generators are scalar, lift their scalar entries integrally. Their residues modulo $p$ have $p$-power order and therefore equal $1$, since $\\mathbb F_p^…","\\label{thm:rank-two-profinite}\nLet $p$ be any prime and let $\\rho_n:\\mathbb Z_p^r\\to\\operatorname{GL}_2(R_n)$ be continuous, with finitely many topological generators as above.\n\\begin{enumerate}\n\\item If $a=n$, all the $A_i$ are scalar and $\\rho_n$ has a continuous integral lift.\n\\item If $a<n$, choose any generator $A_*$ with scalar depth $a$. Take $\\lambda_i$ to be an integer representative of $(A_i)_{11}$ and form $B_i$ modulo $p^{n-a}$. Then\n\\[\n \\boxed{\\rho_n\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [B_*,B_i]=0\\text{ in }M_2(R_{n-a})\\ (1\\leq i\\leq r).}\n\\]\nThe test is finite matrix arithmetic on the \\emph{existing input}. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in \\cref{thm:normalized-cyclic-lift} produces exact commuting integral generator matrices.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf all generators are scalar, lift their scalar entries integrally. Their residues modulo $p$ have $p$-power order and therefore equal $1$, since $\\mathbb F_p^\\times$ has order prime to $p$. The scalar lift…","ofinite}\nLet $p$ be any prime and let $\\rho_n:\\mathbb Z_p^r\\to\\operatorname{GL}_2(R_n)$ be continuous, with finitely many topological generators as above.\n\\begin{enumerate}\n\\item If $a=n$, all the $A_i$ are scalar and $\\rho_n$ has a continuous integral lift.\n\\item If $a<n$, choose any generator $A_*$ with scalar depth $a$. Take $\\lambda_i$ to be an integer representative of $(A_i)_{11}$ and form $B_i$ modulo $p^{n-a}$. Then\n\\[\n \\boxed{\\rho_n\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [B_*,B_i]=0\\text{ in }M_2(R_{n-a})\\ (1\\leq i\\leq r).}\n\\]\nThe test is finite matrix arithmetic on the \\emph{existing input}. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in \\cref{thm:normalized-cyclic-lift} produces exact commuting integral generator matrices.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf all generators are scalar, lift their scalar entries integrally. Their residues modulo $p$ have $p$-power order and therefore equal $1$, since $\\mathbb F_p^\\times$ has order prime to $p$. The scalar lifts are units and define…","$p$ be any prime and let $\\rho_n:\\mathbb Z_p^r\\to\\operatorname{GL}_2(R_n)$ be continuous, with finitely many topological generators as above.\n\\begin{enumerate}\n\\item If $a=n$, all the $A_i$ are scalar and $\\rho_n$ has a continuous integral lift.\n\\item If $a<n$, choose any generator $A_*$ with scalar depth $a$. Take $\\lambda_i$ to be an integer representative of $(A_i)_{11}$ and form $B_i$ modulo $p^{n-a}$. Then\n\\[\n \\boxed{\\rho_n\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [B_*,B_i]=0\\text{ in }M_2(R_{n-a})\\ (1\\leq i\\leq r).}\n\\]\nThe test is finite matrix arithmetic on the \\emph{existing input}. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in \\cref{thm:normalized-cyclic-lift} produces exact commuting integral generator matrices.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf all generators are scalar, lift their scalar entries integrally. Their residues modulo $p$ have $p$-power order and therefore equal $1$, since $\\mathbb F_p^\\times$ has order prime to $p$. The scalar lifts are units and define continuous p…","and let $\\rho_n:\\mathbb Z_p^r\\to\\operatorname{GL}_2(R_n)$ be continuous, with finitely many topological generators as above.\n\\begin{enumerate}\n\\item If $a=n$, all the $A_i$ are scalar and $\\rho_n$ has a continuous integral lift.\n\\item If $a<n$, choose any generator $A_*$ with scalar depth $a$. Take $\\lambda_i$ to be an integer representative of $(A_i)_{11}$ and form $B_i$ modulo $p^{n-a}$. Then\n\\[\n \\boxed{\\rho_n\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [B_*,B_i]=0\\text{ in }M_2(R_{n-a})\\ (1\\leq i\\leq r).}\n\\]\nThe test is finite matrix arithmetic on the \\emph{existing input}. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in \\cref{thm:normalized-cyclic-lift} produces exact commuting integral generator matrices.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf all generators are scalar, lift their scalar entries integrally. Their residues modulo $p$ have $p$-power order and therefore equal $1$, since $\\mathbb F_p^\\times$ has order prime to $p$. The scalar lifts are units and define continuous powers.","]\nThe test is finite matrix arithmetic on the \\emph{existing input}. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in \\cref{thm:normalized-cyclic-lift} produces exact commuting integral generator matrices.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf all generators are scalar, lift their scalar entries integrally. Their residues modulo $p$ have $p$-power order and therefore equal $1$, since $\\mathbb F_p^\\times$ has order prime to $p$. The scalar lifts are units and define continuous powers.","t is finite matrix arithmetic on the \\emph{existing input}. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in \\cref{thm:normalized-cyclic-lift} produces exact commuting integral generator matrices.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf all generators are scalar, lift their scalar entries integrally. Their residues modulo $p$ have $p$-power order and therefore equal $1$, since $\\mathbb F_p^\\times$ has order prime to $p$. The scalar lifts are units and define continuous powers.","\\emph{existing input}. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in \\cref{thm:normalized-cyclic-lift} produces exact commuting integral generator matrices.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf all generators are scalar, lift their scalar entries integrally. Their residues modulo $p$ have $p$-power order and therefore equal $1$, since $\\mathbb F_p^\\times$ has order prime to $p$. The scalar lifts are units and define continuous powers.","ting input}. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in \\cref{thm:normalized-cyclic-lift} produces exact commuting integral generator matrices.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf all generators are scalar, lift their scalar entries integrally. Their residues modulo $p$ have $p$-power order and therefore equal $1$, since $\\mathbb F_p^\\times$ has order prime to $p$. The scalar lifts are units and define continuous powers.","tion of deeper coefficient lifts. When it passes, the polynomial construction in \\cref{thm:normalized-cyclic-lift} produces exact commuting integral generator matrices.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf all generators are scalar, lift their scalar entries integrally. Their residues modulo $p$ have $p$-power order and therefore equal $1$, since $\\mathbb F_p^\\times$ has order prime to $p$. The scalar lifts are units and define continuous powers.","Otherwise every $A_i$ is scalar modulo $p^a$, while $B_*$ is nonscalar modulo $p$ by the definition of $a$. It is cyclic by \\cref{lem:cyclic-centralizer}. Thus every hypothesis of \\cref{thm:normalized-cyclic-lift} holds. In rank two the polynomials have the form $u_i+v_it$ and can be recovered from a two-column cyclic basis by solving one invertible $2\\times2$ linear system modulo $p^{n-a}$. The equivalence with liftability also shows independence of the distinguished generator and of basis choices, although the selected lift itself need not be canonical.\n\\end{proof}","\\begin{corollary}[A sharp uniform rank-two horizon]\\label{cor:rank-two-sharp-horizon}\nFor every $p,n,r$ as above,\n\\[\n \\boxed{\\rho_n\\text{ lifts continuously over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n \\rho_n\\text{ extends continuously to }R_{2n-1}.}\n\\]\nFor $n\\geq2$ and $r\\geq2$, $2n-1$ is optimal as a uniform coefficient level. For $n=1$, all continuous rank-two representations of $\\mathbb Z_p^r$ over $\\mathbb F_p$ lift.\n\\end{corollary}\n\\begin{proof}\nFor a nonscalar tuple, $a\\leq n-1$, so an extension to $2n-1$ restricts to one at $n+a$. Apply \\cref{cor:abelian-cyclic-horizon}. Scalar tuples lift directly. This also proves the assertion at $n=1$.","lary}[A sharp uniform rank-two horizon]\\label{cor:rank-two-sharp-horizon}\nFor every $p,n,r$ as above,\n\\[\n \\boxed{\\rho_n\\text{ lifts continuously over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n \\rho_n\\text{ extends continuously to }R_{2n-1}.}\n\\]\nFor $n\\geq2$ and $r\\geq2$, $2n-1$ is optimal as a uniform coefficient level. For $n=1$, all continuous rank-two representations of $\\mathbb Z_p^r$ over $\\mathbb F_p$ lift.\n\\end{corollary}\n\\begin{proof}\nFor a nonscalar tuple, $a\\leq n-1$, so an extension to $2n-1$ restricts to one at $n+a$. Apply \\cref{cor:abelian-cyclic-horizon}. Scalar tuples lift directly. This also proves the assertion at $n=1$.","For every $p,n,r$ as above,\n\\[\n \\boxed{\\rho_n\\text{ lifts continuously over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n \\rho_n\\text{ extends continuously to }R_{2n-1}.}\n\\]\nFor $n\\geq2$ and $r\\geq2$, $2n-1$ is optimal as a uniform coefficient level. For $n=1$, all continuous rank-two representations of $\\mathbb Z_p^r$ over $\\mathbb F_p$ lift.\n\\end{corollary}\n\\begin{proof}\nFor a nonscalar tuple, $a\\leq n-1$, so an extension to $2n-1$ restricts to one at $n+a$. Apply \\cref{cor:abelian-cyclic-horizon}. Scalar tuples lift directly. This also proves the assertion at $n=1$.","\\boxed{\\rho_n\\text{ lifts continuously over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n \\rho_n\\text{ extends continuously to }R_{2n-1}.}\n\\]\nFor $n\\geq2$ and $r\\geq2$, $2n-1$ is optimal as a uniform coefficient level. For $n=1$, all continuous rank-two representations of $\\mathbb Z_p^r$ over $\\mathbb F_p$ lift.\n\\end{corollary}\n\\begin{proof}\nFor a nonscalar tuple, $a\\leq n-1$, so an extension to $2n-1$ restricts to one at $n+a$. Apply \\cref{cor:abelian-cyclic-horizon}. Scalar tuples lift directly. This also proves the assertion at $n=1$.","continuously over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n \\rho_n\\text{ extends continuously to }R_{2n-1}.}\n\\]\nFor $n\\geq2$ and $r\\geq2$, $2n-1$ is optimal as a uniform coefficient level. For $n=1$, all continuous rank-two representations of $\\mathbb Z_p^r$ over $\\mathbb F_p$ lift.\n\\end{corollary}\n\\begin{proof}\nFor a nonscalar tuple, $a\\leq n-1$, so an extension to $2n-1$ restricts to one at $n+a$. Apply \\cref{cor:abelian-cyclic-horizon}. Scalar tuples lift directly. This also proves the assertion at $n=1$.","\\]\nFor $n\\geq2$ and $r\\geq2$, $2n-1$ is optimal as a uniform coefficient level. For $n=1$, all continuous rank-two representations of $\\mathbb Z_p^r$ over $\\mathbb F_p$ lift.\n\\end{corollary}\n\\begin{proof}\nFor a nonscalar tuple, $a\\leq n-1$, so an extension to $2n-1$ restricts to one at $n+a$. Apply \\cref{cor:abelian-cyclic-horizon}. Scalar tuples lift directly. This also proves the assertion at $n=1$.","For sharpness fix $n\\geq2$ and put\n\\[\n A=I+p^{n-1}E_{12},\\qquad B=I+p^{n-1}E_{21}.\n\\]\nThese matrices commute modulo $p^{2n-2}$ and generate a finite abelian $p$-group there, so they give a continuous extension of their level-$n$ representation to level $2n-2$. However every pair of lifts preserving level $n$ has the form\n$\\widetilde A=A+p^nP$, $\\widetilde B=B+p^nQ$, and\n\\[\n \\widetilde A\\widetilde B-\\widetilde B\\widetilde A\n \\equiv p^{2n-2}\n \\begin{pmatrix}1&0","For sharpness fix $n\\geq2$ and put\n\\[\n A=I+p^{n-1}E_{12},\\qquad B=I+p^{n-1}E_{21}.\n\\]\nThese matrices commute modulo $p^{2n-2}$ and generate a finite abelian $p$-group there, so they give a continuous extension of their level-$n$ representation to level $2n-2$. However every pair of lifts preserving level $n$ has the form\n$\\widetilde A=A+p^nP$, $\\widetilde B=B+p^nQ$, and\n\\[\n \\widetilde A\\widetilde B-\\widetilde B\\widetilde A\n \\equiv p^{2n-2}\n \\begin{pmatrix}1&0\\\\0&-1\\end{pmatrix}\n \\not\\equiv0\\pmod{p^{2n-1}}.\n\\]\nIndeed every correction term has valuation at least $2n-1$. This proves nonextension at $2n-1$ for every prime, including $2$, and hence nonexistence of an integral lift. Add identity generators for $r>2$. No smaller uniform coefficient level can distinguish these inputs from extendable ones by mere existence of an extension.\n\\end{proof}","0&-1\\end{pmatrix}\n \\not\\equiv0\\pmod{p^{2n-1}}.\n\\]\nIndeed every correction term has valuation at least $2n-1$. This proves nonextension at $2n-1$ for every prime, including $2$, and hence nonexistence of an integral lift. Add identity generators for $r>2$. No smaller uniform coefficient level can distinguish these inputs from extendable ones by mere existence of an extension.\n\\end{proof}","\\begin{example}[The first hidden commutator]\\label{ex:abelian-tangent}\nSuppose $n=2$ and all generator matrices are $A_i=I+pX_i$ modulo $p^2$. They always commute at that coefficient level. For rank two, \\cref{thm:rank-two-profinite} gives\n\\[\n \\rho_2\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [X_i,X_j]=0\\text{ in }M_2(\\mathbb F_p)\\text{ for all }i,j.\n\\]\nIf some $X_i$ is nonscalar, its centralizer is commutative and the displayed test agrees with the distinguished-generator criterion; if all are scalar the lift is immediate. For $X_1=E_{12}$ and $X_2=E_{21}$ it fails. Each generator separately has a continuous integral procyclic lift, but the pair does not define an integral action of $\\mathbb Z_p^2$. On the free pro-$p$ source the same pair \\emph{does} l","\\begin{example}[The first hidden commutator]\\label{ex:abelian-tangent}\nSuppose $n=2$ and all generator matrices are $A_i=I+pX_i$ modulo $p^2$. They always commute at that coefficient level. For rank two, \\cref{thm:rank-two-profinite} gives\n\\[\n \\rho_2\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [X_i,X_j]=0\\text{ in }M_2(\\mathbb F_p)\\text{ for all }i,j.\n\\]\nIf some $X_i$ is nonscalar, its centralizer is commutative and the displayed test agrees with the distinguished-generator criterion; if all are scalar the lift is immediate. For $X_1=E_{12}$ and $X_2=E_{21}$ it fails. Each generator separately has a continuous integral procyclic lift, but the pair does not define an integral action of $\\mathbb Z_p^2$. On the free pro-$p$ source the same pair \\emph{does} lift, by \\cref{thm:free-pro-p-lift}. The surv","\\begin{example}[The first hidden commutator]\\label{ex:abelian-tangent}\nSuppose $n=2$ and all generator matrices are $A_i=I+pX_i$ modulo $p^2$. They always commute at that coefficient level. For rank two, \\cref{thm:rank-two-profinite} gives\n\\[\n \\rho_2\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [X_i,X_j]=0\\text{ in }M_2(\\mathbb F_p)\\text{ for all }i,j.\n\\]\nIf some $X_i$ is nonscalar, its centralizer is commutative and the displayed test agrees with the distinguished-generator criterion; if all are scalar the lift is immediate. For $X_1=E_{12}$ and $X_2=E_{21}$ it fails. Each generator separately has a continuous integral procyclic lift, but the pair does not define an integral action of $\\mathbb Z_p^2$. On the free pro-$p$ source the same pair \\emph{does} lift, by \\cref{thm:free-pro-p-lift}. The surviving obstruc","\\begin{example}[The first hidden commutator]\\label{ex:abelian-tangent}\nSuppose $n=2$ and all generator matrices are $A_i=I+pX_i$ modulo $p^2$. They always commute at that coefficient level. For rank two, \\cref{thm:rank-two-profinite} gives\n\\[\n \\rho_2\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [X_i,X_j]=0\\text{ in }M_2(\\mathbb F_p)\\text{ for all }i,j.\n\\]\nIf some $X_i$ is nonscalar, its centralizer is commutative and the displayed test agrees with the distinguished-generator criterion; if all are scalar the lift is immediate. For $X_1=E_{12}$ and $X_2=E_{21}$ it fails. Each generator separately has a continuous integral procyclic lift, but the pair does not define an integral action of $\\mathbb Z_p^2$. On the free pro-$p$ source the same pair \\emph{does} lift, by \\cref{thm:free-pro-p-lift}. The surviving obstruction is therefore an actual source relation, not finite-order torsion imposed by an incidental quotient.\n\\end{example}","commute at that coefficient level. For rank two, \\cref{thm:rank-two-profinite} gives\n\\[\n \\rho_2\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [X_i,X_j]=0\\text{ in }M_2(\\mathbb F_p)\\text{ for all }i,j.\n\\]\nIf some $X_i$ is nonscalar, its centralizer is commutative and the displayed test agrees with the distinguished-generator criterion; if all are scalar the lift is immediate. For $X_1=E_{12}$ and $X_2=E_{21}$ it fails. Each generator separately has a continuous integral procyclic lift, but the pair does not define an integral action of $\\mathbb Z_p^2$. On the free pro-$p$ source the same pair \\emph{does} lift, by \\cref{thm:free-pro-p-lift}. The surviving obstruction is therefore an actual source relation, not finite-order torsion imposed by an incidental quotient.\n\\end{example}","coefficient level. For rank two, \\cref{thm:rank-two-profinite} gives\n\\[\n \\rho_2\\text{ lifts over }\\mathbb Z_p\n \\quad\\Longleftrightarrow\\quad\n [X_i,X_j]=0\\text{ in }M_2(\\mathbb F_p)\\text{ for all }i,j.\n\\]\nIf some $X_i$ is nonscalar, its centralizer is commutative and the displayed test agrees with the distinguished-generator criterion; if all are scalar the lift is immediate. For $X_1=E_{12}$ and $X_2=E_{21}$ it fails. Each generator separately has a continuous integral procyclic lift, but the pair does not define an integral action of $\\mathbb Z_p^2$. On the free pro-$p$ source the same pair \\emph{does} lift, by \\cref{thm:free-pro-p-lift}. The surviving obstruction is therefore an actual source relation, not finite-order torsion imposed by an incidental quotient.\n\\end{example}","[X_i,X_j]=0\\text{ in }M_2(\\mathbb F_p)\\text{ for all }i,j.\n\\]\nIf some $X_i$ is nonscalar, its centralizer is commutative and the displayed test agrees with the distinguished-generator criterion; if all are scalar the lift is immediate. For $X_1=E_{12}$ and $X_2=E_{21}$ it fails. Each generator separately has a continuous integral procyclic lift, but the pair does not define an integral action of $\\mathbb Z_p^2$. On the free pro-$p$ source the same pair \\emph{does} lift, by \\cref{thm:free-pro-p-lift}. The surviving obstruction is therefore an actual source relation, not finite-order torsion imposed by an incidental quotient.\n\\end{example}","F_p)\\text{ for all }i,j.\n\\]\nIf some $X_i$ is nonscalar, its centralizer is commutative and the displayed test agrees with the distinguished-generator criterion; if all are scalar the lift is immediate. For $X_1=E_{12}$ and $X_2=E_{21}$ it fails. Each generator separately has a continuous integral procyclic lift, but the pair does not define an integral action of $\\mathbb Z_p^2$. On the free pro-$p$ source the same pair \\emph{does} lift, by \\cref{thm:free-pro-p-lift}. The surviving obstruction is therefore an actual source relation, not finite-order torsion imposed by an incidental quotient.\n\\end{example}","\\begin{example}[A positive repair that changes only unresolved digits]\\label{ex:abelian-positive-repair}\nWork at $p=3$, $n=3$. Take\n\\[\n A_1=\\begin{pmatrix}1&3","\\begin{example}[A positive repair that changes only unresolved digits]\\label{ex:abelian-positive-repair}\nWork at $p=3$, $n=3$. Take\n\\[\n A_1=\\begin{pmatrix}1&3\\\\9&1\\end{pmatrix},\\qquad\n A_2=\\begin{pmatrix}7&12\\\\9&7\\end{pmatrix}\\pmod{27}.\n\\]\nThey commute modulo $27$ and both are the identity modulo $3$. Choose $a=1$, $\\lambda_1=\\lambda_2=1$. Then\n\\[\n B_1=\\begin{pmatrix}0&1","9&7\\end{pmatrix}\\pmod{27}.\n\\]\nThey commute modulo $27$ and both are the identity modulo $3$. Choose $a=1$, $\\lambda_1=\\lambda_2=1$. Then\n\\[\n B_1=\\begin{pmatrix}0&1","9&7\\end{pmatrix}\\pmod{27}.\n\\]\nThey commute modulo $27$ and both are the identity modulo $3$. Choose $a=1$, $\\lambda_1=\\lambda_2=1$. Then\n\\[\n B_1=\\begin{pmatrix}0&1\\\\3&0\\end{pmatrix},\\qquad\n B_2=\\begin{pmatrix}2&4\\\\3&2\\end{pmatrix}\\pmod9,\n \\qquad B_2=2I+4B_1\\pmod9.\n\\]\nThe vector $e_2$ is cyclic for $B_1$ modulo $3$. The displayed integer representatives of $A_1,A_2$ do not commute exactly: their additive commutator is $\\operatorname{diag}(-81,81)$. The criterion nevertheless passes. Retain $B_1$ integrally and replace $B_2$ by the exact polynomial $2I+4B_1$. This constructs\n\\[\n \\widehat A_1=\\begin{pmatrix}1&3","3&2\\end{pmatrix}\\pmod9,\n \\qquad B_2=2I+4B_1\\pmod9.\n\\]\nThe vector $e_2$ is cyclic for $B_1$ modulo $3$. The displayed integer representatives of $A_1,A_2$ do not commute exactly: their additive commutator is $\\operatorname{diag}(-81,81)$. The criterion nevertheless passes. Retain $B_1$ integrally and replace $B_2$ by the exact polynomial $2I+4B_1$. This constructs\n\\[\n \\widehat A_1=\\begin{pmatrix}1&3","3&2\\end{pmatrix}\\pmod9,\n \\qquad B_2=2I+4B_1\\pmod9.\n\\]\nThe vector $e_2$ is cyclic for $B_1$ modulo $3$. The displayed integer representatives of $A_1,A_2$ do not commute exactly: their additive commutator is $\\operatorname{diag}(-81,81)$. The criterion nevertheless passes. Retain $B_1$ integrally and replace $B_2$ by the exact polynomial $2I+4B_1$. This constructs\n\\[\n \\widehat A_1=\\begin{pmatrix}1&3\\\\9&1\\end{pmatrix},\\qquad\n \\widehat A_2=\\begin{pmatrix}7&12\\\\36&7\\end{pmatrix}.\n\\]\nThey commute exactly, remain units over $\\mathbb Z_3$, and have the original reduction modulo $27$. Their continuous powers give the integral representation. This is an explicit non-destructive repair of a tuple, not a claim that its initial arbitrary integer representatives were already an exact action.\n\\end{example}","36&7\\end{pmatrix}.\n\\]\nThey commute exactly, remain units over $\\mathbb Z_3$, and have the original reduction modulo $27$. Their continuous powers give the integral representation. This is an explicit non-destructive repair of a tuple, not a claim that its initial arbitrary integer representatives were already an exact action.\n\\end{example}","There is a necessary variance distinction. Let $\\mathcal Q_\\Gamma$ have open normal subgroups as objects and an arrow $U\\to U'$ when $U'\\subseteq U$. Write $\\mathscr P_d(n,U)$ for the groupoid of rank-$d$ representations of $\\Gamma/U$ on $R_n^d$, with invertible intertwining matrices as arrows.","\\begin{proposition}[The source--coefficient diagram has mixed variance]\\label{prop:mixed-source-variance}\nReduction of coefficients and inflation of source define a functor\n\\[\n \\mathscr P_d:\\mathbb N_{\\geq1}^{op}\\times\\mathcal Q_\\Gamma\n \\longrightarrow\\mathbf{Gpd}.\n\\]\nEach source-inflation functor is fully faithful, and it commutes with coefficient reduction. An object at $(m,U')$ is a compatible refinement of $\\rho_n$ at $(n,U)$ precisely when its reduction to level $n$ equals, or is supplied with an isomorphism to, the inflation of $\\rho_n$. There is no total source-coarsening functor on arbitrary representations obtained simply by reversing inflation.\n\\end{proposition}\n\\begin{proof}\nCoefficient reduction acts on the matrix entries of representations and intertwiners. Source inflation precomposes them with the surjection $\\Gamma/U'\\to\\Gamma/U$. The two operations commute directly on both objects and arrows. An intertwiner of inflated representations","\\begin{proposition}[The source--coefficient diagram has mixed variance]\\label{prop:mixed-source-variance}\nReduction of coefficients and inflation of source define a functor\n\\[\n \\mathscr P_d:\\mathbb N_{\\geq1}^{op}\\times\\mathcal Q_\\Gamma\n \\longrightarrow\\mathbf{Gpd}.\n\\]\nEach source-inflation functor is fully faithful, and it commutes with coefficient reduction. An object at $(m,U')$ is a compatible refinement of $\\rho_n$ at $(n,U)$ precisely when its reduction to level $n$ equals, or is supplied with an isomorphism to, the inflation of $\\rho_n$. There is no total source-coarsening functor on arbitrary representations obtained simply by reversing inflation.\n\\end{proposition}\n\\begin{proof}\nCoefficient reduction acts on the matrix entries of representations and intertwiners. Source inflation precomposes them with the surjection $\\Gamma/U'\\to\\Gamma/U$. The two operations commute directly on both objects and arrows. An intertwiner of inflated representations commutes with the same matrix image as before inflation, because the source map is onto. Thus inflation is fully fai","ion}[The source--coefficient diagram has mixed variance]\\label{prop:mixed-source-variance}\nReduction of coefficients and inflation of source define a functor\n\\[\n \\mathscr P_d:\\mathbb N_{\\geq1}^{op}\\times\\mathcal Q_\\Gamma\n \\longrightarrow\\mathbf{Gpd}.\n\\]\nEach source-inflation functor is fully faithful, and it commutes with coefficient reduction. An object at $(m,U')$ is a compatible refinement of $\\rho_n$ at $(n,U)$ precisely when its reduction to level $n$ equals, or is supplied with an isomorphism to, the inflation of $\\rho_n$. There is no total source-coarsening functor on arbitrary representations obtained simply by reversing inflation.\n\\end{proposition}\n\\begin{proof}\nCoefficient reduction acts on the matrix entries of representations and intertwiners. Source inflation precomposes them with the surjection $\\Gamma/U'\\to\\Gamma/U$. The two operations commute directly on both objects and arrows. An intertwiner of inflated representations commutes with the same matrix image as before inflation, because the source map is onto. Thus inflation is fully faithful. Finally a representation o…","rce--coefficient diagram has mixed variance]\\label{prop:mixed-source-variance}\nReduction of coefficients and inflation of source define a functor\n\\[\n \\mathscr P_d:\\mathbb N_{\\geq1}^{op}\\times\\mathcal Q_\\Gamma\n \\longrightarrow\\mathbf{Gpd}.\n\\]\nEach source-inflation functor is fully faithful, and it commutes with coefficient reduction. An object at $(m,U')$ is a compatible refinement of $\\rho_n$ at $(n,U)$ precisely when its reduction to level $n$ equals, or is supplied with an isomorphism to, the inflation of $\\rho_n$. There is no total source-coarsening functor on arbitrary representations obtained simply by reversing inflation.\n\\end{proposition}\n\\begin{proof}\nCoefficient reduction acts on the matrix entries of representations and intertwiners. Source inflation precomposes them with the surjection $\\Gamma/U'\\to\\Gamma/U$. The two operations commute directly on both objects and arrows. An intertwiner of inflated representations commutes with the same matrix image as before inflation, because the source map is onto. Thus inflation is fully faithful. Finally a representation of $\\Gamma/U'…","abel{prop:mixed-source-variance}\nReduction of coefficients and inflation of source define a functor\n\\[\n \\mathscr P_d:\\mathbb N_{\\geq1}^{op}\\times\\mathcal Q_\\Gamma\n \\longrightarrow\\mathbf{Gpd}.\n\\]\nEach source-inflation functor is fully faithful, and it commutes with coefficient reduction. An object at $(m,U')$ is a compatible refinement of $\\rho_n$ at $(n,U)$ precisely when its reduction to level $n$ equals, or is supplied with an isomorphism to, the inflation of $\\rho_n$. There is no total source-coarsening functor on arbitrary representations obtained simply by reversing inflation.\n\\end{proposition}\n\\begin{proof}\nCoefficient reduction acts on the matrix entries of representations and intertwiners. Source inflation precomposes them with the surjection $\\Gamma/U'\\to\\Gamma/U$. The two operations commute directly on both objects and arrows. An intertwiner of inflated representations commutes with the same matrix image as before inflation, because the source map is onto. Thus inflation is fully faithful. Finally a representation of $\\Gamma/U'$ descends through $\\Gamma/U$ only if the subg…","tion of source define a functor\n\\[\n \\mathscr P_d:\\mathbb N_{\\geq1}^{op}\\times\\mathcal Q_\\Gamma\n \\longrightarrow\\mathbf{Gpd}.\n\\]\nEach source-inflation functor is fully faithful, and it commutes with coefficient reduction. An object at $(m,U')$ is a compatible refinement of $\\rho_n$ at $(n,U)$ precisely when its reduction to level $n$ equals, or is supplied with an isomorphism to, the inflation of $\\rho_n$. There is no total source-coarsening functor on arbitrary representations obtained simply by reversing inflation.\n\\end{proposition}\n\\begin{proof}\nCoefficient reduction acts on the matrix entries of representations and intertwiners. Source inflation precomposes them with the surjection $\\Gamma/U'\\to\\Gamma/U$. The two operations commute directly on both objects and arrows. An intertwiner of inflated representations commutes with the same matrix image as before inflation, because the source map is onto. Thus inflation is fully faithful. Finally a representation of $\\Gamma/U'$ descends through $\\Gamma/U$ only if the subgroup $U/U'$ acts trivially. This fails for general representations, …","ecisely when its reduction to level $n$ equals, or is supplied with an isomorphism to, the inflation of $\\rho_n$. There is no total source-coarsening functor on arbitrary representations obtained simply by reversing inflation.\n\\end{proposition}\n\\begin{proof}\nCoefficient reduction acts on the matrix entries of representations and intertwiners. Source inflation precomposes them with the surjection $\\Gamma/U'\\to\\Gamma/U$. The two operations commute directly on both objects and arrows. An intertwiner of inflated representations commutes with the same matrix image as before inflation, because the source map is onto. Thus inflation is fully faithful. Finally a representation of $\\Gamma/U'$ descends through $\\Gamma/U$ only if the subgroup $U/U'$ acts trivially. This fails for general representations, as the explicit repaired action in \\cref{ex:inflation-carry} shows at its higher coefficient level.\n\\end{proof}","cts on the matrix entries of representations and intertwiners. Source inflation precomposes them with the surjection $\\Gamma/U'\\to\\Gamma/U$. The two operations commute directly on both objects and arrows. An intertwiner of inflated representations commutes with the same matrix image as before inflation, because the source map is onto. Thus inflation is fully faithful. Finally a representation of $\\Gamma/U'$ descends through $\\Gamma/U$ only if the subgroup $U/U'$ acts trivially. This fails for general representations, as the explicit repaired action in \\cref{ex:inflation-carry} shows at its higher coefficient level.\n\\end{proof}","representations and intertwiners. Source inflation precomposes them with the surjection $\\Gamma/U'\\to\\Gamma/U$. The two operations commute directly on both objects and arrows. An intertwiner of inflated representations commutes with the same matrix image as before inflation, because the source map is onto. Thus inflation is fully faithful. Finally a representation of $\\Gamma/U'$ descends through $\\Gamma/U$ only if the subgroup $U/U'$ acts trivially. This fails for general representations, as the explicit repaired action in \\cref{ex:inflation-carry} shows at its higher coefficient level.\n\\end{proof}",". Source inflation precomposes them with the surjection $\\Gamma/U'\\to\\Gamma/U$. The two operations commute directly on both objects and arrows. An intertwiner of inflated representations commutes with the same matrix image as before inflation, because the source map is onto. Thus inflation is fully faithful. Finally a representation of $\\Gamma/U'$ descends through $\\Gamma/U$ only if the subgroup $U/U'$ acts trivially. This fails for general representations, as the explicit repaired action in \\cref{ex:inflation-carry} shows at its higher coefficient level.\n\\end{proof}","ient search altogether on their stated strata. Scalar-depth normalization reveals the missing relation, the cyclic centralizer supplies a finite polynomial certificate, and continuous matrix powers reconstruct the action. In rank two this yields a complete test at every coefficient precision and every finite source rank. In larger rank it requires the stated cyclic normalized generator. Free pro-$p$ sources furnish a contrasting unrestricted coefficient-lifting case because they impose no additional relations on the generators.","\\begin{theorem}[Torsor residual obstruction]\\label{thm:torsor-obstruction}\nLet $\\mathcal A$ be a sheaf of abelian groups on a space or a site with the fiber products used below, let $P$ be an $\\mathcal A$-torsor, and let $\\{U_i\\to U\\}$ be a cover on which $P$ admits local sections $s_i\\in P(U_i)$. On overlaps there are unique elements\n\\[\na_{ij}\\in\\mathcal A(U_{ij})\n\\]\nsuch that\n\\[\ns_j=s_i+a_{ij}.\n\\]\nThen:\n\\begin{enumerate}\n    \\item $a=(a_{ij})$ is a \\v{C}ech $1$-cocycle;\n    \\item changing the local sections by a $0$-cochain $b=(b_i)$,\n    \\[\n    s_i'=s_i+b_i,\n    \\]\n    changes the residual by the \\v{C}ech coboundary\n    \\[\n    a'_{ij}=a_{ij}+b_j-b_i;\n    \\]\n    \\item the class\n    \\[\n    \\Omega(P;\\{U_i\\})=[a]\\in\\check H^1(\\{U_i\\},\\mathcal A)\n    \\]\n    is independent of the ch","\\begin{theorem}[Torsor residual obstruction]\\label{thm:torsor-obstruction}\nLet $\\mathcal A$ be a sheaf of abelian groups on a space or a site with the fiber products used below, let $P$ be an $\\mathcal A$-torsor, and let $\\{U_i\\to U\\}$ be a cover on which $P$ admits local sections $s_i\\in P(U_i)$. On overlaps there are unique elements\n\\[\na_{ij}\\in\\mathcal A(U_{ij})\n\\]\nsuch that\n\\[\ns_j=s_i+a_{ij}.\n\\]\nThen:\n\\begin{enumerate}\n    \\item $a=(a_{ij})$ is a \\v{C}ech $1$-cocycle;\n    \\item changing the local sections by a $0$-cochain $b=(b_i)$,\n    \\[\n    s_i'=s_i+b_i,\n    \\]\n    changes the residual by the \\v{C}ech coboundary\n    \\[\n    a'_{ij}=a_{ij}+b_j-b_i;\n    \\]\n    \\item the class\n    \\[\n    \\Omega(P;\\{U_i\\})=[a]\\in\\check H^1(\\{U_i\\},\\mathcal A)\n    \\]\n    is independent of the chosen local sections;\n    \\item $\\Omega(P;\\{U_i\\})=0$ if and only if the local sections can be","\\begin{theorem}[Torsor residual obstruction]\\label{thm:torsor-obstruction}\nLet $\\mathcal A$ be a sheaf of abelian groups on a space or a site with the fiber products used below, let $P$ be an $\\mathcal A$-torsor, and let $\\{U_i\\to U\\}$ be a cover on which $P$ admits local sections $s_i\\in P(U_i)$. On overlaps there are unique elements\n\\[\na_{ij}\\in\\mathcal A(U_{ij})\n\\]\nsuch that\n\\[\ns_j=s_i+a_{ij}.\n\\]\nThen:\n\\begin{enumerate}\n    \\item $a=(a_{ij})$ is a \\v{C}ech $1$-cocycle;\n    \\item changing the local sections by a $0$-cochain $b=(b_i)$,\n    \\[\n    s_i'=s_i+b_i,\n    \\]\n    changes the residual by the \\v{C}ech coboundary\n    \\[\n    a'_{ij}=a_{ij}+b_j-b_i;\n    \\]\n    \\item the class\n    \\[\n    \\Omega(P;\\{U_i\\})=[a]\\in\\check H^1(\\{U_i\\},\\mathcal A)\n    \\]\n    is independent of the chosen local sections;\n    \\item $\\Omega(P;\\{U_i\\})=0$ if and only if the local sections can be modified by admissi","\\begin{theorem}[Torsor residual obstruction]\\label{thm:torsor-obstruction}\nLet $\\mathcal A$ be a sheaf of abelian groups on a space or a site with the fiber products used below, let $P$ be an $\\mathcal A$-torsor, and let $\\{U_i\\to U\\}$ be a cover on which $P$ admits local sections $s_i\\in P(U_i)$. On overlaps there are unique elements\n\\[\na_{ij}\\in\\mathcal A(U_{ij})\n\\]\nsuch that\n\\[\ns_j=s_i+a_{ij}.\n\\]\nThen:\n\\begin{enumerate}\n    \\item $a=(a_{ij})$ is a \\v{C}ech $1$-cocycle;\n    \\item changing the local sections by a $0$-cochain $b=(b_i)$,\n    \\[\n    s_i'=s_i+b_i,\n    \\]\n    changes the residual by the \\v{C}ech coboundary\n    \\[\n    a'_{ij}=a_{ij}+b_j-b_i;\n    \\]\n    \\item the class\n    \\[\n    \\Omega(P;\\{U_i\\})=[a]\\in\\check H^1(\\{U_i\\},\\mathcal A)\n    \\]\n    is independent of the chosen local sections;\n    \\item $\\Omega(P;\\{U_i\\})=0$ if and only if the local sections can be modified by admissible local repairs so that they","\\begin{theorem}[Torsor residual obstruction]\\label{thm:torsor-obstruction}\nLet $\\mathcal A$ be a sheaf of abelian groups on a space or a site with the fiber products used below, let $P$ be an $\\mathcal A$-torsor, and let $\\{U_i\\to U\\}$ be a cover on which $P$ admits local sections $s_i\\in P(U_i)$. On overlaps there are unique elements\n\\[\na_{ij}\\in\\mathcal A(U_{ij})\n\\]\nsuch that\n\\[\ns_j=s_i+a_{ij}.\n\\]\nThen:\n\\begin{enumerate}\n    \\item $a=(a_{ij})$ is a \\v{C}ech $1$-cocycle;\n    \\item changing the local sections by a $0$-cochain $b=(b_i)$,\n    \\[\n    s_i'=s_i+b_i,\n    \\]\n    changes the residual by the \\v{C}ech coboundary\n    \\[\n    a'_{ij}=a_{ij}+b_j-b_i;\n    \\]\n    \\item the class\n    \\[\n    \\Omega(P;\\{U_i\\})=[a]\\in\\check H^1(\\{U_i\\},\\mathcal A)\n    \\]\n    is independent of the chosen local sections;\n    \\item $\\Omega(P;\\{U_i\\})=0$ if and only if the local sections can be modified by admissible local repairs so that they glue to a global section","\\begin{theorem}[Torsor residual obstruction]\\label{thm:torsor-obstruction}\nLet $\\mathcal A$ be a sheaf of abelian groups on a space or a site with the fiber products used below, let $P$ be an $\\mathcal A$-torsor, and let $\\{U_i\\to U\\}$ be a cover on which $P$ admits local sections $s_i\\in P(U_i)$. On overlaps there are unique elements\n\\[\na_{ij}\\in\\mathcal A(U_{ij})\n\\]\nsuch that\n\\[\ns_j=s_i+a_{ij}.\n\\]\nThen:\n\\begin{enumerate}\n    \\item $a=(a_{ij})$ is a \\v{C}ech $1$-cocycle;\n    \\item changing the local sections by a $0$-cochain $b=(b_i)$,\n    \\[\n    s_i'=s_i+b_i,\n    \\]\n    changes the residual by the \\v{C}ech coboundary\n    \\[\n    a'_{ij}=a_{ij}+b_j-b_i;\n    \\]\n    \\item the class\n    \\[\n    \\Omega(P;\\{U_i\\})=[a]\\in\\check H^1(\\{U_i\\},\\mathcal A)\n    \\]\n    is independent of the chosen local sections;\n    \\item $\\Omega(P;\\{U_i\\})=0$ if and only if the local sections can be modified by admissible local repairs so that they glue to a global section of $P$ over $U$.\n\\end{enumerate}\n\\end","\\begin{theorem}[Torsor residual obstruction]\\label{thm:torsor-obstruction}\nLet $\\mathcal A$ be a sheaf of abelian groups on a space or a site with the fiber products used below, let $P$ be an $\\mathcal A$-torsor, and let $\\{U_i\\to U\\}$ be a cover on which $P$ admits local sections $s_i\\in P(U_i)$. On overlaps there are unique elements\n\\[\na_{ij}\\in\\mathcal A(U_{ij})\n\\]\nsuch that\n\\[\ns_j=s_i+a_{ij}.\n\\]\nThen:\n\\begin{enumerate}\n    \\item $a=(a_{ij})$ is a \\v{C}ech $1$-cocycle;\n    \\item changing the local sections by a $0$-cochain $b=(b_i)$,\n    \\[\n    s_i'=s_i+b_i,\n    \\]\n    changes the residual by the \\v{C}ech coboundary\n    \\[\n    a'_{ij}=a_{ij}+b_j-b_i;\n    \\]\n    \\item the class\n    \\[\n    \\Omega(P;\\{U_i\\})=[a]\\in\\check H^1(\\{U_i\\},\\mathcal A)\n    \\]\n    is independent of the chosen local sections;\n    \\item $\\Omega(P;\\{U_i\\})=0$ if and only if the local sections can be modified by admissible local repairs so that they glue to a global section of $P$ over $U$.\n\\end{enumerate}\n\\end{theorem}","ction]\\label{thm:torsor-obstruction}\nLet $\\mathcal A$ be a sheaf of abelian groups on a space or a site with the fiber products used below, let $P$ be an $\\mathcal A$-torsor, and let $\\{U_i\\to U\\}$ be a cover on which $P$ admits local sections $s_i\\in P(U_i)$. On overlaps there are unique elements\n\\[\na_{ij}\\in\\mathcal A(U_{ij})\n\\]\nsuch that\n\\[\ns_j=s_i+a_{ij}.\n\\]\nThen:\n\\begin{enumerate}\n    \\item $a=(a_{ij})$ is a \\v{C}ech $1$-cocycle;\n    \\item changing the local sections by a $0$-cochain $b=(b_i)$,\n    \\[\n    s_i'=s_i+b_i,\n    \\]\n    changes the residual by the \\v{C}ech coboundary\n    \\[\n    a'_{ij}=a_{ij}+b_j-b_i;\n    \\]\n    \\item the class\n    \\[\n    \\Omega(P;\\{U_i\\})=[a]\\in\\check H^1(\\{U_i\\},\\mathcal A)\n    \\]\n    is independent of the chosen local sections;\n    \\item $\\Omega(P;\\{U_i\\})=0$ if and only if the local sections can be modified by admissible local repairs so that they glue to a global section of $P$ over $U$.\n\\end{enumerate}\n\\end{theorem}","truction}\nLet $\\mathcal A$ be a sheaf of abelian groups on a space or a site with the fiber products used below, let $P$ be an $\\mathcal A$-torsor, and let $\\{U_i\\to U\\}$ be a cover on which $P$ admits local sections $s_i\\in P(U_i)$. On overlaps there are unique elements\n\\[\na_{ij}\\in\\mathcal A(U_{ij})\n\\]\nsuch that\n\\[\ns_j=s_i+a_{ij}.\n\\]\nThen:\n\\begin{enumerate}\n    \\item $a=(a_{ij})$ is a \\v{C}ech $1$-cocycle;\n    \\item changing the local sections by a $0$-cochain $b=(b_i)$,\n    \\[\n    s_i'=s_i+b_i,\n    \\]\n    changes the residual by the \\v{C}ech coboundary\n    \\[\n    a'_{ij}=a_{ij}+b_j-b_i;\n    \\]\n    \\item the class\n    \\[\n    \\Omega(P;\\{U_i\\})=[a]\\in\\check H^1(\\{U_i\\},\\mathcal A)\n    \\]\n    is independent of the chosen local sections;\n    \\item $\\Omega(P;\\{U_i\\})=0$ if and only if the local sections can be modified by admissible local repairs so that they glue to a global section of $P$ over $U$.\n\\end{enumerate}\n\\end{theorem}","on a space or a site with the fiber products used below, let $P$ be an $\\mathcal A$-torsor, and let $\\{U_i\\to U\\}$ be a cover on which $P$ admits local sections $s_i\\in P(U_i)$. On overlaps there are unique elements\n\\[\na_{ij}\\in\\mathcal A(U_{ij})\n\\]\nsuch that\n\\[\ns_j=s_i+a_{ij}.\n\\]\nThen:\n\\begin{enumerate}\n    \\item $a=(a_{ij})$ is a \\v{C}ech $1$-cocycle;\n    \\item changing the local sections by a $0$-cochain $b=(b_i)$,\n    \\[\n    s_i'=s_i+b_i,\n    \\]\n    changes the residual by the \\v{C}ech coboundary\n    \\[\n    a'_{ij}=a_{ij}+b_j-b_i;\n    \\]\n    \\item the class\n    \\[\n    \\Omega(P;\\{U_i\\})=[a]\\in\\check H^1(\\{U_i\\},\\mathcal A)\n    \\]\n    is independent of the chosen local sections;\n    \\item $\\Omega(P;\\{U_i\\})=0$ if and only if the local sections can be modified by admissible local repairs so that they glue to a global section of $P$ over $U$.\n\\end{enumerate}\n\\end{theorem}","r a site with the fiber products used below, let $P$ be an $\\mathcal A$-torsor, and let $\\{U_i\\to U\\}$ be a cover on which $P$ admits local sections $s_i\\in P(U_i)$. On overlaps there are unique elements\n\\[\na_{ij}\\in\\mathcal A(U_{ij})\n\\]\nsuch that\n\\[\ns_j=s_i+a_{ij}.\n\\]\nThen:\n\\begin{enumerate}\n    \\item $a=(a_{ij})$ is a \\v{C}ech $1$-cocycle;\n    \\item changing the local sections by a $0$-cochain $b=(b_i)$,\n    \\[\n    s_i'=s_i+b_i,\n    \\]\n    changes the residual by the \\v{C}ech coboundary\n    \\[\n    a'_{ij}=a_{ij}+b_j-b_i;\n    \\]\n    \\item the class\n    \\[\n    \\Omega(P;\\{U_i\\})=[a]\\in\\check H^1(\\{U_i\\},\\mathcal A)\n    \\]\n    is independent of the chosen local sections;\n    \\item $\\Omega(P;\\{U_i\\})=0$ if and only if the local sections can be modified by admissible local repairs so that they glue to a global section of $P$ over $U$.\n\\end{enumerate}\n\\end{theorem}","e fiber products used below, let $P$ be an $\\mathcal A$-torsor, and let $\\{U_i\\to U\\}$ be a cover on which $P$ admits local sections $s_i\\in P(U_i)$. On overlaps there are unique elements\n\\[\na_{ij}\\in\\mathcal A(U_{ij})\n\\]\nsuch that\n\\[\ns_j=s_i+a_{ij}.\n\\]\nThen:\n\\begin{enumerate}\n    \\item $a=(a_{ij})$ is a \\v{C}ech $1$-cocycle;\n    \\item changing the local sections by a $0$-cochain $b=(b_i)$,\n    \\[\n    s_i'=s_i+b_i,\n    \\]\n    changes the residual by the \\v{C}ech coboundary\n    \\[\n    a'_{ij}=a_{ij}+b_j-b_i;\n    \\]\n    \\item the class\n    \\[\n    \\Omega(P;\\{U_i\\})=[a]\\in\\check H^1(\\{U_i\\},\\mathcal A)\n    \\]\n    is independent of the chosen local sections;\n    \\item $\\Omega(P;\\{U_i\\})=0$ if and only if the local sections can be modified by admissible local repairs so that they glue to a global section of $P$ over $U$.\n\\end{enumerate}\n\\end{theorem}","}$ be a cover on which $P$ admits local sections $s_i\\in P(U_i)$. On overlaps there are unique elements\n\\[\na_{ij}\\in\\mathcal A(U_{ij})\n\\]\nsuch that\n\\[\ns_j=s_i+a_{ij}.\n\\]\nThen:\n\\begin{enumerate}\n    \\item $a=(a_{ij})$ is a \\v{C}ech $1$-cocycle;\n    \\item changing the local sections by a $0$-cochain $b=(b_i)$,\n    \\[\n    s_i'=s_i+b_i,\n    \\]\n    changes the residual by the \\v{C}ech coboundary\n    \\[\n    a'_{ij}=a_{ij}+b_j-b_i;\n    \\]\n    \\item the class\n    \\[\n    \\Omega(P;\\{U_i\\})=[a]\\in\\check H^1(\\{U_i\\},\\mathcal A)\n    \\]\n    is independent of the chosen local sections;\n    \\item $\\Omega(P;\\{U_i\\})=0$ if and only if the local sections can be modified by admissible local repairs so that they glue to a global section of $P$ over $U$.\n\\end{enumerate}\n\\end{theorem}",")$. On overlaps there are unique elements\n\\[\na_{ij}\\in\\mathcal A(U_{ij})\n\\]\nsuch that\n\\[\ns_j=s_i+a_{ij}.\n\\]\nThen:\n\\begin{enumerate}\n    \\item $a=(a_{ij})$ is a \\v{C}ech $1$-cocycle;\n    \\item changing the local sections by a $0$-cochain $b=(b_i)$,\n    \\[\n    s_i'=s_i+b_i,\n    \\]\n    changes the residual by the \\v{C}ech coboundary\n    \\[\n    a'_{ij}=a_{ij}+b_j-b_i;\n    \\]\n    \\item the class\n    \\[\n    \\Omega(P;\\{U_i\\})=[a]\\in\\check H^1(\\{U_i\\},\\mathcal A)\n    \\]\n    is independent of the chosen local sections;\n    \\item $\\Omega(P;\\{U_i\\})=0$ if and only if the local sections can be modified by admissible local repairs so that they glue to a global section of $P$ over $U$.\n\\end{enumerate}\n\\end{theorem}","ate}\n    \\item $a=(a_{ij})$ is a \\v{C}ech $1$-cocycle;\n    \\item changing the local sections by a $0$-cochain $b=(b_i)$,\n    \\[\n    s_i'=s_i+b_i,\n    \\]\n    changes the residual by the \\v{C}ech coboundary\n    \\[\n    a'_{ij}=a_{ij}+b_j-b_i;\n    \\]\n    \\item the class\n    \\[\n    \\Omega(P;\\{U_i\\})=[a]\\in\\check H^1(\\{U_i\\},\\mathcal A)\n    \\]\n    is independent of the chosen local sections;\n    \\item $\\Omega(P;\\{U_i\\})=0$ if and only if the local sections can be modified by admissible local repairs so that they glue to a global section of $P$ over $U$.\n\\end{enumerate}\n\\end{theorem}","b_i,\n    \\]\n    changes the residual by the \\v{C}ech coboundary\n    \\[\n    a'_{ij}=a_{ij}+b_j-b_i;\n    \\]\n    \\item the class\n    \\[\n    \\Omega(P;\\{U_i\\})=[a]\\in\\check H^1(\\{U_i\\},\\mathcal A)\n    \\]\n    is independent of the chosen local sections;\n    \\item $\\Omega(P;\\{U_i\\})=0$ if and only if the local sections can be modified by admissible local repairs so that they glue to a global section of $P$ over $U$.\n\\end{enumerate}\n\\end{theorem}","\\]\n    changes the residual by the \\v{C}ech coboundary\n    \\[\n    a'_{ij}=a_{ij}+b_j-b_i;\n    \\]\n    \\item the class\n    \\[\n    \\Omega(P;\\{U_i\\})=[a]\\in\\check H^1(\\{U_i\\},\\mathcal A)\n    \\]\n    is independent of the chosen local sections;\n    \\item $\\Omega(P;\\{U_i\\})=0$ if and only if the local sections can be modified by admissible local repairs so that they glue to a global section of $P$ over $U$.\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nOn a triple overlap,\n\\[\ns_k=s_j+a_{jk}=s_i+a_{ij}+a_{jk},\n\\]\nwhile also $s_k=s_i+a_{ik}$. Freeness of the torsor action gives\n\\[\na_{ij}+a_{jk}=a_{ik},\n\\]\nwhich is the \\v{C}ech cocycle identity.","If $s_i'=s_i+b_i$, then\n\\[\ns_j'=s_j+b_j=s_i+a_{ij}+b_j\n=s_i'+(a_{ij}+b_j-b_i),\n\\]\nso the residual changes by the coboundary $\\delta b$. Hence its cohomology class is independent of the chosen local sections.","If $[a]=0$, choose $b$ so that $a+\\delta b=0$. The repaired sections $s_i'=s_i+b_i$ then agree on every overlap, hence glue by the sheaf property to a global section of $P$. Conversely, if a global section exists, its restrictions have zero transition residual. Comparing those restrictions with the original $s_i$ shows that $a$ differs from zero by a coboundary, so $[a]=0$.\n\\end{proof}","In Komorebi language,\n\\[\n\\boxed{\n\\text{overlap residual}\n\\;/\\;\n\\text{local repair}\n\\;=\\;\n\\text{global obstruction class}.}\n\\]\nHere the residual is $a_{ij}$, admissible repair is the change $s_i\\mapsto s_i+b_i$, repairable residual cocycles are coboundaries, and the obstruction is the cohomology class $[a]$. This local repair relation is not being identified with the monoid of exact endofunctors in the stable quotient construction. This example is important because it realizes the proposed obstruction pattern before stabilization and shows that the abstract vocabulary recovers an established local-to-global obstruction mechanism rather than manufacturing one artificially.\nIt is specifically an obstruction to reconstructing a global \\emph{section} (equivalently, to trivializing the torsor), not an obstruction to t","In Komorebi language,\n\\[\n\\boxed{\n\\text{overlap residual}\n\\;/\\;\n\\text{local repair}\n\\;=\\;\n\\text{global obstruction class}.}\n\\]\nHere the residual is $a_{ij}$, admissible repair is the change $s_i\\mapsto s_i+b_i$, repairable residual cocycles are coboundaries, and the obstruction is the cohomology class $[a]$. This local repair relation is not being identified with the monoid of exact endofunctors in the stable quotient construction. This example is important because it realizes the proposed obstruction pattern before stabilization and shows that the abstract vocabulary recovers an established local-to-global obstruction mechanism rather than manufacturing one artificially.\nIt is specifically an obstruction to reconstructing a global \\emph{section} (equivalently, to trivializing the torsor), not an obstruction to the existence of the torsor itse","In Komorebi language,\n\\[\n\\boxed{\n\\text{overlap residual}\n\\;/\\;\n\\text{local repair}\n\\;=\\;\n\\text{global obstruction class}.}\n\\]\nHere the residual is $a_{ij}$, admissible repair is the change $s_i\\mapsto s_i+b_i$, repairable residual cocycles are coboundaries, and the obstruction is the cohomology class $[a]$. This local repair relation is not being identified with the monoid of exact endofunctors in the stable quotient construction. This example is important because it realizes the proposed obstruction pattern before stabilization and shows that the abstract vocabulary recovers an established local-to-global obstruction mechanism rather than manufacturing one artificially.\nIt is specifically an obstruction to reconstructing a global \\emph{section} (equivalently, to trivializing the torsor), not an obstruction to the existence of the torsor itself.","Let $(\\mathcal C,J)$ be a site and let $\\mathscr K$ assign a Komorebi category to each object of the site, with restriction functors on overlaps.","Given a cover $\\{U_i\\to U\\}$, local objects $X_i\\in\\mathscr K(U_i)$ carry descent isomorphisms\n\\[\n\\phi_{ij}:X_i|_{U_{ij}}\\xrightarrow{\\sim}X_j|_{U_{ij}}\n\\]\nsatisfying the cocycle condition on triple overlaps.","\\begin{theorem}[Descent transfer lemma]\\label{thm:descent-transfer}\nLet $A^\\bullet,B^\\bullet:\\Delta\\to\\Cat_\\infty$ be small \\v{C}ech descent diagrams for a fixed cover. Suppose $\\Lambda^\\bullet:A^\\bullet\\to B^\\bullet$ is a natural transformation that is an equivalence at every degree, including every overlap level. Then\n\\[\n\\lim_{[n]\\in\\Delta} A^n\\simeq\\lim_{[n]\\in\\Delta} B^n.\n\\]\nFor ordinary category-valued descent, use the corresponding coherent pseudolimits rather than strict equalizers of object sets.\n\\end{theorem}","\\begin{proof}\nEquivalences in a functor category are detected objectwise. Hence $\\Lambda^\\bullet$ is an equivalence of diagrams. Applying the limit functor preserves this equivalence. The ordinary category formulation is the same argument with coherent descent categories and their quasi-inverses.\n\\end{proof}","This is the abstract form of the principle:\n\\[\n\\boxed{\n\\text{local equivalence}\n+\n\\text{restriction compatibility}\n+\n\\text{effective descent}\n\\Longrightarrow\n\\text{global equivalence}.\n}\n\\]","Let\n\\[\n\\mathbb K_{\\mathrm{arith}}\n\\qquad\\text{and}\\qquad\n\\mathbb K_{\\mathrm{aut}}\n\\]\nbe arithmetic and automorphic seeded two-axis calculi.","A strong correspondence should include:\n\\begin{align*}\n\\Lambda(TX)&\\simeq T'\\Lambda(X),\\\\\n\\Lambda(CX)&\\simeq C'\\Lambda(X),\\\\\n\\Lambda(\\Delta_{T,C}X)&\\simeq\\Delta_{T',C'}\\Lambda(X),\\\\\n\\bar\\Lambda(\\Obs X)&\\simeq\\Obs'\\Lambda(X).\n\\end{align*}","Here $\\bar\\Lambda$ is the induced functor on the designated obstruction quotients. The last two requirements are stronger than matching ordinary local invariants. They demand that the correspondence preserve the structure of failed exchange and failed reconstruction.","\\paragraph{Interpretation.}\nThis reframes a possible route to local-global reciprocity as:\n\\[\n\\boxed{\n\\text{identify the globalization obstruction}\n\\quad\\text{and prove that local correspondence transports it.}\n}\n\\]\nIf\n\\[\n\\Lambda_*(\\Omega_{\\mathrm{arith}})=\\Omega_{\\mathrm{aut}},\n\\]\nand $\\Lambda_*$ preserves and reflects the zero object, then vanishing of one obstruction is equivalent to vanishing of the other. An equivalence, or a conservative exact functor, suffices for this extra detection condition. A general exact functor only preserves vanishing. Effective reconstruction then gives the stated global conclusion. Neither a sheaf/stack model for arbitrary local Langlands data nor automorphic existence follows from the abstract categorical lemmas. Places of a number field are not automatically a covering family for an effective descent problem. Those arithmetic identifications remain obligations.","\\subsection{\\texorpdfstring{$p$}{p}-adic congruence resolution}","Let the resolution level be $n\\ge1$ and take\n\\[\nX_n=\\mathbb Z/p^n\\mathbb Z.\n\\]\nFor $m\\le n$, coarsening is reduction\n\\[\nC_{n\\to m}:\\mathbb Z/p^n\\mathbb Z\\to\\mathbb Z/p^m\\mathbb Z.\n\\]\nFor any polynomial $f\\in\\mathbb Z[x]$, the induced transformation satisfies\n\\[\nC_{n\\to m}(f(x))\n=\nf(C_{n\\to m}(x)).\n\\]\nThus the transformation-resolution square commutes exactly. This is a set-level equality, not a cofiber calculation. Regarding the sets as discrete categories gives an identity comparison, while a stable residual requires a separately specified stable realization.\nThis is a simple instance where arithmetic transformation is resolution-natural.","By contrast, operations involving division by $p$ need not descend uniformly. Their domain of definition and lift multiplicity depend on valuation. This gives a natural source of nontrivial resolution obstruction.","Let a fine state be\n\\[\nx=(x_1,x_2)\\in\\mathbb R^2\n\\]\nand let coarse resolution record only the average\n\\[\nC(x)=\\frac{x_1+x_2}{2}.\n\\]\nLet the fine transformation be\n\\[\nA_1(x_1,x_2)=(ax_1,bx_2)\n\\]\nand the coarse transformation be multiplication by $c$:\n\\[\nA_0(y)=cy.\n\\]\nThen\n\\[\nC(A_1x)-A_0(Cx)\n=\n\\frac{(a-c)x_1+(b-c)x_2}{2}.\n\\]\nChoosing\n\\[\nc=\\frac{a+b}{2}\n\\]\ngives\n\\[\nC(A_1x)-A_0(Cx)\n=\n\\frac{a-b}{4}(x_1-x_2).\n\\]\nThe defect depends only on the antisymmetric fine mode $x_1-x_2$, precisely the information erased by averaging. This is a concrete model of a mixed residual exposing structure invisible at coarse resolution.","resolution record only the average\n\\[\nC(x)=\\frac{x_1+x_2}{2}.\n\\]\nLet the fine transformation be\n\\[\nA_1(x_1,x_2)=(ax_1,bx_2)\n\\]\nand the coarse transformation be multiplication by $c$:\n\\[\nA_0(y)=cy.\n\\]\nThen\n\\[\nC(A_1x)-A_0(Cx)\n=\n\\frac{(a-c)x_1+(b-c)x_2}{2}.\n\\]\nChoosing\n\\[\nc=\\frac{a+b}{2}\n\\]\ngives\n\\[\nC(A_1x)-A_0(Cx)\n=\n\\frac{a-b}{4}(x_1-x_2).\n\\]\nThe defect depends only on the antisymmetric fine mode $x_1-x_2$, precisely the information erased by averaging. This is a concrete model of a mixed residual exposing structure invisible at coarse resolution.","Fix $X\\in\\Tcat_s$. First record the exact commutation profile\n\\[\n\\mathcal E_A(X;s)=\\{r\\le s:\\Delta^A_{s,r}(X)\\simeq0\\}.\n\\]\nNo interval or monotonicity property is assumed. For $R=\\mathbb N$ and finite $s$, a well-defined consecutive coarsening depth is\n\\[\n\\kappa_A(X;s):=\\max\\{k\\in\\{0,\\ldots,s\\}:\\Delta^A_{s,r}(X)\\simeq0\n\\text{ for all }s-k\\le r\\le s\\}.\n\\]\nThe set is nonempty because $\\Delta^A_{s,s}(X)\\simeq0$. A larger value means that more consecutive coarsenings preserve this exchange. The preceding version's supremum over exact target levels was uninformative on a fixed finite tower because the identity level $r=s$ is always exact.","consecutive coarsening depth is\n\\[\n\\kappa_A(X;s):=\\max\\{k\\in\\{0,\\ldots,s\\}:\\Delta^A_{s,r}(X)\\simeq0\n\\text{ for all }s-k\\le r\\le s\\}.\n\\]\nThe set is nonempty because $\\Delta^A_{s,s}(X)\\simeq0$. A larger value means that more consecutive coarsenings preserve this exchange. The preceding version's supremum over exact target levels was uninformative on a fixed finite tower because the identity level $r=s$ is always exact.","When a numerical size function $\\|\\cdot\\|$ is available on the fiber, define\n\\[\n\\mathcal E_A(s,r;X):=\\|\\Delta^A_{s,r}(X)\\|.\n\\]\nPossible choices include dimension, rank, norm, entropy, categorical amplitude, support size, or proof-complexity measures.","For a chain $r_0<r_1<\\cdots<r_n$ and chosen lifts $X_i\\in\\Tcat_{r_i}$ with $C_{r_{i+1}\\to r_i}X_{i+1}\\cong X_i$, record\n\\[\nB_X=(b_{r_0\\to r_1}(X_0),\\ldots,b_{r_{n-1}\\to r_n}(X_{n-1})).\n\\]\nThis profile measures how hidden multiplicity emerges under refinement.","Instead of a single growth function, separate growth by generator type:\n\\[\ng_T(n),\\qquad g_R(n),\\qquad g_\\Delta(n).\n\\]\nThis may distinguish theories in which most complexity comes from representation changes, resolution branching, or residual generation.","\\begin{axiom}[Two-axis presentation]\nEvery admissible structured state is represented in a system\n\\[\n\\pi:\\K\\to\\Rcat\n\\]\nwhose fibers encode transformations and whose base encodes resolution.\n\\end{axiom}","\\begin{axiom}[Resolution compositionality]\nCoarsening is functorial:\n\\[\nC_{t\\to r}=C_{s\\to r}C_{t\\to s}.\n\\]\n\\end{axiom}","\\begin{axiom}[Exchange data]\nEvery designated transformation family carries a coherent comparison\n\\[\nA_rC_{s\\to r}\\Longrightarrow C_{s\\to r}A_s.\n\\]\n\\end{axiom}","These are axioms for a class of structures, not proposed additional axioms of ZFC. Two-axis existence alone imposes no substantive restriction: every category $\\mathcal C$ has the degenerate presentation over the one-object resolution category with fiber $\\mathcal C$. A foundational claim needs nontrivial, semantically specified resolutions and a theorem relating them to the intended mathematics.","The seed calculus can be interpreted as a search grammar. A machine state may contain\n\\[\n\\mathsf S=\n(\\Seeds,\\mathscr T,\\mathscr R,\\Delta,\\Omega,\\mathsf{Cert}).\n\\]\nFor a target problem $P$, search can occur jointly over transformation and resolution:\n\\[\n(T^\\ast,R^\\ast)\n\\in\n\\operatorname*{arg\\,min}_{T,R}\n\\mathcal C\\bigl(\\Omega(T,R;P)\\bigr),\n\\]\nwhere $\\mathcal C$ is a problem-dependent residual or obstruction cost. This expression describes a search objective. It does not assert that a minimizer exists or that the search is decidable.","This changes the computational question from\n\\[\n\\text{``Which operation solves the problem?''}\n\\]\nto\n\\[\n\\boxed{\n\\text{``At which resolution and in which representation does the obstruction become simplest?''}\n}\n\\]","A generic discovery loop is:\n\\[\n\\boxed{\n\\text{seed}\n\\to\n\\text{generate}\n\\to\n\\text{normalize}\n\\to\n\\text{measure residual}\n\\to\n\\text{promote residual}\n\\to\n\\text{expand zoo}\n\\to\n\\text{certify}.\n}\n\\]","\\begin{definition}[Seeded two-axis double computad]\\label{def:twoaxis-computad}\nA \\emph{seeded two-axis double computad} $\\mathbb S$ consists of a small double computad together with:\n\\begin{enumerate}\n    \\item a distinguished set $\\Seeds\\subseteq \\mathbb S_0$ of seed objects;\n    \\item a designated class of horizontal generating $1$-cells, interpreted as transformation generators;\n    \\item a designated class of vertical generating $1$-cells, interpreted as resolution generators;\n    \\item designated generating $2$-cells, interpreted as exchange or compatibility cells.\n\\end{enumerate}\nGenerator boundaries may be well-typed words, but no composition of the generators is identified with another generator at the signature level. An unlabelled realization of $\\mathbb S$ in a strict double category $\\mathbb D$ is a map of","\\begin{definition}[Seeded two-axis double computad]\\label{def:twoaxis-computad}\nA \\emph{seeded two-axis double computad} $\\mathbb S$ consists of a small double computad together with:\n\\begin{enumerate}\n    \\item a distinguished set $\\Seeds\\subseteq \\mathbb S_0$ of seed objects;\n    \\item a designated class of horizontal generating $1$-cells, interpreted as transformation generators;\n    \\item a designated class of vertical generating $1$-cells, interpreted as resolution generators;\n    \\item designated generating $2$-cells, interpreted as exchange or compatibility cells.\n\\end{enumerate}\nGenerator boundaries may be well-typed words, but no composition of the generators is identified with another generator at the signature level. An unlabelled realization of $\\mathbb S$ in a strict double category $\\mathbb D$ is a map of the underlying double computads. A label-preserving realization additionally requires matching distinguished subsets in the targe","\\begin{definition}[Seeded two-axis double computad]\\label{def:twoaxis-computad}\nA \\emph{seeded two-axis double computad} $\\mathbb S$ consists of a small double computad together with:\n\\begin{enumerate}\n    \\item a distinguished set $\\Seeds\\subseteq \\mathbb S_0$ of seed objects;\n    \\item a designated class of horizontal generating $1$-cells, interpreted as transformation generators;\n    \\item a designated class of vertical generating $1$-cells, interpreted as resolution generators;\n    \\item designated generating $2$-cells, interpreted as exchange or compatibility cells.\n\\end{enumerate}\nGenerator boundaries may be well-typed words, but no composition of the generators is identified with another generator at the signature level. An unlabelled realization of $\\mathbb S$ in a strict double category $\\mathbb D$ is a map of the underlying double computads. A label-preserving realization additionally requires matching distinguished subsets in the target and preservation of those designations.\n\\end{definition}","le computad]\\label{def:twoaxis-computad}\nA \\emph{seeded two-axis double computad} $\\mathbb S$ consists of a small double computad together with:\n\\begin{enumerate}\n    \\item a distinguished set $\\Seeds\\subseteq \\mathbb S_0$ of seed objects;\n    \\item a designated class of horizontal generating $1$-cells, interpreted as transformation generators;\n    \\item a designated class of vertical generating $1$-cells, interpreted as resolution generators;\n    \\item designated generating $2$-cells, interpreted as exchange or compatibility cells.\n\\end{enumerate}\nGenerator boundaries may be well-typed words, but no composition of the generators is identified with another generator at the signature level. An unlabelled realization of $\\mathbb S$ in a strict double category $\\mathbb D$ is a map of the underlying double computads. A label-preserving realization additionally requires matching distinguished subsets in the target and preservation of those designations.\n\\end{definition}","$\\mathbb S$ consists of a small double computad together with:\n\\begin{enumerate}\n    \\item a distinguished set $\\Seeds\\subseteq \\mathbb S_0$ of seed objects;\n    \\item a designated class of horizontal generating $1$-cells, interpreted as transformation generators;\n    \\item a designated class of vertical generating $1$-cells, interpreted as resolution generators;\n    \\item designated generating $2$-cells, interpreted as exchange or compatibility cells.\n\\end{enumerate}\nGenerator boundaries may be well-typed words, but no composition of the generators is identified with another generator at the signature level. An unlabelled realization of $\\mathbb S$ in a strict double category $\\mathbb D$ is a map of the underlying double computads. A label-preserving realization additionally requires matching distinguished subsets in the target and preservation of those designations.\n\\end{definition}","ation generators;\n    \\item a designated class of vertical generating $1$-cells, interpreted as resolution generators;\n    \\item designated generating $2$-cells, interpreted as exchange or compatibility cells.\n\\end{enumerate}\nGenerator boundaries may be well-typed words, but no composition of the generators is identified with another generator at the signature level. An unlabelled realization of $\\mathbb S$ in a strict double category $\\mathbb D$ is a map of the underlying double computads. A label-preserving realization additionally requires matching distinguished subsets in the target and preservation of those designations.\n\\end{definition}","class of vertical generating $1$-cells, interpreted as resolution generators;\n    \\item designated generating $2$-cells, interpreted as exchange or compatibility cells.\n\\end{enumerate}\nGenerator boundaries may be well-typed words, but no composition of the generators is identified with another generator at the signature level. An unlabelled realization of $\\mathbb S$ in a strict double category $\\mathbb D$ is a map of the underlying double computads. A label-preserving realization additionally requires matching distinguished subsets in the target and preservation of those designations.\n\\end{definition}","\\begin{theorem}[Free seeded two-axis double category]\\label{thm:free-double}\nFor every small seeded two-axis double computad $\\mathbb S$, there exists a strict double category $\\mathbb D_{\\mathbb S}$ and a canonical map of seeded double computads\n\\[\n\\iota_{\\mathbb S}:\\mathbb S\\longrightarrow U\\mathbb D_{\\mathbb S}\n\\]\nsuch that for every strict double category $\\mathbb D$, restriction along $\\iota_{\\mathbb S}$ induces a natural bijection\n\\[\n\\operatorname{DblCat}(\\mathbb D_{\\mathbb S},\\mathbb D)\n\\cong\n\\operatorname{DblCptd}(\\mathbb S,U\\mathbb D),\n\\]\nfor the underlying computadic data. If the target carries distinguished subsets for the same labels, restriction gives the analogous bijection with label-preservation imposed on both sides.\n\\end{theorem}","category]\\label{thm:free-double}\nFor every small seeded two-axis double computad $\\mathbb S$, there exists a strict double category $\\mathbb D_{\\mathbb S}$ and a canonical map of seeded double computads\n\\[\n\\iota_{\\mathbb S}:\\mathbb S\\longrightarrow U\\mathbb D_{\\mathbb S}\n\\]\nsuch that for every strict double category $\\mathbb D$, restriction along $\\iota_{\\mathbb S}$ induces a natural bijection\n\\[\n\\operatorname{DblCat}(\\mathbb D_{\\mathbb S},\\mathbb D)\n\\cong\n\\operatorname{DblCptd}(\\mathbb S,U\\mathbb D),\n\\]\nfor the underlying computadic data. If the target carries distinguished subsets for the same labels, restriction gives the analogous bijection with label-preservation imposed on both sides.\n\\end{theorem}","The residual layer becomes canonical after passing to stable categories. Let $\\mathbf{Cat}^{\\mathrm{add}}$ denote the $\\infty$-category of small additive $\\infty$-categories and additive functors, and let $\\mathbf{Cat}^{\\mathrm{st}}$ denote small stable $\\infty$-categories and exact functors. Ordinary additive categories are included by their nerves. The forgetful functor retains the underlying additive $\\infty$-category, not merely its homotopy category.","dual layer becomes canonical after passing to stable categories. Let $\\mathbf{Cat}^{\\mathrm{add}}$ denote the $\\infty$-category of small additive $\\infty$-categories and additive functors, and let $\\mathbf{Cat}^{\\mathrm{st}}$ denote small stable $\\infty$-categories and exact functors. Ordinary additive categories are included by their nerves. The forgetful functor retains the underlying additive $\\infty$-category, not merely its homotopy category.","\\begin{definition}[Additive two-axis system]\nAn \\emph{additive two-axis system} over a small resolution category $\\Rcat$ is a functor\n\\[\nF:\\Rcat^{op}\\longrightarrow \\mathbf{Cat}^{\\mathrm{add}}.\n\\]\nThus every fiber $F(r)$ is additive and every coarsening functor $C_{s\\to r}$ is additive.\n\\end{definition}","For a small additive category $\\mathcal A$, write $\\operatorname{St}^{\\mathrm{add}}(\\mathcal A)$ for its free stable envelope. We use the established adjunction\n\\[\n\\operatorname{St}^{\\mathrm{add}}:\n\\mathbf{Cat}^{\\mathrm{add}}\n\\rightleftarrows\n\\mathbf{Cat}^{\\mathrm{st}}:U,\n\\]\ncharacterized by\n\\[\n\\operatorname{Fun}^{\\mathrm{ex}}\n\\bigl(\\operatorname{St}^{\\mathrm{add}}(\\mathcal A),\\mathcal C\\bigr)\n\\simeq\n\\operatorname{Fun}^{\\oplus}(\\mathcal A,U\\mathcal C)\n\\]\nfor every stable $\\mathcal C$; see Saunier--Winges~\\cite{SaunierWinges2026}.","{add}}(\\mathcal A)$ for its free stable envelope. We use the established adjunction\n\\[\n\\operatorname{St}^{\\mathrm{add}}:\n\\mathbf{Cat}^{\\mathrm{add}}\n\\rightleftarrows\n\\mathbf{Cat}^{\\mathrm{st}}:U,\n\\]\ncharacterized by\n\\[\n\\operatorname{Fun}^{\\mathrm{ex}}\n\\bigl(\\operatorname{St}^{\\mathrm{add}}(\\mathcal A),\\mathcal C\\bigr)\n\\simeq\n\\operatorname{Fun}^{\\oplus}(\\mathcal A,U\\mathcal C)\n\\]\nfor every stable $\\mathcal C$; see Saunier--Winges~\\cite{SaunierWinges2026}.","\\begin{theorem}[Pointwise stable completion]\\label{thm:pointwise-stable}\nLet $\\Rcat$ be small. Postcomposition with $\\operatorname{St}^{\\mathrm{add}}$ defines a functor\n\\[\n\\operatorname{St}^{\\mathrm{add}}_{\\Rcat}:\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{add}})\n\\longrightarrow\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{st}}),\n\\qquad\nF\\longmapsto \\operatorname{St}^{\\mathrm{add}}\\circ F.\n\\]\nIt is left adjoint to pointwise forgetting\n\\[\nU_{\\Rcat}:\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{st}})\n\\longrightarrow\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{add}}).\n\\]\nEquivalently, for every additive two-axis system $F$ and stable two-axis system $G$ there is a natural equivalence\n\\[\n\\operatorname{Map}\n\\bigl(\\operatorname{St}^{\\mathrm{add}}_{","\\begin{theorem}[Pointwise stable completion]\\label{thm:pointwise-stable}\nLet $\\Rcat$ be small. Postcomposition with $\\operatorname{St}^{\\mathrm{add}}$ defines a functor\n\\[\n\\operatorname{St}^{\\mathrm{add}}_{\\Rcat}:\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{add}})\n\\longrightarrow\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{st}}),\n\\qquad\nF\\longmapsto \\operatorname{St}^{\\mathrm{add}}\\circ F.\n\\]\nIt is left adjoint to pointwise forgetting\n\\[\nU_{\\Rcat}:\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{st}})\n\\longrightarrow\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{add}}).\n\\]\nEquivalently, for every additive two-axis system $F$ and stable two-axis system $G$ there is a natural equivalence\n\\[\n\\operatorname{Map}\n\\bigl(\\operatorname{St}^{\\mathrm{add}}_{\\Rcat}F,G\\bigr)\n\\simeq\n\\operatorname{Map}\n\\bigl(F,U_{\\Rcat}G\\bigr)","\\begin{theorem}[Pointwise stable completion]\\label{thm:pointwise-stable}\nLet $\\Rcat$ be small. Postcomposition with $\\operatorname{St}^{\\mathrm{add}}$ defines a functor\n\\[\n\\operatorname{St}^{\\mathrm{add}}_{\\Rcat}:\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{add}})\n\\longrightarrow\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{st}}),\n\\qquad\nF\\longmapsto \\operatorname{St}^{\\mathrm{add}}\\circ F.\n\\]\nIt is left adjoint to pointwise forgetting\n\\[\nU_{\\Rcat}:\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{st}})\n\\longrightarrow\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{add}}).\n\\]\nEquivalently, for every additive two-axis system $F$ and stable two-axis system $G$ there is a natural equivalence\n\\[\n\\operatorname{Map}\n\\bigl(\\operatorname{St}^{\\mathrm{add}}_{\\Rcat}F,G\\bigr)\n\\simeq\n\\operatorname{Map}\n\\bigl(F,U_{\\Rcat}G\\bigr).\n\\]\n\\end{theorem}","intwise-stable}\nLet $\\Rcat$ be small. Postcomposition with $\\operatorname{St}^{\\mathrm{add}}$ defines a functor\n\\[\n\\operatorname{St}^{\\mathrm{add}}_{\\Rcat}:\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{add}})\n\\longrightarrow\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{st}}),\n\\qquad\nF\\longmapsto \\operatorname{St}^{\\mathrm{add}}\\circ F.\n\\]\nIt is left adjoint to pointwise forgetting\n\\[\nU_{\\Rcat}:\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{st}})\n\\longrightarrow\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{add}}).\n\\]\nEquivalently, for every additive two-axis system $F$ and stable two-axis system $G$ there is a natural equivalence\n\\[\n\\operatorname{Map}\n\\bigl(\\operatorname{St}^{\\mathrm{add}}_{\\Rcat}F,G\\bigr)\n\\simeq\n\\operatorname{Map}\n\\bigl(F,U_{\\Rcat}G\\bigr).\n\\]\n\\end{theorem}","^{\\mathrm{add}})\n\\longrightarrow\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{st}}),\n\\qquad\nF\\longmapsto \\operatorname{St}^{\\mathrm{add}}\\circ F.\n\\]\nIt is left adjoint to pointwise forgetting\n\\[\nU_{\\Rcat}:\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{st}})\n\\longrightarrow\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{add}}).\n\\]\nEquivalently, for every additive two-axis system $F$ and stable two-axis system $G$ there is a natural equivalence\n\\[\n\\operatorname{Map}\n\\bigl(\\operatorname{St}^{\\mathrm{add}}_{\\Rcat}F,G\\bigr)\n\\simeq\n\\operatorname{Map}\n\\bigl(F,U_{\\Rcat}G\\bigr).\n\\]\n\\end{theorem}","w\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{st}}),\n\\qquad\nF\\longmapsto \\operatorname{St}^{\\mathrm{add}}\\circ F.\n\\]\nIt is left adjoint to pointwise forgetting\n\\[\nU_{\\Rcat}:\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{st}})\n\\longrightarrow\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{add}}).\n\\]\nEquivalently, for every additive two-axis system $F$ and stable two-axis system $G$ there is a natural equivalence\n\\[\n\\operatorname{Map}\n\\bigl(\\operatorname{St}^{\\mathrm{add}}_{\\Rcat}F,G\\bigr)\n\\simeq\n\\operatorname{Map}\n\\bigl(F,U_{\\Rcat}G\\bigr).\n\\]\n\\end{theorem}","hbf{Cat}^{\\mathrm{st}}),\n\\qquad\nF\\longmapsto \\operatorname{St}^{\\mathrm{add}}\\circ F.\n\\]\nIt is left adjoint to pointwise forgetting\n\\[\nU_{\\Rcat}:\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{st}})\n\\longrightarrow\n\\operatorname{Fun}(\\Rcat^{op},\\mathbf{Cat}^{\\mathrm{add}}).\n\\]\nEquivalently, for every additive two-axis system $F$ and stable two-axis system $G$ there is a natural equivalence\n\\[\n\\operatorname{Map}\n\\bigl(\\operatorname{St}^{\\mathrm{add}}_{\\Rcat}F,G\\bigr)\n\\simeq\n\\operatorname{Map}\n\\bigl(F,U_{\\Rcat}G\\bigr).\n\\]\n\\end{theorem}","\\begin{proof}\nAny adjunction $L\\dashv U$ induces an adjunction on functor categories by postcomposition. Concretely, a natural transformation\n\\[\n\\alpha:L\\circ F\\Rightarrow G\n\\]\nhas pointwise adjoint mates\n\\[\n\\alpha_r^{\\flat}:F(r)\\longrightarrow U(G(r)).\n\\]\nNaturality of the unit and counit implies that the family $\\{\\alpha_r^{\\flat}\\}_r$ is natural in $r$. The mate construction is invertible and respects higher coherence. Applying this to $L=\\operatorname{St}^{\\mathrm{add}}$ gives the result.\n\\end{proof}","uces an adjunction on functor categories by postcomposition. Concretely, a natural transformation\n\\[\n\\alpha:L\\circ F\\Rightarrow G\n\\]\nhas pointwise adjoint mates\n\\[\n\\alpha_r^{\\flat}:F(r)\\longrightarrow U(G(r)).\n\\]\nNaturality of the unit and counit implies that the family $\\{\\alpha_r^{\\flat}\\}_r$ is natural in $r$. The mate construction is invertible and respects higher coherence. Applying this to $L=\\operatorname{St}^{\\mathrm{add}}$ gives the result.\n\\end{proof}","\\begin{definition}[Coherent additive transformation family]\nLet $F:\\Rcat^{op}\\to\\mathbf{Cat}^{\\mathrm{add}}$. A coherent additive transformation family $A$ consists of additive endofunctors\n\\[\nA_r:F(r)\\to F(r)\n\\]\nand, for each $r\\le s$, a natural transformation\n\\[\n\\eta^A_{s,r}:A_rC_{s\\to r}\\Longrightarrow C_{s\\to r}A_s,\n\\]\nsatisfying the identity and transitivity coherences for chains $r\\le s\\le t$.\n\\end{definition}","The universal property above is an equivalence of functor $\\infty$-categories, not only of their cores. This distinction is essential for noninvertible exchange maps. To extend $\\eta:f\\Rightarrow g$ between additive functors $\\mathcal A\\to\\mathcal B$, compose it with $j_{\\mathcal B}:\\mathcal B\\to\\operatorname{St}^{\\mathrm{add}}(\\mathcal B)$ and lift the resulting morphism through\n\\[\n\\operatorname{Fun}^{\\mathrm{ex}}(\\operatorname{St}^{\\mathrm{add}}(\\mathcal A),\\operatorname{St}^{\\mathrm{add}}(\\mathcal B))\n\\simeq\\operatorname{Fun}^{\\oplus}(\\mathcal A,\\operatorname{St}^{\\mathrm{add}}(\\mathcal B)).\n\\]\nFull faithfulness transports the supplied composition coherences as well. Accordingly $A_r$, $C_{s\\to r}$, and their coherently supplied exchange cells extend,","The universal property above is an equivalence of functor $\\infty$-categories, not only of their cores. This distinction is essential for noninvertible exchange maps. To extend $\\eta:f\\Rightarrow g$ between additive functors $\\mathcal A\\to\\mathcal B$, compose it with $j_{\\mathcal B}:\\mathcal B\\to\\operatorname{St}^{\\mathrm{add}}(\\mathcal B)$ and lift the resulting morphism through\n\\[\n\\operatorname{Fun}^{\\mathrm{ex}}(\\operatorname{St}^{\\mathrm{add}}(\\mathcal A),\\operatorname{St}^{\\mathrm{add}}(\\mathcal B))\n\\simeq\\operatorname{Fun}^{\\oplus}(\\mathcal A,\\operatorname{St}^{\\mathrm{add}}(\\mathcal B)).\n\\]\nFull faithfulness transports the supplied composition coherences as well. Accordingly $A_r$, $C_{s\\to r}$, and their coherently supplied exchange cells extend, essentially uniquely relative to the input, to exact functors and natural transformations\n\\[\n\\widehat A_r,\n\\qquad\n\\widehat C_{s\\to r","The universal property above is an equivalence of functor $\\infty$-categories, not only of their cores. This distinction is essential for noninvertible exchange maps. To extend $\\eta:f\\Rightarrow g$ between additive functors $\\mathcal A\\to\\mathcal B$, compose it with $j_{\\mathcal B}:\\mathcal B\\to\\operatorname{St}^{\\mathrm{add}}(\\mathcal B)$ and lift the resulting morphism through\n\\[\n\\operatorname{Fun}^{\\mathrm{ex}}(\\operatorname{St}^{\\mathrm{add}}(\\mathcal A),\\operatorname{St}^{\\mathrm{add}}(\\mathcal B))\n\\simeq\\operatorname{Fun}^{\\oplus}(\\mathcal A,\\operatorname{St}^{\\mathrm{add}}(\\mathcal B)).\n\\]\nFull faithfulness transports the supplied composition coherences as well. Accordingly $A_r$, $C_{s\\to r}$, and their coherently supplied exchange cells extend, essentially uniquely relative to the input, to exact functors and natural transformations\n\\[\n\\widehat A_r,\n\\qquad\n\\widehat C_{s\\to r},\n\\qquad\n\\widehat\\eta^A_{s,r}:\n\\widehat A_r\\widehat","The universal property above is an equivalence of functor $\\infty$-categories, not only of their cores. This distinction is essential for noninvertible exchange maps. To extend $\\eta:f\\Rightarrow g$ between additive functors $\\mathcal A\\to\\mathcal B$, compose it with $j_{\\mathcal B}:\\mathcal B\\to\\operatorname{St}^{\\mathrm{add}}(\\mathcal B)$ and lift the resulting morphism through\n\\[\n\\operatorname{Fun}^{\\mathrm{ex}}(\\operatorname{St}^{\\mathrm{add}}(\\mathcal A),\\operatorname{St}^{\\mathrm{add}}(\\mathcal B))\n\\simeq\\operatorname{Fun}^{\\oplus}(\\mathcal A,\\operatorname{St}^{\\mathrm{add}}(\\mathcal B)).\n\\]\nFull faithfulness transports the supplied composition coherences as well. Accordingly $A_r$, $C_{s\\to r}$, and their coherently supplied exchange cells extend, essentially uniquely relative to the input, to exact functors and natural transformations\n\\[\n\\widehat A_r,\n\\qquad\n\\widehat C_{s\\to r},\n\\qquad\n\\widehat\\eta^A_{s,r}:\n\\widehat A_r\\widehat C_{s\\to r}\n\\Longrightarrow\n\\widehat C_{s\\to r}\\widehat A_s.\n\\]","B\\to\\operatorname{St}^{\\mathrm{add}}(\\mathcal B)$ and lift the resulting morphism through\n\\[\n\\operatorname{Fun}^{\\mathrm{ex}}(\\operatorname{St}^{\\mathrm{add}}(\\mathcal A),\\operatorname{St}^{\\mathrm{add}}(\\mathcal B))\n\\simeq\\operatorname{Fun}^{\\oplus}(\\mathcal A,\\operatorname{St}^{\\mathrm{add}}(\\mathcal B)).\n\\]\nFull faithfulness transports the supplied composition coherences as well. Accordingly $A_r$, $C_{s\\to r}$, and their coherently supplied exchange cells extend, essentially uniquely relative to the input, to exact functors and natural transformations\n\\[\n\\widehat A_r,\n\\qquad\n\\widehat C_{s\\to r},\n\\qquad\n\\widehat\\eta^A_{s,r}:\n\\widehat A_r\\widehat C_{s\\to r}\n\\Longrightarrow\n\\widehat C_{s\\to r}\\widehat A_s.\n\\]","peratorname{St}^{\\mathrm{add}}(\\mathcal B)$ and lift the resulting morphism through\n\\[\n\\operatorname{Fun}^{\\mathrm{ex}}(\\operatorname{St}^{\\mathrm{add}}(\\mathcal A),\\operatorname{St}^{\\mathrm{add}}(\\mathcal B))\n\\simeq\\operatorname{Fun}^{\\oplus}(\\mathcal A,\\operatorname{St}^{\\mathrm{add}}(\\mathcal B)).\n\\]\nFull faithfulness transports the supplied composition coherences as well. Accordingly $A_r$, $C_{s\\to r}$, and their coherently supplied exchange cells extend, essentially uniquely relative to the input, to exact functors and natural transformations\n\\[\n\\widehat A_r,\n\\qquad\n\\widehat C_{s\\to r},\n\\qquad\n\\widehat\\eta^A_{s,r}:\n\\widehat A_r\\widehat C_{s\\to r}\n\\Longrightarrow\n\\widehat C_{s\\to r}\\widehat A_s.\n\\]","cal A),\\operatorname{St}^{\\mathrm{add}}(\\mathcal B))\n\\simeq\\operatorname{Fun}^{\\oplus}(\\mathcal A,\\operatorname{St}^{\\mathrm{add}}(\\mathcal B)).\n\\]\nFull faithfulness transports the supplied composition coherences as well. Accordingly $A_r$, $C_{s\\to r}$, and their coherently supplied exchange cells extend, essentially uniquely relative to the input, to exact functors and natural transformations\n\\[\n\\widehat A_r,\n\\qquad\n\\widehat C_{s\\to r},\n\\qquad\n\\widehat\\eta^A_{s,r}:\n\\widehat A_r\\widehat C_{s\\to r}\n\\Longrightarrow\n\\widehat C_{s\\to r}\\widehat A_s.\n\\]","\\begin{definition}[Universal stable mixed residual]\\label{def:universal-residual}\nFor $X\\in\\operatorname{St}^{\\mathrm{add}}(F(s))$, define\n\\[\n\\Delta^A_{s,r}(X)\n:=\n\\Cofib\\!\\left(\n\\widehat\\eta^A_{s,r,X}\n\\right)\n\\in\n\\operatorname{St}^{\\mathrm{add}}(F(r)).\n\\]\n\\end{definition}","\\begin{theorem}[Universal residual preservation]\\label{thm:residual-preservation}\nLet $G:\\Rcat^{op}\\to\\mathbf{Cat}^{\\mathrm{st}}$ be a stable two-axis system and let\n\\[\n\\Phi:\\operatorname{St}^{\\mathrm{add}}_{\\Rcat}F\\longrightarrow G\n\\]\nbe an exact natural transformation which intertwines the chosen transformation families and exchange cells. Then for every $r\\le s$ and every $X$ there is a canonical equivalence\n\\[\n\\Phi_r\\bigl(\\Delta^A_{s,r}(X)\\bigr)\n\\simeq\n\\Cofib\\!\\left(\n\\Phi_r(\\widehat\\eta^A_{s,r,X})\n\\right).\n\\]\nIn particular, if the exchange data in $G$ are denoted by $\\eta^B$, then\n\\[\n\\Phi_r\\bigl(\\Delta^A_{s,r}(X)\\bigr)\n\\simeq\n\\Delta^B_{s,r}(\\Phi_sX).\n\\]\n\\end{theorem}","niversal residual preservation]\\label{thm:residual-preservation}\nLet $G:\\Rcat^{op}\\to\\mathbf{Cat}^{\\mathrm{st}}$ be a stable two-axis system and let\n\\[\n\\Phi:\\operatorname{St}^{\\mathrm{add}}_{\\Rcat}F\\longrightarrow G\n\\]\nbe an exact natural transformation which intertwines the chosen transformation families and exchange cells. Then for every $r\\le s$ and every $X$ there is a canonical equivalence\n\\[\n\\Phi_r\\bigl(\\Delta^A_{s,r}(X)\\bigr)\n\\simeq\n\\Cofib\\!\\left(\n\\Phi_r(\\widehat\\eta^A_{s,r,X})\n\\right).\n\\]\nIn particular, if the exchange data in $G$ are denoted by $\\eta^B$, then\n\\[\n\\Phi_r\\bigl(\\Delta^A_{s,r}(X)\\bigr)\n\\simeq\n\\Delta^B_{s,r}(\\Phi_sX).\n\\]\n\\end{theorem}","nd let\n\\[\n\\Phi:\\operatorname{St}^{\\mathrm{add}}_{\\Rcat}F\\longrightarrow G\n\\]\nbe an exact natural transformation which intertwines the chosen transformation families and exchange cells. Then for every $r\\le s$ and every $X$ there is a canonical equivalence\n\\[\n\\Phi_r\\bigl(\\Delta^A_{s,r}(X)\\bigr)\n\\simeq\n\\Cofib\\!\\left(\n\\Phi_r(\\widehat\\eta^A_{s,r,X})\n\\right).\n\\]\nIn particular, if the exchange data in $G$ are denoted by $\\eta^B$, then\n\\[\n\\Phi_r\\bigl(\\Delta^A_{s,r}(X)\\bigr)\n\\simeq\n\\Delta^B_{s,r}(\\Phi_sX).\n\\]\n\\end{theorem}","ratorname{St}^{\\mathrm{add}}_{\\Rcat}F\\longrightarrow G\n\\]\nbe an exact natural transformation which intertwines the chosen transformation families and exchange cells. Then for every $r\\le s$ and every $X$ there is a canonical equivalence\n\\[\n\\Phi_r\\bigl(\\Delta^A_{s,r}(X)\\bigr)\n\\simeq\n\\Cofib\\!\\left(\n\\Phi_r(\\widehat\\eta^A_{s,r,X})\n\\right).\n\\]\nIn particular, if the exchange data in $G$ are denoted by $\\eta^B$, then\n\\[\n\\Phi_r\\bigl(\\Delta^A_{s,r}(X)\\bigr)\n\\simeq\n\\Delta^B_{s,r}(\\Phi_sX).\n\\]\n\\end{theorem}","mathrm{add}}_{\\Rcat}F\\longrightarrow G\n\\]\nbe an exact natural transformation which intertwines the chosen transformation families and exchange cells. Then for every $r\\le s$ and every $X$ there is a canonical equivalence\n\\[\n\\Phi_r\\bigl(\\Delta^A_{s,r}(X)\\bigr)\n\\simeq\n\\Cofib\\!\\left(\n\\Phi_r(\\widehat\\eta^A_{s,r,X})\n\\right).\n\\]\nIn particular, if the exchange data in $G$ are denoted by $\\eta^B$, then\n\\[\n\\Phi_r\\bigl(\\Delta^A_{s,r}(X)\\bigr)\n\\simeq\n\\Delta^B_{s,r}(\\Phi_sX).\n\\]\n\\end{theorem}","\\begin{proof}\nEach $\\Phi_r$ is exact, hence preserves finite colimits and therefore cofiber sequences. Apply $\\Phi_r$ to the defining cofiber sequence\n\\[\n\\widehat A_r\\widehat C_{s\\to r}X\n\\xrightarrow{\\widehat\\eta^A_{s,r,X}}\n\\widehat C_{s\\to r}\\widehat A_sX\n\\longrightarrow\n\\Delta^A_{s,r}(X).\n\\]\nThe result is a cofiber sequence in $G(r)$. Compatibility of $\\Phi$ with the transformation and exchange data identifies its first morphism with the target exchange morphism, giving the stated equivalence.\n\\end{proof}","\\begin{definition}[Stable seeded zoo]\nFix seed sets $\\Seeds_r\\subseteq\\Ob(F(r))$. A \\emph{stable seeded sub-zoo} of $\\operatorname{St}^{\\mathrm{add}}_{\\Rcat}F$ is a family of full stable subcategories\n\\[\n\\mathcal Z_r\\subseteq\\operatorname{St}^{\\mathrm{add}}(F(r))\n\\]\nsuch that:\n\\begin{enumerate}\n    \\item $\\mathcal Z_r$ contains the images of all seeds in $\\Seeds_r$;\n    \\item $\\widehat A_r(\\mathcal Z_r)\\subseteq\\mathcal Z_r$ for every designated transformation generator;\n    \\item $\\widehat C_{s\\to r}(\\mathcal Z_s)\\subseteq\\mathcal Z_r$ for every $r\\le s$.\n\\end{enumerate}\nThe \\emph{minimal stable zoo} is the intersection of all such families.\n\\end{definition}","ph{stable seeded sub-zoo} of $\\operatorname{St}^{\\mathrm{add}}_{\\Rcat}F$ is a family of full stable subcategories\n\\[\n\\mathcal Z_r\\subseteq\\operatorname{St}^{\\mathrm{add}}(F(r))\n\\]\nsuch that:\n\\begin{enumerate}\n    \\item $\\mathcal Z_r$ contains the images of all seeds in $\\Seeds_r$;\n    \\item $\\widehat A_r(\\mathcal Z_r)\\subseteq\\mathcal Z_r$ for every designated transformation generator;\n    \\item $\\widehat C_{s\\to r}(\\mathcal Z_s)\\subseteq\\mathcal Z_r$ for every $r\\le s$.\n\\end{enumerate}\nThe \\emph{minimal stable zoo} is the intersection of all such families.\n\\end{definition}","at}F$ is a family of full stable subcategories\n\\[\n\\mathcal Z_r\\subseteq\\operatorname{St}^{\\mathrm{add}}(F(r))\n\\]\nsuch that:\n\\begin{enumerate}\n    \\item $\\mathcal Z_r$ contains the images of all seeds in $\\Seeds_r$;\n    \\item $\\widehat A_r(\\mathcal Z_r)\\subseteq\\mathcal Z_r$ for every designated transformation generator;\n    \\item $\\widehat C_{s\\to r}(\\mathcal Z_s)\\subseteq\\mathcal Z_r$ for every $r\\le s$.\n\\end{enumerate}\nThe \\emph{minimal stable zoo} is the intersection of all such families.\n\\end{definition}","\\begin{proof}\nThe family of entire fibers $\\operatorname{St}^{\\mathrm{add}}(F(r))$ is a stable seeded sub-zoo, so the defining class is nonempty. Arbitrary intersections of full stable subcategories are again full stable subcategories, and invariance under the designated exact functors is preserved under intersection. Hence the intersection is the least stable seeded sub-zoo.\n\\end{proof}","\\begin{theorem}[Automatic residual promotion]\\label{thm:auto-promotion}\nLet $\\mathcal Z$ be the minimal stable zoo. If $X\\in\\mathcal Z_s$, then every mixed residual\n\\[\n\\Delta^A_{s,r}(X)\n\\]\nbelongs to $\\mathcal Z_r$.\n\\end{theorem}","\\begin{proof}\nBy invariance,\n\\[\n\\widehat A_r\\widehat C_{s\\to r}X\n\\quad\\text{and}\\quad\n\\widehat C_{s\\to r}\\widehat A_sX\n\\]\nare objects of $\\mathcal Z_r$. Since $\\mathcal Z_r$ is stable, it is closed under cofibers. Therefore the cofiber of the exchange morphism between these two objects, namely $\\Delta^A_{s,r}(X)$, also lies in $\\mathcal Z_r$.\n\\end{proof}","\\begin{lemma}[Residual formation is exact in its input]\\label{lem:exact-residual-functor}\nLet $P,Q:\\mathcal C\\to\\mathcal D$ be exact functors between small stable $\\infty$-categories, and let $\\eta:P\\Rightarrow Q$ be a natural transformation. Then\n\\[\n\\Delta_\\eta(X):=\\Cofib(\\eta_X)\n\\]\nis an exact functor $\\mathcal C\\to\\mathcal D$. If $q:\\mathcal D\\to\\mathcal O$ is exact, the subcategory\n\\[\n\\{X:q\\Delta_\\eta(X)\\simeq0\\}\n\\]\nis thick.\n\\end{lemma}","\\begin{proof}\nCofibers in the functor category are computed pointwise. The full subcategory of exact functors is closed under these finite colimits: a finite colimit of finite-colimit-preserving functors preserves finite colimits, and its value at zero is zero. Hence $\\Delta_\\eta$ is exact. The composite $q\\Delta_\\eta$ is exact as well. Its kernel is closed under shifts, cofibers, and retracts, which proves the last assertion.\n\\end{proof}","\\begin{definition}[Operation orbit of the seeds]\nIn a stable indexed system, let $\\mathscr P$ be the designated exact transformations and coarsenings. For each $r$, let $\\mathcal G_r$ contain every image at level $r$ of a seed under a finite well-typed word in $\\mathscr P$, including the empty word. Define\n\\[\n\\mathcal Z_r^{\\mathrm{thick}}:=\\operatorname{Thick}(\\mathcal G_r).\n\\]\nRefinement is included here only when an actual exact refinement functor has been separately specified. A multivalued lift relation is not silently treated as an exact functor.\n\\end{definition}","\\begin{proposition}[Orbit generation of the invariant thick zoo]\\label{prop:orbit-zoo}\nThe family $\\{\\mathcal Z_r^{\\mathrm{thick}}\\}_r$ is the least family of thick subcategories containing the seeds and preserved by all operations in $\\mathscr P$.\n\\end{proposition}","of}\nEvery operation sends an orbit generator to an orbit generator at the target level. Exactness and preservation of retracts therefore imply that it sends the thick closure of the source orbit into the thick closure of the target orbit. Thus the displayed family is invariant. Any invariant thick family containing the seeds contains all finite operation words applied to them, hence contains all $\\mathcal G_r$ and their thick closures. This proves minimality.\n\\end{proof}","\\begin{theorem}[Orbit-generator test for an exchange obstruction]\\label{thm:generator-test}\nFix an exchange $\\eta^A_{s,r}$ between exact functors, and an exact quotient $q_r$ of the target fiber. If\n\\[\nq_r\\Delta^A_{s,r}(g)\\simeq0\n\\quad\\text{for every }g\\in\\mathcal G_s,\n\\]\nthen\n\\[\nq_r\\Delta^A_{s,r}(X)\\simeq0\n\\quad\\text{for every }X\\in\\mathcal Z_s^{\\mathrm{thick}}.\n\\]\nThe same assertion holds for $q_r=\\id$, in which case the exchange is an equivalence on the entire generated thick zoo.\n\\end{theorem}","\\begin{proof}\nBy \\cref{lem:exact-residual-functor}, the objects on which $q_r\\Delta^A_{s,r}$ vanishes form a thick subcategory. By hypothesis it contains $\\mathcal G_s$, hence it contains $\\operatorname{Thick}(\\mathcal G_s)$. Vanishing of the unquotiented cofiber is equivalent to invertibility of the exchange map.\n\\end{proof}","K35 & Principal-ideal reconstruction & Reduced excisive diagrams classified by diagrams on $P$, \\cref{thm:principal-reconstruction}.","K55 & Finite-group invariant residual & Identified with $H^1[p^n]$, with group-order bound, \\cref{thm:invariant-bockstein}.","K60 & One-step action-lifting obstruction & The finite cyclic action cannot lift from $p^2$ to $p^3$, \\cref{thm:one-step-group-lift}.","K62 & Complete one-step relation obstruction & $H^2$ class vanishes exactly for a next-level lift, \\cref{thm:relation-H2}.","K63 & Repair ambiguity and certificates & $H^1$, $H^0$, and exact finite linear witnesses, \\cref{prop:repair-H1}.","K66 & Convergent action repair & Fixed finite group and $m>2v_p(|G|)$, \\cref{thm:convergent-action-repair}.","K67 & Finite action-lifting horizon & Necessary and sufficient level $\\max\\{n+e,2e+1\\}$, \\cref{thm:finite-action-horizon}.","K70 & Cyclic relation Jacobian & Affine next-level certificate with $d^2$ variables, \\cref{prop:cyclic-jacobian}.","K74 & Continuous tower reconstruction & Vanishing $H^2$ or a compatible tower; finite-generation compactness, \\cref{thm:continuous-tower}.","K75 & Free pro-$p$ coefficient lifting & Arbitrary integral generator lifts extend continuously, \\cref{thm:free-pro-p-lift}.","K76 & Procyclic matrix powers & Exact residual $p$-power-order criterion, including $p=2$, \\cref{lem:procyclic-powers}.","K79 & Cyclic-stratum horizon & Level $n+a$ suffices under the cyclic hypothesis, \\cref{cor:abelian-cyclic-horizon}.","K80 & Complete rank-two profinite test & All continuous $\\mathbb Z_p^r$ inputs, no extra deformation conditions, \\cref{thm:rank-two-profinite}.","K81 & Sharp rank-two horizon & Level $2n-1$, optimal for $n,r\\geq2$, \\cref{cor:rank-two-sharp-horizon}.","The general ordered-filtration independence target and the converse of the single derived-invariant test remain refuted. Fixed-finite-group lifting, continuous free pro-$p$ lifting, and rank-two lifting for $\\mathbb Z_p^r$ now have constructive solutions. Higher-rank abelian lifting is covered on the specified cyclic normalized stratum. Arbitrary profinite relations, geometric local conditions, and automorphic globalization remain separate obligations.","\\item \\textbf{Beyond strict pure exchange.} Extend K32 to a specified non-strict enhanced realization or a rewrite signature with additional critical overlaps. Identify the missing coherent cells explicitly rather than assume that homotopy-category diamonds supply them.\n\\item \\textbf{Noncommutative and constrained lifting.} K72--K82 handle continuous source refinement and give complete rank-two $\\mathbb Z_p^r$ reconstruction. Next specify a noncommutative profinite presentation or a noncyclic higher-rank centralizer. Construct its actual relation obstruction and a sufficient repair criterion. Prescribed determinants and geometric local conditions must be added as equations with their own compatibility checks. No uniform averaging bound over growing finite quotients is assumed.\n\\item \\textbf{Arithmetic existence beyond lattices.} K51 closes a concrete lattice case of K15. For the intended K17 application, construct the generic arithmetic or automorphic data and the required identifications, rather than importing them as gluing hypotheses. A globally glued lattice with supplied operato…","\\subsection{Resolution completion}\nA resolution tower may admit an inverse limit\n\\[\nX_\\infty\\simeq\\varprojlim_r X_r.\n\\]\nOne can ask whether the seed zoo commutes with completion:\n\\[\n\\Zoo(X_\\infty)\n\\stackrel{?}{\\simeq}\n\\varprojlim_r\\Zoo(X_r).\n\\]\nThis is only a proposed comparison. A common category and an actual comparison map must first be constructed before a residual is defined.","\\subsection{Bi-completion}\nThere are then two completion operations:\n\\[\n\\widehat{(-)}_T,\n\\qquad\n\\widehat{(-)}_R.\n\\]\nAfter both completions and the comparison have been defined in a common category, the mixed question is whether\n\\[\n\\widehat{\\widehat X_T}_R\n\\simeq\n\\widehat{\\widehat X_R}_T.\n\\]\nGiven the specified comparison, its defect is a candidate completion-level mixed residual in a setting where such defects are defined. The comparison is now explicit in \\cref{thm:completion-repair-defect}: even compatible finite-level repairs can fail to commute with completion, with a nonzero generic-fiber defect. An unconditional bi-completion claim is therefore false.","\\subsection{Resolution spectrum}\nFor a fixed $X\\in\\Tcat_s$ and a specified detection predicate $P_r$ at each level $r\\le s$, define:\n\\[\n\\operatorname{Spec}_R(P;X)\n=\n\\{r\\le s:P_r(C_{s\\to r}X)\\}.\n\\]\nThis could behave like a support or spectrum for distinguishability.","\\subsection{Residual genealogy}\nResidual promotion creates a directed graph\n\\[\ns\\to\\Delta_1\\to\\Delta_2\\to\\cdots.\n\\]\nClassify whether genealogies terminate, become periodic, stabilize, or grow indefinitely. A terminating genealogy would correspond to finite obstruction resolution. Periodic genealogy may encode a persistent but structured defect.","\\subsection{Seed minimality}\nGiven a generated zoo $\\Zoo$, ask for the smallest seed set $\\Seeds$ with\n\\[\n\\Zoo(\\Seeds)=\\Zoo.\n\\]\nThis is a mathematical compression problem. Nonuniqueness of minimal seeds could itself be meaningful.","\\subsection{Dual zoos}\nIf the calculus carries a duality $D$, study\n\\[\nD\\Zoo(\\Seeds)\n\\quad\\text{versus}\\quad\n\\Zoo(D\\Seeds).\n\\]\nThe residual of dualization against zoo generation measures whether the generative grammar is duality-complete.","This suggests the strengthened ontological phrase:\n\\[\n\\boxed{\n\\text{to remain identifiable is to remain coherent under both transformation and resolution.}\n}\n\\]\nThe mathematical content of the project is not this sentence itself. The mathematical content is the attempt to make ``coherent,'' ``transformation,'' ``resolution,'' ``residual,'' and ``reconstruction'' precise enough that such a sentence can be tested rather than merely asserted.","The first formalization supports a concrete picture:\n\\[\n\\boxed{\n\\begin{array}{c}\n\\text{seeds generate states},\\\\\n\\text{transformations generate representations},\\\\\n\\text{resolutions generate distinguishability},\\\\\n\\text{exchange maps relate the axes},\\\\\n\\text{residuals measure failed exchange},\\\\\n\\text{residuals become new seeds},\\\\\n\\text{obstructions classify persistent failure},\\\\\n\\text{descent reconstructs compatible local structure}.\n\\end{array}\n}\n\\]\nThe retained categorical core supplies explicit constructions under fixed hypotheses. Stable completion provides functorial cofibers, exact realizations preserve residuals, and the invariant stable zoo contains them automatically. Each normalization path carries a residual filtration, but the filtration is genuinely history-sensitive. The corrected intrinsic object is an ideal-indexed residual diagram for a specified coherent exchange system, whose terminal residual is independent of schedule.","on. A finite-quotient obstruction can disappear when more of the source is allowed to act, but an actual commutativity relation cannot be discarded. Scalar-depth normalization exposes that relation, and a cyclic centralizer constructs compatible integral matrices from finite polynomial data. In rank two the cyclic condition is automatic after normalization, yielding a complete test and the sharp $2n-1$ horizon. This proves reconstruction without eliminating the coefficient category, while the mixed-variance diagram states exactly how the two resolution coordinates interact.","\\bibitem{SaunierWinges2026}\nV.~Saunier and C.~Winges,\n\\emph{On exact categories and their stable envelopes},\narXiv:2502.03408v2, 28 November 2025.\nDefinition 3.1 and the following universal property give the free stable envelope $\\operatorname{St}^{\\mathrm{add}}$ and its adjunction with the forgetful functor from stable to additive categories.\n\\url{https://arxiv.org/abs/2502.03408v2}.","\\bibitem{StacksKoszul}\nThe Stacks Project Authors,\n\\emph{The Koszul complex}, Section 15.29, Tag 0621,\nDefinitions 15.29.1--2 and Lemmas 15.29.4, 15.29.7--8, 15.29.12,\nchecked 26 September 2026.\nThe cited section uses homological grading. This manuscript puts the exterior\n$k$-th term in cohomological degree $-k$ and states all differentials explicitly.\n\\url{https://stacks.math.columbia.edu/tag/0621}.","\\bibitem{MilneCFT2020}\nJ.~S. Milne, \\emph{Class Field Theory}, version 4.03, 6 August 2020.\nChapter II, 1.9 (coefficient long exact sequence), 1.18--1.21 (crossed homomorphisms, cyclic $H^1$, and explicit boundaries), and 3.4 (cyclic periodic resolution). Primary source checked 27 September 2026, including PDF pages 70, 75, and 90, corresponding to printed pages 61, 66, and 81.\n\\url{https://www.jmilne.org/math/CourseNotes/CFT.pdf}.","\\begin{align*}\n\\text{Resolution system:}\\quad\n&F:\\Rcat^{op}\\to\\Cat.\\\\[0.4em]\n\\text{Fiber:}\\quad\n&\\Tcat_r=F(r).\\\\[0.4em]\n\\text{Coarsening:}\\quad\n&C_{s\\to r}:\\Tcat_s\\to\\Tcat_r.\\\\[0.4em]\n\\text{Total category:}\\quad\n&\\K=\\int_{\\Rcat}F.\\\\[0.4em]\n\\text{Resolution equivalence:}\\quad\n&X\\sim_rY\\iff C_{t\\to r}X\\cong C_{t\\to r}Y.\\\\[0.4em]\n\\text{Lift set:}\\quad\n&\\Lift_{r\\to s}(X)=\\{Y:C_{s\\to r}Y\\cong X\\}/\\cong.\\\\[0.4em]\n\\text{Exchange map:}\\quad\n&\\eta^A_{s,r}:A_rC_{s\\to r}\\Rightarrow C_{s\\to r}A_s.\\\\[0.4em]\n\\text{Stable residual:}\\quad\n&\\Delta^A_{s,r}(X)=\\Cofib(\\eta^A_{s,r,X}).\\\\[0.4em]\n\\text{Stable obstruction quotient:}\\quad\n&\\mathcal O_{\\mathscr M}=\\mathcal Z/\\operatorname{Thick}(\\mathcal K_{\\mathscr M}).\\\\[0.4em]\n\\text{Invariant obstruction quotient:}\\quad\n&\\mathcal O^{\\mathrm{inv}}_{\\mathscr M}=\\mathcal Z/\\mathcal J_{\\mathscr M}.\\\\[0.4em]\n\\text{Event residual diagram:}\\quad\n&\\mathcal F_X(I)=\\Cofib(X_{\\varnothing}\\to X_I).\\\\[0.4em]\n\\text{Obstruction class:}\\quad\n&\\Obs_{\\mathscr M}(X)=q(X).\\\\[0.4em]\n\\text{Seed zoo:}\\quad\n&\\Zoo(\\Seeds_0)=\\mu A.\\bigl(\\Seeds_0\\cup\\Phi(A)\\bigr).\\\\[0.4em]\n\\text{Res…","\\subsection{Mathematical corrections}\n\\begin{enumerate}\n\\item \\textbf{Resolution:} fixed a common state set and a universal zeroth equivalence relation in the ultrametric construction. Distinguished isomorphism classes of lifts from framed lift groupoids. Excluded $\\log 0$ from ordinary branching information.\n\\item \\textbf{Exchange:} distinguished invertible comparison maps from exact functors. Required complete local exchange rules for the syntactic normal-form statement. Replaced the split cofiber identification by the correct shift $\\Delta\\simeq E[1]$. Removed an unsupported recursive termination claim for residual branches.\n\\item \\textbf{Coherence:} separated a normal object, its comparison map, and a filtration. Noninvertible exchange cells are extended using the full functor-category stable-envelope universal property, not just a mapping-space adjunction.\n\\item \\textbf{Quotients:} distinguished killability, thick killability, and constructor-invariant","relation in the ultrametric construction. Distinguished isomorphism classes of lifts from framed lift groupoids. Excluded $\\log 0$ from ordinary branching information.\n\\item \\textbf{Exchange:} distinguished invertible comparison maps from exact functors. Required complete local exchange rules for the syntactic normal-form statement. Replaced the split cofiber identification by the correct shift $\\Delta\\simeq E[1]$. Removed an unsupported recursive termination claim for residual branches.\n\\item \\textbf{Coherence:} separated a normal object, its comparison map, and a filtration. Noninvertible exchange cells are extended using the full functor-category stable-envelope universal property, not just a mapping-space adjunction.\n\\item \\textbf{Quotients:} distinguished killability, thick killability, and constructor-invariant thick killability. Stated universal uniqueness in the enhanced categorical sense. Corrected the postcomposition orientation in the directed condition.\n\\item \\textbf{Descent and scope:} included all overlap degrees and descent of morphisms. Required reflection as well as …","\\subsection{Exact finite regression}\nThe companion script \\texttt{checks/check\\_models.py} uses integer matrices, rational Gaussian elimination, and finite labelled words. It contains no floating-point tests. The recorded run uses Python 3.13.5 and checks every two-sort word of length $0$ through $10$, including the empty word and words containing only one sort.","The word test constructs reachable labelled words and order ideals independently, then compares their states and edges. The constructor test confirms the non-thick killability and invariant-collapse examples, together with a nonzero directed projection model. The cone tests compare the ordered dimensions $(0,1,3)$ and $(0,2,3)$, verify the split projection's shifted cofiber, and distinguish the identity and zero maps with equal endpoints.","The three executed routes were strict semantic realization, interaction/excision reconstruction, and integral quotient classification. They produced K32--K41 under their written hypotheses. The arithmetic test bed uses perfect complexes over $\\mathbb Z$ and explicit finite integer data. It does not discharge a Langlands-facing existence hypothesis.","y for the specified pure grammar. Weak cells and additional rewrite relations retain separate coherence obligations.\n\\item A square's commutativity is not its excision. The second cofiber vanishes precisely for a homotopy pushout, not for every commuting square.\n\\item Context-independent event residuals do not identify ordered filtrations. Nor do they split extensions along dependent events: the $\\mathbb Z/p^2$ chain is an explicit counterexample.\n\\item The total normalization residual $C(N)$ and a cube's total cofiber $K(n_1,\\ldots,n_m)$ are different objects. The $(6,10,15)$ example refutes an inference from highest interaction vanishing to all lower interactions vanishing.\n\\item The integral noncollapse theorem uses positive scalar labels. Two zero labels give an interaction with a free unit summand and collapse the entire quotient. The matrix criterion identifies the analogous free-cofiber failure.\n\\item Equality of the kernel with that of localization is proved. No unproved equivalence of complete quotient categories or idempotent completions is substituted for it.\n\\end{enumerat…","e coherence obligations.\n\\item A square's commutativity is not its excision. The second cofiber vanishes precisely for a homotopy pushout, not for every commuting square.\n\\item Context-independent event residuals do not identify ordered filtrations. Nor do they split extensions along dependent events: the $\\mathbb Z/p^2$ chain is an explicit counterexample.\n\\item The total normalization residual $C(N)$ and a cube's total cofiber $K(n_1,\\ldots,n_m)$ are different objects. The $(6,10,15)$ example refutes an inference from highest interaction vanishing to all lower interactions vanishing.\n\\item The integral noncollapse theorem uses positive scalar labels. Two zero labels give an interaction with a free unit summand and collapse the entire quotient. The matrix criterion identifies the analogous free-cofiber failure.\n\\item Equality of the kernel with that of localization is proved. No unproved equivalence of complete quotient categories or idempotent completions is substituted for it.\n\\end{enumerate}","quare's commutativity is not its excision. The second cofiber vanishes precisely for a homotopy pushout, not for every commuting square.\n\\item Context-independent event residuals do not identify ordered filtrations. Nor do they split extensions along dependent events: the $\\mathbb Z/p^2$ chain is an explicit counterexample.\n\\item The total normalization residual $C(N)$ and a cube's total cofiber $K(n_1,\\ldots,n_m)$ are different objects. The $(6,10,15)$ example refutes an inference from highest interaction vanishing to all lower interactions vanishing.\n\\item The integral noncollapse theorem uses positive scalar labels. Two zero labels give an interaction with a free unit summand and collapse the entire quotient. The matrix criterion identifies the analogous free-cofiber failure.\n\\item Equality of the kernel with that of localization is proved. No unproved equivalence of complete quotient categories or idempotent completions is substituted for it.\n\\end{enumerate}","second cofiber vanishes precisely for a homotopy pushout, not for every commuting square.\n\\item Context-independent event residuals do not identify ordered filtrations. Nor do they split extensions along dependent events: the $\\mathbb Z/p^2$ chain is an explicit counterexample.\n\\item The total normalization residual $C(N)$ and a cube's total cofiber $K(n_1,\\ldots,n_m)$ are different objects. The $(6,10,15)$ example refutes an inference from highest interaction vanishing to all lower interactions vanishing.\n\\item The integral noncollapse theorem uses positive scalar labels. Two zero labels give an interaction with a free unit summand and collapse the entire quotient. The matrix criterion identifies the analogous free-cofiber failure.\n\\item Equality of the kernel with that of localization is proved. No unproved equivalence of complete quotient categories or idempotent completions is substituted for it.\n\\end{enumerate}","Cohomology dimensions over $\\mathbb F_2,\\mathbb F_3,\\mathbb F_5,\\mathbb F_7$ & 29,856","The matrix family includes identity, zero, rank-deficient, triangular, and torsion examples. Its 121 pairs are exhaustive for that listed family, not for all integer matrices. Nine pairs have two free cone components. Additional signed/zero-vector checks, the excluded all-zero pair, and the named $(6,10)$, $(6,10,15)$, and nonsplit prime-chain examples are recorded separately. No primary case within the stated bounds was skipped.","\\subsection{Adversarial checks}\n\\begin{enumerate}\n\\item Derived reduction is exact, but naive tensor on arbitrary modules is not. The infinite periodic resolution over $\\mathbb Z/p^2$ prevents replacing a perfect two-term cone by its degree-zero cohomology.\n\\item Canonical tensor/base-change exchanges have zero residual. Nonzero scalar cells are specified additional data, not an unexplained consequence of coarsening.\n\\item Prime-power interactions can generate the whole perfect category even for nonzero labels. A multi-prime quotient removes whole components and can still collapse if every prime is selected.\n\\item Coherent finite reductions do not imply commutation with an infinite inverse limit. The field factors in K49 exhibit the precise missing information.\n\\item The based tower theorem requires common bounded ranks and compatible matrices. Arbitrary incompatible le","patible matrices. Arbitrary incompatible level choices are not claimed to lift.\n\\item The fracture square reconstructs a supplied complex. K51 separately proves effective existence for independent lattice data, but requires the generic identifications explicitly. Operator gluing requires exact intertwining, not only matching characteristic polynomials.\n\\item Generic-fiber defects are computed in $D(\\widehat B)$, not asserted to be perfect objects over the completed ring. Their cofiber is the $[1]$ shift of the stated fiber.\n\\end{enumerate}","square reconstructs a supplied complex. K51 separately proves effective existence for independent lattice data, but requires the generic identifications explicitly. Operator gluing requires exact intertwining, not only matching characteristic polynomials.\n\\item Generic-fiber defects are computed in $D(\\widehat B)$, not asserted to be perfect objects over the completed ring. Their cofiber is the $[1]$ shift of the stated fiber.\n\\end{enumerate}","The new suite checks all 1,080 ordered coefficient pairs in the eight listed rings $\\mathbb Z/q$ with $q\\in\\{2,3,4,5,8,9,16,25\\}$. It enumerates kernels and images to check three cohomology degrees and verifies middle-cohomology elementary divisors by torsion counts. It also checks 160 annihilator-transition tuples, 1,728 exchange coherences, 72 prime-component projections, 2,304 positive-label pair instances, all 30 pairs of subspaces of $\\mathbb F_2^2$ and $\\mathbb F_3^2$, 2,430 operator-preservation instances, 6,100 rank-one rational lattice membership tests, and 972 principal-part reconstruction witnesses. Exact bounds and nonclaims are in the JSON report.","icient pairs in the eight listed rings $\\mathbb Z/q$ with $q\\in\\{2,3,4,5,8,9,16,25\\}$. It enumerates kernels and images to check three cohomology degrees and verifies middle-cohomology elementary divisors by torsion counts. It also checks 160 annihilator-transition tuples, 1,728 exchange coherences, 72 prime-component projections, 2,304 positive-label pair instances, all 30 pairs of subspaces of $\\mathbb F_2^2$ and $\\mathbb F_3^2$, 2,430 operator-preservation instances, 6,100 rank-one rational lattice membership tests, and 972 principal-part reconstruction witnesses. Exact bounds and nonclaims are in the JSON report.","eight listed rings $\\mathbb Z/q$ with $q\\in\\{2,3,4,5,8,9,16,25\\}$. It enumerates kernels and images to check three cohomology degrees and verifies middle-cohomology elementary divisors by torsion counts. It also checks 160 annihilator-transition tuples, 1,728 exchange coherences, 72 prime-component projections, 2,304 positive-label pair instances, all 30 pairs of subspaces of $\\mathbb F_2^2$ and $\\mathbb F_3^2$, 2,430 operator-preservation instances, 6,100 rank-one rational lattice membership tests, and 972 principal-part reconstruction witnesses. Exact bounds and nonclaims are in the JSON report.","These are finite regression checks, not computational proofs of universal categorical statements or assertions about every $p$-adic input. The limiting and gluing theorems have separate written proofs. The arithmetic-square argument and nilpotent-cone argument use classical mechanisms; no historical novelty is inferred from their packaging here.","\\subsection{Proof boundaries checked}\n\\begin{enumerate}\n\\item The boundary uses torsion-freeness of the source lattices and of the target of the linear constraint. The residual at precision $n$ is an annihilator group, with transition multiplication by $p^{m-n}$, not an arbitrarily identified constant system.\n\\item Sharp lifting delay concerns solutions for the same fixed integral constraints. It does not grant a lift of constraints or group actions that were supplied only at finite level.\n\\item Ordinary invariants are not an exact functor. Their cokernel residual is not silently substituted into an exact-functor theorem.\n\\item Fully derived invariants commute with a coefficient module perfect over the source. At finite singular coefficients that perfectness can fail. The exact counterexample is calculated in enhanced unbounded derived categories, not inside perfect fibers.\n\\item","\\subsection{Proof boundaries checked}\n\\begin{enumerate}\n\\item The boundary uses torsion-freeness of the source lattices and of the target of the linear constraint. The residual at precision $n$ is an annihilator group, with transition multiplication by $p^{m-n}$, not an arbitrarily identified constant system.\n\\item Sharp lifting delay concerns solutions for the same fixed integral constraints. It does not grant a lift of constraints or group actions that were supplied only at finite level.\n\\item Ordinary invariants are not an exact functor. Their cokernel residual is not silently substituted into an exact-functor theorem.\n\\item Fully derived invariants commute with a coefficient module perfect over the source. At finite singular coefficients that perfectness can fail. The exact counterexample is calculated in enhanced unbounded derived categories, not inside perfect fibers.\n\\item The $\\mathbb Z/p^2$ example assumes $p$ odd and $J$ invertible modul","em Ordinary invariants are not an exact functor. Their cokernel residual is not silently substituted into an exact-functor theorem.\n\\item Fully derived invariants commute with a coefficient module perfect over the source. At finite singular coefficients that perfectness can fail. The exact counterexample is calculated in enhanced unbounded derived categories, not inside perfect fibers.\n\\item The $\\mathbb Z/p^2$ example assumes $p$ odd and $J$ invertible modulo $p$. Explicit tests outside both hypotheses have nonzero positive cohomology and do not satisfy its asserted quasi-isomorphism.\n\\item The source of the derived comparison is computed from a bounded-above free resolution of its degree-zero invariant module. Tensoring the original nonnegative invariant complex termwise would give the wrong answer.\n\\item The vanishing degree-zero comparison follows from naturality of evaluation. The cofiber's cohomology in all integer degrees follows from its long exact sequence, not extrapolation of a truncated numerical calculation.\n\\item Nonzero derived exchange obstructs a coefficient lift. It…","n exact functor. Their cokernel residual is not silently substituted into an exact-functor theorem.\n\\item Fully derived invariants commute with a coefficient module perfect over the source. At finite singular coefficients that perfectness can fail. The exact counterexample is calculated in enhanced unbounded derived categories, not inside perfect fibers.\n\\item The $\\mathbb Z/p^2$ example assumes $p$ odd and $J$ invertible modulo $p$. Explicit tests outside both hypotheses have nonzero positive cohomology and do not satisfy its asserted quasi-isomorphism.\n\\item The source of the derived comparison is computed from a bounded-above free resolution of its degree-zero invariant module. Tensoring the original nonnegative invariant complex termwise would give the wrong answer.\n\\item The vanishing degree-zero comparison follows from naturality of evaluation. The cofiber's cohomology in all integer degrees follows from its long exact sequence, not extrapolation of a truncated numerical calculation.\n\\item Nonzero derived exchange obstructs a coefficient lift. Its converse is not proved. The se…","tor. Their cokernel residual is not silently substituted into an exact-functor theorem.\n\\item Fully derived invariants commute with a coefficient module perfect over the source. At finite singular coefficients that perfectness can fail. The exact counterexample is calculated in enhanced unbounded derived categories, not inside perfect fibers.\n\\item The $\\mathbb Z/p^2$ example assumes $p$ odd and $J$ invertible modulo $p$. Explicit tests outside both hypotheses have nonzero positive cohomology and do not satisfy its asserted quasi-isomorphism.\n\\item The source of the derived comparison is computed from a bounded-above free resolution of its degree-zero invariant module. Tensoring the original nonnegative invariant complex termwise would give the wrong answer.\n\\item The vanishing degree-zero comparison follows from naturality of evaluation. The cofiber's cohomology in all integer degrees follows from its long exact sequence, not extrapolation of a truncated numerical calculation.\n\\item Nonzero derived exchange obstructs a coefficient lift. Its converse is not proved. The separate matri…","sidual is not silently substituted into an exact-functor theorem.\n\\item Fully derived invariants commute with a coefficient module perfect over the source. At finite singular coefficients that perfectness can fail. The exact counterexample is calculated in enhanced unbounded derived categories, not inside perfect fibers.\n\\item The $\\mathbb Z/p^2$ example assumes $p$ odd and $J$ invertible modulo $p$. Explicit tests outside both hypotheses have nonzero positive cohomology and do not satisfy its asserted quasi-isomorphism.\n\\item The source of the derived comparison is computed from a bounded-above free resolution of its degree-zero invariant module. Tensoring the original nonnegative invariant complex termwise would give the wrong answer.\n\\item The vanishing degree-zero comparison follows from naturality of evaluation. The cofiber's cohomology in all integer degrees follows from its long exact sequence, not extrapolation of a truncated numerical calculation.\n\\item Nonzero derived exchange obstructs a coefficient lift. Its converse is not proved. The separate matrix relation calculation","n. The cofiber's cohomology in all integer degrees follows from its long exact sequence, not extrapolation of a truncated numerical calculation.\n\\item Nonzero derived exchange obstructs a coefficient lift. Its converse is not proved. The separate matrix relation calculation rules out the specified lift from level two to level three.\n\\item Killing the unbounded target residual kills every perfect $k$-complex. To conclude that it kills all of $D(k)$, the exact functor must also preserve coproducts.\n\\end{enumerate}","rees follows from its long exact sequence, not extrapolation of a truncated numerical calculation.\n\\item Nonzero derived exchange obstructs a coefficient lift. Its converse is not proved. The separate matrix relation calculation rules out the specified lift from level two to level three.\n\\item Killing the unbounded target residual kills every perfect $k$-complex. To conclude that it kills all of $D(k)$, the exact functor must also preserve coproducts.\n\\end{enumerate}","\\midrule\nAll $2\\times2$ matrices with entries in $[-2,2]$ & 625","Norm-quotient Smith certificates, $2\\leq m\\leq12$ & 11","Invertible $2\\times2$ action matrices over $\\mathbb F_p$, $p=3,5,7$ & 2,544","The kernel levels are $(p,n)=(2,1),(2,2),(2,3),(3,1),(3,2)$. The sharpness and lifting loops use the precisely computed valuation bounds. No matrix in that family is skipped. The full group-action enumeration is exactly $\\operatorname{GL}_2(\\mathbb F_3)$, $\\operatorname{GL}_2(\\mathbb F_5)$, and $\\operatorname{GL}_2(\\mathbb F_7)$. The next-level lifts are all matrices $A+p^2B$ for the chosen swap $J$ and $B\\in M_2(\\mathbb F_p)$ at those three primes.","el levels are $(p,n)=(2,1),(2,2),(2,3),(3,1),(3,2)$. The sharpness and lifting loops use the precisely computed valuation bounds. No matrix in that family is skipped. The full group-action enumeration is exactly $\\operatorname{GL}_2(\\mathbb F_3)$, $\\operatorname{GL}_2(\\mathbb F_5)$, and $\\operatorname{GL}_2(\\mathbb F_7)$. The next-level lifts are all matrices $A+p^2B$ for the chosen swap $J$ and $B\\in M_2(\\mathbb F_p)$ at those three primes.","\\subsection{Executed routes and decisive outcomes}\nThree routes were pursued. First, the normalized multiplicativity defect gives a complete next-level obstruction in $H^2(G,\\operatorname{Ad}\\bar\\rho)$, with explicit linear repair and nonexistence certificates. Second, a central sign factor makes the derived-invariant test blind to an action that still cannot lift. This refutes sufficiency of that one diagnostic. Third, finite averaging gives a convergent correction whenever defect depth exceeds twice the valuation of the group order. Its finite-horizon corollary supplies a necessary-and-sufficient existence test for a fixed finite group.","\\subsection{Proof boundaries examined}\n\\begin{enumerate}\n\\item The ideal $p^nR_{n+1}$ is square-zero even at $n=1$. Matrix corrections use left multiplication, which fixes the signs in the cocycle equation and strict-conjugacy action.\n\\item The next-level obstruction is a class, not the chosen raw cocycle. The affine equation is necessary and sufficient for the entire fixed group multiplication table. A failed choice of representative matrices is not by itself an obstruction.\n\\item The approximate conjugations $\\alpha_g$ are not an action. The averaging proof starts with an exact associativity identity and controls its quadratic remainder before averaging. Division by $|G|$ loses $e=v_p(|G|)$ digits, explicitly accounted for at each step.\n\\item The convergence threshold is","\\subsection{Proof boundaries examined}\n\\begin{enumerate}\n\\item The ideal $p^nR_{n+1}$ is square-zero even at $n=1$. Matrix corrections use left multiplication, which fixes the signs in the cocycle equation and strict-conjugacy action.\n\\item The next-level obstruction is a class, not the chosen raw cocycle. The affine equation is necessary and sufficient for the entire fixed group multiplication table. A failed choice of representative matrices is not by itself an obstruction.\n\\item The approximate conjugations $\\alpha_g$ are not an action. The averaging proof starts with an exact associativity identity and controls its quadratic remainder before averaging. Division by $|G|$ loses $e=v_p(|G|)$ digits, explicitly accounted for at each step.\n\\item The convergence threshold is strictly $m>2e$. The finite w","atrix corrections use left multiplication, which fixes the signs in the cocycle equation and strict-conjugacy action.\n\\item The next-level obstruction is a class, not the chosen raw cocycle. The affine equation is necessary and sufficient for the entire fixed group multiplication table. A failed choice of representative matrices is not by itself an obstruction.\n\\item The approximate conjugations $\\alpha_g$ are not an action. The averaging proof starts with an exact associativity identity and controls its quadratic remainder before averaging. Division by $|G|$ loses $e=v_p(|G|)$ digits, explicitly accounted for at each step.\n\\item The convergence threshold is strictly $m>2e$. The finite witness must extend the requested $\\rho_n$, but the reconstructed integral action need only preserve that level, not the witness's top $e$ digits. The example $3\\pmod8$ for $C_2$ verifies that this distinction is real.\n\\item The central-sign example uses odd $p$, so its idempotents are defined. It is a counterexample for a fixed finite group; its restriction to $C_p$ supplies the impossible relation. N…",", not the chosen raw cocycle. The affine equation is necessary and sufficient for the entire fixed group multiplication table. A failed choice of representative matrices is not by itself an obstruction.\n\\item The approximate conjugations $\\alpha_g$ are not an action. The averaging proof starts with an exact associativity identity and controls its quadratic remainder before averaging. Division by $|G|$ loses $e=v_p(|G|)$ digits, explicitly accounted for at each step.\n\\item The convergence threshold is strictly $m>2e$. The finite witness must extend the requested $\\rho_n$, but the reconstructed integral action need only preserve that level, not the witness's top $e$ digits. The example $3\\pmod8$ for $C_2$ verifies that this distinction is real.\n\\item The central-sign example uses odd $p$, so its idempotents are defined. It is a counterexample for a fixed finite group; its restriction to $C_p$ supplies the impossible relation. No identification of its entire exchange cone with the adjoint $H^2$ class is made.\n\\item Finite catalogue means a finite set of isomorphism classes. It does not …","osen raw cocycle. The affine equation is necessary and sufficient for the entire fixed group multiplication table. A failed choice of representative matrices is not by itself an obstruction.\n\\item The approximate conjugations $\\alpha_g$ are not an action. The averaging proof starts with an exact associativity identity and controls its quadratic remainder before averaging. Division by $|G|$ loses $e=v_p(|G|)$ digits, explicitly accounted for at each step.\n\\item The convergence threshold is strictly $m>2e$. The finite witness must extend the requested $\\rho_n$, but the reconstructed integral action need only preserve that level, not the witness's top $e$ digits. The example $3\\pmod8$ for $C_2$ verifies that this distinction is real.\n\\item The central-sign example uses odd $p$, so its idempotents are defined. It is a counterexample for a fixed finite group; its restriction to $C_p$ supplies the impossible relation. No identification of its entire exchange cone with the adjoint $H^2$ class is made.\n\\item Finite catalogue means a finite set of isomorphism classes. It does not claim finite…","ion table. A failed choice of representative matrices is not by itself an obstruction.\n\\item The approximate conjugations $\\alpha_g$ are not an action. The averaging proof starts with an exact associativity identity and controls its quadratic remainder before averaging. Division by $|G|$ loses $e=v_p(|G|)$ digits, explicitly accounted for at each step.\n\\item The convergence threshold is strictly $m>2e$. The finite witness must extend the requested $\\rho_n$, but the reconstructed integral action need only preserve that level, not the witness's top $e$ digits. The example $3\\pmod8$ for $C_2$ verifies that this distinction is real.\n\\item The central-sign example uses odd $p$, so its idempotents are defined. It is a counterexample for a fixed finite group; its restriction to $C_p$ supplies the impossible relation. No identification of its entire exchange cone with the adjoint $H^2$ class is made.\n\\item Finite catalogue means a finite set of isomorphism classes. It does not claim finite automorphism groups or a category equivalence under reduction. Strict framed matrices need not be uniqu…","is not by itself an obstruction.\n\\item The approximate conjugations $\\alpha_g$ are not an action. The averaging proof starts with an exact associativity identity and controls its quadratic remainder before averaging. Division by $|G|$ loses $e=v_p(|G|)$ digits, explicitly accounted for at each step.\n\\item The convergence threshold is strictly $m>2e$. The finite witness must extend the requested $\\rho_n$, but the reconstructed integral action need only preserve that level, not the witness's top $e$ digits. The example $3\\pmod8$ for $C_2$ verifies that this distinction is real.\n\\item The central-sign example uses odd $p$, so its idempotents are defined. It is a counterexample for a fixed finite group; its restriction to $C_p$ supplies the impossible relation. No identification of its entire exchange cone with the adjoint $H^2$ class is made.\n\\item Finite catalogue means a finite set of isomorphism classes. It does not claim finite automorphism groups or a category equivalence under reduction. Strict framed matrices need not be unique.\n\\item The profinite example changes the source from…","e averaging proof starts with an exact associativity identity and controls its quadratic remainder before averaging. Division by $|G|$ loses $e=v_p(|G|)$ digits, explicitly accounted for at each step.\n\\item The convergence threshold is strictly $m>2e$. The finite witness must extend the requested $\\rho_n$, but the reconstructed integral action need only preserve that level, not the witness's top $e$ digits. The example $3\\pmod8$ for $C_2$ verifies that this distinction is real.\n\\item The central-sign example uses odd $p$, so its idempotents are defined. It is a counterexample for a fixed finite group; its restriction to $C_p$ supplies the impossible relation. No identification of its entire exchange cone with the adjoint $H^2$ class is made.\n\\item Finite catalogue means a finite set of isomorphism classes. It does not claim finite automorphism groups or a category equivalence under reduction. Strict framed matrices need not be unique.\n\\item The profinite example changes the source from $C_p$ to $\\mathbb Z_p$. The latter has an integral lift because it does not impose the same finite-…","with an exact associativity identity and controls its quadratic remainder before averaging. Division by $|G|$ loses $e=v_p(|G|)$ digits, explicitly accounted for at each step.\n\\item The convergence threshold is strictly $m>2e$. The finite witness must extend the requested $\\rho_n$, but the reconstructed integral action need only preserve that level, not the witness's top $e$ digits. The example $3\\pmod8$ for $C_2$ verifies that this distinction is real.\n\\item The central-sign example uses odd $p$, so its idempotents are defined. It is a counterexample for a fixed finite group; its restriction to $C_p$ supplies the impossible relation. No identification of its entire exchange cone with the adjoint $H^2$ class is made.\n\\item Finite catalogue means a finite set of isomorphism classes. It does not claim finite automorphism groups or a category equivalence under reduction. Strict framed matrices need not be unique.\n\\item The profinite example changes the source from $C_p$ to $\\mathbb Z_p$. The latter has an integral lift because it does not impose the same finite-order relation. This prev…","associativity identity and controls its quadratic remainder before averaging. Division by $|G|$ loses $e=v_p(|G|)$ digits, explicitly accounted for at each step.\n\\item The convergence threshold is strictly $m>2e$. The finite witness must extend the requested $\\rho_n$, but the reconstructed integral action need only preserve that level, not the witness's top $e$ digits. The example $3\\pmod8$ for $C_2$ verifies that this distinction is real.\n\\item The central-sign example uses odd $p$, so its idempotents are defined. It is a counterexample for a fixed finite group; its restriction to $C_p$ supplies the impossible relation. No identification of its entire exchange cone with the adjoint $H^2$ class is made.\n\\item Finite catalogue means a finite set of isomorphism classes. It does not claim finite automorphism groups or a category equivalence under reduction. Strict framed matrices need not be unique.\n\\item The profinite example changes the source from $C_p$ to $\\mathbb Z_p$. The latter has an integral lift because it does not impose the same finite-order relation. This prevents applying …","n by $|G|$ loses $e=v_p(|G|)$ digits, explicitly accounted for at each step.\n\\item The convergence threshold is strictly $m>2e$. The finite witness must extend the requested $\\rho_n$, but the reconstructed integral action need only preserve that level, not the witness's top $e$ digits. The example $3\\pmod8$ for $C_2$ verifies that this distinction is real.\n\\item The central-sign example uses odd $p$, so its idempotents are defined. It is a counterexample for a fixed finite group; its restriction to $C_p$ supplies the impossible relation. No identification of its entire exchange cone with the adjoint $H^2$ class is made.\n\\item Finite catalogue means a finite set of isomorphism classes. It does not claim finite automorphism groups or a category equivalence under reduction. Strict framed matrices need not be unique.\n\\item The profinite example changes the source from $C_p$ to $\\mathbb Z_p$. The latter has an integral lift because it does not impose the same finite-order relation. This prevents applying a fixed-finite-quotient nonlifting verdict indiscriminately to a Galois representatio…","old is strictly $m>2e$. The finite witness must extend the requested $\\rho_n$, but the reconstructed integral action need only preserve that level, not the witness's top $e$ digits. The example $3\\pmod8$ for $C_2$ verifies that this distinction is real.\n\\item The central-sign example uses odd $p$, so its idempotents are defined. It is a counterexample for a fixed finite group; its restriction to $C_p$ supplies the impossible relation. No identification of its entire exchange cone with the adjoint $H^2$ class is made.\n\\item Finite catalogue means a finite set of isomorphism classes. It does not claim finite automorphism groups or a category equivalence under reduction. Strict framed matrices need not be unique.\n\\item The profinite example changes the source from $C_p$ to $\\mathbb Z_p$. The latter has an integral lift because it does not impose the same finite-order relation. This prevents applying a fixed-finite-quotient nonlifting verdict indiscriminately to a Galois representation.\n\\end{enumerate}","tegral action need only preserve that level, not the witness's top $e$ digits. The example $3\\pmod8$ for $C_2$ verifies that this distinction is real.\n\\item The central-sign example uses odd $p$, so its idempotents are defined. It is a counterexample for a fixed finite group; its restriction to $C_p$ supplies the impossible relation. No identification of its entire exchange cone with the adjoint $H^2$ class is made.\n\\item Finite catalogue means a finite set of isomorphism classes. It does not claim finite automorphism groups or a category equivalence under reduction. Strict framed matrices need not be unique.\n\\item The profinite example changes the source from $C_p$ to $\\mathbb Z_p$. The latter has an integral lift because it does not impose the same finite-order relation. This prevents applying a fixed-finite-quotient nonlifting verdict indiscriminately to a Galois representation.\n\\end{enumerate}","group; its restriction to $C_p$ supplies the impossible relation. No identification of its entire exchange cone with the adjoint $H^2$ class is made.\n\\item Finite catalogue means a finite set of isomorphism classes. It does not claim finite automorphism groups or a category equivalence under reduction. Strict framed matrices need not be unique.\n\\item The profinite example changes the source from $C_p$ to $\\mathbb Z_p$. The latter has an integral lift because it does not impose the same finite-order relation. This prevents applying a fixed-finite-quotient nonlifting verdict indiscriminately to a Galois representation.\n\\end{enumerate}","s restriction to $C_p$ supplies the impossible relation. No identification of its entire exchange cone with the adjoint $H^2$ class is made.\n\\item Finite catalogue means a finite set of isomorphism classes. It does not claim finite automorphism groups or a category equivalence under reduction. Strict framed matrices need not be unique.\n\\item The profinite example changes the source from $C_p$ to $\\mathbb Z_p$. The latter has an integral lift because it does not impose the same finite-order relation. This prevents applying a fixed-finite-quotient nonlifting verdict indiscriminately to a Galois representation.\n\\end{enumerate}","\\subsection{Finite computation contract}\nAll five exact regression scripts passed, including reruns of the four inherited suites. The new script uses integer and modular arithmetic only in mathematical checks. It examines eight specified cochain fixtures, including the nonabelian group $S_3$, with 593 cocycle triples and 56 changes of section. Positive solutions and negative left-nullspace witnesses are checked directly against their equations.","The cyclic suite enumerates all 367 valid rank-two actions and all 40,987 next-level generator matrices in its six explicitly listed $(p,n,q)$ cases. The averaging suite performs 330 corrections in 110 deterministic fixtures, including primes dividing and not dividing the group order. Every use of division tracks finite working precision, and the exact associativity identity is checked before the predicted new defect depth. There are also 36 rank-one horizon comparisons, 98 conjugacy-average checks, and all 3,107 next-level lifts of three selected non-scalar nonliftable matrices. The sign and profinite power calculations are supplementary finite checks.","These are bounded tests of the implementation. They do not establish $p$-adic convergence, exhaustive classification for arbitrary groups, or historical novelty. Those conclusions either have their written proofs above or are not claimed. Full iterators, exclusions, counts, interpreter version, wall times, command lines, hashes before and after execution, and report hashes are stored in the package. Each suite has a 40-second subprocess timeout. No memory cap was set. Numerical-library thread limits are cooperative.","The sufficient existence question is closed here only for a fixed finite source group over $\\mathbb Z_p$. Continuous lifting with growing finite images, geometric local conditions, and automorphic realization remain separate obligations. A useful continuation must address those actual hypotheses rather than merely rename the already-classical finite-group obstruction or averaging argument.","The new results have written proofs and an authoring-session self-review. They have not received an independent mathematical review or Lean certification. Lean and Lake are not available on the runtime path. No claim of historical novelty is made for continuous cohomology, free pro-$p$ groups, or cyclic centralizers, nor is novelty inferred for their combination from a limited search.","n{Executed routes and mathematical boundaries}\nThe continuous-obstruction route retains the actual profinite source and proves the finite-quotient inflation criterion, including an explicit carry cocycle killed by source refinement. It does not infer that all finite-quotient obstructions eventually vanish. The free-source route constructs lifts without source relations beyond those of a free pro-$p$ group. The abelian route retains commutativity, exposes its normalized residual, and gives a complete criterion on the cyclic normalized stratum, including every rank-two input. The example with $E_{12},E_{21}$ proves the uniform $2n-1$ horizon sharp for $n,r\\geq2$.","ycle killed by source refinement. It does not infer that all finite-quotient obstructions eventually vanish. The free-source route constructs lifts without source relations beyond those of a free pro-$p$ group. The abelian route retains commutativity, exposes its normalized residual, and gives a complete criterion on the cyclic normalized stratum, including every rank-two input. The example with $E_{12},E_{21}$ proves the uniform $2n-1$ horizon sharp for $n,r\\geq2$.","t does not infer that all finite-quotient obstructions eventually vanish. The free-source route constructs lifts without source relations beyond those of a free pro-$p$ group. The abelian route retains commutativity, exposes its normalized residual, and gives a complete criterion on the cyclic normalized stratum, including every rank-two input. The example with $E_{12},E_{21}$ proves the uniform $2n-1$ horizon sharp for $n,r\\geq2$.","finite-quotient obstructions eventually vanish. The free-source route constructs lifts without source relations beyond those of a free pro-$p$ group. The abelian route retains commutativity, exposes its normalized residual, and gives a complete criterion on the cyclic normalized stratum, including every rank-two input. The example with $E_{12},E_{21}$ proves the uniform $2n-1$ horizon sharp for $n,r\\geq2$.","The self-review checked the following distinctions.\n\\begin{enumerate}\n\\item Continuous $H^2$ here has a finite discrete coefficient module. No statement about arbitrary topological coefficient theories is substituted for that definition.\n\\item A continuous repair cochain may require a finer finite source quotient. It is not required to factor through the initial quotient.\n\\item The cohomology-vanishing theorem gives a sufficient condition and the finitely branching tree gives a compactness criterion. Neither supplies a general numerical horizon for all profinite groups.\n\\item In the cyclic test, the normalized generator must be cyclic modulo $p$. Rank two makes that condition automatic after minimal scalar-depth normalization. No such assertion is made for arbitrary higher rank.\n\\","nuous repair cochain may require a finer finite source quotient. It is not required to factor through the initial quotient.\n\\item The cohomology-vanishing theorem gives a sufficient condition and the finitely branching tree gives a compactness criterion. Neither supplies a general numerical horizon for all profinite groups.\n\\item In the cyclic test, the normalized generator must be cyclic modulo $p$. Rank two makes that condition automatic after minimal scalar-depth normalization. No such assertion is made for arbitrary higher rank.\n\\item Cancellation of $p^{2a}$ takes place in torsion-free integral matrices for necessity, or is translated into the stated lower-modulus congruence for a finite witness. It is not division by a unit in $R_n$.\n\\item The positive construction preserves the given reduction, but not extra determinant, ramification, or geometric conditions. The constructed matrices are units over $\\mathbb Z_p$, not necessarily unimodular over $\\mathbb Z$.\n\\item The continuous-powers argument includes $p=2$. The explicit carry example retains its separate odd-prime hypothesis…","gives a sufficient condition and the finitely branching tree gives a compactness criterion. Neither supplies a general numerical horizon for all profinite groups.\n\\item In the cyclic test, the normalized generator must be cyclic modulo $p$. Rank two makes that condition automatic after minimal scalar-depth normalization. No such assertion is made for arbitrary higher rank.\n\\item Cancellation of $p^{2a}$ takes place in torsion-free integral matrices for necessity, or is translated into the stated lower-modulus congruence for a finite witness. It is not division by a unit in $R_n$.\n\\item The positive construction preserves the given reduction, but not extra determinant, ramification, or geometric conditions. The constructed matrices are units over $\\mathbb Z_p$, not necessarily unimodular over $\\mathbb Z$.\n\\item The continuous-powers argument includes $p=2$. The explicit carry example retains its separate odd-prime hypothesis.\n\\item Source inflation and coefficient reduction have opposite variance. A preliminary draft description treating both as unrestricted coarsenings was replaced b…","c test, the normalized generator must be cyclic modulo $p$. Rank two makes that condition automatic after minimal scalar-depth normalization. No such assertion is made for arbitrary higher rank.\n\\item Cancellation of $p^{2a}$ takes place in torsion-free integral matrices for necessity, or is translated into the stated lower-modulus congruence for a finite witness. It is not division by a unit in $R_n$.\n\\item The positive construction preserves the given reduction, but not extra determinant, ramification, or geometric conditions. The constructed matrices are units over $\\mathbb Z_p$, not necessarily unimodular over $\\mathbb Z$.\n\\item The continuous-powers argument includes $p=2$. The explicit carry example retains its separate odd-prime hypothesis.\n\\item Source inflation and coefficient reduction have opposite variance. A preliminary draft description treating both as unrestricted coarsenings was replaced by \\cref{prop:mixed-source-variance}; reversing inflation would require descent through the smaller quotient.\n\\end{enumerate}","trary higher rank.\n\\item Cancellation of $p^{2a}$ takes place in torsion-free integral matrices for necessity, or is translated into the stated lower-modulus congruence for a finite witness. It is not division by a unit in $R_n$.\n\\item The positive construction preserves the given reduction, but not extra determinant, ramification, or geometric conditions. The constructed matrices are units over $\\mathbb Z_p$, not necessarily unimodular over $\\mathbb Z$.\n\\item The continuous-powers argument includes $p=2$. The explicit carry example retains its separate odd-prime hypothesis.\n\\item Source inflation and coefficient reduction have opposite variance. A preliminary draft description treating both as unrestricted coarsenings was replaced by \\cref{prop:mixed-source-variance}; reversing inflation would require descent through the smaller quotient.\n\\end{enumerate}","}$ takes place in torsion-free integral matrices for necessity, or is translated into the stated lower-modulus congruence for a finite witness. It is not division by a unit in $R_n$.\n\\item The positive construction preserves the given reduction, but not extra determinant, ramification, or geometric conditions. The constructed matrices are units over $\\mathbb Z_p$, not necessarily unimodular over $\\mathbb Z$.\n\\item The continuous-powers argument includes $p=2$. The explicit carry example retains its separate odd-prime hypothesis.\n\\item Source inflation and coefficient reduction have opposite variance. A preliminary draft description treating both as unrestricted coarsenings was replaced by \\cref{prop:mixed-source-variance}; reversing inflation would require descent through the smaller quotient.\n\\end{enumerate}","ity, or is translated into the stated lower-modulus congruence for a finite witness. It is not division by a unit in $R_n$.\n\\item The positive construction preserves the given reduction, but not extra determinant, ramification, or geometric conditions. The constructed matrices are units over $\\mathbb Z_p$, not necessarily unimodular over $\\mathbb Z$.\n\\item The continuous-powers argument includes $p=2$. The explicit carry example retains its separate odd-prime hypothesis.\n\\item Source inflation and coefficient reduction have opposite variance. A preliminary draft description treating both as unrestricted coarsenings was replaced by \\cref{prop:mixed-source-variance}; reversing inflation would require descent through the smaller quotient.\n\\end{enumerate}","The source-inflation checks cover 495 cocycle triples, all 3,107 refined-quotient pairs for $p=3,5,7$, and 83 repaired matrix values. The centralizer checks cover 1,017 cyclic matrices and 982,827 candidate centralizer pairs in the explicitly listed finite rings and companion-matrix family. Complete rank-two tables contain all 10, 33, 832, and 29,889 valid continuous two-generator actions over $\\mathbb F_2$, $\\mathbb F_3$, $\\mathbb Z/4$, and $\\mathbb Z/9$, respectively. Each positive case is checked against an exactly commuting integral polynomial certificate.","flation checks cover 495 cocycle triples, all 3,107 refined-quotient pairs for $p=3,5,7$, and 83 repaired matrix values. The centralizer checks cover 1,017 cyclic matrices and 982,827 candidate centralizer pairs in the explicitly listed finite rings and companion-matrix family. Complete rank-two tables contain all 10, 33, 832, and 29,889 valid continuous two-generator actions over $\\mathbb F_2$, $\\mathbb F_3$, $\\mathbb Z/4$, and $\\mathbb Z/9$, respectively. Each positive case is checked against an exactly commuting integral polynomial certificate.","cover 495 cocycle triples, all 3,107 refined-quotient pairs for $p=3,5,7$, and 83 repaired matrix values. The centralizer checks cover 1,017 cyclic matrices and 982,827 candidate centralizer pairs in the explicitly listed finite rings and companion-matrix family. Complete rank-two tables contain all 10, 33, 832, and 29,889 valid continuous two-generator actions over $\\mathbb F_2$, $\\mathbb F_3$, $\\mathbb Z/4$, and $\\mathbb Z/9$, respectively. Each positive case is checked against an exactly commuting integral polynomial certificate.","riples, all 3,107 refined-quotient pairs for $p=3,5,7$, and 83 repaired matrix values. The centralizer checks cover 1,017 cyclic matrices and 982,827 candidate centralizer pairs in the explicitly listed finite rings and companion-matrix family. Complete rank-two tables contain all 10, 33, 832, and 29,889 valid continuous two-generator actions over $\\mathbb F_2$, $\\mathbb F_3$, $\\mathbb Z/4$, and $\\mathbb Z/9$, respectively. Each positive case is checked against an exactly commuting integral polynomial certificate.","An orthogonal enumeration of all 1,048,576 ordered pairs of admissible generator matrices modulo $8$ gives 59,392 commuting pairs. Their 664 distinct reductions modulo $4$ agree exactly with the positive set from the normalized criterion. All 4,096 rank-two tangent triples over $\\mathbb F_2$ are also checked. The sharpness suite includes 24 prime/precision families and all 72,353 correction pairs in three stated finite fixtures. Higher-rank checks use 54 constructed three-generator fixtures, not an exhaustive higher-rank population, with 162 recovered polynomial certificates and 648 finite continuity checks. Exact ranges and nonclaims are in \\texttt{checks/profinite\\_results.json}.","The included command-line tool \\texttt{tools/lift\\_abelian\\_rank2.py} validates the matrix input and returns either exact integral generators or a nonzero normalized-commutator certificate. It applies only to the unconstrained rank-two $\\mathbb Z_p^r$ problem. Its positive and negative examples are preserved in \\texttt{examples/}. The universal lifting and convergence statements follow from the written proofs, not from these bounded computations.","\\item[v0.11.0] Proves continuous one-step obstruction and source-inflation criteria, continuous tower reconstruction, and free pro-$p$ coefficient lifting. Gives a normalized cyclic-centralizer lifting criterion for $\\mathbb Z_p^r$, a complete rank-two test, and the sharp uniform horizon $2n-1$. Correctly separates source inflation from coefficient reduction by mixed variance. Adds exact positive and negative examples, a certificate CLI, and a sixth regression suite. Preserves the entire v0.10.0 package. Written proofs and self-review only.\n\\item[v0.10.0] Adds a complete $H^2$ next-level obstruction, cochain repair certificates, and a central-sign counterexample to the sufficiency of derived-invariant exchange. Gives a direct averaging proof of known finite-group ultrametric stability, an exact finite lifting horizon, con","\\item[v0.11.0] Proves continuous one-step obstruction and source-inflation criteria, continuous tower reconstruction, and free pro-$p$ coefficient lifting. Gives a normalized cyclic-centralizer lifting criterion for $\\mathbb Z_p^r$, a complete rank-two test, and the sharp uniform horizon $2n-1$. Correctly separates source inflation from coefficient reduction by mixed variance. Adds exact positive and negative examples, a certificate CLI, and a sixth regression suite. Preserves the entire v0.10.0 package. Written proofs and self-review only.\n\\item[v0.10.0] Adds a complete $H^2$ next-level obstruction, cochain repair certificates, and a central-sign counterexample to the sufficiency of derived-invariant exchange. Gives a direct averaging proof of known finite-group ultrametric stability, an exact finite lifting horizon, controlled conjugacy, a finite class catalogue, a cyclic Jacobian certificate, and an explicit profi","\\item[v0.11.0] Proves continuous one-step obstruction and source-inflation criteria, continuous tower reconstruction, and free pro-$p$ coefficient lifting. Gives a normalized cyclic-centralizer lifting criterion for $\\mathbb Z_p^r$, a complete rank-two test, and the sharp uniform horizon $2n-1$. Correctly separates source inflation from coefficient reduction by mixed variance. Adds exact positive and negative examples, a certificate CLI, and a sixth regression suite. Preserves the entire v0.10.0 package. Written proofs and self-review only.\n\\item[v0.10.0] Adds a complete $H^2$ next-level obstruction, cochain repair certificates, and a central-sign counterexample to the sufficiency of derived-invariant exchange. Gives a direct averaging proof of known finite-group ultrametric stability, an exact finite lifting horizon, controlled conjugacy, a finite class catalogue, a cyclic Jacobian certificate, and an explicit profinite scope boundary. Five exact suites pass. Preserves every inh","\\item[v0.10.0] Adds a complete $H^2$ next-level obstruction, cochain repair certificates, and a central-sign counterexample to the sufficiency of derived-invariant exchange. Gives a direct averaging proof of known finite-group ultrametric stability, an exact finite lifting horizon, controlled conjugacy, a finite class catalogue, a cyclic Jacobian certificate, and an explicit profinite scope boundary. Five exact suites pass. Preserves every inherited theorem/proof block and the full v0.9.0 package. Written proofs and self-review only.\n\\item[v0.9.0] Adds intrinsic kernel and invariant comparisons, explicit cohomological residuals, sharp precision-window lifting, equivariant morphism correction, and norm-quotient examples. Prove","\\item[v0.4.0] Adds a direct nonstable obstruction test bed: local sections of an abelian torsor produce overlap residuals, local changes of trivialization act as repairs, and the resulting \\v{C}ech $H^1$ class vanishes exactly when the local sections can be repaired to reconstruct a global section. This closes the first descent test-bed obligation and shows that the residual/repair/obstruction pattern recovers a classical cohomological mechanism.\n\\item[v0.3.0] Proves the stable residual filtration along a chosen normalization path by iterated octahedral cofiber sequences. Constructs the universal stable obstruction quotient by the thick closure of operationally killable residuals, proves exact detection under a directed common-annihilator hypothesis, and proves functorial transport of obstruction quotients under exact correspondences. K11 and K12 are closed in these stable forms; path-inde"]}