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The fine and coarse state spaces are linked by a surjective linear map $C$. Each homogeneous term in a local analytic vector field is resolved into packet interactions. Linear terms give ordinary directed edges, while higher order terms give directed hyperedges. We prove that a multilinear interaction of order $m$ descends exactly when its coarse output vanishes whenever any input lies in $\\Ker C$. For block sum aggregations, this becomes a direct coefficient test. Once the fine output coefficients are summed within a coarse fiber, the resulting value must depend only on the coarse input fibers.","arse description of a nonlinear system is exact rather than fitted after the fact. The fine and coarse state spaces are linked by a surjective linear map $C$. Each homogeneous term in a local analytic vector field is resolved into packet interactions. Linear terms give ordinary directed edges, while higher order terms give directed hyperedges. We prove that a multilinear interaction of order $m$ descends exactly when its coarse output vanishes whenever any input lies in $\\Ker C$. For block sum aggregations, this becomes a direct coefficient test. Once the fine output coefficients are summed within a coarse fiber, the resulting value must depend only on the coarse input fibers.","t. The fine and coarse state spaces are linked by a surjective linear map $C$. Each homogeneous term in a local analytic vector field is resolved into packet interactions. Linear terms give ordinary directed edges, while higher order terms give directed hyperedges. We prove that a multilinear interaction of order $m$ descends exactly when its coarse output vanishes whenever any input lies in $\\Ker C$. For block sum aggregations, this becomes a direct coefficient test. Once the fine output coefficients are summed within a coarse fiber, the resulting value must depend only on the coarse input fibers.","\\subsection{The quotient problem}\nLet $X_{\\fine}$ be a fine state space, let $X_{\\coarse}$ be a coarse state space, and let\n\\[\n C:X_{\\fine}\\twoheadrightarrow X_{\\coarse}\n\\]\nbe a surjective linear map. A fine trajectory satisfies\n\\begin{equation}\\label{eq:fine-flow-intro}\n \\dot x(t)=F_{\\fine}(x(t)),\n \\qquad x(0)=x_0,\n\\end{equation}\nand the coarse observable of interest is\n\\[\n z(t)=Cx(t).\n\\]\nThe first question is whether $z(t)$ obeys a closed equation of its own. Equivalently, can we define a vector field $F_{\\coarse}$ such that\n\\begin{equation}\\label{eq:vector-field-quotient-intro}\n C F_{\\fine}=F_{\\coarse}C?\n\\end{equation}\nWhen this identity holds and both initial value problems have unique solutions, aggregation commutes with evolu","\\subsection{The quotient problem}\nLet $X_{\\fine}$ be a fine state space, let $X_{\\coarse}$ be a coarse state space, and let\n\\[\n C:X_{\\fine}\\twoheadrightarrow X_{\\coarse}\n\\]\nbe a surjective linear map. A fine trajectory satisfies\n\\begin{equation}\\label{eq:fine-flow-intro}\n \\dot x(t)=F_{\\fine}(x(t)),\n \\qquad x(0)=x_0,\n\\end{equation}\nand the coarse observable of interest is\n\\[\n z(t)=Cx(t).\n\\]\nThe first question is whether $z(t)$ obeys a closed equation of its own. Equivalently, can we define a vector field $F_{\\coarse}$ such that\n\\begin{equation}\\label{eq:vector-field-quotient-intro}\n C F_{\\fine}=F_{\\coarse}C?\n\\end{equation}\nWhen this identity holds and both initial value problems have unique solutions, aggregation commutes with evolution:\n\\begin{equation}\\label{eq:flow-quot","\\subsection{The quotient problem}\nLet $X_{\\fine}$ be a fine state space, let $X_{\\coarse}$ be a coarse state space, and let\n\\[\n C:X_{\\fine}\\twoheadrightarrow X_{\\coarse}\n\\]\nbe a surjective linear map. A fine trajectory satisfies\n\\begin{equation}\\label{eq:fine-flow-intro}\n \\dot x(t)=F_{\\fine}(x(t)),\n \\qquad x(0)=x_0,\n\\end{equation}\nand the coarse observable of interest is\n\\[\n z(t)=Cx(t).\n\\]\nThe first question is whether $z(t)$ obeys a closed equation of its own. Equivalently, can we define a vector field $F_{\\coarse}$ such that\n\\begin{equation}\\label{eq:vector-field-quotient-intro}\n C F_{\\fine}=F_{\\coarse}C?\n\\end{equation}\nWhen this identity holds and both initial value problems have unique solutions, aggregation commutes with evolution:\n\\begin{equation}\\label{eq:flow-quotient-intro}\n C\\Phi_t^{\\fine}=\\Phi_t^{\\coarse}C.\n\\end{equation}\nThe coarse traject","\\subsection{The quotient problem}\nLet $X_{\\fine}$ be a fine state space, let $X_{\\coarse}$ be a coarse state space, and let\n\\[\n C:X_{\\fine}\\twoheadrightarrow X_{\\coarse}\n\\]\nbe a surjective linear map. A fine trajectory satisfies\n\\begin{equation}\\label{eq:fine-flow-intro}\n \\dot x(t)=F_{\\fine}(x(t)),\n \\qquad x(0)=x_0,\n\\end{equation}\nand the coarse observable of interest is\n\\[\n z(t)=Cx(t).\n\\]\nThe first question is whether $z(t)$ obeys a closed equation of its own. Equivalently, can we define a vector field $F_{\\coarse}$ such that\n\\begin{equation}\\label{eq:vector-field-quotient-intro}\n C F_{\\fine}=F_{\\coarse}C?\n\\end{equation}\nWhen this identity holds and both initial value problems have unique solutions, aggregation commutes with evolution:\n\\begin{equation}\\label{eq:flow-quotient-intro}\n C\\Phi_t^{\\fine}=\\Phi_t^{\\coarse}C.\n\\end{equation}\nThe coarse trajectory is then an exact quotient of the fine one. No fitting step is involved.","lem}\nLet $X_{\\fine}$ be a fine state space, let $X_{\\coarse}$ be a coarse state space, and let\n\\[\n C:X_{\\fine}\\twoheadrightarrow X_{\\coarse}\n\\]\nbe a surjective linear map. A fine trajectory satisfies\n\\begin{equation}\\label{eq:fine-flow-intro}\n \\dot x(t)=F_{\\fine}(x(t)),\n \\qquad x(0)=x_0,\n\\end{equation}\nand the coarse observable of interest is\n\\[\n z(t)=Cx(t).\n\\]\nThe first question is whether $z(t)$ obeys a closed equation of its own. Equivalently, can we define a vector field $F_{\\coarse}$ such that\n\\begin{equation}\\label{eq:vector-field-quotient-intro}\n C F_{\\fine}=F_{\\coarse}C?\n\\end{equation}\nWhen this identity holds and both initial value problems have unique solutions, aggregation commutes with evolution:\n\\begin{equation}\\label{eq:flow-quotient-intro}\n C\\Phi_t^{\\fine}=\\Phi_t^{\\coarse}C.\n\\end{equation}\nThe coarse trajectory is then an exact quotient of the fine one. No fitting step is involved.","and let\n\\[\n C:X_{\\fine}\\twoheadrightarrow X_{\\coarse}\n\\]\nbe a surjective linear map. A fine trajectory satisfies\n\\begin{equation}\\label{eq:fine-flow-intro}\n \\dot x(t)=F_{\\fine}(x(t)),\n \\qquad x(0)=x_0,\n\\end{equation}\nand the coarse observable of interest is\n\\[\n z(t)=Cx(t).\n\\]\nThe first question is whether $z(t)$ obeys a closed equation of its own. Equivalently, can we define a vector field $F_{\\coarse}$ such that\n\\begin{equation}\\label{eq:vector-field-quotient-intro}\n C F_{\\fine}=F_{\\coarse}C?\n\\end{equation}\nWhen this identity holds and both initial value problems have unique solutions, aggregation commutes with evolution:\n\\begin{equation}\\label{eq:flow-quotient-intro}\n C\\Phi_t^{\\fine}=\\Phi_t^{\\coarse}C.\n\\end{equation}\nThe coarse trajectory is then an exact quotient of the fine one. No fitting step is involved.","row X_{\\coarse}\n\\]\nbe a surjective linear map. A fine trajectory satisfies\n\\begin{equation}\\label{eq:fine-flow-intro}\n \\dot x(t)=F_{\\fine}(x(t)),\n \\qquad x(0)=x_0,\n\\end{equation}\nand the coarse observable of interest is\n\\[\n z(t)=Cx(t).\n\\]\nThe first question is whether $z(t)$ obeys a closed equation of its own. Equivalently, can we define a vector field $F_{\\coarse}$ such that\n\\begin{equation}\\label{eq:vector-field-quotient-intro}\n C F_{\\fine}=F_{\\coarse}C?\n\\end{equation}\nWhen this identity holds and both initial value problems have unique solutions, aggregation commutes with evolution:\n\\begin{equation}\\label{eq:flow-quotient-intro}\n C\\Phi_t^{\\fine}=\\Phi_t^{\\coarse}C.\n\\end{equation}\nThe coarse trajectory is then an exact quotient of the fine one. No fitting step is involved.","rest is\n\\[\n z(t)=Cx(t).\n\\]\nThe first question is whether $z(t)$ obeys a closed equation of its own. Equivalently, can we define a vector field $F_{\\coarse}$ such that\n\\begin{equation}\\label{eq:vector-field-quotient-intro}\n C F_{\\fine}=F_{\\coarse}C?\n\\end{equation}\nWhen this identity holds and both initial value problems have unique solutions, aggregation commutes with evolution:\n\\begin{equation}\\label{eq:flow-quotient-intro}\n C\\Phi_t^{\\fine}=\\Phi_t^{\\coarse}C.\n\\end{equation}\nThe coarse trajectory is then an exact quotient of the fine one. No fitting step is involved.","\\subsection{Why hypergraphs appear}\nSuppose a local analytic vector field has the expansion\n\\begin{equation}\\label{eq:analytic-expansion-intro}\n F_{\\fine}(x)=\\sum_{m\\geq0}A_m^{\\fine}[x,\\ldots,x].\n\\end{equation}\nThe linear coefficient $A_1^{\\fine}$ moves one input packet into one output packet. The quadratic coefficient $A_2^{\\fine}$ combines two input packets before producing an output. In general, $A_m^{\\fine}$ combines $m$ inputs. We record this incidence by the directed hyperedge\n\\[\n (j_1,\\ldots,j_m)\\longrightarrow i.\n\\]","ction{Why hypergraphs appear}\nSuppose a local analytic vector field has the expansion\n\\begin{equation}\\label{eq:analytic-expansion-intro}\n F_{\\fine}(x)=\\sum_{m\\geq0}A_m^{\\fine}[x,\\ldots,x].\n\\end{equation}\nThe linear coefficient $A_1^{\\fine}$ moves one input packet into one output packet. The quadratic coefficient $A_2^{\\fine}$ combines two input packets before producing an output. In general, $A_m^{\\fine}$ combines $m$ inputs. We record this incidence by the directed hyperedge\n\\[\n (j_1,\\ldots,j_m)\\longrightarrow i.\n\\]","ppear}\nSuppose a local analytic vector field has the expansion\n\\begin{equation}\\label{eq:analytic-expansion-intro}\n F_{\\fine}(x)=\\sum_{m\\geq0}A_m^{\\fine}[x,\\ldots,x].\n\\end{equation}\nThe linear coefficient $A_1^{\\fine}$ moves one input packet into one output packet. The quadratic coefficient $A_2^{\\fine}$ combines two input packets before producing an output. In general, $A_m^{\\fine}$ combines $m$ inputs. We record this incidence by the directed hyperedge\n\\[\n (j_1,\\ldots,j_m)\\longrightarrow i.\n\\]","egin{equation}\\label{eq:analytic-expansion-intro}\n F_{\\fine}(x)=\\sum_{m\\geq0}A_m^{\\fine}[x,\\ldots,x].\n\\end{equation}\nThe linear coefficient $A_1^{\\fine}$ moves one input packet into one output packet. The quadratic coefficient $A_2^{\\fine}$ combines two input packets before producing an output. In general, $A_m^{\\fine}$ combines $m$ inputs. We record this incidence by the directed hyperedge\n\\[\n (j_1,\\ldots,j_m)\\longrightarrow i.\n\\]","At order $m$, the central question is\n\\begin{equation}\\label{eq:arity-question}\n \\exists A_m^{\\coarse}\n \\quad\\text{such that}\\quad\n C A_m^{\\fine}=A_m^{\\coarse}C^{\\times m}?\n\\end{equation}\nIf this question has an answer at every order, the quotient relation can be carried from individual interactions to the full field, its derivatives, its interaction trees, and its flow.","\\subsection{Why Bayesian observations add another nonlinear operation}\nPrediction is only one part of a filter. Given a probability vector $\\mu$ and a positive likelihood $\\lambda$, Bayesian conditioning is\n\\begin{equation}\\label{eq:bayes-intro}\n \\Bup_{\\lambda}(\\mu)\n =\n \\frac{\\lambda\\odot\\mu}{\\lambda^T\\mu}.\n\\end{equation}\nLikelihood multiplication is linear, but the evidence in the denominator couples all packets. The full update is therefore nonlinear even when the likelihood acts separately on each state.","Classical sufficiency theory asks whether a statistic preserves all information relevant to a chosen observation family \\cite{HalmosSavage1949,Blackwell1953}. Here the operator question is\n\\begin{equation}\\label{eq:bayes-question}\n C\\Bup_{\\lambda^{\\fine}}\n =\n \\Bup_{\\lambda^{\\coarse}}C.\n\\end{equation}\nWhen both the flow identity \\eqref{eq:flow-quotient-intro} and the Bayesian identity \\eqref{eq:bayes-question} hold at each observation time, the whole filtering sequence commutes with aggregation. Hidden Markov filtering is the linear unary case \\cite{BaumPetrie1966}. The same argument applies when the prediction step comes from a nonlinear quotient flow.","\\begin{enumerate}[label=\\textbf{C\\arabic*.},leftmargin=3.5em]\n\\item \\textbf{A quotient criterion for every interaction order.}\nFor a surjective map $C$, a multilinear fine interaction descends exactly when $CA_m^{\\fine}$ vanishes whenever one input lies in $\\Ker C$. Equivalently, its linearization on $X_{\\fine}^{\\otimes m}$ factors through $C^{\\otimes m}$.","The proof strategy can be summarized by the chain\n\\[\n\\boxed{\n\\begin{gathered}\n \\text{multilinear quotient}\n \\Longrightarrow\n \\text{interaction tree quotient}\n \\Longrightarrow\n \\text{flow quotient}\n \\\\\n \\Longrightarrow\n \\text{Bayesian quotient}\n \\Longrightarrow\n \\text{filtering cocycle quotient}\n \\Longrightarrow\n \\text{posterior certificate}.\n\\end{gathered}}\n\\]","\\subsection{Finite packet resolutions}\nThe main algebraic results are stated for vector spaces of finite dimension over $\\R$ or $\\C$. The quotient and tensor factorizations are then purely algebraic. The later analytic results add the stated convergence conditions and the usual assumptions for local uniqueness.","\\begin{definition}[Packet resolution]\\label{def:packet-resolution}\nA packet resolution of a vector space $X$ is a finite family of projections $(P_i)_{i\\in I}$ satisfying\n\\[\n P_iP_j=\\delta_{ij}P_i,\n \\qquad\n \\sum_{i\\in I}P_i=I_X.\n\\]\nThus\n\\[\n X=\\bigoplus_{i\\in I}X_i,\n \\qquad X_i=P_iX.\n\\]\nWhen $X$ is a Hilbert space, the principal norm identities will assume that the $P_i$ are orthogonal.\n\\end{definition}","Let $(P_j)_{j\\in J}$ resolve a source $X$ and $(Q_i)_{i\\in I}$ resolve a target $Y$.","\\begin{definition}[Resolved multilinear block and hyperedge]\\label{def:resolved-block}\nFor a multilinear map $A:X^m\\to Y$ of order $m$, define\n\\begin{equation}\\label{eq:resolved-block}\n A_{i\\mid j_1,\\ldots,j_m}\n =\n Q_iA(P_{j_1}\\,\\cdot,\\ldots,P_{j_m}\\,\\cdot).\n\\end{equation}\nThe ordered incidence\n\\[\n (j_1,\\ldots,j_m)\\longrightarrow i\n\\]\nis the corresponding directed interaction hyperedge. We call the hyperedge active when the block in \\eqref{eq:resolved-block} is nonzero.\n\\end{definition}","When $m=1$, the active blocks form an ordinary directed transfer graph. When $m=2$, each active block joins a pair of input packets. For general $m$, the block records one interaction with $m$ inputs.","\\begin{theorem}[Exact multilinear packet reconstruction]\\label{thm:packet-reconstruction}\nEvery multilinear map $A:X^m\\to Y$ of order $m$ satisfies\n\\begin{equation}\\label{eq:packet-reconstruction}\n A(x_1,\\ldots,x_m)\n =\n \\sum_{i,j_1,\\ldots,j_m}\n A_{i\\mid j_1,\\ldots,j_m}\n (P_{j_1}x_1,\\ldots,P_{j_m}x_m).\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nInsert $I_X=\\sum_jP_j$ in every input slot and $I_Y=\\sum_iQ_i$ after applying $A$. Multilinearity distributes all finite sums and gives \\eqref{eq:packet-reconstruction}.\n\\end{proof}","\\subsection{Coarse maps, sections, and hidden directions}\nLet\n\\[\n C:X_{\\fine}\\twoheadrightarrow X_{\\coarse}\n\\]\nbe a surjective linear coarse map. Finite dimensionality guarantees a linear right inverse\n\\[\n E:X_{\\coarse}\\to X_{\\fine},\n \\qquad CE=I_{X_{\\coarse}}.\n\\]\nDefine\n\\begin{equation}\\label{eq:representative-hidden-proj}\n P_C=EC,\n \\qquad\n Q_C=I_{X_{\\fine}}-EC.\n\\end{equation}\nThen\n\\[\n P_C^2=P_C,\n \\qquad\n Q_C^2=Q_C,\n \\qquad\n \\Ran Q_C=\\Ker C.\n\\]\nEvery fine state has the exact decomposition\n\\begin{equation}\\label{eq:representative-hidden-decomposition}\n x=ECx+Q_Cx.\n\\end{equation}\nThe first term is the chosen fine representative of the coarse state. The second lies in $\\Ker C$, the hidden subspace. Exact quotienting means that motion inside this subspace cannot change the aggregated in","\\subsection{Coarse maps, sections, and hidden directions}\nLet\n\\[\n C:X_{\\fine}\\twoheadrightarrow X_{\\coarse}\n\\]\nbe a surjective linear coarse map. Finite dimensionality guarantees a linear right inverse\n\\[\n E:X_{\\coarse}\\to X_{\\fine},\n \\qquad CE=I_{X_{\\coarse}}.\n\\]\nDefine\n\\begin{equation}\\label{eq:representative-hidden-proj}\n P_C=EC,\n \\qquad\n Q_C=I_{X_{\\fine}}-EC.\n\\end{equation}\nThen\n\\[\n P_C^2=P_C,\n \\qquad\n Q_C^2=Q_C,\n \\qquad\n \\Ran Q_C=\\Ker C.\n\\]\nEvery fine state has the exact decomposition\n\\begin{equation}\\label{eq:representative-hidden-decomposition}\n x=ECx+Q_Cx.\n\\end{equation}\nThe first term is the chosen fine representative of the coarse state. The second lies in $\\Ker C$, the hidden subspace. Exact quotienting means that motion inside this subspace cannot change the aggregated interaction output.","tive linear coarse map. Finite dimensionality guarantees a linear right inverse\n\\[\n E:X_{\\coarse}\\to X_{\\fine},\n \\qquad CE=I_{X_{\\coarse}}.\n\\]\nDefine\n\\begin{equation}\\label{eq:representative-hidden-proj}\n P_C=EC,\n \\qquad\n Q_C=I_{X_{\\fine}}-EC.\n\\end{equation}\nThen\n\\[\n P_C^2=P_C,\n \\qquad\n Q_C^2=Q_C,\n \\qquad\n \\Ran Q_C=\\Ker C.\n\\]\nEvery fine state has the exact decomposition\n\\begin{equation}\\label{eq:representative-hidden-decomposition}\n x=ECx+Q_Cx.\n\\end{equation}\nThe first term is the chosen fine representative of the coarse state. The second lies in $\\Ker C$, the hidden subspace. Exact quotienting means that motion inside this subspace cannot change the aggregated interaction output.","n}\\label{eq:representative-hidden-proj}\n P_C=EC,\n \\qquad\n Q_C=I_{X_{\\fine}}-EC.\n\\end{equation}\nThen\n\\[\n P_C^2=P_C,\n \\qquad\n Q_C^2=Q_C,\n \\qquad\n \\Ran Q_C=\\Ker C.\n\\]\nEvery fine state has the exact decomposition\n\\begin{equation}\\label{eq:representative-hidden-decomposition}\n x=ECx+Q_Cx.\n\\end{equation}\nThe first term is the chosen fine representative of the coarse state. The second lies in $\\Ker C$, the hidden subspace. Exact quotienting means that motion inside this subspace cannot change the aggregated interaction output.","\\begin{definition}[Compatible packet aggregation]\\label{def:packet-aggregation}\nSuppose $(P_i)_{i\\in I_{\\fine}}$ and $(\\overline P_a)_{a\\in I_{\\coarse}}$ resolve the fine and coarse spaces. A surjection\n\\[\n \\pi:I_{\\fine}\\twoheadrightarrow I_{\\coarse}\n\\]\nis compatible with $C$ when\n\\begin{equation}\\label{eq:packet-aggregation-compatibility}\n \\overline P_aC\n =\n C\\sum_{\\pi(i)=a}P_i\n \\qquad(a\\in I_{\\coarse}).\n\\end{equation}\n\\end{definition}","In the scalar coordinate model used later, each $P_i$ selects one coordinate and $C$ sums all coordinates in the same fiber of $\\pi$.","\\begin{definition}[Exact interaction quotient]\\label{def:exact-interaction-quotient}\nLet $A_m^{\\fine}:X_{\\fine}^m\\to X_{\\fine}$ and $A_m^{\\coarse}:X_{\\coarse}^m\\to X_{\\coarse}$ be multilinear maps of order $m$. We say that $A_m^{\\fine}$ \\emph{descends through $C$} to $A_m^{\\coarse}$, and write\n\\[\n A_m^{\\fine}\\Downarrow_C A_m^{\\coarse},\n\\]\nwhen\n\\begin{equation}\\label{eq:exact-interaction-quotient}\n C A_m^{\\fine}\n =\n A_m^{\\coarse}C^{\\times m}.\n\\end{equation}\nFor $m=0$, descent means $CA_0^{\\fine}=A_0^{\\coarse}$.\n\\end{definition}","ion-quotient}\nLet $A_m^{\\fine}:X_{\\fine}^m\\to X_{\\fine}$ and $A_m^{\\coarse}:X_{\\coarse}^m\\to X_{\\coarse}$ be multilinear maps of order $m$. We say that $A_m^{\\fine}$ \\emph{descends through $C$} to $A_m^{\\coarse}$, and write\n\\[\n A_m^{\\fine}\\Downarrow_C A_m^{\\coarse},\n\\]\nwhen\n\\begin{equation}\\label{eq:exact-interaction-quotient}\n C A_m^{\\fine}\n =\n A_m^{\\coarse}C^{\\times m}.\n\\end{equation}\nFor $m=0$, descent means $CA_0^{\\fine}=A_0^{\\coarse}$.\n\\end{definition}","m^{\\fine}:X_{\\fine}^m\\to X_{\\fine}$ and $A_m^{\\coarse}:X_{\\coarse}^m\\to X_{\\coarse}$ be multilinear maps of order $m$. We say that $A_m^{\\fine}$ \\emph{descends through $C$} to $A_m^{\\coarse}$, and write\n\\[\n A_m^{\\fine}\\Downarrow_C A_m^{\\coarse},\n\\]\nwhen\n\\begin{equation}\\label{eq:exact-interaction-quotient}\n C A_m^{\\fine}\n =\n A_m^{\\coarse}C^{\\times m}.\n\\end{equation}\nFor $m=0$, descent means $CA_0^{\\fine}=A_0^{\\coarse}$.\n\\end{definition}","Here\n\\[\n C^{\\times m}(x_1,\\ldots,x_m)\n =(Cx_1,\\ldots,Cx_m).\n\\]\nWe write $C^{\\otimes m}$ for the induced linear map between algebraic tensor powers.","\\begin{definition}[Interaction defect]\\label{def:interaction-defect}\nFor a proposed coarse tensor $B_m$, define\n\\begin{equation}\\label{eq:interaction-defect}\n \\Delta_m[B_m]\n =\n CA_m^{\\fine}-B_mC^{\\times m}.\n\\end{equation}\nExact descent is the condition $\\Delta_m[B_m]=0$.\n\\end{definition}","\\subsection{Norm conventions}\nFor a multilinear map $A:X^m\\to Y$ of order $m$, let\n\\[\n \\norm{A}\n =\n \\sup_{\\norm{x_r}\\leq1}\n \\norm{A(x_1,\\ldots,x_m)}.\n\\]\nIf $E$ is a fixed right inverse of $C$, then any exact coarse tensor necessarily satisfies\n\\begin{equation}\\label{eq:coarse-tensor-bound-preview}\n A_m^{\\coarse}(z_1,\\ldots,z_m)\n =\n CA_m^{\\fine}(Ez_1,\\ldots,Ez_m),\n\\end{equation}\nso\n\\[\n \\norm{A_m^{\\coarse}}\n \\leq\n \\norm{C}\\norm{A_m^{\\fine}}\\norm{E}^m.\n\\]\nThe formula also makes the later uniqueness argument immediate.","\\begin{theorem}[Universal multilinear quotient criterion]\\label{thm:universal-quotient}\nLet $C:X_{\\fine}\\twoheadrightarrow X_{\\coarse}$ be surjective and let\n\\[\n A_m^{\\fine}:X_{\\fine}^m\\to X_{\\fine}\n\\]\nbe a multilinear map of order $m$, where $m\\geq1$. The following statements are equivalent.","\\begin{enumerate}[label=(\\roman*),leftmargin=2.8em]\n\\item There exists a multilinear map $A_m^{\\coarse}:X_{\\coarse}^m\\to X_{\\coarse}$ of order $m$ such that\n\\[\n CA_m^{\\fine}=A_m^{\\coarse}C^{\\times m}.\n\\]","\\item For every slot $r\\in\\{1,\\ldots,m\\}$,\n\\begin{equation}\\label{eq:slot-kernel-condition}\n CA_m^{\\fine}(x_1,\\ldots,x_{r-1},k,x_{r+1},\\ldots,x_m)=0\n\\end{equation}\nwhenever $k\\in\\Ker C$.","\\item The map $CA_m^{\\fine}$ is constant on products of fibers: if $Cx_r=Cx_r'$ for all $r$, then\n\\begin{equation}\\label{eq:fiber-constancy-abstract}\n CA_m^{\\fine}(x_1,\\ldots,x_m)\n =\n CA_m^{\\fine}(x_1',\\ldots,x_m').\n\\end{equation}","\\item Let $\\widehat A_m^{\\fine}:X_{\\fine}^{\\otimes m}\\to X_{\\coarse}$ be the linearization of $CA_m^{\\fine}$. Then\n\\begin{equation}\\label{eq:tensor-kernel-condition}\n \\Ker(C^{\\otimes m})\n \\subseteq\n \\Ker(\\widehat A_m^{\\fine}).\n\\end{equation}\n\\end{enumerate}","When these conditions hold, the coarse tensor is unique and, for every right inverse $E$ of $C$,\n\\begin{equation}\\label{eq:coarse-via-section}\n A_m^{\\coarse}(z_1,\\ldots,z_m)\n =\n CA_m^{\\fine}(Ez_1,\\ldots,Ez_m).\n\\end{equation}\nThe right side is independent of the chosen right inverse.\n\\end{theorem}","\\begin{proof}\nAssume (i). If $k\\in\\Ker C$, then inserting $k$ into any input slot gives\n\\[\n CA_m^{\\fine}(x_1,\\ldots,k,\\ldots,x_m)\n =\n A_m^{\\coarse}(Cx_1,\\ldots,0,\\ldots,Cx_m)=0,\n\\]\nso (ii) holds.","Assume (ii). If $Cx_r=Cx_r'$, then $x_r'-x_r\\in\\Ker C$. Replace the inputs one at a time. Multilinearity and \\eqref{eq:slot-kernel-condition} show that each replacement leaves $CA_m^{\\fine}$ unchanged. Hence (iii) holds.","Assume (iii). For $z_r\\in X_{\\coarse}$ choose arbitrary lifts $x_r$ with $Cx_r=z_r$ and set\n\\[\n A_m^{\\coarse}(z_1,\\ldots,z_m)\n =\n CA_m^{\\fine}(x_1,\\ldots,x_m).\n\\]\nFiber constancy makes the definition independent of the lifts. Multilinearity is inherited from $A_m^{\\fine}$, and the defining identity gives (i). Surjectivity of $C$ also gives uniqueness: every coarse input tuple has a fine lift.","To compare (ii) and (iv), identify a multilinear map of order $m$ with its linearization on the algebraic tensor product. Since $C$ is surjective,\n\\[\n \\Ker(C^{\\otimes m})\n =\n \\sum_{r=1}^m\n X_{\\fine}^{\\otimes(r-1)}\\otimes\\Ker C\\otimes\n X_{\\fine}^{\\otimes(m-r)}.\n\\]\nChoose a splitting $X_{\\fine}=E(X_{\\coarse})\\oplus\\Ker C$ and expand the tensor power. Vanishing of $\\widehat A_m^{\\fine}$ on this sum is exactly the slot condition. Thus (ii) and (iv) are equivalent. Formula \\eqref{eq:coarse-via-section} follows by choosing $x_r=Ez_r$.\n\\end{proof}","is surjective,\n\\[\n \\Ker(C^{\\otimes m})\n =\n \\sum_{r=1}^m\n X_{\\fine}^{\\otimes(r-1)}\\otimes\\Ker C\\otimes\n X_{\\fine}^{\\otimes(m-r)}.\n\\]\nChoose a splitting $X_{\\fine}=E(X_{\\coarse})\\oplus\\Ker C$ and expand the tensor power. Vanishing of $\\widehat A_m^{\\fine}$ on this sum is exactly the slot condition. Thus (ii) and (iv) are equivalent. Formula \\eqref{eq:coarse-via-section} follows by choosing $x_r=Ez_r$.\n\\end{proof}","\\begin{remark}[Nonlinear congruence interpretation]\\label{rem:nonlinear-congruence}\nFor a general map $F:X_{\\fine}\\to X_{\\fine}$, descent through $C$ is equivalent to\n\\[\n Cx=Cy\n \\quad\\Longrightarrow\\quad\n CF(x)=CF(y).\n\\]\nThus the relation $x\\sim_Cy$, defined by $Cx=Cy$, must be a congruence for the coarse output of $F$. \\Cref{thm:universal-quotient} turns that statement into a condition on each multilinear coefficient.\n\\end{remark}","\\begin{corollary}[Bounded coarse tensor]\\label{cor:bounded-coarse-tensor}\nUnder the hypotheses of \\Cref{thm:universal-quotient}, for every bounded right inverse $E$,\n\\[\n \\norm{A_m^{\\coarse}}\n \\leq\n \\norm{C}\\norm{A_m^{\\fine}}\\norm{E}^m.\n\\]\n\\end{corollary}","Let\n\\[\n I_{\\fine}=\\{1,\\ldots,n\\},\n \\qquad\n I_{\\coarse}=\\{1,\\ldots,q\\},\n\\]\nand let\n\\[\n \\pi:I_{\\fine}\\twoheadrightarrow I_{\\coarse}.\n\\]\nWrite $(e_i)$ and $(f_a)$ for the standard bases of $\\R^n$ and $\\R^q$. Define\n\\begin{equation}\\label{eq:block-sum-C}\n Ce_i=f_{\\pi(i)},\n \\qquad\n (Cx)_a=\\sum_{\\pi(i)=a}x_i.\n\\end{equation}\nSuppose\n\\begin{equation}\\label{eq:fine-coordinates}\n A_m^{\\fine}(e_{j_1},\\ldots,e_{j_m})\n =\n \\sum_{i=1}^n\n a^{\\fine,(m)}_{i;j_1,\\ldots,j_m}e_i.\n\\end{equation}\nFor each coarse output packet $a$, sum the fine coefficients over its output fiber and define\n\\begin{equation}\\label{eq:aggregated-coefficient}\n g^{(m)}_{a;j_1,\\ldots,j_m}\n =\n \\sum_{i:\\pi(i)=a}\n a^{\\fine,(m)}_{i;j_1,\\ldots,j_m}.\n\\end{equation}","I_{\\coarse}.\n\\]\nWrite $(e_i)$ and $(f_a)$ for the standard bases of $\\R^n$ and $\\R^q$. Define\n\\begin{equation}\\label{eq:block-sum-C}\n Ce_i=f_{\\pi(i)},\n \\qquad\n (Cx)_a=\\sum_{\\pi(i)=a}x_i.\n\\end{equation}\nSuppose\n\\begin{equation}\\label{eq:fine-coordinates}\n A_m^{\\fine}(e_{j_1},\\ldots,e_{j_m})\n =\n \\sum_{i=1}^n\n a^{\\fine,(m)}_{i;j_1,\\ldots,j_m}e_i.\n\\end{equation}\nFor each coarse output packet $a$, sum the fine coefficients over its output fiber and define\n\\begin{equation}\\label{eq:aggregated-coefficient}\n g^{(m)}_{a;j_1,\\ldots,j_m}\n =\n \\sum_{i:\\pi(i)=a}\n a^{\\fine,(m)}_{i;j_1,\\ldots,j_m}.\n\\end{equation}","$\\R^q$. Define\n\\begin{equation}\\label{eq:block-sum-C}\n Ce_i=f_{\\pi(i)},\n \\qquad\n (Cx)_a=\\sum_{\\pi(i)=a}x_i.\n\\end{equation}\nSuppose\n\\begin{equation}\\label{eq:fine-coordinates}\n A_m^{\\fine}(e_{j_1},\\ldots,e_{j_m})\n =\n \\sum_{i=1}^n\n a^{\\fine,(m)}_{i;j_1,\\ldots,j_m}e_i.\n\\end{equation}\nFor each coarse output packet $a$, sum the fine coefficients over its output fiber and define\n\\begin{equation}\\label{eq:aggregated-coefficient}\n g^{(m)}_{a;j_1,\\ldots,j_m}\n =\n \\sum_{i:\\pi(i)=a}\n a^{\\fine,(m)}_{i;j_1,\\ldots,j_m}.\n\\end{equation}","\\begin{theorem}[Coefficient criterion for every interaction order]\\label{thm:packet-coordinate-criterion}\nThe fine tensor $A_m^{\\fine}$ descends through the block sum map \\eqref{eq:block-sum-C} if and only if, for every coarse output $a$,\n\\begin{equation}\\label{eq:packet-fiber-constancy}\n g^{(m)}_{a;j_1,\\ldots,j_m}\n =\n g^{(m)}_{a;j_1',\\ldots,j_m'}\n\\end{equation}\nwhenever\n\\[\n \\pi(j_r)=\\pi(j_r')\n \\qquad(r=1,\\ldots,m).\n\\]\nWhen this condition holds, the unique coarse coefficients are\n\\begin{equation}\\label{eq:coarse-coordinate-formula}\n a^{\\coarse,(m)}_{a;b_1,\\ldots,b_m}\n =\n g^{(m)}_{a;j_1,\\ldots,j_m},\n \\qquad\n \\pi(j_r)=b_r.\n\\end{equation}\n\\end{theorem}","The fine tensor $A_m^{\\fine}$ descends through the block sum map \\eqref{eq:block-sum-C} if and only if, for every coarse output $a$,\n\\begin{equation}\\label{eq:packet-fiber-constancy}\n g^{(m)}_{a;j_1,\\ldots,j_m}\n =\n g^{(m)}_{a;j_1',\\ldots,j_m'}\n\\end{equation}\nwhenever\n\\[\n \\pi(j_r)=\\pi(j_r')\n \\qquad(r=1,\\ldots,m).\n\\]\nWhen this condition holds, the unique coarse coefficients are\n\\begin{equation}\\label{eq:coarse-coordinate-formula}\n a^{\\coarse,(m)}_{a;b_1,\\ldots,b_m}\n =\n g^{(m)}_{a;j_1,\\ldots,j_m},\n \\qquad\n \\pi(j_r)=b_r.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nApplying $C$ to \\eqref{eq:fine-coordinates} gives\n\\[\n CA_m^{\\fine}(e_{j_1},\\ldots,e_{j_m})\n =\n \\sum_{a=1}^q\n g^{(m)}_{a;j_1,\\ldots,j_m}f_a.\n\\]\nIf a coarse tensor exists, this vector equals\n\\[\n A_m^{\\coarse}(f_{\\pi(j_1)},\\ldots,f_{\\pi(j_m)}),\n\\]\nwhich depends only on the coarse input labels. This proves necessity. Conversely, constancy on each fiber makes \\eqref{eq:coarse-coordinate-formula} well defined. Equality on every tuple of fine basis vectors then gives\n\\[\n CA_m^{\\fine}=A_m^{\\coarse}C^{\\times m}\n\\]\nby multilinearity.\n\\end{proof}",",\\ldots,e_{j_m})\n =\n \\sum_{a=1}^q\n g^{(m)}_{a;j_1,\\ldots,j_m}f_a.\n\\]\nIf a coarse tensor exists, this vector equals\n\\[\n A_m^{\\coarse}(f_{\\pi(j_1)},\\ldots,f_{\\pi(j_m)}),\n\\]\nwhich depends only on the coarse input labels. This proves necessity. Conversely, constancy on each fiber makes \\eqref{eq:coarse-coordinate-formula} well defined. Equality on every tuple of fine basis vectors then gives\n\\[\n CA_m^{\\fine}=A_m^{\\coarse}C^{\\times m}\n\\]\nby multilinearity.\n\\end{proof}","For $m=2$, the criterion has two useful views. The hypergraph records which input fibers interact, while the coefficient matrix shows whether those interactions are constant on each product fiber. The exact synthetic tensor used in the certificate run has the required block structure to floating point precision.","\\begin{corollary}[Unary aggregate rate criterion]\\label{cor:unary-criterion}\nFor $m=1$, exact descent is equivalent to\n\\begin{equation}\\label{eq:unary-aggregate-rate}\n \\sum_{i:\\pi(i)=a}a^{\\fine,(1)}_{i;j}\n =\n \\sum_{i:\\pi(i)=a}a^{\\fine,(1)}_{i;j'}\n\\end{equation}\nfor all coarse outputs $a$ whenever $\\pi(j)=\\pi(j')$.\n\\end{corollary}","\\begin{proof}\nSet $m=1$ in \\Cref{thm:packet-coordinate-criterion}. The condition then involves one input fiber and becomes \\eqref{eq:unary-aggregate-rate}.\n\\end{proof}","Let $Q_{\\fine}$ be the generator of a continuous time Markov chain, using the column vector convention. The left side of \\eqref{eq:unary-aggregate-rate} is the total transition rate from fine state $j$ into coarse packet $a$. Corollary~\\ref{cor:unary-criterion} is therefore the standard criterion for strong lumpability.","\\begin{corollary}[Quadratic hypergraph criterion]\\label{cor:quadratic-criterion}\nFor $m=2$, exact descent is equivalent to\n\\[\n \\sum_{i:\\pi(i)=a}a^{\\fine,(2)}_{i;j,k}\n =\n \\sum_{i:\\pi(i)=a}a^{\\fine,(2)}_{i;j',k'}\n\\]\nwhenever\n\\[\n \\pi(j)=\\pi(j'),\n \\qquad\n \\pi(k)=\\pi(k').\n\\]\nThe resulting coarse interaction is the directed quadratic hyperedge\n\\[\n (\\pi(j),\\pi(k))\\longrightarrow a.\n\\]\n\\end{corollary}","\\begin{proof}\nSet $m=2$ in \\Cref{thm:packet-coordinate-criterion}. Fiber constancy in the two input slots gives the displayed equality, and \\eqref{eq:coarse-coordinate-formula} assigns its common value to the coarse hyperedge.\n\\end{proof}","\\begin{example}[A quotient with invisible internal interactions]\\label{ex:invisible-interactions}\nLet $X_{\\coarse}=\\R^q$, $X_{\\fine}=\\R^{2q}$, and\n\\[\n C=\\begin{bmatrix}I_q&I_q\\end{bmatrix},\n \\qquad\n E=\\frac12\\begin{bmatrix}I_q","\\begin{example}[A quotient with invisible internal interactions]\\label{ex:invisible-interactions}\nLet $X_{\\coarse}=\\R^q$, $X_{\\fine}=\\R^{2q}$, and\n\\[\n C=\\begin{bmatrix}I_q&I_q\\end{bmatrix},\n \\qquad\n E=\\frac12\\begin{bmatrix}I_q\\\\I_q\\end{bmatrix}.\n\\]\nLet $A_m^{\\coarse}$ be arbitrary and let $K_m:X_{\\fine}^m\\to X_{\\fine}$ satisfy\n\\[\n CK_m=0.\n\\]\nThen\n\\begin{equation}\\label{eq:exact-lift-construction}\n A_m^{\\fine}\n =\n EA_m^{\\coarse}C^{\\times m}+K_m\n\\end{equation}\nsatisfies\n\\[\n CA_m^{\\fine}=A_m^{\\coarse}C^{\\times m}.\n\\]\nThe tensor $K_m$ may contain many fine hyperedges, so the fine model can remain complicated. Exact quotienting asks only that every hidden contribution disappear after applying $C$.\n\\end{example}","I_q\\end{bmatrix}.\n\\]\nLet $A_m^{\\coarse}$ be arbitrary and let $K_m:X_{\\fine}^m\\to X_{\\fine}$ satisfy\n\\[\n CK_m=0.\n\\]\nThen\n\\begin{equation}\\label{eq:exact-lift-construction}\n A_m^{\\fine}\n =\n EA_m^{\\coarse}C^{\\times m}+K_m\n\\end{equation}\nsatisfies\n\\[\n CA_m^{\\fine}=A_m^{\\coarse}C^{\\times m}.\n\\]\nThe tensor $K_m$ may contain many fine hyperedges, so the fine model can remain complicated. Exact quotienting asks only that every hidden contribution disappear after applying $C$.\n\\end{example}","^m\\to X_{\\fine}$ satisfy\n\\[\n CK_m=0.\n\\]\nThen\n\\begin{equation}\\label{eq:exact-lift-construction}\n A_m^{\\fine}\n =\n EA_m^{\\coarse}C^{\\times m}+K_m\n\\end{equation}\nsatisfies\n\\[\n CA_m^{\\fine}=A_m^{\\coarse}C^{\\times m}.\n\\]\nThe tensor $K_m$ may contain many fine hyperedges, so the fine model can remain complicated. Exact quotienting asks only that every hidden contribution disappear after applying $C$.\n\\end{example}","\\begin{proposition}[Symmetry descends]\\label{prop:symmetry-descends}\nIf $A_m^{\\fine}$ is symmetric in its input slots and descends through $C$, then the unique coarse tensor $A_m^{\\coarse}$ is symmetric.\n\\end{proposition}","\\begin{proof}\nChoose lifts of the coarse inputs. Permuting the coarse inputs permutes their lifts, and symmetry of $A_m^{\\fine}$ leaves \\eqref{eq:coarse-via-section} unchanged.\n\\end{proof}","\\begin{proposition}[Quotient support]\\label{prop:quotient-support}\nUnder the coordinate criterion, every active coarse hyperedge\n\\[\n (b_1,\\ldots,b_m)\\longrightarrow a\n\\]\nis supported by at least one fine hyperedge whose input and output labels map to that coarse incidence. The converse can fail because several active fine hyperedges may cancel after aggregation.\n\\end{proposition}","\\subsection{Composition across refinement levels}\nLet\n\\[\n X_2\\xrightarrow{C_{21}}X_1\\xrightarrow{C_{10}}X_0,\n \\qquad\n C_{20}=C_{10}C_{21}.\n\\]","\\begin{proposition}[Transitivity of exact interaction quotients]\\label{prop:quotient-transitivity}\nIf\n\\[\n A_m^{(2)}\\Downarrow_{C_{21}}A_m^{(1)}\n \\qquad\\text{and}\\qquad\n A_m^{(1)}\\Downarrow_{C_{10}}A_m^{(0)},\n\\]\nthen\n\\[\n A_m^{(2)}\\Downarrow_{C_{20}}A_m^{(0)}.\n\\]\n\\end{proposition}","\\begin{proof}\nCompose the two identities:\n\\[\n C_{20}A_m^{(2)}\n =C_{10}A_m^{(1)}C_{21}^{\\times m}\n =A_m^{(0)}C_{10}^{\\times m}C_{21}^{\\times m}\n =A_m^{(0)}C_{20}^{\\times m}.\n\\]\n\\end{proof}","\\begin{theorem}[Interaction defect across two levels]\\label{thm:multilevel-defect}\nDefine\n\\[\n \\Delta_m^{21}\n =C_{21}A_m^{(2)}-A_m^{(1)}C_{21}^{\\times m},\n\\]\n\\[\n \\Delta_m^{10}\n =C_{10}A_m^{(1)}-A_m^{(0)}C_{10}^{\\times m},\n\\]\nand\n\\[\n \\Delta_m^{20}\n =C_{20}A_m^{(2)}-A_m^{(0)}C_{20}^{\\times m}.\n\\]\nThen\n\\begin{equation}\\label{eq:multilevel-defect}\n \\boxed{\n \\Delta_m^{20}\n =\n C_{10}\\Delta_m^{21}\n +\n \\Delta_m^{10}C_{21}^{\\times m}.}\n\\end{equation}\nConsequently,\n\\begin{equation}\\label{eq:multilevel-defect-norm}\n \\norm{\\Delta_m^{20}}\n \\leq\n \\norm{C_{10}}\\norm{\\Delta_m^{21}}\n +\n \\norm{\\Delta_m^{10}}\\norm{C_{21}}^m.\n\\end{equation}\n\\end{theorem}","ne\n\\[\n \\Delta_m^{21}\n =C_{21}A_m^{(2)}-A_m^{(1)}C_{21}^{\\times m},\n\\]\n\\[\n \\Delta_m^{10}\n =C_{10}A_m^{(1)}-A_m^{(0)}C_{10}^{\\times m},\n\\]\nand\n\\[\n \\Delta_m^{20}\n =C_{20}A_m^{(2)}-A_m^{(0)}C_{20}^{\\times m}.\n\\]\nThen\n\\begin{equation}\\label{eq:multilevel-defect}\n \\boxed{\n \\Delta_m^{20}\n =\n C_{10}\\Delta_m^{21}\n +\n \\Delta_m^{10}C_{21}^{\\times m}.}\n\\end{equation}\nConsequently,\n\\begin{equation}\\label{eq:multilevel-defect-norm}\n \\norm{\\Delta_m^{20}}\n \\leq\n \\norm{C_{10}}\\norm{\\Delta_m^{21}}\n +\n \\norm{\\Delta_m^{10}}\\norm{C_{21}}^m.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nAdd and subtract $C_{10}A_m^{(1)}C_{21}^{\\times m}$:\n\\begin{align*}\n\\Delta_m^{20}\n&=C_{10}C_{21}A_m^{(2)}-A_m^{(0)}C_{10}^{\\times m}C_{21}^{\\times m}","\\begin{proof}\nAdd and subtract $C_{10}A_m^{(1)}C_{21}^{\\times m}$:\n\\begin{align*}\n\\Delta_m^{20}\n&=C_{10}C_{21}A_m^{(2)}-A_m^{(0)}C_{10}^{\\times m}C_{21}^{\\times m}\\\\\n&=C_{10}\\bigl(C_{21}A_m^{(2)}-A_m^{(1)}C_{21}^{\\times m}\\bigr)\\\\\n&\\quad+\n\\bigl(C_{10}A_m^{(1)}-A_m^{(0)}C_{10}^{\\times m}\\bigr)C_{21}^{\\times m}.\n\\end{align*}\nThis proves \\eqref{eq:multilevel-defect}. Taking norms and using submultiplicativity yields the estimate.\n\\end{proof}","\\subsection{Descent of polynomial and analytic vector fields}\nLet\n\\begin{equation}\\label{eq:fine-analytic-field}\n F_{\\fine}(x)\n =\n \\sum_{m=0}^{\\infty}A_m^{\\fine}[x^{\\otimes m}]\n\\end{equation}\nbe locally analytic, where $A_m^{\\fine}$ is a symmetric multilinear map of order $m$ for $m\\geq1$ and $A_0^{\\fine}\\in X_{\\fine}$. Throughout,\n\\[\n A_m[x^{\\otimes m}]\n =A_m(x,\\ldots,x).\n\\]","\\begin{theorem}[Analytic interaction descent]\\label{thm:analytic-descent}\nLet $C:X_{\\fine}\\twoheadrightarrow X_{\\coarse}$ have a bounded right inverse $E$. Suppose the coefficient series converges normally on every smaller ball,\n\\begin{equation}\\label{eq:fine-normal-summability}\n \\sum_{m=0}^{\\infty}\\norm{A_m^{\\fine}}r^m<\\infty\n \\qquad(0\\leq r0$.\n\\end{definition}","\\begin{theorem}[Naturality of normalized transfer]\\label{thm:normalized-transfer-naturality}\nLet $L_{\\fine}:X_{\\fine}\\to X_{\\fine}$ and $L_{\\coarse}:X_{\\coarse}\\to X_{\\coarse}$ be positive linear operators. Suppose\n\\begin{equation}\\label{eq:positive-intertwining}\n CL_{\\fine}=sL_{\\coarse}C\n\\end{equation}\nfor some $s>0$, and suppose \\eqref{eq:mass-preserving-C} holds. Then, on the common domain of normalization,\n\\begin{equation}\\label{eq:normalized-transfer-naturality}\n \\boxed{\n C\\Nrm_{\\ell_{\\fine}}(L_{\\fine})\n =\n \\Nrm_{\\ell_{\\coarse}}(L_{\\coarse})C.}\n\\end{equation}\n\\end{theorem}","malized transfer]\\label{thm:normalized-transfer-naturality}\nLet $L_{\\fine}:X_{\\fine}\\to X_{\\fine}$ and $L_{\\coarse}:X_{\\coarse}\\to X_{\\coarse}$ be positive linear operators. Suppose\n\\begin{equation}\\label{eq:positive-intertwining}\n CL_{\\fine}=sL_{\\coarse}C\n\\end{equation}\nfor some $s>0$, and suppose \\eqref{eq:mass-preserving-C} holds. Then, on the common domain of normalization,\n\\begin{equation}\\label{eq:normalized-transfer-naturality}\n \\boxed{\n C\\Nrm_{\\ell_{\\fine}}(L_{\\fine})\n =\n \\Nrm_{\\ell_{\\coarse}}(L_{\\coarse})C.}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nFor a positive state $x$,\n\\begin{align*}\nC\\Nrm_{\\ell_{\\fine}}(L_{\\fine})(x)\n&=\n\\frac{CL_{\\fine}x}{\\ell_{\\fine}(L_{\\fine}x)}","\\begin{proof}\nFor a positive state $x$,\n\\begin{align*}\nC\\Nrm_{\\ell_{\\fine}}(L_{\\fine})(x)\n&=\n\\frac{CL_{\\fine}x}{\\ell_{\\fine}(L_{\\fine}x)}\\\\\n&=\n\\frac{sL_{\\coarse}Cx}{\\ell_{\\coarse}(CL_{\\fine}x)}\\\\\n&=\n\\frac{sL_{\\coarse}Cx}{s\\ell_{\\coarse}(L_{\\coarse}Cx)}\\\\\n&=\n\\Nrm_{\\ell_{\\coarse}}(L_{\\coarse})(Cx).\n\\end{align*}\n\\end{proof}","\\begin{remark}[Gauge invariance]\nNormalization removes the scale $s$. At the operator level, this is the familiar fact that a positive rescaling of a likelihood leaves the posterior unchanged.\n\\end{remark}","\\begin{proposition}[Composition law for normalized transfers]\\label{prop:normalized-composition}\nFor positive operators $L_1,L_2$ on the same ordered state space,\n\\begin{equation}\\label{eq:normalized-composition}\n \\Nrm_{\\ell}(L_2)\n \\circ\n \\Nrm_{\\ell}(L_1)\n =\n \\Nrm_{\\ell}(L_2L_1)\n\\end{equation}\nwherever both sides are defined.\n\\end{proposition}","\\subsection{Classical Bayesian update}\nOn the finite simplex, let\n\\[\n L_{\\lambda}=\\diag(\\lambda_1,\\ldots,\\lambda_n),\n \\qquad\n \\lambda_i>0.\n\\]\nWith $\\ell(x)=\\one^Tx$,\n\\[\n \\Nrm_{\\ell}(L_{\\lambda})(\\mu)\n =\n \\frac{\\lambda\\odot\\mu}{\\lambda^T\\mu}\n =\\Bup_{\\lambda}(\\mu).\n\\]\nThe evidence is\n\\[\n Z_{\\lambda}(\\mu)=\\lambda^T\\mu.\n\\]\nThe composition law becomes\n\\begin{equation}\\label{eq:bayes-composition}\n \\Bup_{\\lambda_2}\\circ\\Bup_{\\lambda_1}\n =\n \\Bup_{\\lambda_1\\odot\\lambda_2}.\n\\end{equation}","\\subsection{Exact Bayesian refinement}\nLet $C$ be the block sum aggregation associated with a surjection $\\pi:\\{1,\\ldots,n\\}\\twoheadrightarrow\\{1,\\ldots,q\\}$.","\\begin{theorem}[Sufficient condition for Bayesian refinement]\\label{thm:bayes-sufficient}\nLet $\\lambda^{\\fine}\\in\\R_{>0}^n$ and $\\lambda^{\\coarse}\\in\\R_{>0}^q$. If\n\\begin{equation}\\label{eq:fiber-likelihood-sufficient}\n \\lambda^{\\fine}=sC^T\\lambda^{\\coarse}\n\\end{equation}\nfor some $s>0$, then\n\\begin{equation}\\label{eq:exact-bayes-refinement}\n \\boxed{\n C\\Bup_{\\lambda^{\\fine}}\n =\n \\Bup_{\\lambda^{\\coarse}}C.}\n\\end{equation}\nMoreover,\n\\begin{equation}\\label{eq:evidence-scaling}\n Z_{\\lambda^{\\fine}}(\\mu)\n =\n sZ_{\\lambda^{\\coarse}}(C\\mu).\n\\end{equation}\n\\end{theorem}","ent]\\label{thm:bayes-sufficient}\nLet $\\lambda^{\\fine}\\in\\R_{>0}^n$ and $\\lambda^{\\coarse}\\in\\R_{>0}^q$. If\n\\begin{equation}\\label{eq:fiber-likelihood-sufficient}\n \\lambda^{\\fine}=sC^T\\lambda^{\\coarse}\n\\end{equation}\nfor some $s>0$, then\n\\begin{equation}\\label{eq:exact-bayes-refinement}\n \\boxed{\n C\\Bup_{\\lambda^{\\fine}}\n =\n \\Bup_{\\lambda^{\\coarse}}C.}\n\\end{equation}\nMoreover,\n\\begin{equation}\\label{eq:evidence-scaling}\n Z_{\\lambda^{\\fine}}(\\mu)\n =\n sZ_{\\lambda^{\\coarse}}(C\\mu).\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nEquation \\eqref{eq:fiber-likelihood-sufficient} is equivalent to\n\\[\n CL_{\\lambda^{\\fine}}\n =sL_{\\lambda^{\\coarse}}C.\n\\]\nApply \\Cref{thm:normalized-transfer-naturality}. The evidence identity follows from\n\\[\n (\\lambda^{\\fine})^T\\mu\n =s(\\lambda^{\\coarse})^TC\\mu.\n\\]\n\\end{proof}","\\begin{theorem}[Necessary condition for Bayesian refinement]\\label{thm:bayes-necessity}\nAssume $q\\geq2$, every fiber of $\\pi$ is nonempty, and $\\lambda^{\\fine},\\lambda^{\\coarse}$ are strictly positive. If\n\\begin{equation}\\label{eq:bayes-equality-all-priors}\n C\\Bup_{\\lambda^{\\fine}}(\\mu)\n =\n \\Bup_{\\lambda^{\\coarse}}(C\\mu)\n\\end{equation}\nfor every $\\mu$ in the interior of $\\Delta_n$, then there exists $s>0$ such that\n\\[\n \\lambda^{\\fine}=sC^T\\lambda^{\\coarse}.\n\\]\n\\end{theorem}","heorem}[Necessary condition for Bayesian refinement]\\label{thm:bayes-necessity}\nAssume $q\\geq2$, every fiber of $\\pi$ is nonempty, and $\\lambda^{\\fine},\\lambda^{\\coarse}$ are strictly positive. If\n\\begin{equation}\\label{eq:bayes-equality-all-priors}\n C\\Bup_{\\lambda^{\\fine}}(\\mu)\n =\n \\Bup_{\\lambda^{\\coarse}}(C\\mu)\n\\end{equation}\nfor every $\\mu$ in the interior of $\\Delta_n$, then there exists $s>0$ such that\n\\[\n \\lambda^{\\fine}=sC^T\\lambda^{\\coarse}.\n\\]\n\\end{theorem}","condition for Bayesian refinement]\\label{thm:bayes-necessity}\nAssume $q\\geq2$, every fiber of $\\pi$ is nonempty, and $\\lambda^{\\fine},\\lambda^{\\coarse}$ are strictly positive. If\n\\begin{equation}\\label{eq:bayes-equality-all-priors}\n C\\Bup_{\\lambda^{\\fine}}(\\mu)\n =\n \\Bup_{\\lambda^{\\coarse}}(C\\mu)\n\\end{equation}\nfor every $\\mu$ in the interior of $\\Delta_n$, then there exists $s>0$ such that\n\\[\n \\lambda^{\\fine}=sC^T\\lambda^{\\coarse}.\n\\]\n\\end{theorem}","\\begin{proof}\nBoth sides of \\eqref{eq:bayes-equality-all-priors} extend continuously to any prior with nonzero evidence. Strict positivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets $a$ and $b$. Choose $i\\in\\pi^{-1}(a)$ and $k\\in\\pi^{-1}(b)$, then consider a prior supported on $\\{i,k\\}$ with masses $u,v>0$. On the fine side, the ratio of the posterior masses of packets $a$ and $b$ is\n\\[\n \\frac{\\lambda_i^{\\fine}u}{\\lambda_k^{\\fine}v},\n\\]\nwhereas the coarse posterior ratio is\n\\[\n \\frac{\\lambda_a^{\\coarse}u}{\\lambda_b^{\\coarse}v}.\n\\]\nEquality for all $u,v>0$ gives\n\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The","\\begin{proof}\nBoth sides of \\eqref{eq:bayes-equality-all-priors} extend continuously to any prior with nonzero evidence. Strict positivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets $a$ and $b$. Choose $i\\in\\pi^{-1}(a)$ and $k\\in\\pi^{-1}(b)$, then consider a prior supported on $\\{i,k\\}$ with masses $u,v>0$. On the fine side, the ratio of the posterior masses of packets $a$ and $b$ is\n\\[\n \\frac{\\lambda_i^{\\fine}u}{\\lambda_k^{\\fine}v},\n\\]\nwhereas the coarse posterior ratio is\n\\[\n \\frac{\\lambda_a^{\\coarse}u}{\\lambda_b^{\\coarse}v}.\n\\]\nEquality for all $u,v>0$ gives\n\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine lik","\\begin{proof}\nBoth sides of \\eqref{eq:bayes-equality-all-priors} extend continuously to any prior with nonzero evidence. Strict positivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets $a$ and $b$. Choose $i\\in\\pi^{-1}(a)$ and $k\\in\\pi^{-1}(b)$, then consider a prior supported on $\\{i,k\\}$ with masses $u,v>0$. On the fine side, the ratio of the posterior masses of packets $a$ and $b$ is\n\\[\n \\frac{\\lambda_i^{\\fine}u}{\\lambda_k^{\\fine}v},\n\\]\nwhereas the coarse posterior ratio is\n\\[\n \\frac{\\lambda_a^{\\coarse}u}{\\lambda_b^{\\coarse}v}.\n\\]\nEquality for all $u,v>0$ gives\n\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore consta","\\begin{proof}\nBoth sides of \\eqref{eq:bayes-equality-all-priors} extend continuously to any prior with nonzero evidence. Strict positivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets $a$ and $b$. Choose $i\\in\\pi^{-1}(a)$ and $k\\in\\pi^{-1}(b)$, then consider a prior supported on $\\{i,k\\}$ with masses $u,v>0$. On the fine side, the ratio of the posterior masses of packets $a$ and $b$ is\n\\[\n \\frac{\\lambda_i^{\\fine}u}{\\lambda_k^{\\fine}v},\n\\]\nwhereas the coarse posterior ratio is\n\\[\n \\frac{\\lambda_a^{\\coarse}u}{\\lambda_b^{\\coarse}v}.\n\\]\nEquality for all $u,v>0$ gives\n\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Wri","\\begin{proof}\nBoth sides of \\eqref{eq:bayes-equality-all-priors} extend continuously to any prior with nonzero evidence. Strict positivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets $a$ and $b$. Choose $i\\in\\pi^{-1}(a)$ and $k\\in\\pi^{-1}(b)$, then consider a prior supported on $\\{i,k\\}$ with masses $u,v>0$. On the fine side, the ratio of the posterior masses of packets $a$ and $b$ is\n\\[\n \\frac{\\lambda_i^{\\fine}u}{\\lambda_k^{\\fine}v},\n\\]\nwhereas the coarse posterior ratio is\n\\[\n \\frac{\\lambda_a^{\\coarse}u}{\\lambda_b^{\\coarse}v}.\n\\]\nEquality for all $u,v>0$ gives\n\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda","\\begin{proof}\nBoth sides of \\eqref{eq:bayes-equality-all-priors} extend continuously to any prior with nonzero evidence. Strict positivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets $a$ and $b$. Choose $i\\in\\pi^{-1}(a)$ and $k\\in\\pi^{-1}(b)$, then consider a prior supported on $\\{i,k\\}$ with masses $u,v>0$. On the fine side, the ratio of the posterior masses of packets $a$ and $b$ is\n\\[\n \\frac{\\lambda_i^{\\fine}u}{\\lambda_k^{\\fine}v},\n\\]\nwhereas the coarse posterior ratio is\n\\[\n \\frac{\\lambda_a^{\\coarse}u}{\\lambda_b^{\\coarse}v}.\n\\]\nEquality for all $u,v>0$ gives\n\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda_a$. Equation \\eqre","of \\eqref{eq:bayes-equality-all-priors} extend continuously to any prior with nonzero evidence. Strict positivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets $a$ and $b$. Choose $i\\in\\pi^{-1}(a)$ and $k\\in\\pi^{-1}(b)$, then consider a prior supported on $\\{i,k\\}$ with masses $u,v>0$. On the fine side, the ratio of the posterior masses of packets $a$ and $b$ is\n\\[\n \\frac{\\lambda_i^{\\fine}u}{\\lambda_k^{\\fine}v},\n\\]\nwhereas the coarse posterior ratio is\n\\[\n \\frac{\\lambda_a^{\\coarse}u}{\\lambda_b^{\\coarse}v}.\n\\]\nEquality for all $u,v>0$ gives\n\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda_a$. Equation \\eqref{eq:likelihood-ratio-necessity} now gives\n\\[\n \\frac{\\overline\\lamb…","f{eq:bayes-equality-all-priors} extend continuously to any prior with nonzero evidence. Strict positivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets $a$ and $b$. Choose $i\\in\\pi^{-1}(a)$ and $k\\in\\pi^{-1}(b)$, then consider a prior supported on $\\{i,k\\}$ with masses $u,v>0$. On the fine side, the ratio of the posterior masses of packets $a$ and $b$ is\n\\[\n \\frac{\\lambda_i^{\\fine}u}{\\lambda_k^{\\fine}v},\n\\]\nwhereas the coarse posterior ratio is\n\\[\n \\frac{\\lambda_a^{\\coarse}u}{\\lambda_b^{\\coarse}v}.\n\\]\nEquality for all $u,v>0$ gives\n\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda_a$. Equation \\eqref{eq:likelihood-ratio-necessity} now gives\n\\[\n \\frac{\\overline\\lambda_a}{\\o…","es-equality-all-priors} extend continuously to any prior with nonzero evidence. Strict positivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets $a$ and $b$. Choose $i\\in\\pi^{-1}(a)$ and $k\\in\\pi^{-1}(b)$, then consider a prior supported on $\\{i,k\\}$ with masses $u,v>0$. On the fine side, the ratio of the posterior masses of packets $a$ and $b$ is\n\\[\n \\frac{\\lambda_i^{\\fine}u}{\\lambda_k^{\\fine}v},\n\\]\nwhereas the coarse posterior ratio is\n\\[\n \\frac{\\lambda_a^{\\coarse}u}{\\lambda_b^{\\coarse}v}.\n\\]\nEquality for all $u,v>0$ gives\n\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda_a$. Equation \\eqref{eq:likelihood-ratio-necessity} now gives\n\\[\n \\frac{\\overline\\lambda_a}{\\overline\\…","ivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets $a$ and $b$. Choose $i\\in\\pi^{-1}(a)$ and $k\\in\\pi^{-1}(b)$, then consider a prior supported on $\\{i,k\\}$ with masses $u,v>0$. On the fine side, the ratio of the posterior masses of packets $a$ and $b$ is\n\\[\n \\frac{\\lambda_i^{\\fine}u}{\\lambda_k^{\\fine}v},\n\\]\nwhereas the coarse posterior ratio is\n\\[\n \\frac{\\lambda_a^{\\coarse}u}{\\lambda_b^{\\coarse}v}.\n\\]\nEquality for all $u,v>0$ gives\n\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda_a$. Equation \\eqref{eq:likelihood-ratio-necessity} now gives\n\\[\n \\frac{\\overline\\lambda_a}{\\overline\\lambda_b}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}\n\\]\nfor every pair $a,b$. Hen…","istinct coarse packets $a$ and $b$. Choose $i\\in\\pi^{-1}(a)$ and $k\\in\\pi^{-1}(b)$, then consider a prior supported on $\\{i,k\\}$ with masses $u,v>0$. On the fine side, the ratio of the posterior masses of packets $a$ and $b$ is\n\\[\n \\frac{\\lambda_i^{\\fine}u}{\\lambda_k^{\\fine}v},\n\\]\nwhereas the coarse posterior ratio is\n\\[\n \\frac{\\lambda_a^{\\coarse}u}{\\lambda_b^{\\coarse}v}.\n\\]\nEquality for all $u,v>0$ gives\n\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda_a$. Equation \\eqref{eq:likelihood-ratio-necessity} now gives\n\\[\n \\frac{\\overline\\lambda_a}{\\overline\\lambda_b}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}\n\\]\nfor every pair $a,b$. Hence $\\overline\\lambda=s\\lambda^{\\coarse}$ for one $s>0$, proving the result.…","d $b$. Choose $i\\in\\pi^{-1}(a)$ and $k\\in\\pi^{-1}(b)$, then consider a prior supported on $\\{i,k\\}$ with masses $u,v>0$. On the fine side, the ratio of the posterior masses of packets $a$ and $b$ is\n\\[\n \\frac{\\lambda_i^{\\fine}u}{\\lambda_k^{\\fine}v},\n\\]\nwhereas the coarse posterior ratio is\n\\[\n \\frac{\\lambda_a^{\\coarse}u}{\\lambda_b^{\\coarse}v}.\n\\]\nEquality for all $u,v>0$ gives\n\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda_a$. Equation \\eqref{eq:likelihood-ratio-necessity} now gives\n\\[\n \\frac{\\overline\\lambda_a}{\\overline\\lambda_b}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}\n\\]\nfor every pair $a,b$. Hence $\\overline\\lambda=s\\lambda^{\\coarse}$ for one $s>0$, proving the result.\n\\end{proof}","of packets $a$ and $b$ is\n\\[\n \\frac{\\lambda_i^{\\fine}u}{\\lambda_k^{\\fine}v},\n\\]\nwhereas the coarse posterior ratio is\n\\[\n \\frac{\\lambda_a^{\\coarse}u}{\\lambda_b^{\\coarse}v}.\n\\]\nEquality for all $u,v>0$ gives\n\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda_a$. Equation \\eqref{eq:likelihood-ratio-necessity} now gives\n\\[\n \\frac{\\overline\\lambda_a}{\\overline\\lambda_b}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}\n\\]\nfor every pair $a,b$. Hence $\\overline\\lambda=s\\lambda^{\\coarse}$ for one $s>0$, proving the result.\n\\end{proof}","s\n\\[\n \\frac{\\lambda_i^{\\fine}u}{\\lambda_k^{\\fine}v},\n\\]\nwhereas the coarse posterior ratio is\n\\[\n \\frac{\\lambda_a^{\\coarse}u}{\\lambda_b^{\\coarse}v}.\n\\]\nEquality for all $u,v>0$ gives\n\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda_a$. Equation \\eqref{eq:likelihood-ratio-necessity} now gives\n\\[\n \\frac{\\overline\\lambda_a}{\\overline\\lambda_b}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}\n\\]\nfor every pair $a,b$. Hence $\\overline\\lambda=s\\lambda^{\\coarse}$ for one $s>0$, proving the result.\n\\end{proof}","mbda_k^{\\fine}v},\n\\]\nwhereas the coarse posterior ratio is\n\\[\n \\frac{\\lambda_a^{\\coarse}u}{\\lambda_b^{\\coarse}v}.\n\\]\nEquality for all $u,v>0$ gives\n\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda_a$. Equation \\eqref{eq:likelihood-ratio-necessity} now gives\n\\[\n \\frac{\\overline\\lambda_a}{\\overline\\lambda_b}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}\n\\]\nfor every pair $a,b$. Hence $\\overline\\lambda=s\\lambda^{\\coarse}$ for one $s>0$, proving the result.\n\\end{proof}","\\fine}v},\n\\]\nwhereas the coarse posterior ratio is\n\\[\n \\frac{\\lambda_a^{\\coarse}u}{\\lambda_b^{\\coarse}v}.\n\\]\nEquality for all $u,v>0$ gives\n\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda_a$. Equation \\eqref{eq:likelihood-ratio-necessity} now gives\n\\[\n \\frac{\\overline\\lambda_a}{\\overline\\lambda_b}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}\n\\]\nfor every pair $a,b$. Hence $\\overline\\lambda=s\\lambda^{\\coarse}$ for one $s>0$, proving the result.\n\\end{proof}","\\]\nwhereas the coarse posterior ratio is\n\\[\n \\frac{\\lambda_a^{\\coarse}u}{\\lambda_b^{\\coarse}v}.\n\\]\nEquality for all $u,v>0$ gives\n\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda_a$. Equation \\eqref{eq:likelihood-ratio-necessity} now gives\n\\[\n \\frac{\\overline\\lambda_a}{\\overline\\lambda_b}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}\n\\]\nfor every pair $a,b$. Hence $\\overline\\lambda=s\\lambda^{\\coarse}$ for one $s>0$, proving the result.\n\\end{proof}","$ gives\n\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda_a$. Equation \\eqref{eq:likelihood-ratio-necessity} now gives\n\\[\n \\frac{\\overline\\lambda_a}{\\overline\\lambda_b}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}\n\\]\nfor every pair $a,b$. Hence $\\overline\\lambda=s\\lambda^{\\coarse}$ for one $s>0$, proving the result.\n\\end{proof}","\\begin{equation}\\label{eq:likelihood-ratio-necessity}\n \\frac{\\lambda_i^{\\fine}}{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda_a$. Equation \\eqref{eq:likelihood-ratio-necessity} now gives\n\\[\n \\frac{\\overline\\lambda_a}{\\overline\\lambda_b}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}\n\\]\nfor every pair $a,b$. Hence $\\overline\\lambda=s\\lambda^{\\coarse}$ for one $s>0$, proving the result.\n\\end{proof}","{\\lambda_k^{\\fine}}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}.\n\\end{equation}\nTake another state $j$ in the same fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda_a$. Equation \\eqref{eq:likelihood-ratio-necessity} now gives\n\\[\n \\frac{\\overline\\lambda_a}{\\overline\\lambda_b}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}\n\\]\nfor every pair $a,b$. Hence $\\overline\\lambda=s\\lambda^{\\coarse}$ for one $s>0$, proving the result.\n\\end{proof}","me fiber as $i$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda_a$. Equation \\eqref{eq:likelihood-ratio-necessity} now gives\n\\[\n \\frac{\\overline\\lambda_a}{\\overline\\lambda_b}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}\n\\]\nfor every pair $a,b$. Hence $\\overline\\lambda=s\\lambda^{\\coarse}$ for one $s>0$, proving the result.\n\\end{proof}","$. Applying the same argument to $j$ and $k$ gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda_a$. Equation \\eqref{eq:likelihood-ratio-necessity} now gives\n\\[\n \\frac{\\overline\\lambda_a}{\\overline\\lambda_b}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}\n\\]\nfor every pair $a,b$. Hence $\\overline\\lambda=s\\lambda^{\\coarse}$ for one $s>0$, proving the result.\n\\end{proof}","gives $\\lambda_j^{\\fine}=\\lambda_i^{\\fine}$. The fine likelihood is therefore constant on each fiber. Write its value on fiber $a$ as $\\overline\\lambda_a$. Equation \\eqref{eq:likelihood-ratio-necessity} now gives\n\\[\n \\frac{\\overline\\lambda_a}{\\overline\\lambda_b}\n =\n \\frac{\\lambda_a^{\\coarse}}{\\lambda_b^{\\coarse}}\n\\]\nfor every pair $a,b$. Hence $\\overline\\lambda=s\\lambda^{\\coarse}$ for one $s>0$, proving the result.\n\\end{proof}","\\begin{remark}[The case of one coarse packet]\nIf $q=1$, both sides of \\eqref{eq:bayes-equality-all-priors} equal the unique coarse state, regardless of the fine likelihood. Necessity therefore requires a nontrivial coarse state space.\n\\end{remark}","\\begin{proposition}[Bayesian differential]\\label{prop:bayes-differential}\nAt an interior prior $\\mu$,\n\\begin{equation}\\label{eq:bayes-differential}\n D\\Bup_{\\lambda}(\\mu)[h]\n =\n \\frac{\\lambda\\odot h}{Z_{\\lambda}(\\mu)}\n -\n \\Bup_{\\lambda}(\\mu)\n \\frac{\\lambda^Th}{Z_{\\lambda}(\\mu)}.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nDifferentiate the quotient $L_{\\lambda}\\mu/(\\one^TL_{\\lambda}\\mu)$.\n\\end{proof}","\\begin{corollary}[Bayesian jet naturality]\\label{cor:bayes-jet-naturality}\nUnder the hypotheses of \\Cref{thm:bayes-sufficient}, for every $r\\geq1$ and every interior fine prior,\n\\begin{equation}\\label{eq:bayes-jet-naturality}\n C D^r\\Bup_{\\lambda^{\\fine}}(\\mu)\n [h_1,\\ldots,h_r]\n =\n D^r\\Bup_{\\lambda^{\\coarse}}(C\\mu)\n [Ch_1,\\ldots,Ch_r].\n\\end{equation}\n\\end{corollary}","\\subsection{Finite observation sequences}\nLet $0=t_00).\n\\]\nThen\n\\begin{equation}\\label{eq:step-evidence-factor}\n z_k^{\\fine}=s_kz_k^{\\coarse}\n\\end{equation}\nfor every step, and the likelihood of the observation sequence satisfies\n\\begin{equation}\\label{eq:total-evidence-factor}\n \\boxed{\n Z_{1:N}^{\\fine}\n =\n \\left(\\prod_{k=1}^Ns_k\\right)\n Z_{1:N}^{\\coarse}.}\n\\end{equation}\nUnder the gauge $s_k=1$, the fine and coarse sequential evidences agree exactly.\n\\end{theorem}",":evidence-factorization}\nAssume the hypotheses of \\Cref{thm:exact-cocycle}, let\n\\[\n \\mu_0^{\\coarse}=C\\mu_0^{\\fine},\n\\]\nand suppose\n\\[\n \\lambda_{y_k}^{\\fine}\n =s_kC^T\\lambda_{y_k}^{\\coarse}\n \\qquad(s_k>0).\n\\]\nThen\n\\begin{equation}\\label{eq:step-evidence-factor}\n z_k^{\\fine}=s_kz_k^{\\coarse}\n\\end{equation}\nfor every step, and the likelihood of the observation sequence satisfies\n\\begin{equation}\\label{eq:total-evidence-factor}\n \\boxed{\n Z_{1:N}^{\\fine}\n =\n \\left(\\prod_{k=1}^Ns_k\\right)\n Z_{1:N}^{\\coarse}.}\n\\end{equation}\nUnder the gauge $s_k=1$, the fine and coarse sequential evidences agree exactly.\n\\end{theorem}","{\\fine}\n =s_kC^T\\lambda_{y_k}^{\\coarse}\n \\qquad(s_k>0).\n\\]\nThen\n\\begin{equation}\\label{eq:step-evidence-factor}\n z_k^{\\fine}=s_kz_k^{\\coarse}\n\\end{equation}\nfor every step, and the likelihood of the observation sequence satisfies\n\\begin{equation}\\label{eq:total-evidence-factor}\n \\boxed{\n Z_{1:N}^{\\fine}\n =\n \\left(\\prod_{k=1}^Ns_k\\right)\n Z_{1:N}^{\\coarse}.}\n\\end{equation}\nUnder the gauge $s_k=1$, the fine and coarse sequential evidences agree exactly.\n\\end{theorem}","\\begin{proof}\nInitial compatibility gives $C\\mu_0^{\\fine}=\\mu_0^{\\coarse}$. For later steps, \\Cref{cor:exact-posterior} gives the same identity after every update. Flow naturality therefore yields\n\\[\n C\\widehat\\mu_k^{\\fine}=\\widehat\\mu_k^{\\coarse}.\n\\]\nTherefore\n\\[\n z_k^{\\fine}\n =s_k(\\lambda_{y_k}^{\\coarse})^TC\\widehat\\mu_k^{\\fine}\n =s_kz_k^{\\coarse}.\n\\]\nThe likelihood of the complete observation sequence is the product of the conditional evidence terms. Multiplying the identities gives \\eqref{eq:total-evidence-factor}.\n\\end{proof}","\\begin{corollary}[Hidden Markov lumpability]\\label{cor:hmm-lumpability}\nLet $M_{\\fine}$ and $M_{\\coarse}$ be column stochastic matrices satisfying\n\\[\n CM_{\\fine}=M_{\\coarse}C,\n\\]\nand let every observation likelihood satisfy\n\\[\n \\lambda_y^{\\fine}=s_yC^T\\lambda_y^{\\coarse}.\n\\]\nThen, for every observation sequence $y_{1:N}$,\n\\begin{align*}\n&C\\bigl(\n\\Bup_{\\lambda_{y_N}^{\\fine}}\\circ M_{\\fine}\n\\circ\\cdots\\circ\n\\Bup_{\\lambda_{y_1}^{\\fine}}\\circ M_{\\fine}\n\\bigr)","\\begin{corollary}[Hidden Markov lumpability]\\label{cor:hmm-lumpability}\nLet $M_{\\fine}$ and $M_{\\coarse}$ be column stochastic matrices satisfying\n\\[\n CM_{\\fine}=M_{\\coarse}C,\n\\]\nand let every observation likelihood satisfy\n\\[\n \\lambda_y^{\\fine}=s_yC^T\\lambda_y^{\\coarse}.\n\\]\nThen, for every observation sequence $y_{1:N}$,\n\\begin{align*}\n&C\\bigl(\n\\Bup_{\\lambda_{y_N}^{\\fine}}\\circ M_{\\fine}\n\\circ\\cdots\\circ\n\\Bup_{\\lambda_{y_1}^{\\fine}}\\circ M_{\\fine}\n\\bigr)\\\\\n&\\qquad=\n\\bigl(\n\\Bup_{\\lambda_{y_N}^{\\coarse}}\\circ M_{\\coarse}\n\\circ\\cdots\\circ\n\\Bup_{\\lambda_{y_1}^{\\coarse}}\\circ M_{\\coarse}\n\\bigr)C.\n\\end{align*}\n\\end{corollary}","\\begin{proof}\nApply \\Cref{thm:exact-cocycle} with the linear prediction maps $M_{\\fine}$ and $M_{\\coarse}$.\n\\end{proof}","\\begin{corollary}[Nonlinear quotient filtering]\\label{cor:nonlinear-filtering}\nSuppose the fine and coarse prediction maps are flow segments of locally Lipschitz vector fields satisfying $CF_{\\fine}=F_{\\coarse}C$, and suppose all observation likelihoods are constant on each fiber up to scale. Then the nonlinear filtering cocycle commutes with $C$ exactly.\n\\end{corollary}","\\begin{proof}\nIntertwining of the vector fields gives $C\\Phi_k^{\\fine}=\\Phi_k^{\\coarse}C$ by \\Cref{thm:exact-quotient-flow}. Likelihoods that are constant on each fiber give $C\\Bup_k^{\\fine}=\\Bup_k^{\\coarse}C$ by \\Cref{thm:bayes-sufficient}. Apply \\Cref{thm:exact-cocycle}.\n\\end{proof}","\\begin{boundary}[Sufficiency for the chosen observation family]\nThe condition $\\lambda_y^{\\fine}=s_yC^T\\lambda_y^{\\coarse}$ is sufficient for the specified finite likelihood family. It does not assert that $C$ is sufficient for every possible future observation model. Adding a likelihood that distinguishes two states inside one fiber can invalidate the coarse filter without changing the dynamics.\n\\end{boundary}","Exact quotienting is an algebraic yes or no question. A useful model also needs to say how far it misses when the answer is no. We separate three sources of error:\n\\[\n \\text{interaction mismatch},\n \\qquad\n \\text{dependence on hidden directions},\n \\qquad\n \\text{dynamical amplification}.\n\\]","\\subsection{Best coarse interaction and quotient defect}\nLet\n\\[\n \\mathfrak Q_m(C)\n =\n \\{B_mC^{\\times m}:B_m\\in\\cL^m(X_{\\coarse};X_{\\coarse})\\}\n\\]\nbe the subspace of outputs from multilinear maps of order $m$ that factor through the coarse space.","\\begin{definition}[Best quotient error]\\label{def:best-quotient-error}\nFor a fine interaction $A_m^{\\fine}$, define\n\\begin{equation}\\label{eq:best-quotient-error}\n \\varepsilon_m^\\star\n =\n \\inf_{B_m\\in\\cL^m(X_{\\coarse};X_{\\coarse})}\n \\norm{CA_m^{\\fine}-B_mC^{\\times m}}.\n\\end{equation}\n\\end{definition}","\\begin{proof}\nThe map\n\\[\n B_m\\longmapsto B_mC^{\\times m}\n\\]\nis injective because $C$ is surjective. Its range is a finite dimensional linear subspace of $\\cL^m(X_{\\fine};X_{\\coarse})$, hence it is closed. In a finite dimensional normed space, the distance to a nonempty closed subspace attains its minimum. A nearest point in this range therefore exists. Any such point has a unique preimage under the displayed map, and that preimage is a minimizing coarse tensor.\n\\end{proof}","The minimum value depends on the chosen norm. A minimizing tensor need not be unique when that norm is not strictly convex. Exact lumpability is independent of the norm because it is the condition $\\varepsilon_m^\\star=0$.","\\begin{definition}[Coarse tensor induced by a section]\\label{def:section-coarse-tensor}\nFix $E$ with $CE=I$. Define\n\\begin{equation}\\label{eq:section-coarse-tensor}\n A_m^{\\coarse,E}\n =\n CA_m^{\\fine}E^{\\times m}.\n\\end{equation}\n\\end{definition}","Let $P_C=EC$ and $Q_C=I-EC$ as in \\eqref{eq:representative-hidden-proj}.","\\begin{proposition}[Expansion of the section defect by hidden inputs]\\label{prop:hidden-slot-expansion}\nThe defect of \\eqref{eq:section-coarse-tensor} is\n\\begin{equation}\\label{eq:section-defect}\n \\Delta_m^E(x_1,\\ldots,x_m)\n =\n CA_m^{\\fine}(x_1,\\ldots,x_m)\n -\n CA_m^{\\fine}(P_Cx_1,\\ldots,P_Cx_m),\n\\end{equation}\nand admits the exact expansion\n\\begin{equation}\\label{eq:hidden-slot-sum}\n \\boxed{\n \\Delta_m^E(x_1,\\ldots,x_m)\n =\n \\sum_{\\varnothing\\neq S\\subseteq\\{1,\\ldots,m\\}}\n CA_m^{\\fine}(y_1^S,\\ldots,y_m^S),}\n\\end{equation}\nwhere\n\\[\n y_r^S=\n \\begin{cases}\n Q_Cx_r,&r\\in S,","ensor} is\n\\begin{equation}\\label{eq:section-defect}\n \\Delta_m^E(x_1,\\ldots,x_m)\n =\n CA_m^{\\fine}(x_1,\\ldots,x_m)\n -\n CA_m^{\\fine}(P_Cx_1,\\ldots,P_Cx_m),\n\\end{equation}\nand admits the exact expansion\n\\begin{equation}\\label{eq:hidden-slot-sum}\n \\boxed{\n \\Delta_m^E(x_1,\\ldots,x_m)\n =\n \\sum_{\\varnothing\\neq S\\subseteq\\{1,\\ldots,m\\}}\n CA_m^{\\fine}(y_1^S,\\ldots,y_m^S),}\n\\end{equation}\nwhere\n\\[\n y_r^S=\n \\begin{cases}\n Q_Cx_r,&r\\in S,\\\\\n P_Cx_r,&r\\notin S.\n \\end{cases}\n\\]\n\\end{proposition}","\\begin{proof}\nEquation \\eqref{eq:section-defect} follows from\n\\[\n A_m^{\\coarse,E}(Cx_1,\\ldots,Cx_m)\n =CA_m^{\\fine}(ECx_1,\\ldots,ECx_m).\n\\]\nNow expand every input as $x_r=P_Cx_r+Q_Cx_r$. The term with no $Q_C$ factor is the term subtracted in \\eqref{eq:section-defect}. Every remaining term has at least one hidden input and appears exactly once.\n\\end{proof}","\\begin{corollary}[Exactness as disappearance of hidden interactions]\\label{cor:hidden-interactions-exactness}\nThe tensor defined from the chosen section descends exactly if and only if the sum in \\eqref{eq:hidden-slot-sum} vanishes for all inputs. The stronger slotwise condition\n\\[\n CA_m^{\\fine}(x_1,\\ldots,Q_Cx_r,\\ldots,x_m)=0\n\\]\nfor each slot is equivalent to exact descent and makes every summand vanish separately.\n\\end{corollary}","\\begin{proof}\nThe first statement is \\eqref{eq:hidden-slot-sum} with $\\Delta_m^E=0$. Since $\\Ran Q_C=\\Ker C$, the condition in each slot is exactly the kernel condition in \\Cref{thm:universal-quotient}. It is therefore equivalent to exact descent. Every term indexed by a nonempty subset contains at least one hidden input, so each term vanishes.\n\\end{proof}","\\subsection{Defect majorants for the full field}\nSuppose fine and proposed coarse fields have expansions\n\\[\n F_{\\fine}(x)=\\sum_{m\\geq0}A_m^{\\fine}[x^{\\otimes m}],\n \\qquad\n F_{\\coarse}(z)=\\sum_{m\\geq0}A_m^{\\coarse}[z^{\\otimes m}].\n\\]\nDefine the field defect\n\\begin{equation}\\label{eq:field-defect}\n \\delta_F(x)\n =\n CF_{\\fine}(x)-F_{\\coarse}(Cx).\n\\end{equation}\nWhenever both series converge absolutely,\n\\begin{equation}\\label{eq:field-defect-expansion}\n \\delta_F(x)\n =\n \\sum_{m\\geq0}\\Delta_m[x^{\\otimes m}].\n\\end{equation}","ants for the full field}\nSuppose fine and proposed coarse fields have expansions\n\\[\n F_{\\fine}(x)=\\sum_{m\\geq0}A_m^{\\fine}[x^{\\otimes m}],\n \\qquad\n F_{\\coarse}(z)=\\sum_{m\\geq0}A_m^{\\coarse}[z^{\\otimes m}].\n\\]\nDefine the field defect\n\\begin{equation}\\label{eq:field-defect}\n \\delta_F(x)\n =\n CF_{\\fine}(x)-F_{\\coarse}(Cx).\n\\end{equation}\nWhenever both series converge absolutely,\n\\begin{equation}\\label{eq:field-defect-expansion}\n \\delta_F(x)\n =\n \\sum_{m\\geq0}\\Delta_m[x^{\\otimes m}].\n\\end{equation}","\\begin{proposition}[Analytic defect majorant]\\label{prop:defect-majorant}\nIf $\\norm{\\Delta_m}\\leq\\varepsilon_m$ and the scalar series\n\\[\n \\mathfrak d_F(R)=\\sum_{m\\geq0}\\varepsilon_mR^m\n\\]\nconverges, then\n\\begin{equation}\\label{eq:field-defect-majorant}\n \\boxed{\n \\sup_{\\norm{x}\\leq R}\\norm{\\delta_F(x)}\n \\leq\n \\mathfrak d_F(R).}\n\\end{equation}\n\\end{proposition}","The sequence $(\\varepsilon_m)$ shows which interaction orders obstruct the quotient. A model may be exactly lumpable through order $r$ and fail for the first time at order $r+1$.","\\begin{theorem}[Approximate quotient flow bound]\\label{thm:approx-flow-bound}\nLet\n\\[\n \\dot x=F_{\\fine}(x),\n \\qquad\n \\dot z=F_{\\coarse}(z),\n\\]\nand suppose both solutions remain in domains on which\n\\[\n \\norm{F_{\\coarse}(u)-F_{\\coarse}(v)}\n \\leq L\\norm{u-v}.\n\\]\nSet\n\\[\n e(t)=\\norm{Cx(t)-z(t)}.\n\\]\nThen\n\\begin{equation}\\label{eq:approx-flow-general}\n \\boxed{\n e(t)\n \\leq\n e^{Lt}e(0)\n +\n \\int_0^t e^{L(t-s)}\\norm{\\delta_F(x(s))}\\dd s.}\n\\end{equation}\nIf $e(0)=0$ and $\\norm{\\delta_F(x(s))}\\leq\\varepsilon_F$ on $[0,t]$, then\n\\begin{equation}\\label{eq:approx-flow-uniform}\n e(t)\n \\leq\n \\begin{cases}\n \\displaystyle\n \\varepsilon_F\\frac{e^{Lt}-1}{L},&L>0,","rox-flow-bound}\nLet\n\\[\n \\dot x=F_{\\fine}(x),\n \\qquad\n \\dot z=F_{\\coarse}(z),\n\\]\nand suppose both solutions remain in domains on which\n\\[\n \\norm{F_{\\coarse}(u)-F_{\\coarse}(v)}\n \\leq L\\norm{u-v}.\n\\]\nSet\n\\[\n e(t)=\\norm{Cx(t)-z(t)}.\n\\]\nThen\n\\begin{equation}\\label{eq:approx-flow-general}\n \\boxed{\n e(t)\n \\leq\n e^{Lt}e(0)\n +\n \\int_0^t e^{L(t-s)}\\norm{\\delta_F(x(s))}\\dd s.}\n\\end{equation}\nIf $e(0)=0$ and $\\norm{\\delta_F(x(s))}\\leq\\varepsilon_F$ on $[0,t]$, then\n\\begin{equation}\\label{eq:approx-flow-uniform}\n e(t)\n \\leq\n \\begin{cases}\n \\displaystyle\n \\varepsilon_F\\frac{e^{Lt}-1}{L},&L>0,","d}\nLet\n\\[\n \\dot x=F_{\\fine}(x),\n \\qquad\n \\dot z=F_{\\coarse}(z),\n\\]\nand suppose both solutions remain in domains on which\n\\[\n \\norm{F_{\\coarse}(u)-F_{\\coarse}(v)}\n \\leq L\\norm{u-v}.\n\\]\nSet\n\\[\n e(t)=\\norm{Cx(t)-z(t)}.\n\\]\nThen\n\\begin{equation}\\label{eq:approx-flow-general}\n \\boxed{\n e(t)\n \\leq\n e^{Lt}e(0)\n +\n \\int_0^t e^{L(t-s)}\\norm{\\delta_F(x(s))}\\dd s.}\n\\end{equation}\nIf $e(0)=0$ and $\\norm{\\delta_F(x(s))}\\leq\\varepsilon_F$ on $[0,t]$, then\n\\begin{equation}\\label{eq:approx-flow-uniform}\n e(t)\n \\leq\n \\begin{cases}\n \\displaystyle\n \\varepsilon_F\\frac{e^{Lt}-1}{L},&L>0,","\\dot z=F_{\\coarse}(z),\n\\]\nand suppose both solutions remain in domains on which\n\\[\n \\norm{F_{\\coarse}(u)-F_{\\coarse}(v)}\n \\leq L\\norm{u-v}.\n\\]\nSet\n\\[\n e(t)=\\norm{Cx(t)-z(t)}.\n\\]\nThen\n\\begin{equation}\\label{eq:approx-flow-general}\n \\boxed{\n e(t)\n \\leq\n e^{Lt}e(0)\n +\n \\int_0^t e^{L(t-s)}\\norm{\\delta_F(x(s))}\\dd s.}\n\\end{equation}\nIf $e(0)=0$ and $\\norm{\\delta_F(x(s))}\\leq\\varepsilon_F$ on $[0,t]$, then\n\\begin{equation}\\label{eq:approx-flow-uniform}\n e(t)\n \\leq\n \\begin{cases}\n \\displaystyle\n \\varepsilon_F\\frac{e^{Lt}-1}{L},&L>0,","suppose both solutions remain in domains on which\n\\[\n \\norm{F_{\\coarse}(u)-F_{\\coarse}(v)}\n \\leq L\\norm{u-v}.\n\\]\nSet\n\\[\n e(t)=\\norm{Cx(t)-z(t)}.\n\\]\nThen\n\\begin{equation}\\label{eq:approx-flow-general}\n \\boxed{\n e(t)\n \\leq\n e^{Lt}e(0)\n +\n \\int_0^t e^{L(t-s)}\\norm{\\delta_F(x(s))}\\dd s.}\n\\end{equation}\nIf $e(0)=0$ and $\\norm{\\delta_F(x(s))}\\leq\\varepsilon_F$ on $[0,t]$, then\n\\begin{equation}\\label{eq:approx-flow-uniform}\n e(t)\n \\leq\n \\begin{cases}\n \\displaystyle\n \\varepsilon_F\\frac{e^{Lt}-1}{L},&L>0,\\\\[1.1ex]\n t\\varepsilon_F,&L=0.\n \\end{cases}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nLet $y(t)=Cx(t)$. From \\eqref{eq:field-defect},\n\\[\n \\dot y(t)=F_{\\coarse}(y(t))+\\delta_F(x(t)).\n\\]\nSubtract the integral equations for $y$ and $z$:\n\\[\n y(t)-z(t)\n =y(0)-z(0)\n +\\int_0^t\n \\bigl(F_{\\coarse}(y(s))-F_{\\coarse}(z(s))\\bigr)\\dd s\n +\\int_0^t\\delta_F(x(s))\\dd s.\n\\]\nTaking norms and applying the Grönwall inequality yields \\eqref{eq:approx-flow-general}. The uniform cases follow by evaluating the exponential integral.\n\\end{proof}","\\begin{corollary}[Coefficient certificate for a flow segment]\\label{cor:coefficient-flow-certificate}\nIf $\\norm{x(s)}\\leq R$ and Proposition~\\ref{prop:defect-majorant} applies, then a flow interval of length $T$ satisfies\n\\[\n \\norm{C\\Phi_T^{\\fine}(x_0)-\\Phi_T^{\\coarse}(Cx_0)}\n \\leq\n \\mathfrak d_F(R)\\frac{e^{LT}-1}{L}\n\\]\nfor $L>0$, with $T\\mathfrak d_F(R)$ at $L=0$.\n\\end{corollary}","\\begin{proof}\nBy Proposition~\\ref{prop:defect-majorant}, $\\norm{\\delta_F(x(s))}\\leq\\mathfrak d_F(R)$ throughout the segment. Apply \\Cref{thm:approx-flow-bound} with compatible initial data, so $e(0)=0$, and evaluate the exponential integral.\n\\end{proof}","\\subsection{Refinement towers}\nLet\n\\[\n X_L\\xrightarrow{C_{L,L-1}}X_{L-1}\n \\longrightarrow\\cdots\\longrightarrow\n X_1\\xrightarrow{C_{10}}X_0.\n\\]\nFor $a>b$, write\n\\[\n C_{ab}=C_{b+1,b}\\cdots C_{a,a-1}.\n\\]\nLet $\\Delta_m^{ab}$ denote the interaction defect for the composite map $C_{ab}$. Iterating \\Cref{thm:multilevel-defect} yields a telescoping formula.","\\begin{corollary}[Telescoping interaction defect]\\label{cor:tower-defect}\nFor an interaction of order $m$ assigned at every level,\n\\begin{equation}\\label{eq:tower-defect}\n \\Delta_m^{L0}\n =\n \\sum_{\\ell=0}^{L-1}\n C_{\\ell0}\\,\n \\Delta_m^{\\ell+1,\\ell}\n C_{L,\\ell+1}^{\\times m},\n\\end{equation}\nwhere $C_{00}=I$ and $C_{LL}=I$. Consequently,\n\\begin{equation}\\label{eq:tower-defect-bound}\n \\norm{\\Delta_m^{L0}}\n \\leq\n \\sum_{\\ell=0}^{L-1}\n \\norm{C_{\\ell0}}\n \\norm{\\Delta_m^{\\ell+1,\\ell}}\n \\norm{C_{L,\\ell+1}}^m.\n\\end{equation}\n\\end{corollary}","\\begin{lemma}[Normalization perturbation]\\label{lem:normalization-perturbation}\nLet $u,v\\in\\R_+^q$ have positive total mass. Then\n\\begin{equation}\\label{eq:normalization-perturbation}\n \\boxed{\n \\norm{\n \\frac{u}{\\one^Tu}\n -\n \\frac{v}{\\one^Tv}\n }_1\n \\leq\n \\frac{2\\norm{u-v}_1}\n {\\max\\{\\one^Tu,\\one^Tv\\}}.}\n\\end{equation}\n\\end{lemma}","\\begin{proof}\nWrite $a=\\one^Tu$ and $b=\\one^Tv$. Then\n\\begin{align*}\n\\norm{\\frac{u}{a}-\\frac{v}{b}}_1\n&\\leq\n\\frac{\\norm{u-v}_1}{a}\n+\n\\norm{v}_1\\abs{\\frac1a-\\frac1b}","\\begin{proof}\nWrite $a=\\one^Tu$ and $b=\\one^Tv$. Then\n\\begin{align*}\n\\norm{\\frac{u}{a}-\\frac{v}{b}}_1\n&\\leq\n\\frac{\\norm{u-v}_1}{a}\n+\n\\norm{v}_1\\abs{\\frac1a-\\frac1b}\\\\\n&=\n\\frac{\\norm{u-v}_1}{a}\n+\n\\frac{\\abs{a-b}}{a}\\\\\n&\\leq\n\\frac{2\\norm{u-v}_1}{a}.\n\\end{align*}\nInterchanging $u$ and $v$ gives the same inequality with $b$ in the denominator. Taking the smaller of the two upper bounds yields the denominator $\\max\\{a,b\\}$.\n\\end{proof}","&\\leq\n\\frac{2\\norm{u-v}_1}{a}.\n\\end{align*}\nInterchanging $u$ and $v$ gives the same inequality with $b$ in the denominator. Taking the smaller of the two upper bounds yields the denominator $\\max\\{a,b\\}$.\n\\end{proof}","\\subsection{Defect in the observation intertwining law}\nLet $L_{\\fine}$ and $L_{\\coarse}$ be positive linear observation operators, and choose a scale $s>0$. On a finite simplex, we write $\\Nrm(L)$ for normalization with respect to the total mass $\\one^T$. Define\n\\begin{equation}\\label{eq:observation-linear-defect}\n R_L\n =\n CL_{\\fine}-sL_{\\coarse}C.\n\\end{equation}","\\begin{theorem}[Approximate normalized transfer naturality]\\label{thm:approx-normalized-transfer}\nOn finite simplices with block sum $C$, let\n\\[\n u=CL_{\\fine}\\mu,\n \\qquad\n v=sL_{\\coarse}C\\mu.\n\\]\nIf $u$ and $v$ are nonnegative with positive masses, then\n\\begin{equation}\\label{eq:approx-normalized-transfer}\n \\boxed{\n \\norm{\n C\\Nrm(L_{\\fine})(\\mu)\n -\n \\Nrm(L_{\\coarse})(C\\mu)\n }_1\n \\leq\n \\frac{2\\norm{R_L\\mu}_1}\n {\\max\\{\\one^Tu,\\one^Tv\\}}.}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nBecause $C$ preserves total mass,\n\\[\n C\\Nrm(L_{\\fine})(\\mu)\n =\\frac{u}{\\one^Tu}.\n\\]\nMultiplication of $L_{\\coarse}$ by $s$ does not change its normalized output, so\n\\[\n \\Nrm(L_{\\coarse})(C\\mu)\n =\\frac{v}{\\one^Tv}.\n\\]\nNow $u-v=R_L\\mu$, and Lemma~\\ref{lem:normalization-perturbation} applies.\n\\end{proof}","\\subsection{Likelihood variation inside a fiber}\nFor diagonal likelihood operators, write\n\\begin{equation}\\label{eq:likelihood-residual}\n \\lambda^{\\fine}\n =\n sC^T\\lambda^{\\coarse}+r_{\\lambda}.\n\\end{equation}\nThen\n\\[\n R_L\\mu=C(r_{\\lambda}\\odot\\mu).\n\\]\nSince block sum aggregation is an $\\ell^1$ contraction on signed vectors,\n\\begin{equation}\\label{eq:likelihood-residual-l1}\n \\norm{R_L\\mu}_1\n \\leq\n \\sum_i\\mu_i\\abs{r_{\\lambda,i}}\n \\leq\n \\norm{r_{\\lambda}}_{\\infty}.\n\\end{equation}","\\begin{corollary}[Bayesian refinement defect]\\label{cor:bayes-refinement-defect}\nLet\n\\[\n z_{\\fine}=Z_{\\lambda^{\\fine}}(\\mu),\n \\qquad\n z_{\\coarse}=sZ_{\\lambda^{\\coarse}}(C\\mu).\n\\]\nThen\n\\begin{equation}\\label{eq:bayes-refinement-defect}\n \\boxed{\n \\norm{\n C\\Bup_{\\lambda^{\\fine}}(\\mu)\n -\n \\Bup_{\\lambda^{\\coarse}}(C\\mu)\n }_1\n \\leq\n \\frac{2\\sum_i\\mu_i\\abs{r_{\\lambda,i}}}\n {\\max\\{z_{\\fine},z_{\\coarse}\\}}.}\n\\end{equation}\nIf both evidences are at least $z_*>0$, then\n\\begin{equation}\\label{eq:bayes-refinement-uniform}\n d_{\\Bup}\n \\leq\n \\frac{2\\norm{r_{\\lambda}}_{\\infty}}{z_*}.\n\\end{equation}\n\\end{corollary}","refinement-defect}\nLet\n\\[\n z_{\\fine}=Z_{\\lambda^{\\fine}}(\\mu),\n \\qquad\n z_{\\coarse}=sZ_{\\lambda^{\\coarse}}(C\\mu).\n\\]\nThen\n\\begin{equation}\\label{eq:bayes-refinement-defect}\n \\boxed{\n \\norm{\n C\\Bup_{\\lambda^{\\fine}}(\\mu)\n -\n \\Bup_{\\lambda^{\\coarse}}(C\\mu)\n }_1\n \\leq\n \\frac{2\\sum_i\\mu_i\\abs{r_{\\lambda,i}}}\n {\\max\\{z_{\\fine},z_{\\coarse}\\}}.}\n\\end{equation}\nIf both evidences are at least $z_*>0$, then\n\\begin{equation}\\label{eq:bayes-refinement-uniform}\n d_{\\Bup}\n \\leq\n \\frac{2\\norm{r_{\\lambda}}_{\\infty}}{z_*}.\n\\end{equation}\n\\end{corollary}","_{\\fine}=Z_{\\lambda^{\\fine}}(\\mu),\n \\qquad\n z_{\\coarse}=sZ_{\\lambda^{\\coarse}}(C\\mu).\n\\]\nThen\n\\begin{equation}\\label{eq:bayes-refinement-defect}\n \\boxed{\n \\norm{\n C\\Bup_{\\lambda^{\\fine}}(\\mu)\n -\n \\Bup_{\\lambda^{\\coarse}}(C\\mu)\n }_1\n \\leq\n \\frac{2\\sum_i\\mu_i\\abs{r_{\\lambda,i}}}\n {\\max\\{z_{\\fine},z_{\\coarse}\\}}.}\n\\end{equation}\nIf both evidences are at least $z_*>0$, then\n\\begin{equation}\\label{eq:bayes-refinement-uniform}\n d_{\\Bup}\n \\leq\n \\frac{2\\norm{r_{\\lambda}}_{\\infty}}{z_*}.\n\\end{equation}\n\\end{corollary}","\\begin{definition}[Fiber oscillation]\\label{def:fiber-oscillation}\nFor a fine likelihood, define\n\\[\n \\osc_a(\\lambda^{\\fine})\n =\n \\max_{i:\\pi(i)=a}\\lambda_i^{\\fine}\n -\n \\min_{i:\\pi(i)=a}\\lambda_i^{\\fine},\n\\]\nand\n\\[\n \\osc_C(\\lambda^{\\fine})\n =\\max_a\\osc_a(\\lambda^{\\fine}).\n\\]\n\\end{definition}","When $q\\geq2$ and commutation is required for every interior prior, exact Bayesian refinement is equivalent to $\\osc_C(\\lambda^{\\fine})=0$ once the coarse likelihood and global scale have been chosen consistently. In an approximate model, $\\lambda_a^{\\coarse}$ may be the midpoint, the ordinary mean, or a mean chosen to reflect the task on fiber $a$. The midpoint minimizes the largest absolute residual in that fiber and gives\n\\[\n \\norm{r_{\\lambda}}_{\\infty}\n =\\frac12\\osc_C(\\lambda^{\\fine})\n\\]\nwhen $s=1$.","required for every interior prior, exact Bayesian refinement is equivalent to $\\osc_C(\\lambda^{\\fine})=0$ once the coarse likelihood and global scale have been chosen consistently. In an approximate model, $\\lambda_a^{\\coarse}$ may be the midpoint, the ordinary mean, or a mean chosen to reflect the task on fiber $a$. The midpoint minimizes the largest absolute residual in that fiber and gives\n\\[\n \\norm{r_{\\lambda}}_{\\infty}\n =\\frac12\\osc_C(\\lambda^{\\fine})\n\\]\nwhen $s=1$.","to $\\osc_C(\\lambda^{\\fine})=0$ once the coarse likelihood and global scale have been chosen consistently. In an approximate model, $\\lambda_a^{\\coarse}$ may be the midpoint, the ordinary mean, or a mean chosen to reflect the task on fiber $a$. The midpoint minimizes the largest absolute residual in that fiber and gives\n\\[\n \\norm{r_{\\lambda}}_{\\infty}\n =\\frac12\\osc_C(\\lambda^{\\fine})\n\\]\nwhen $s=1$.","\\begin{proposition}[Lipschitz bound under an evidence floor]\\label{prop:bayes-lipschitz}\nLet $\\lambda\\in\\R_{>0}^q$. If\n\\[\n Z_{\\lambda}(\\mu),Z_{\\lambda}(\\nu)\\geq z_*>0,\n\\]\nthen\n\\begin{equation}\\label{eq:bayes-lipschitz}\n \\boxed{\n \\norm{\\Bup_{\\lambda}(\\mu)-\\Bup_{\\lambda}(\\nu)}_1\n \\leq\n \\frac{2\\norm{\\lambda}_{\\infty}}{z_*}\n \\norm{\\mu-\\nu}_1.}\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nApply Lemma~\\ref{lem:normalization-perturbation} to\n\\[\n u=\\lambda\\odot\\mu,\n \\qquad\n v=\\lambda\\odot\\nu.\n\\]\nThen\n\\[\n \\norm{u-v}_1\n \\leq\\norm{\\lambda}_{\\infty}\\norm{\\mu-\\nu}_1.\n\\]\n\\end{proof}","\\begin{remark}[Scale invariance of the bound]\nReplacing $\\lambda$ by $s\\lambda$ multiplies both $\\norm{\\lambda}_{\\infty}$ and $z_*$ by $s$, leaving the ratio unchanged.\n\\end{remark}","\\subsection{Prediction and update errors}\nLet the fine recursion be\n\\begin{equation}\\label{eq:fine-filter-recursion}\n \\widehat\\mu_k^{\\fine}\n =\\Phi_k^{\\fine}(\\mu_{k-1}^{\\fine}),\n \\qquad\n \\mu_k^{\\fine}\n =\\Bup_k^{\\fine}(\\widehat\\mu_k^{\\fine}),\n\\end{equation}\nand the coarse recursion be\n\\begin{equation}\\label{eq:coarse-filter-recursion}\n \\widehat\\mu_k^{\\coarse}\n =\\Phi_k^{\\coarse}(\\mu_{k-1}^{\\coarse}),\n \\qquad\n \\mu_k^{\\coarse}\n =\\Bup_k^{\\coarse}(\\widehat\\mu_k^{\\coarse}).\n\\end{equation}\nDefine the discrepancies before and after each update by\n\\begin{equation}\\label{eq:pre-post-errors}\n e_k^-\n =\\norm{C\\widehat\\mu_k^{\\fine}-\\widehat\\mu_k^{\\coarse}}_1,\n \\qquad\n e_k^+\n =\\norm{C\\mu_k^{\\fine}-\\mu_k^{\\coarse}}_1.\n\\end{equation}\nAt the initial time, set\n\\[\n e_0^+=\\norm{C\\mu_0^{\\fine}-\\mu_0^{\\coarse}}_1.\n\\]","{\\fine}),\n \\qquad\n \\mu_k^{\\fine}\n =\\Bup_k^{\\fine}(\\widehat\\mu_k^{\\fine}),\n\\end{equation}\nand the coarse recursion be\n\\begin{equation}\\label{eq:coarse-filter-recursion}\n \\widehat\\mu_k^{\\coarse}\n =\\Phi_k^{\\coarse}(\\mu_{k-1}^{\\coarse}),\n \\qquad\n \\mu_k^{\\coarse}\n =\\Bup_k^{\\coarse}(\\widehat\\mu_k^{\\coarse}).\n\\end{equation}\nDefine the discrepancies before and after each update by\n\\begin{equation}\\label{eq:pre-post-errors}\n e_k^-\n =\\norm{C\\widehat\\mu_k^{\\fine}-\\widehat\\mu_k^{\\coarse}}_1,\n \\qquad\n e_k^+\n =\\norm{C\\mu_k^{\\fine}-\\mu_k^{\\coarse}}_1.\n\\end{equation}\nAt the initial time, set\n\\[\n e_0^+=\\norm{C\\mu_0^{\\fine}-\\mu_0^{\\coarse}}_1.\n\\]","=\\Phi_k^{\\coarse}(\\mu_{k-1}^{\\coarse}),\n \\qquad\n \\mu_k^{\\coarse}\n =\\Bup_k^{\\coarse}(\\widehat\\mu_k^{\\coarse}).\n\\end{equation}\nDefine the discrepancies before and after each update by\n\\begin{equation}\\label{eq:pre-post-errors}\n e_k^-\n =\\norm{C\\widehat\\mu_k^{\\fine}-\\widehat\\mu_k^{\\coarse}}_1,\n \\qquad\n e_k^+\n =\\norm{C\\mu_k^{\\fine}-\\mu_k^{\\coarse}}_1.\n\\end{equation}\nAt the initial time, set\n\\[\n e_0^+=\\norm{C\\mu_0^{\\fine}-\\mu_0^{\\coarse}}_1.\n\\]","Assume the coarse prediction is Lipschitz with constant $L_{\\Phi,k}$ and that its quotient defect is bounded by\n\\begin{equation}\\label{eq:prediction-map-defect}\n \\sup_{\\mu}\n \\norm{C\\Phi_k^{\\fine}(\\mu)-\\Phi_k^{\\coarse}(C\\mu)}_1\n \\leq d_{\\Phi,k}.\n\\end{equation}\nAssume the coarse update is Lipschitz with constant $L_{\\Bup,k}$ and that its direct refinement defect is bounded by\n\\begin{equation}\\label{eq:update-map-defect}\n \\sup_{\\mu}\n \\norm{C\\Bup_k^{\\fine}(\\mu)-\\Bup_k^{\\coarse}(C\\mu)}_1\n \\leq d_{\\Bup,k}.\n\\end{equation}\nThe suprema in \\eqref{eq:prediction-map-defect} and \\eqref{eq:update-map-defect} are taken over a declared admissible set that contains both recursions.","\\begin{lemma}[Prediction bound for one step]\\label{lem:prediction-step}\nUnder \\eqref{eq:prediction-map-defect},\n\\begin{equation}\\label{eq:prediction-step}\n e_k^-\n \\leq\n L_{\\Phi,k}e_{k-1}^+\n +d_{\\Phi,k}.\n\\end{equation}\n\\end{lemma}","\\begin{proof}\nInsert and subtract $\\Phi_k^{\\coarse}(C\\mu_{k-1}^{\\fine})$. The quotient defect bounds one difference, while the Lipschitz estimate bounds the other by $L_{\\Phi,k}e_{k-1}^+$.\n\\end{proof}","\\begin{lemma}[Update bound for one step]\\label{lem:update-step}\nUnder \\eqref{eq:update-map-defect},\n\\begin{equation}\\label{eq:update-step}\n e_k^+\n \\leq\n L_{\\Bup,k}e_k^-\n +d_{\\Bup,k}.\n\\end{equation}\n\\end{lemma}","\\begin{proof}\nInsert and subtract $\\Bup_k^{\\coarse}(C\\widehat\\mu_k^{\\fine})$. The direct refinement defect bounds one difference, while the Lipschitz estimate bounds the other by $L_{\\Bup,k}e_k^-$.\n\\end{proof}","Combining the two inequalities gives\n\\begin{equation}\\label{eq:combined-step}\n e_k^+\n \\leq\n \\alpha_ke_{k-1}^++\\beta_k,\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:alpha-beta}\n \\alpha_k=L_{\\Bup,k}L_{\\Phi,k},\n \\qquad\n \\beta_k=L_{\\Bup,k}d_{\\Phi,k}+d_{\\Bup,k}.\n\\end{equation}","\\begin{theorem}[Certified approximate filtering]\\label{thm:posterior-certificate}\nUnder the assumptions above,\n\\begin{equation}\\label{eq:posterior-certificate}\n \\boxed{\n e_N^+\n \\leq\n \\left(\\prod_{j=1}^N\\alpha_j\\right)e_0^+\n +\n \\sum_{k=1}^N\n \\left(\\prod_{j=k+1}^N\\alpha_j\\right)\n \\beta_k.}\n\\end{equation}\nHere and below, an empty product equals one. With exact initial aggregation $e_0^+=0$,\n\\begin{equation}\\label{eq:posterior-certificate-exact-initial}\n e_N^+\n \\leq\n \\sum_{k=1}^N\n \\left(\\prod_{j=k+1}^N\\alpha_j\\right)\n \\bigl(L_{\\Bup,k}d_{\\Phi,k}+d_{\\Bup,k}\\bigr).\n\\end{equation}\n\\end{theorem}","Certified approximate filtering]\\label{thm:posterior-certificate}\nUnder the assumptions above,\n\\begin{equation}\\label{eq:posterior-certificate}\n \\boxed{\n e_N^+\n \\leq\n \\left(\\prod_{j=1}^N\\alpha_j\\right)e_0^+\n +\n \\sum_{k=1}^N\n \\left(\\prod_{j=k+1}^N\\alpha_j\\right)\n \\beta_k.}\n\\end{equation}\nHere and below, an empty product equals one. With exact initial aggregation $e_0^+=0$,\n\\begin{equation}\\label{eq:posterior-certificate-exact-initial}\n e_N^+\n \\leq\n \\sum_{k=1}^N\n \\left(\\prod_{j=k+1}^N\\alpha_j\\right)\n \\bigl(L_{\\Bup,k}d_{\\Phi,k}+d_{\\Bup,k}\\bigr).\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nRepeated substitution in the affine recurrence \\eqref{eq:combined-step} gives the stated formula. The initial error is multiplied by every later amplification factor. A defect introduced at step $k$ is multiplied only by the factors from steps $k+1$ through $N$.\n\\end{proof}","\\begin{corollary}[Uniform stability regimes]\\label{cor:filter-stability-regimes}\nSuppose $0\\leq\\alpha_k\\leq\\alpha$ and $0\\leq\\beta_k\\leq\\beta$.","\\begin{enumerate}[label=(\\roman*),leftmargin=2.8em]\n\\item If $0\\leq\\alpha<1$, then\n\\begin{equation}\\label{eq:stable-filter-bound}\n e_N^+\n \\leq\n \\alpha^Ne_0^+\n +\n \\beta\\frac{1-\\alpha^N}{1-\\alpha},\n \\qquad\n \\limsup_{N\\to\\infty}e_N^+\n \\leq\\frac{\\beta}{1-\\alpha}.\n\\end{equation}","\\item If $\\alpha=1$, then\n\\[\n e_N^+\\leq e_0^++N\\beta.\n\\]","\\item If $\\alpha>1$, then\n\\[\n e_N^+\n \\leq\n \\alpha^Ne_0^+\n +\n \\beta\\frac{\\alpha^N-1}{\\alpha-1}.\n\\]\n\\end{enumerate}\n\\end{corollary}","\\begin{proof}\nBound the products in \\eqref{eq:posterior-certificate} by powers of $\\alpha$ and sum the geometric series.\n\\end{proof}","This trichotomy describes the certificate, not necessarily the observed filter. A stable trajectory can still come with a bound for which $\\alpha>1$. In that case, the chosen global norm estimates simply fail to certify contraction.","\\subsection{From interaction coefficients to posterior error}\nFor a flow interval of length $T_k$, suppose the fine trajectory remains in a ball of radius $R_k$ and the coarse field has Lipschitz constant $L_{F,k}$. By Corollary~\\ref{cor:coefficient-flow-certificate}, one may take\n\\begin{equation}\\label{eq:dphi-coefficient}\n d_{\\Phi,k}\n =\n \\mathfrak d_{F,k}(R_k)\n \\frac{e^{L_{F,k}T_k}-1}{L_{F,k}},\n\\end{equation}\nwith the continuous value $T_k\\mathfrak d_{F,k}(R_k)$ at $L_{F,k}=0$.\nFor the observation, Corollary~\\ref{cor:bayes-refinement-defect} gives\n\\begin{equation}\\label{eq:dbayes-coefficient}\n d_{\\Bup,k}\n \\leq\n \\frac{2\\norm{r_{\\lambda,k}}_{\\infty}}{z_{*,k}}.\n\\end{equation}\nSubstituting these bounds into \\eqref{eq:posterior-certificate-exact-initial} gives a certificate expres","\\subsection{From interaction coefficients to posterior error}\nFor a flow interval of length $T_k$, suppose the fine trajectory remains in a ball of radius $R_k$ and the coarse field has Lipschitz constant $L_{F,k}$. By Corollary~\\ref{cor:coefficient-flow-certificate}, one may take\n\\begin{equation}\\label{eq:dphi-coefficient}\n d_{\\Phi,k}\n =\n \\mathfrak d_{F,k}(R_k)\n \\frac{e^{L_{F,k}T_k}-1}{L_{F,k}},\n\\end{equation}\nwith the continuous value $T_k\\mathfrak d_{F,k}(R_k)$ at $L_{F,k}=0$.\nFor the observation, Corollary~\\ref{cor:bayes-refinement-defect} gives\n\\begin{equation}\\label{eq:dbayes-coefficient}\n d_{\\Bup,k}\n \\leq\n \\frac{2\\norm{r_{\\lambda,k}}_{\\infty}}{z_{*,k}}.\n\\end{equation}\nSubstituting these bounds into \\eqref{eq:posterior-certificate-exact-initial} gives a certificate expressed directly in terms of interaction tensor defects and likelih","\\subsection{From interaction coefficients to posterior error}\nFor a flow interval of length $T_k$, suppose the fine trajectory remains in a ball of radius $R_k$ and the coarse field has Lipschitz constant $L_{F,k}$. By Corollary~\\ref{cor:coefficient-flow-certificate}, one may take\n\\begin{equation}\\label{eq:dphi-coefficient}\n d_{\\Phi,k}\n =\n \\mathfrak d_{F,k}(R_k)\n \\frac{e^{L_{F,k}T_k}-1}{L_{F,k}},\n\\end{equation}\nwith the continuous value $T_k\\mathfrak d_{F,k}(R_k)$ at $L_{F,k}=0$.\nFor the observation, Corollary~\\ref{cor:bayes-refinement-defect} gives\n\\begin{equation}\\label{eq:dbayes-coefficient}\n d_{\\Bup,k}\n \\leq\n \\frac{2\\norm{r_{\\lambda,k}}_{\\infty}}{z_{*,k}}.\n\\end{equation}\nSubstituting these bounds into \\eqref{eq:posterior-certificate-exact-initial} gives a certificate expressed directly in terms of interaction tensor defects and likelihood residuals within each fiber.","sterior error}\nFor a flow interval of length $T_k$, suppose the fine trajectory remains in a ball of radius $R_k$ and the coarse field has Lipschitz constant $L_{F,k}$. By Corollary~\\ref{cor:coefficient-flow-certificate}, one may take\n\\begin{equation}\\label{eq:dphi-coefficient}\n d_{\\Phi,k}\n =\n \\mathfrak d_{F,k}(R_k)\n \\frac{e^{L_{F,k}T_k}-1}{L_{F,k}},\n\\end{equation}\nwith the continuous value $T_k\\mathfrak d_{F,k}(R_k)$ at $L_{F,k}=0$.\nFor the observation, Corollary~\\ref{cor:bayes-refinement-defect} gives\n\\begin{equation}\\label{eq:dbayes-coefficient}\n d_{\\Bup,k}\n \\leq\n \\frac{2\\norm{r_{\\lambda,k}}_{\\infty}}{z_{*,k}}.\n\\end{equation}\nSubstituting these bounds into \\eqref{eq:posterior-certificate-exact-initial} gives a certificate expressed directly in terms of interaction tensor defects and likelihood residuals within each fiber.","val of length $T_k$, suppose the fine trajectory remains in a ball of radius $R_k$ and the coarse field has Lipschitz constant $L_{F,k}$. By Corollary~\\ref{cor:coefficient-flow-certificate}, one may take\n\\begin{equation}\\label{eq:dphi-coefficient}\n d_{\\Phi,k}\n =\n \\mathfrak d_{F,k}(R_k)\n \\frac{e^{L_{F,k}T_k}-1}{L_{F,k}},\n\\end{equation}\nwith the continuous value $T_k\\mathfrak d_{F,k}(R_k)$ at $L_{F,k}=0$.\nFor the observation, Corollary~\\ref{cor:bayes-refinement-defect} gives\n\\begin{equation}\\label{eq:dbayes-coefficient}\n d_{\\Bup,k}\n \\leq\n \\frac{2\\norm{r_{\\lambda,k}}_{\\infty}}{z_{*,k}}.\n\\end{equation}\nSubstituting these bounds into \\eqref{eq:posterior-certificate-exact-initial} gives a certificate expressed directly in terms of interaction tensor defects and likelihood residuals within each fiber.","ield has Lipschitz constant $L_{F,k}$. By Corollary~\\ref{cor:coefficient-flow-certificate}, one may take\n\\begin{equation}\\label{eq:dphi-coefficient}\n d_{\\Phi,k}\n =\n \\mathfrak d_{F,k}(R_k)\n \\frac{e^{L_{F,k}T_k}-1}{L_{F,k}},\n\\end{equation}\nwith the continuous value $T_k\\mathfrak d_{F,k}(R_k)$ at $L_{F,k}=0$.\nFor the observation, Corollary~\\ref{cor:bayes-refinement-defect} gives\n\\begin{equation}\\label{eq:dbayes-coefficient}\n d_{\\Bup,k}\n \\leq\n \\frac{2\\norm{r_{\\lambda,k}}_{\\infty}}{z_{*,k}}.\n\\end{equation}\nSubstituting these bounds into \\eqref{eq:posterior-certificate-exact-initial} gives a certificate expressed directly in terms of interaction tensor defects and likelihood residuals within each fiber.","\\begin{corollary}[Coarse observable error]\\label{cor:observable-error}\nFor any coarse observable $g\\in\\R^q$,\n\\[\n \\abs{g^TC\\mu_N^{\\fine}-g^T\\mu_N^{\\coarse}}\n \\leq\n \\norm{g}_{\\infty}e_N^+.\n\\]\nIn particular,\n\\[\n \\TV(C\\mu_N^{\\fine},\\mu_N^{\\coarse})\n =\\frac12e_N^+.\n\\]\n\\end{corollary}","\\begin{proof}\nApply the duality between $\\ell^1$ and $\\ell^\\infty$. The total variation identity is standard for finite probability vectors.\n\\end{proof}","\\subsection{Sequential evidence error}\nFor brevity, write $\\lambda_k^{\\fine}=\\lambda_{y_k}^{\\fine}$ and $\\lambda_k^{\\coarse}=\\lambda_{y_k}^{\\coarse}$, and decompose\n\\begin{equation}\\label{eq:step-likelihood-residual}\n \\lambda_k^{\\fine}\n =s_kC^T\\lambda_k^{\\coarse}+r_k.\n\\end{equation}\nThis gives\n\\begin{align}\n z_k^{\\fine}-s_kz_k^{\\coarse}\n &=\n s_k(\\lambda_k^{\\coarse})^T\n (C\\widehat\\mu_k^{\\fine}-\\widehat\\mu_k^{\\coarse})\n +r_k^T\\widehat\\mu_k^{\\fine}.\n \\label{eq:evidence-error-identity}\n\\end{align}","\\begin{proposition}[Evidence certificate for one step]\\label{prop:evidence-error}\nAssume $\\widehat\\mu_k^{\\fine}$ is a probability vector. Under \\eqref{eq:step-likelihood-residual},\n\\begin{equation}\\label{eq:evidence-error-bound}\n \\boxed{\n \\abs{z_k^{\\fine}-s_kz_k^{\\coarse}}\n \\leq\n s_k\\norm{\\lambda_k^{\\coarse}}_{\\infty}e_k^-\n +\\norm{r_k}_{\\infty}.}\n\\end{equation}\nIf $z_k^{\\fine},s_kz_k^{\\coarse}\\geq z_{*,k}>0$, then\n\\begin{equation}\\label{eq:log-evidence-step}\n \\boxed{\n \\abs{\n \\log z_k^{\\fine}\n -\\log(s_kz_k^{\\coarse})}\n \\leq\n \\frac{\n s_k\\norm{\\lambda_k^{\\coarse}}_{\\infty}e_k^-\n +\\norm{r_k}_{\\infty}\n }{z_{*,k}}.}\n\\end{equation}\n\\end{proposition}","for one step]\\label{prop:evidence-error}\nAssume $\\widehat\\mu_k^{\\fine}$ is a probability vector. Under \\eqref{eq:step-likelihood-residual},\n\\begin{equation}\\label{eq:evidence-error-bound}\n \\boxed{\n \\abs{z_k^{\\fine}-s_kz_k^{\\coarse}}\n \\leq\n s_k\\norm{\\lambda_k^{\\coarse}}_{\\infty}e_k^-\n +\\norm{r_k}_{\\infty}.}\n\\end{equation}\nIf $z_k^{\\fine},s_kz_k^{\\coarse}\\geq z_{*,k}>0$, then\n\\begin{equation}\\label{eq:log-evidence-step}\n \\boxed{\n \\abs{\n \\log z_k^{\\fine}\n -\\log(s_kz_k^{\\coarse})}\n \\leq\n \\frac{\n s_k\\norm{\\lambda_k^{\\coarse}}_{\\infty}e_k^-\n +\\norm{r_k}_{\\infty}\n }{z_{*,k}}.}\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nApply Hölder's inequality to \\eqref{eq:evidence-error-identity}. Since $\\widehat\\mu_k^{\\fine}$ is a probability vector, $\\norm{\\widehat\\mu_k^{\\fine}}_1=1$. For the logarithms, the mean value theorem gives\n\\[\n \\abs{\\log a-\\log b}\n \\leq\\frac{\\abs{a-b}}{\\min\\{a,b\\}}.\n\\]\n\\end{proof}","\\begin{corollary}[Certificate for cumulative log evidence]\\label{cor:cumulative-evidence}\nLet $Z_{1:N}^{\\fine}=\\prod_kz_k^{\\fine}$ and $Z_{1:N}^{\\coarse}=\\prod_kz_k^{\\coarse}$. Then\n\\begin{equation}\\label{eq:cumulative-evidence}\n \\boxed{\n \\abs{\n \\log Z_{1:N}^{\\fine}\n -\\sum_{k=1}^N\\log s_k\n -\\log Z_{1:N}^{\\coarse}}\n \\leq\n \\sum_{k=1}^N\n \\frac{\n s_k\\norm{\\lambda_k^{\\coarse}}_{\\infty}e_k^-\n +\\norm{r_k}_{\\infty}\n }{z_{*,k}}.}\n\\end{equation}\n\\end{corollary}","\\subsection{Hilbert Schmidt tensor blocks}\nLet $X$ and $Y$ be finite dimensional Hilbert spaces with orthogonal packet projectors $(P_j)$ and $(Q_i)$. Identify a multilinear Hilbert Schmidt tensor of order $m$ with the Hilbert Schmidt linear map\n\\[\n \\widehat A:X^{\\otimes m}\\to Y.\n\\]\nIts packet block is\n\\[\n \\widehat A_{i\\mid j_1,\\ldots,j_m}\n =\n Q_i\\widehat A(P_{j_1}\\otimes\\cdots\\otimes P_{j_m}).\n\\]","\\begin{theorem}[Orthogonal hyperedge Parseval identity]\\label{thm:hyperedge-parseval}\nDistinct packet blocks are orthogonal in Hilbert Schmidt inner product, and\n\\begin{equation}\\label{eq:hyperedge-parseval}\n \\boxed{\n \\norm{\\widehat A}_{\\mathrm{HS}}^2\n =\n \\sum_{i,j_1,\\ldots,j_m}\n \\norm{\\widehat A_{i\\mid j_1,\\ldots,j_m}}_{\\mathrm{HS}}^2.}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nThe tensor product input projectors\n\\[\n P_{j_1}\\otimes\\cdots\\otimes P_{j_m}\n\\]\nare mutually orthogonal, as are the output projectors $Q_i$. For two distinct block labels, orthogonality on at least one side forces the Hilbert Schmidt inner product to vanish. Exact reconstruction is therefore an orthogonal sum, and Pythagoras gives \\eqref{eq:hyperedge-parseval}.\n\\end{proof}","\\subsection{Retained field and named residual}\nChoose a retained hyperedge set $\\cE_m^{\\mathrm{keep}}$. Define\n\\begin{equation}\\label{eq:pruned-tensor}\n \\widehat A_m^{\\mathrm{keep}}\n =\n \\sum_{h\\in\\cE_m^{\\mathrm{keep}}}\n \\widehat A_{m,h},\n \\qquad\n R_m\n =\n \\widehat A_m-\\widehat A_m^{\\mathrm{keep}}.\n\\end{equation}","\\begin{corollary}[Exact pruning residual]\\label{cor:pruning-residual}\nUnder the hypotheses of \\Cref{thm:hyperedge-parseval},\n\\begin{equation}\\label{eq:pruning-residual}\n \\boxed{\n \\norm{R_m}_{\\mathrm{HS}}^2\n =\n \\sum_{h\\notin\\cE_m^{\\mathrm{keep}}}\n \\norm{\\widehat A_{m,h}}_{\\mathrm{HS}}^2.}\n\\end{equation}\n\\end{corollary}","\\begin{proof}\nThe retained and omitted block families are orthogonal by \\Cref{thm:hyperedge-parseval}. Since $R_m$ is exactly the sum of the omitted blocks, Pythagoras gives \\eqref{eq:pruning-residual}.\n\\end{proof}","\\subsection{\\texorpdfstring{Retaining the $k$ largest energy hyperedges}{Retaining the k largest energy hyperedges}}\nLet\n\\[\n e_h=\\norm{\\widehat A_{m,h}}_{\\mathrm{HS}}^2\n\\]\nand order the energies\n\\[\n e_{h_1}\\geq e_{h_2}\\geq\\cdots\\geq e_{h_M}.\n\\]","\\begin{theorem}[Optimal retention of $k$ orthogonal hyperedges]\\label{thm:best-k-hyperedges}\nAmong all retained sets of cardinality $k$, the set\n\\[\n \\{h_1,\\ldots,h_k\\}\n\\]\nminimizes the Hilbert Schmidt residual. The minimum is\n\\begin{equation}\\label{eq:best-k-residual}\n \\boxed{\n \\min_{|\\cE|=k}\n \\norm{R_m(\\cE)}_{\\mathrm{HS}}^2\n =\n \\sum_{r>k}e_{h_r}.}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nBy \\eqref{eq:pruning-residual}, minimizing the omitted energy is equivalent to maximizing the retained energy. For a fixed cardinality, this is achieved by selecting the $k$ largest nonnegative terms.\n\\end{proof}","\\subsection{Hyperedge entropy}\nFor nonzero total interaction energy, define\n\\[\n p_h=\\frac{e_h}{\\sum_{\\eta}e_{\\eta}},\n \\qquad\n H_m=-\\sum_hp_h\\log p_h,\n\\]\nand the effective hyperedge count\n\\[\n N_{\\mathrm{eff},m}=e^{H_m}.\n\\]\nLow entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The quantity deliberately depends on the resolution. It evaluates a representation rather than an invariant of the unresolved tensor alone.","\\subsection{Combined quotient and pruning defect}\nLet the full fine coefficient split as\n\\[\n A_m^{\\fine}\n =A_m^{\\mathrm{keep}}+R_m^{\\fine}.\n\\]\nFor a proposed coarse coefficient $A_m^{\\coarse}$, define\n\\[\n \\Delta_m^{\\mathrm{total}}\n =CA_m^{\\fine}-A_m^{\\coarse}C^{\\times m}.\n\\]\nThen\n\\begin{align}\n \\Delta_m^{\\mathrm{total}}\n &=\n \\underbrace{CA_m^{\\mathrm{keep}}-A_m^{\\coarse}C^{\\times m}}\n _{\\Delta_m^{\\mathrm{keep}}}\n +CR_m^{\\fine}.\n \\label{eq:combined-pruning-quotient}\n\\end{align}\nTherefore\n\\begin{equation}\\label{eq:combined-pruning-bound}\n \\boxed{\n \\norm{\\Delta_m^{\\mathrm{total}}}\n \\leq\n \\norm{\\Delta_m^{\\mathrm{keep}}}\n +\\norm{C}\\norm{R_m^{\\fine}}.}\n\\end{equation}","hrm{keep}}+R_m^{\\fine}.\n\\]\nFor a proposed coarse coefficient $A_m^{\\coarse}$, define\n\\[\n \\Delta_m^{\\mathrm{total}}\n =CA_m^{\\fine}-A_m^{\\coarse}C^{\\times m}.\n\\]\nThen\n\\begin{align}\n \\Delta_m^{\\mathrm{total}}\n &=\n \\underbrace{CA_m^{\\mathrm{keep}}-A_m^{\\coarse}C^{\\times m}}\n _{\\Delta_m^{\\mathrm{keep}}}\n +CR_m^{\\fine}.\n \\label{eq:combined-pruning-quotient}\n\\end{align}\nTherefore\n\\begin{equation}\\label{eq:combined-pruning-bound}\n \\boxed{\n \\norm{\\Delta_m^{\\mathrm{total}}}\n \\leq\n \\norm{\\Delta_m^{\\mathrm{keep}}}\n +\\norm{C}\\norm{R_m^{\\fine}}.}\n\\end{equation}","For analytic fields on a ball of radius $R$,\n\\begin{equation}\\label{eq:combined-field-majorant}\n \\mathfrak d_F(R)\n \\leq\n \\sum_m\n \\norm{\\Delta_m^{\\mathrm{keep}}}R^m\n +\n \\norm{C}\\sum_m\\norm{R_m^{\\fine}}R^m.\n\\end{equation}\nThe flow and posterior bounds use this combined quantity. A pruning decision is justified only when the omitted part remains explicit and is carried through the later estimates like any other model error.","\\subsection{A budgeted design problem}\nThese terms lead to a practical design problem. The objective below records the tradeoff without claiming that it can always be solved efficiently:\n\\begin{equation}\\label{eq:budgeted-design}\n \\min_{C,\\{A_m^{\\coarse}\\},\\{\\cE_m^{\\mathrm{keep}}\\}}\n \\left[\n \\sum_mw_m\\norm{\\Delta_m^{\\mathrm{total}}}\n +\\lambda\\sum_m\\operatorname{Cost}(\\cE_m^{\\mathrm{keep}})\n +\\gamma\\sum_y d_{\\Bup,y}\n \\right].\n\\end{equation}\nThe minimization is subject to any required positivity, mass, rank, and interpretability conditions on $C$. The three terms measure failure of dynamical closure, computational cost, and loss of observational information. Their weights and the admissible packet families must be chosen for the application.","an always be solved efficiently:\n\\begin{equation}\\label{eq:budgeted-design}\n \\min_{C,\\{A_m^{\\coarse}\\},\\{\\cE_m^{\\mathrm{keep}}\\}}\n \\left[\n \\sum_mw_m\\norm{\\Delta_m^{\\mathrm{total}}}\n +\\lambda\\sum_m\\operatorname{Cost}(\\cE_m^{\\mathrm{keep}})\n +\\gamma\\sum_y d_{\\Bup,y}\n \\right].\n\\end{equation}\nThe minimization is subject to any required positivity, mass, rank, and interpretability conditions on $C$. The three terms measure failure of dynamical closure, computational cost, and loss of observational information. Their weights and the admissible packet families must be chosen for the application.","\\subsection{Exact test for every interaction order}\nFor a tensor of order $m$ in scalar packet coordinates, compute the aggregated coefficient array\n\\[\n g^{(m)}_{a;j_1,\\ldots,j_m}\n =\n \\sum_{i:\\pi(i)=a}\n a^{\\fine,(m)}_{i;j_1,\\ldots,j_m}.\n\\]\nFor every coarse input tuple $\\mathbf b=(b_1,\\ldots,b_m)$, define the fine fiber product\n\\[\n \\cG_{\\mathbf b}\n =\n \\{(j_1,\\ldots,j_m):\\pi(j_r)=b_r\\text{ for all }r\\}.\n\\]","\\begin{algorithm}[Exact packet coordinate lumpability test]\\label{alg:exact-lumpability}\nFor every coarse output $a$ and coarse input tuple $\\mathbf b$:\n\\begin{enumerate}[label=\\arabic*.,leftmargin=2.3em]\n\\item Evaluate $g^{(m)}_{a;\\mathbf j}$ for all $\\mathbf j\\in\\cG_{\\mathbf b}$.\n\\item In exact arithmetic, test whether all values are equal. With floating point input, compare their spread with a declared tolerance and record that tolerance.\n\\item When the exact test passes, assign the common value to $a^{\\coarse,(m)}_{a;\\mathbf b}$. When only the numerical tolerance is met, assign a declared representative such as the mean or the midrange.\n\\item Otherwise, return a witness $(a,\\mathbf j,\\mathbf j')$ with unequal aggregate coefficients.\n\\end{enumerate}\n\\end{algorithm}","put $a$ and coarse input tuple $\\mathbf b$:\n\\begin{enumerate}[label=\\arabic*.,leftmargin=2.3em]\n\\item Evaluate $g^{(m)}_{a;\\mathbf j}$ for all $\\mathbf j\\in\\cG_{\\mathbf b}$.\n\\item In exact arithmetic, test whether all values are equal. With floating point input, compare their spread with a declared tolerance and record that tolerance.\n\\item When the exact test passes, assign the common value to $a^{\\coarse,(m)}_{a;\\mathbf b}$. When only the numerical tolerance is met, assign a declared representative such as the mean or the midrange.\n\\item Otherwise, return a witness $(a,\\mathbf j,\\mathbf j')$ with unequal aggregate coefficients.\n\\end{enumerate}\n\\end{algorithm}","em In exact arithmetic, test whether all values are equal. With floating point input, compare their spread with a declared tolerance and record that tolerance.\n\\item When the exact test passes, assign the common value to $a^{\\coarse,(m)}_{a;\\mathbf b}$. When only the numerical tolerance is met, assign a declared representative such as the mean or the midrange.\n\\item Otherwise, return a witness $(a,\\mathbf j,\\mathbf j')$ with unequal aggregate coefficients.\n\\end{enumerate}\n\\end{algorithm}","A dense implementation requires $O(n^{m+1})$ arithmetic to aggregate a tensor of order $m$ on $n$ states. If the fine tensor is stored as $E_m$ active scalar hyperedges, the aggregation pass is $O(E_m)$, followed by grouping and consistency checks over the represented coarse incidences.","\\subsection{Approximate coarse coefficients}\nWhen exact fiber constancy fails, the approximation depends on a chosen loss and, when relevant, a task distribution. Under scalar squared loss with nonnegative weights $w_{\\mathbf j}$ on a fiber product, assume their total weight is positive. The least squares representative is\n\\begin{equation}\\label{eq:least-squares-coarse-coeff}\n a^{\\coarse,\\mathrm{LS}}_{a;\\mathbf b}\n =\n \\frac{\n \\sum_{\\mathbf j\\in\\cG_{\\mathbf b}}\n w_{\\mathbf j}g_{a;\\mathbf j}\n }{\n \\sum_{\\mathbf j\\in\\cG_{\\mathbf b}}w_{\\mathbf j}\n }.\n\\end{equation}\nFor scalar worst case loss, a minimax representative is the midrange\n\\begin{equation}\\label{eq:minimax-coarse-coeff}\n a^{\\coarse,\\infty}_{a;\\mathbf b}\n =\n \\frac12\n \\left(\n \\max_{\\mathbf j\\in\\cG_{\\mathbf b}}g_{a;\\mathbf j}\n +\n \\min_{\\mathbf j\\in\\cG_{\\mathbf b}}g_{a;\\mathbf j}\n \\right),\n\\end{equation}\nwith optimal scalar fiber error\n\\begin{equa","\\subsection{Approximate coarse coefficients}\nWhen exact fiber constancy fails, the approximation depends on a chosen loss and, when relevant, a task distribution. Under scalar squared loss with nonnegative weights $w_{\\mathbf j}$ on a fiber product, assume their total weight is positive. The least squares representative is\n\\begin{equation}\\label{eq:least-squares-coarse-coeff}\n a^{\\coarse,\\mathrm{LS}}_{a;\\mathbf b}\n =\n \\frac{\n \\sum_{\\mathbf j\\in\\cG_{\\mathbf b}}\n w_{\\mathbf j}g_{a;\\mathbf j}\n }{\n \\sum_{\\mathbf j\\in\\cG_{\\mathbf b}}w_{\\mathbf j}\n }.\n\\end{equation}\nFor scalar worst case loss, a minimax representative is the midrange\n\\begin{equation}\\label{eq:minimax-coarse-coeff}\n a^{\\coarse,\\infty}_{a;\\mathbf b}\n =\n \\frac12\n \\left(\n \\max_{\\mathbf j\\in\\cG_{\\mathbf b}}g_{a;\\mathbf j}\n +\n \\min_{\\mathbf j\\in\\cG_{\\mathbf b}}g_{a;\\mathbf j}\n \\right),\n\\end{equation}\nwith optimal scalar fiber error\n\\begin{equation}\\label{eq:minimax-fiber-error}\n \\varepsilon^{\\infty}_{a;\\mathbf b}\n =\n \\frac12\n \\left(\n \\max_{\\mathbf j\\in\\cG_{\\mathbf b}}g_{a;\\mathbf j}\n -\n \\min_{\\mathbf j\\i…","r total weight is positive. The least squares representative is\n\\begin{equation}\\label{eq:least-squares-coarse-coeff}\n a^{\\coarse,\\mathrm{LS}}_{a;\\mathbf b}\n =\n \\frac{\n \\sum_{\\mathbf j\\in\\cG_{\\mathbf b}}\n w_{\\mathbf j}g_{a;\\mathbf j}\n }{\n \\sum_{\\mathbf j\\in\\cG_{\\mathbf b}}w_{\\mathbf j}\n }.\n\\end{equation}\nFor scalar worst case loss, a minimax representative is the midrange\n\\begin{equation}\\label{eq:minimax-coarse-coeff}\n a^{\\coarse,\\infty}_{a;\\mathbf b}\n =\n \\frac12\n \\left(\n \\max_{\\mathbf j\\in\\cG_{\\mathbf b}}g_{a;\\mathbf j}\n +\n \\min_{\\mathbf j\\in\\cG_{\\mathbf b}}g_{a;\\mathbf j}\n \\right),\n\\end{equation}\nwith optimal scalar fiber error\n\\begin{equation}\\label{eq:minimax-fiber-error}\n \\varepsilon^{\\infty}_{a;\\mathbf b}\n =\n \\frac12\n \\left(\n \\max_{\\mathbf j\\in\\cG_{\\mathbf b}}g_{a;\\mathbf j}\n -\n \\min_{\\mathbf j\\in\\cG_{\\mathbf b}}g_{a;\\mathbf j}\n \\right).\n\\end{equation}\nThese formulas are local coefficient summaries. A global operator norm can couple several fibers and may require a joint optimization.","athbf b}}w_{\\mathbf j}\n }.\n\\end{equation}\nFor scalar worst case loss, a minimax representative is the midrange\n\\begin{equation}\\label{eq:minimax-coarse-coeff}\n a^{\\coarse,\\infty}_{a;\\mathbf b}\n =\n \\frac12\n \\left(\n \\max_{\\mathbf j\\in\\cG_{\\mathbf b}}g_{a;\\mathbf j}\n +\n \\min_{\\mathbf j\\in\\cG_{\\mathbf b}}g_{a;\\mathbf j}\n \\right),\n\\end{equation}\nwith optimal scalar fiber error\n\\begin{equation}\\label{eq:minimax-fiber-error}\n \\varepsilon^{\\infty}_{a;\\mathbf b}\n =\n \\frac12\n \\left(\n \\max_{\\mathbf j\\in\\cG_{\\mathbf b}}g_{a;\\mathbf j}\n -\n \\min_{\\mathbf j\\in\\cG_{\\mathbf b}}g_{a;\\mathbf j}\n \\right).\n\\end{equation}\nThese formulas are local coefficient summaries. A global operator norm can couple several fibers and may require a joint optimization.","\\subsection{Certificate pipeline}\nThe reference implementation follows the dependency chain\n\\[\n\\begin{array}{c}\n\\text{interaction tensors}\\\\\n\\downarrow\\\\\n\\text{fiber constancy and kernel tests}\\\\\n\\downarrow\\\\\n\\text{coarse tensor and }\\Delta_m\\\\\n\\downarrow\\\\\n\\mathfrak d_F(R)\\text{ and }d_{\\Phi,k}\\\\\n\\downarrow\\\\\n r_{\\lambda,k},z_{*,k},d_{\\Bup,k}\\\\\n\\downarrow\\\\\n e_k^-,e_k^+,\\Err_N.\n\\end{array}\n\\]\nEach numerical record stores the theorem identifier, the inputs, the residual, the tolerance, and the outcome. The suite includes examples built to satisfy the exact identities and perturbations built to violate them.","\\subsection{Deterministic generator for exact examples}\nA coarse tensor can be lifted to a fine tensor by\n\\begin{equation}\\label{eq:exact-instance-generator}\n A_m^{\\fine}\n =\n EA_m^{\\coarse}C^{\\times m}+K_m,\n \\qquad\n CK_m=0.\n\\end{equation}\nThe term $K_m$ can contain complicated hidden interactions without changing the quotient. This makes the construction useful for testing because the correct coarse tensor is known before any numerical calculation.","\\begin{proposition}[Correctness of the exact example generator]\\label{prop:exact-instance-generator}\nEvery tensor constructed by \\eqref{eq:exact-instance-generator} satisfies\n\\[\n CA_m^{\\fine}=A_m^{\\coarse}C^{\\times m}.\n\\]\nConversely, every exact fine tensor admits such a representation for every right inverse $E$, with\n\\[\n K_m=A_m^{\\fine}-EA_m^{\\coarse}C^{\\times m}.\n\\]\n\\end{proposition}","\\begin{proof}\nThe forward identity follows from $CE=I$ and $CK_m=0$. For the converse, exact descent gives\n\\[\n CK_m\n =CA_m^{\\fine}-CEA_m^{\\coarse}C^{\\times m}=0.\n\\]\n\\end{proof}","\\subsection{A nonlinear fine system with four states and a quotient with two states}\nLet\n\\[\n x=(x_1,x_2,x_3,x_4)^T\\in\\Delta_4,\n \\qquad\n C=\n \\begin{bmatrix}\n 1&1&0&0\\\\\n 0&0&1&1\n \\end{bmatrix},\n\\]\nso\n\\[\n z=Cx=(z_1,z_2)^T,\n \\qquad\n z_1=x_1+x_2,\n \\quad\n z_2=x_3+x_4.\n\\]\nThe coarse dynamics are\n\\begin{equation}\\label{eq:coarse-two-state}\n \\dot z_1=-az_1+bz_2,\n \\qquad\n \\dot z_2=az_1-bz_2,\n \\qquad a,b>0.\n\\end{equation}\nDefine nonlinear internal exchange channels\n\\begin{align}\n h_{12}(x)\n &=\n (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,\n \\label{eq:h12}","0&0&1&1\n \\end{bmatrix},\n\\]\nso\n\\[\n z=Cx=(z_1,z_2)^T,\n \\qquad\n z_1=x_1+x_2,\n \\quad\n z_2=x_3+x_4.\n\\]\nThe coarse dynamics are\n\\begin{equation}\\label{eq:coarse-two-state}\n \\dot z_1=-az_1+bz_2,\n \\qquad\n \\dot z_2=az_1-bz_2,\n \\qquad a,b>0.\n\\end{equation}\nDefine nonlinear internal exchange channels\n\\begin{align}\n h_{12}(x)\n &=\n (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,\n \\label{eq:h12}","0&0&1&1\n \\end{bmatrix},\n\\]\nso\n\\[\n z=Cx=(z_1,z_2)^T,\n \\qquad\n z_1=x_1+x_2,\n \\quad\n z_2=x_3+x_4.\n\\]\nThe coarse dynamics are\n\\begin{equation}\\label{eq:coarse-two-state}\n \\dot z_1=-az_1+bz_2,\n \\qquad\n \\dot z_2=az_1-bz_2,\n \\qquad a,b>0.\n\\end{equation}\nDefine nonlinear internal exchange channels\n\\begin{align}\n h_{12}(x)\n &=\n (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,\n \\label{eq:h12}\\\\\n h_{34}(x)\n &=\n (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3,\n \\label{eq:h34}\n\\end{align}\nwith all coefficients nonnegative. The fine system is\n\\begin{equation}\\label{eq:fine-four-state}\n\\begin{aligned}\n \\dot x_1&=-ax_1+bx_3+h_{12}(x),","h_{34}(x)\n &=\n (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3,\n \\label{eq:h34}\n\\end{align}\nwith all coefficients nonnegative. The fine system is\n\\begin{equation}\\label{eq:fine-four-state}\n\\begin{aligned}\n \\dot x_1&=-ax_1+bx_3+h_{12}(x),\\\\\n \\dot x_2&=-ax_2+bx_4-h_{12}(x),\\\\\n \\dot x_3&= ax_1-bx_3+h_{34}(x),\\\\\n \\dot x_4&= ax_2-bx_4-h_{34}(x).\n\\end{aligned}\n\\end{equation}","\\begin{proposition}[Exact quotient with nonlinear hidden exchange]\\label{prop:hidden-exchange-quotient}\nThe system \\eqref{eq:fine-four-state} preserves total mass and the nonnegative orthant. After aggregation by $C$, it satisfies the coarse system \\eqref{eq:coarse-two-state} exactly.\n\\end{proposition}","\\begin{proof}\nSumming the four equations gives zero. To check positivity, set one coordinate equal to zero. Every term entering its derivative is then nonnegative. This holds for $x_1$, $x_2$, $x_3$, and $x_4$, so the nonnegative orthant is forward invariant. Finally,\n\\begin{align*}\n \\dot x_1+\\dot x_2\n &=-a(x_1+x_2)+b(x_3+x_4),","\\begin{proof}\nSumming the four equations gives zero. To check positivity, set one coordinate equal to zero. Every term entering its derivative is then nonnegative. This holds for $x_1$, $x_2$, $x_3$, and $x_4$, so the nonnegative orthant is forward invariant. Finally,\n\\begin{align*}\n \\dot x_1+\\dot x_2\n &=-a(x_1+x_2)+b(x_3+x_4),\\\\\n \\dot x_3+\\dot x_4\n &=a(x_1+x_2)-b(x_3+x_4),\n\\end{align*}\nso the internal nonlinear channels cancel under $C$ and the aggregate obeys \\eqref{eq:coarse-two-state}.\n\\end{proof}","\\dot x_3+\\dot x_4\n &=a(x_1+x_2)-b(x_3+x_4),\n\\end{align*}\nso the internal nonlinear channels cancel under $C$ and the aggregate obeys \\eqref{eq:coarse-two-state}.\n\\end{proof}","The quadratic terms in \\eqref{eq:h12} and \\eqref{eq:h34} are genuine fine hyperedges. Their output vectors lie in the internal directions of the two fibers,\n\\[\n (1,-1,0,0)^T,\n \\qquad\n (0,0,1,-1)^T,\n\\]\nwhich are contained in $\\Ker C$. They alter the fine distribution while remaining invisible to the coarse state. \\Cref{fig:exact-quotient-flow} shows this distinction. The two fine states inside the first fiber move differently, yet their sum follows the coarse trajectory.","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.82\\linewidth]{exact_quotient_flow.pdf}\n\\caption{Hidden fine motion under an exact quotient. The microstates $x_1$ and $x_2$ evolve separately. Their sum agrees with the coarse coordinate $z_1$ throughout the trajectory. Sparse markers on the coarse curve make the overlap visible.}\n\\label{fig:exact-quotient-flow}\n\\end{figure}","\\subsection{Exact nonlinear filtering}\nChoose a coarse likelihood\n\\[\n \\lambda^{\\coarse}=(\\lambda_1,\\lambda_2)^T>0\n\\]\nand lift it by\n\\[\n \\lambda^{\\fine}=C^T\\lambda^{\\coarse}\n =(\\lambda_1,\\lambda_1,\\lambda_2,\\lambda_2)^T.\n\\]\nBy Proposition~\\ref{prop:hidden-exchange-quotient} together with \\Cref{thm:bayes-sufficient,thm:exact-cocycle}, every finite sequence of flow segments and such observations satisfies\n\\[\n C\\mu_k^{\\fine}=\\mu_k^{\\coarse}.\n\\]\nThe fine filter may continue to distinguish $x_1$ from $x_2$ and $x_3$ from $x_4$, but those distinctions do not affect the exact coarse posterior.","{Exact nonlinear filtering}\nChoose a coarse likelihood\n\\[\n \\lambda^{\\coarse}=(\\lambda_1,\\lambda_2)^T>0\n\\]\nand lift it by\n\\[\n \\lambda^{\\fine}=C^T\\lambda^{\\coarse}\n =(\\lambda_1,\\lambda_1,\\lambda_2,\\lambda_2)^T.\n\\]\nBy Proposition~\\ref{prop:hidden-exchange-quotient} together with \\Cref{thm:bayes-sufficient,thm:exact-cocycle}, every finite sequence of flow segments and such observations satisfies\n\\[\n C\\mu_k^{\\fine}=\\mu_k^{\\coarse}.\n\\]\nThe fine filter may continue to distinguish $x_1$ from $x_2$ and $x_3$ from $x_4$, but those distinctions do not affect the exact coarse posterior.","bda_1,\\lambda_2)^T>0\n\\]\nand lift it by\n\\[\n \\lambda^{\\fine}=C^T\\lambda^{\\coarse}\n =(\\lambda_1,\\lambda_1,\\lambda_2,\\lambda_2)^T.\n\\]\nBy Proposition~\\ref{prop:hidden-exchange-quotient} together with \\Cref{thm:bayes-sufficient,thm:exact-cocycle}, every finite sequence of flow segments and such observations satisfies\n\\[\n C\\mu_k^{\\fine}=\\mu_k^{\\coarse}.\n\\]\nThe fine filter may continue to distinguish $x_1$ from $x_2$ and $x_3$ from $x_4$, but those distinctions do not affect the exact coarse posterior.","da_2)^T>0\n\\]\nand lift it by\n\\[\n \\lambda^{\\fine}=C^T\\lambda^{\\coarse}\n =(\\lambda_1,\\lambda_1,\\lambda_2,\\lambda_2)^T.\n\\]\nBy Proposition~\\ref{prop:hidden-exchange-quotient} together with \\Cref{thm:bayes-sufficient,thm:exact-cocycle}, every finite sequence of flow segments and such observations satisfies\n\\[\n C\\mu_k^{\\fine}=\\mu_k^{\\coarse}.\n\\]\nThe fine filter may continue to distinguish $x_1$ from $x_2$ and $x_3$ from $x_4$, but those distinctions do not affect the exact coarse posterior.","\\]\nand lift it by\n\\[\n \\lambda^{\\fine}=C^T\\lambda^{\\coarse}\n =(\\lambda_1,\\lambda_1,\\lambda_2,\\lambda_2)^T.\n\\]\nBy Proposition~\\ref{prop:hidden-exchange-quotient} together with \\Cref{thm:bayes-sufficient,thm:exact-cocycle}, every finite sequence of flow segments and such observations satisfies\n\\[\n C\\mu_k^{\\fine}=\\mu_k^{\\coarse}.\n\\]\nThe fine filter may continue to distinguish $x_1$ from $x_2$ and $x_3$ from $x_4$, but those distinctions do not affect the exact coarse posterior.","it by\n\\[\n \\lambda^{\\fine}=C^T\\lambda^{\\coarse}\n =(\\lambda_1,\\lambda_1,\\lambda_2,\\lambda_2)^T.\n\\]\nBy Proposition~\\ref{prop:hidden-exchange-quotient} together with \\Cref{thm:bayes-sufficient,thm:exact-cocycle}, every finite sequence of flow segments and such observations satisfies\n\\[\n C\\mu_k^{\\fine}=\\mu_k^{\\coarse}.\n\\]\nThe fine filter may continue to distinguish $x_1$ from $x_2$ and $x_3$ from $x_4$, but those distinctions do not affect the exact coarse posterior.","\\begin{certificate}[Exact nonlinear filtering residuals]\\label{cert:exact-filter-residuals}\nFor the deterministic run with four states, the largest aggregate prediction residual was $\\ExactPredictionResidual$. The largest posterior residual after an update was $\\ExactPosteriorResidual$, and the largest residual in a single evidence factor was $\\ExactEvidenceResidual$. The small nonzero values come from floating point integration and arithmetic in the finite example.\n\\end{certificate}","\\subsection{A controlled dynamical violation}\nAdd a directed transition between fine states\n\\begin{equation}\\label{eq:dynamical-perturbation}\n \\varepsilon x_1\n \\begin{bmatrix}\n -1\\\\0\\\\1\\\\0\n \\end{bmatrix}\n\\end{equation}\nto the fine vector field. This perturbation preserves nonnegativity and total mass, but the added transfer between fibers is activated only by the mass in state $x_1$. The aggregate field defect is\n\\begin{equation}\\label{eq:perturbation-field-defect}\n \\delta_F(x)\n =\n \\varepsilon x_1\n \\begin{bmatrix}-1","0\n \\end{bmatrix}\n\\end{equation}\nto the fine vector field. This perturbation preserves nonnegativity and total mass, but the added transfer between fibers is activated only by the mass in state $x_1$. The aggregate field defect is\n\\begin{equation}\\label{eq:perturbation-field-defect}\n \\delta_F(x)\n =\n \\varepsilon x_1\n \\begin{bmatrix}-1","0\n \\end{bmatrix}\n\\end{equation}\nto the fine vector field. This perturbation preserves nonnegativity and total mass, but the added transfer between fibers is activated only by the mass in state $x_1$. The aggregate field defect is\n\\begin{equation}\\label{eq:perturbation-field-defect}\n \\delta_F(x)\n =\n \\varepsilon x_1\n \\begin{bmatrix}-1\\\\1\\end{bmatrix},\n\\end{equation}\nso, on the simplex,\n\\begin{equation}\\label{eq:perturbation-defect-bound}\n \\norm{\\delta_F(x)}_1\n =2\\varepsilon x_1\n \\leq2\\varepsilon.\n\\end{equation}\nThe coarse Markov flow on two states is an $\\ell^1$ contraction. Over one interval, this gives\n\\[\n \\norm{C\\Phi_T^{\\fine,\\varepsilon}(x)-\\Phi_T^{\\coarse}(Cx)}_1\n \\leq2\\varepsilon T.\n\\]","1\\end{bmatrix},\n\\end{equation}\nso, on the simplex,\n\\begin{equation}\\label{eq:perturbation-defect-bound}\n \\norm{\\delta_F(x)}_1\n =2\\varepsilon x_1\n \\leq2\\varepsilon.\n\\end{equation}\nThe coarse Markov flow on two states is an $\\ell^1$ contraction. Over one interval, this gives\n\\[\n \\norm{C\\Phi_T^{\\fine,\\varepsilon}(x)-\\Phi_T^{\\coarse}(Cx)}_1\n \\leq2\\varepsilon T.\n\\]","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.78\\linewidth]{flow_perturbation_scaling.pdf}\n\\caption{Aggregate flow error across a logarithmic perturbation sweep. The measured discrepancy has first order scaling and remains below the certificate $2\\varepsilon T$ from \\eqref{eq:perturbation-defect-bound}.}\n\\label{fig:flow-perturbation-scaling}\n\\end{figure}","\\subsection{A controlled observation violation}\nPerturb the fine likelihood inside one fiber:\n\\begin{equation}\\label{eq:likelihood-perturbation-example}\n \\lambda^{\\fine,\\eta}\n =\n C^T\\lambda^{\\coarse}\n +\n \\eta(1,-1,0,0)^T.\n\\end{equation}\nAssume $\\abs{\\eta}<\\lambda_1^{\\coarse}$, so every component of $\\lambda^{\\fine,\\eta}$ remains strictly positive. The fiber oscillation is $2\\abs{\\eta}$ and\n\\[\n \\norm{r_{\\lambda}}_{\\infty}=\\abs{\\eta}.\n\\]\nIf the evidence is bounded below by $z_*>0$, then\n\\[\n d_{\\Bup}\n \\leq\n \\frac{2\\abs{\\eta}}{z_*}.\n\\]\nThe sweep in \\Cref{fig:bayesian-perturbation-scaling} varies only the observation model and isolates this effect at one interior prior. The measured discrepancy is first order in $\\eta$ and remains below the likelihood certificate.","subsection{A controlled observation violation}\nPerturb the fine likelihood inside one fiber:\n\\begin{equation}\\label{eq:likelihood-perturbation-example}\n \\lambda^{\\fine,\\eta}\n =\n C^T\\lambda^{\\coarse}\n +\n \\eta(1,-1,0,0)^T.\n\\end{equation}\nAssume $\\abs{\\eta}<\\lambda_1^{\\coarse}$, so every component of $\\lambda^{\\fine,\\eta}$ remains strictly positive. The fiber oscillation is $2\\abs{\\eta}$ and\n\\[\n \\norm{r_{\\lambda}}_{\\infty}=\\abs{\\eta}.\n\\]\nIf the evidence is bounded below by $z_*>0$, then\n\\[\n d_{\\Bup}\n \\leq\n \\frac{2\\abs{\\eta}}{z_*}.\n\\]\nThe sweep in \\Cref{fig:bayesian-perturbation-scaling} varies only the observation model and isolates this effect at one interior prior. The measured discrepancy is first order in $\\eta$ and remains below the likelihood certificate.","ne fiber:\n\\begin{equation}\\label{eq:likelihood-perturbation-example}\n \\lambda^{\\fine,\\eta}\n =\n C^T\\lambda^{\\coarse}\n +\n \\eta(1,-1,0,0)^T.\n\\end{equation}\nAssume $\\abs{\\eta}<\\lambda_1^{\\coarse}$, so every component of $\\lambda^{\\fine,\\eta}$ remains strictly positive. The fiber oscillation is $2\\abs{\\eta}$ and\n\\[\n \\norm{r_{\\lambda}}_{\\infty}=\\abs{\\eta}.\n\\]\nIf the evidence is bounded below by $z_*>0$, then\n\\[\n d_{\\Bup}\n \\leq\n \\frac{2\\abs{\\eta}}{z_*}.\n\\]\nThe sweep in \\Cref{fig:bayesian-perturbation-scaling} varies only the observation model and isolates this effect at one interior prior. The measured discrepancy is first order in $\\eta$ and remains below the likelihood certificate.","n{equation}\\label{eq:likelihood-perturbation-example}\n \\lambda^{\\fine,\\eta}\n =\n C^T\\lambda^{\\coarse}\n +\n \\eta(1,-1,0,0)^T.\n\\end{equation}\nAssume $\\abs{\\eta}<\\lambda_1^{\\coarse}$, so every component of $\\lambda^{\\fine,\\eta}$ remains strictly positive. The fiber oscillation is $2\\abs{\\eta}$ and\n\\[\n \\norm{r_{\\lambda}}_{\\infty}=\\abs{\\eta}.\n\\]\nIf the evidence is bounded below by $z_*>0$, then\n\\[\n d_{\\Bup}\n \\leq\n \\frac{2\\abs{\\eta}}{z_*}.\n\\]\nThe sweep in \\Cref{fig:bayesian-perturbation-scaling} varies only the observation model and isolates this effect at one interior prior. The measured discrepancy is first order in $\\eta$ and remains below the likelihood certificate.","emains strictly positive. The fiber oscillation is $2\\abs{\\eta}$ and\n\\[\n \\norm{r_{\\lambda}}_{\\infty}=\\abs{\\eta}.\n\\]\nIf the evidence is bounded below by $z_*>0$, then\n\\[\n d_{\\Bup}\n \\leq\n \\frac{2\\abs{\\eta}}{z_*}.\n\\]\nThe sweep in \\Cref{fig:bayesian-perturbation-scaling} varies only the observation model and isolates this effect at one interior prior. The measured discrepancy is first order in $\\eta$ and remains below the likelihood certificate.","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.78\\linewidth]{bayesian_perturbation_scaling.pdf}\n\\caption{Posterior discrepancy caused by variation of the likelihood inside one aggregation fiber. The error scales linearly with $\\eta$ in the tested range and remains below the bound obtained from the normalization estimate.}\n\\label{fig:bayesian-perturbation-scaling}\n\\end{figure}","Combining this observation perturbation with \\eqref{eq:dynamical-perturbation} gives a posterior discrepancy that is first order in $(\\varepsilon,\\eta)$ over a fixed finite horizon. \\Cref{thm:posterior-certificate} supplies a conservative upper bound.","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.78\\linewidth]{posterior_error_certificate.pdf}\n\\caption{Joint dynamical and likelihood perturbation sweep with $\\varepsilon=\\eta=\\rho$. For small $\\rho$, the final posterior discrepancy grows linearly, while the recursive certificate remains above every measured value.}\n\\label{fig:posterior-error-certificate}\n\\end{figure}","\\begin{certificate}[Perturbation scaling]\\label{cert:perturbation-scaling}\nThe experiment used logarithmically spaced values with $\\varepsilon=\\eta$. The fitted slope of the final posterior discrepancy was $\\PerturbationSlope$. Every measured error remained below its recursive certificate.\n\\end{certificate}","\\subsection{Random exact tensors through order three}\nThe implementation draws random coarse tensors of orders $m=1,2,3$, lifts them through \\eqref{eq:exact-instance-generator}, and adds random hidden output tensors $K_m$ with $CK_m=0$. It then checks:\n\\[\n CA_m^{\\fine}-A_m^{\\coarse}C^{\\times m}=0,\n\\]\nannihilation of hidden inputs, constancy on packet fibers, and recovery of the original coarse tensor. A single coefficient perturbation provides a negative control that must violate fiber constancy.","\\subsection{Concentrated hyperedge field}\nA synthetic interaction field with decreasing orthogonal block energies illustrates \\Cref{thm:best-k-hyperedges}. The computed hyperedge entropy is $\\HyperedgeEntropyValue$, corresponding to an effective count of approximately $\\EffectiveHyperedgeCount$ hyperedges despite a larger ambient support. The ranked spectrum in \\Cref{fig:hyperedge-energy-spectrum} shows why the field is compressible in this representation.","\\begin{figure}[H]\n\\centering\n\\begin{subfigure}[t]{0.48\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{hyperedge_energy_spectrum.pdf}\n\\caption{Ranked normalized hyperedge energy.}\n\\label{fig:hyperedge-energy-spectrum}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[t]{0.48\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{hyperedge_pruning.pdf}\n\\caption{Residual after retaining the $k$ largest energy hyperedges.}\n\\label{fig:hyperedge-pruning}\n\\end{subfigure}\n\\caption{Energy concentration and certified pruning in the synthetic interaction field. A small leading set carries most of the energy, and the residual decreases to zero when the complete support is restored.}\n\\label{fig:hyperedge-compression}\n\\end{figure}","\\subsection{Interpretation for reaction networks}\nConsider a polynomial reaction system\n\\[\n \\dot x=S v(x),\n \\qquad\n v_r(x)=k_r\\prod_i x_i^{\\alpha_{ir}}.\n\\]\nEach reaction monomial contributes one interaction hyperedge to every species changed by the corresponding stoichiometric column. The inputs are the reactants. A block sum aggregation is exact precisely when the monomial coefficients, after summation over each output fiber, are constant on the product fibers specified by \\Cref{thm:packet-coordinate-criterion}. This gives a coefficient test at every interaction order. Specialized reductions of reaction networks may also use stoichiometric, kinetic, or bisimulation structure \\cite{Cardelli2015,Cardelli2017,BaezPollard2017}.","\\begin{boundary}[Physical interpretation]\nThe quotient theorem is algebraic. It does not by itself decide whether a packet represents a species class, spatial region, symmetry sector, latent state, or learned feature. The scientific meaning of $C$ must come from the application and is not created by exact factorization alone.\n\\end{boundary}","\\subsection{Markov lumpability and Markov functions}\nFor unary stochastic dynamics, the identity\n\\[\n CQ_{\\fine}=Q_{\\coarse}C\n\\]\nis a standard exact lumping condition. It implies\n\\[\n Ce^{tQ_{\\fine}}=e^{tQ_{\\coarse}}C\n\\]\nand means that the coarse process is Markov for every fine initial distribution compatible with the quotient. Classical treatments distinguish several forms of strong, ordinary, exact, and weak lumpability, depending on the admissible initial laws and the direction of the stochastic convention \\cite{KemenySnell1960,RogersPitman1981,Buchholz1994}. Here we fix one column vector convention and extend the coefficient criterion from unary edges to multilinear hyperedges.","\\subsection{Polynomial differential equivalence}\nExact aggregation of polynomial vector fields has been developed through differential equivalences, bisimulations, and algorithms based on successive partition refinement, including applications to chemical reaction networks \\cite{Cardelli2015,Cardelli2017}. The universal factorization theorem here is elementary multilinear algebra once $C$ is fixed. Its value is that it exposes every interaction order, makes the quotient condition independent of a particular polynomial syntax, and connects the same coefficient criterion to rooted tree composition and Bayesian updates. We do not claim an algorithmic advantage over specialized methods based on partition refinement.","\\begin{boundary}[A prescribed coarse map]\nThe principal theorems assume that $C$ is supplied. They test and quantify whether a proposed aggregation is valid. Discovering an optimal $C$ from data or from a large interaction tensor is a separate problem and can be combinatorial, nonconvex, or statistically nonidentifiable.\n\\end{boundary}","\\subsection{Coarsest exact joint partition}\nFor a finite interaction family $(A_m^{\\fine})_{m\\leq M}$ and likelihood family $(\\lambda_y^{\\fine})_{y\\in\\cY}$, seek\n\\begin{equation}\\label{eq:coarsest-joint-partition}\n \\min_{\\pi}|I_{\\coarse}|\n\\end{equation}\nsubject to\n\\[\n CA_m^{\\fine}=A_m^{\\coarse}C^{\\times m}\n \\quad(m\\leq M),\n\\]\nand\n\\[\n \\lambda_y^{\\fine}=s_yC^T\\lambda_y^{\\coarse}\n \\quad(y\\in\\cY).\n\\]\nThe partition determined by the observations alone groups states with identical likelihood profiles. The dynamical condition also depends on the partition because changing the partition changes the output aggregation. A terminating refinement algorithm, together with a useful complexity bound, remains an open problem.","\\subsection{Approximate joint design}\nA finite approximate problem is\n\\[\n \\min_{C,\\{A_m^{\\coarse}\\},\\{\\lambda_y^{\\coarse}\\}}\n \\left[\n \\sum_mw_m\\norm{CA_m^{\\fine}-A_m^{\\coarse}C^{\\times m}}\n +\\sum_yv_yd_{\\Bup,y}\n +\\gamma\\operatorname{Complexity}(C)\n \\right].\n\\]\nThe main questions concern existence, tractable relaxations, and stability of the minimizers. Another question is whether the task weights can be learned without invalidating the certificate after a distribution shift.","\\subsection{Posterior distributions over resolutions}\nInstead of selecting one $C$, place a posterior on a finite family of candidate resolutions:\n\\[\n p(C\\mid D)\n \\propto\n p(D\\mid C)p(C).\n\\]\nPredictions then average both parameter and topology uncertainty:\n\\[\n p(z_t\\mid D)\n =\n \\sum_Cp(z_t\\mid C,D)p(C\\mid D).\n\\]\nA fuller model should distinguish uncertainty in the coefficients of a fixed hypergraph from uncertainty about the hypergraph and the quotient map themselves.","\\subsection{Stochastic interaction quotients}\nFor a diffusion\n\\[\n \\dd X_t=b(X_t)\\dd t+\\sigma(X_t)\\dd W_t,\n\\]\nthe generator is\n\\[\n \\cL\\varphi\n =D\\varphi[b]\n +\\frac12\\tr\\bigl(\\sigma\\sigma^TD^2\\varphi\\bigr).\n\\]\nPacket resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let $\\cC$ denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition is generator intertwining\n\\[\n \\cC\\cL_{\\fine}=\\cL_{\\coarse}\\cC\n\\]\nfor the pullback or averaging map appropriate to observables. A complete theorem would characterize closure of the drift and covariance, prove positivity of the induced semigroup, and carry generator defects into filtering error bounds.","\\dd X_t=b(X_t)\\dd t+\\sigma(X_t)\\dd W_t,\n\\]\nthe generator is\n\\[\n \\cL\\varphi\n =D\\varphi[b]\n +\\frac12\\tr\\bigl(\\sigma\\sigma^TD^2\\varphi\\bigr).\n\\]\nPacket resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let $\\cC$ denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition is generator intertwining\n\\[\n \\cC\\cL_{\\fine}=\\cL_{\\coarse}\\cC\n\\]\nfor the pullback or averaging map appropriate to observables. A complete theorem would characterize closure of the drift and covariance, prove positivity of the induced semigroup, and carry generator defects into filtering error bounds.","\\subsection{Coarse dynamics with memory}\nEliminating hidden packets generally produces memory. The target form is\n\\[\n \\dot z(t)\n =A_{PP}z(t)\n +\\int_0^tK(t-s)z(s)\\dd s\n +\\eta(t).\n\\]\nThe quotient theory should allow a fine system without memory to produce coarse dynamics with memory. Volterra hyperedges and fractional kernels would then be treated as native coarse interactions rather than as evidence that quotienting failed.","\\subsection{Sharper filtering metrics}\nThe $\\ell^1$ normalization bounds are robust but conservative. Positive operators often contract projective metrics, relative entropy, or other information geometric divergences more sharply. One goal is to replace the factor\n\\[\n \\frac{2\\norm{\\lambda}_{\\infty}}{z_*}\n\\]\nby a contraction coefficient adapted to the trajectory or to the operator, while retaining an auditable finite certificate.","\\subsection{Identifiability of hidden dynamics}\nFor fixed $C$ and coarse field $F_{\\coarse}$, the exact fine realizations form the affine family\n\\[\n \\{F_{\\fine}:CF_{\\fine}=F_{\\coarse}C\\}.\n\\]\nAny two exact fine models differ by a map into $\\Ker C$. Coarse observations alone cannot identify that hidden component. Additional fine observations, intervention assumptions, regularity, or minimum complexity principles are required. A statistical theory should characterize precisely which hidden hyperedges are identifiable under a given observation and intervention design.","\\section{Conclusion}\nAn exact aggregation is more than a useful summary of a fine model. It is a quotient. Fine states with the same coarse image must produce the same coarse output. For a multilinear interaction, this reduces to a complete finite test. After applying $C$, the interaction must vanish whenever any input lies in $\\Ker C$. In packet coordinates, the coefficients summed over each output fiber must be constant on every product of input fibers. The familiar unary criterion for Markov lumpability is the first order case of this general condition.","The theory does not show that an optimal quotient can always be found cheaply. Coarse observations alone also cannot determine hidden dynamics that lie in $\\Ker C$. Those questions remain open. The result proved here is narrower and exact:\n\\begin{center}\n\\fbox{\\begin{minipage}{0.88\\linewidth}\nA coarse model is exact when its interactions and observations descend through the same quotient. When they do not, each failure can be measured at the stage where it occurs and carried forward to the final posterior or observable by an explicit certificate.\n\\end{minipage}}\n\\end{center}","\\section{Formulation on the quotient space}\\label{app:quotient}\nLet\n\\[\n q:X_{\\fine}\\to X_{\\fine}/\\Ker C\n\\]\nbe the canonical quotient map. Since $C$ is surjective, there is a unique vector space isomorphism\n\\[\n \\widetilde C:X_{\\fine}/\\Ker C\\to X_{\\coarse}\n\\]\nsatisfying\n\\[\n C=\\widetilde Cq.\n\\]\nThe multilinear map $CA_m^{\\fine}:X_{\\fine}^m\\to X_{\\coarse}$ descends if and only if its value is unchanged when any one input is altered by an element of $\\Ker C$. The induced quotient tensor is\n\\[\n \\overline A_m\n ([x_1],\\ldots,[x_m])\n =CA_m^{\\fine}(x_1,\\ldots,x_m),\n\\]\nand\n\\[\n A_m^{\\coarse}\n =\n \\widetilde C\\,\n \\overline A_m\n (\\widetilde C^{-1},\\ldots,\\widetilde C^{-1}).\n\\]\nThis formulation shows that \\Cref{thm:universal-quotient} is independent of coordinates and of the chosen right inverse.","quotient}\nLet\n\\[\n q:X_{\\fine}\\to X_{\\fine}/\\Ker C\n\\]\nbe the canonical quotient map. Since $C$ is surjective, there is a unique vector space isomorphism\n\\[\n \\widetilde C:X_{\\fine}/\\Ker C\\to X_{\\coarse}\n\\]\nsatisfying\n\\[\n C=\\widetilde Cq.\n\\]\nThe multilinear map $CA_m^{\\fine}:X_{\\fine}^m\\to X_{\\coarse}$ descends if and only if its value is unchanged when any one input is altered by an element of $\\Ker C$. The induced quotient tensor is\n\\[\n \\overline A_m\n ([x_1],\\ldots,[x_m])\n =CA_m^{\\fine}(x_1,\\ldots,x_m),\n\\]\nand\n\\[\n A_m^{\\coarse}\n =\n \\widetilde C\\,\n \\overline A_m\n (\\widetilde C^{-1},\\ldots,\\widetilde C^{-1}).\n\\]\nThis formulation shows that \\Cref{thm:universal-quotient} is independent of coordinates and of the chosen right inverse.","/\\Ker C\n\\]\nbe the canonical quotient map. Since $C$ is surjective, there is a unique vector space isomorphism\n\\[\n \\widetilde C:X_{\\fine}/\\Ker C\\to X_{\\coarse}\n\\]\nsatisfying\n\\[\n C=\\widetilde Cq.\n\\]\nThe multilinear map $CA_m^{\\fine}:X_{\\fine}^m\\to X_{\\coarse}$ descends if and only if its value is unchanged when any one input is altered by an element of $\\Ker C$. The induced quotient tensor is\n\\[\n \\overline A_m\n ([x_1],\\ldots,[x_m])\n =CA_m^{\\fine}(x_1,\\ldots,x_m),\n\\]\nand\n\\[\n A_m^{\\coarse}\n =\n \\widetilde C\\,\n \\overline A_m\n (\\widetilde C^{-1},\\ldots,\\widetilde C^{-1}).\n\\]\nThis formulation shows that \\Cref{thm:universal-quotient} is independent of coordinates and of the chosen right inverse.","que vector space isomorphism\n\\[\n \\widetilde C:X_{\\fine}/\\Ker C\\to X_{\\coarse}\n\\]\nsatisfying\n\\[\n C=\\widetilde Cq.\n\\]\nThe multilinear map $CA_m^{\\fine}:X_{\\fine}^m\\to X_{\\coarse}$ descends if and only if its value is unchanged when any one input is altered by an element of $\\Ker C$. The induced quotient tensor is\n\\[\n \\overline A_m\n ([x_1],\\ldots,[x_m])\n =CA_m^{\\fine}(x_1,\\ldots,x_m),\n\\]\nand\n\\[\n A_m^{\\coarse}\n =\n \\widetilde C\\,\n \\overline A_m\n (\\widetilde C^{-1},\\ldots,\\widetilde C^{-1}).\n\\]\nThis formulation shows that \\Cref{thm:universal-quotient} is independent of coordinates and of the chosen right inverse.","\\section{A finite observational partition lemma}\\label{app:observation-partition}\nLet $\\cY$ be a finite observation alphabet and let each state $i$ have a strictly positive likelihood profile\n\\[\n \\Lambda(i)\n =\n (\\lambda_y(i))_{y\\in\\cY}.\n\\]\nDefine\n\\[\n i\\sim_{\\mathrm{obs}}j\n \\quad\\Longleftrightarrow\\quad\n \\Lambda(i)=\\Lambda(j).\n\\]","\\begin{proposition}[Coarsest deterministic observation partition]\\label{prop:coarsest-observation-partition}\nThe equivalence classes of $\\sim_{\\mathrm{obs}}$ form the coarsest partition for which every likelihood in the family is constant on every block. Consequently, this partition is the coarsest deterministic block sum aggregation satisfying exact Bayesian refinement for all $y\\in\\cY$, up to one global positive scale for each observation.\n\\end{proposition}","\\begin{proof}\nEvery likelihood is constant on an equivalence class by definition. Any partition on which every likelihood is constant on each block can place $i$ and $j$ in the same block only if all profile coordinates agree, so it refines the equivalence class partition. The Bayesian statement follows from \\Cref{thm:bayes-sufficient,thm:bayes-necessity}.\n\\end{proof}","\\section{Certificate schemas and reproducibility}\\label{app:schemas}\nA minimal interaction certificate stores\n\\begin{equation}\\label{eq:lumpability-certificate-schema}\n \\mathsf L_m\n =\n (m,C,A_m^{\\fine},A_m^{\\coarse},\\Delta_m,\n \\norm{\\Delta_m},\\text{norm},\\text{tolerance}).\n\\end{equation}\nA Bayesian certificate stores\n\\begin{equation}\\label{eq:bayes-certificate-schema}\n \\mathsf B_y\n =\n (\\lambda_y^{\\fine},\\lambda_y^{\\coarse},s_y,r_y,\n z_{*,y},d_{\\Bup,y}).\n\\end{equation}\nA filtering certificate stores\n\\begin{equation}\\label{eq:filter-certificate-schema}\n \\mathsf F_N\n =\n (e_0^+,\\{L_{\\Phi,k}\\},\\{L_{\\Bup,k}\\},\n \\{d_{\\Phi,k}\\},\\{d_{\\Bup,k}\\},\n \\{e_k^-,e_k^+\\}).\n\\end{equation}\nThe release serializes the executed finite ledger as JSON files and CSV tables. A certificate records a computation under specified inputs. The ledger provides cryptographic authenticity only when it is paired with a separate signed digest layer.","abel{eq:lumpability-certificate-schema}\n \\mathsf L_m\n =\n (m,C,A_m^{\\fine},A_m^{\\coarse},\\Delta_m,\n \\norm{\\Delta_m},\\text{norm},\\text{tolerance}).\n\\end{equation}\nA Bayesian certificate stores\n\\begin{equation}\\label{eq:bayes-certificate-schema}\n \\mathsf B_y\n =\n (\\lambda_y^{\\fine},\\lambda_y^{\\coarse},s_y,r_y,\n z_{*,y},d_{\\Bup,y}).\n\\end{equation}\nA filtering certificate stores\n\\begin{equation}\\label{eq:filter-certificate-schema}\n \\mathsf F_N\n =\n (e_0^+,\\{L_{\\Phi,k}\\},\\{L_{\\Bup,k}\\},\n \\{d_{\\Phi,k}\\},\\{d_{\\Bup,k}\\},\n \\{e_k^-,e_k^+\\}).\n\\end{equation}\nThe release serializes the executed finite ledger as JSON files and CSV tables. A certificate records a computation under specified inputs. The ledger provides cryptographic authenticity only when it is paired with a separate signed digest layer.","PRIQ-026 & Optimal approximation with $k$ hyperedges & PRIQ-025","\\midrule\n\\endhead\n$X_{\\fine},X_{\\coarse}$ & fine and coarse state spaces","$P_i,\\overline P_a$ & fine and coarse packet projectors","$\\pi$ & map from fine packet labels to coarse labels","$C$ & surjective aggregation or quotient map","$E$ & right inverse selecting a fine representative of each coarse state","$P_C=EC$ & representative projection","$Q_C=I-EC$ & projection onto the hidden directions, with range $\\Ker C$","$A_m^{\\fine},A_m^{\\coarse}$ & fine and coarse interaction tensors of order $m$","$\\Delta_m$ & quotient defect at interaction order $m$","$F_{\\fine},F_{\\coarse}$ & fine and coarse vector fields","$\\Phi_t^{\\fine},\\Phi_t^{\\coarse}$ & fine and coarse flows or prediction maps","$L_{\\fine},L_{\\coarse}$ & positive observation transfer operators","$\\Bup_{\\lambda}$ & Bayesian normalized likelihood update","$\\Gamma_k$ & single step prediction and update map","$\\Psi_{1:N}$ & cocycle for a finite observation sequence","$d_{\\Phi,k}$ & direct prediction quotient defect at step $k$","$d_{\\Bup,k}$ & direct Bayesian refinement defect at step $k$","$e_k^-,e_k^+$ & aggregate errors before and after an update","$z_{*,k}$ & positive evidence floor","$R_m$ & omitted hyperedge residual tensor","$H_m$ & entropy of the normalized hyperedge energy distribution at order $m$","\\[\nCA_m^f=A_m^cC^{\\times m}\n\\Rightarrow\nCF_f=F_cC\n\\Rightarrow\nC\\Phi_t^f=\\Phi_t^cC.\n\\]","\\[\nCL_y^f=s_yL_y^cC\n\\Rightarrow\nC\\mathsf B_y^f=\\mathsf B_y^cC\n\\Rightarrow\nC\\Psi_{1:N}^f=\\Psi_{1:N}^cC.\n\\]","For a surjective coarse map \\(C:X_f\\to X_c\\), the paper studies","\\[\nCA_m^f=A_m^cC^{\\times m}\n\\Longrightarrow\nCF_f=F_cC\n\\Longrightarrow\nC\\Phi_t^f=\\Phi_t^cC.\n\\]","\\[\nCL_f=sL_cC\n\\Longrightarrow\nC\\mathsf B_f=\\mathsf B_cC.\n\\]","\\[\nC\\Psi_{1:N}^f=\\Psi_{1:N}^cC.\n\\]","\\midrule\n\\endhead\nQ1D & order 1 quotient defect & $1.19\\times 10^{-16}$ & $\\leq 10^{-10}$ & PASS","Q1K & order 1 hidden input residual & $5.38\\times 10^{-16}$ & $\\leq 10^{-10}$ & PASS","Q1R & order 1 coarse tensor recovery & $8.41\\times 10^{-17}$ & $\\leq 10^{-10}$ & PASS","Q2D & order 2 quotient defect & $3.76\\times 10^{-16}$ & $\\leq 10^{-10}$ & PASS","Q2K & order 2 hidden input residual & $5.71\\times 10^{-16}$ & $\\leq 10^{-10}$ & PASS","Q2R & order 2 coarse tensor recovery & $5.55\\times 10^{-17}$ & $\\leq 10^{-10}$ & PASS","Q3D & order 3 quotient defect & $8.39\\times 10^{-16}$ & $\\leq 10^{-10}$ & PASS","Q3K & order 3 hidden input residual & $1.15\\times 10^{-15}$ & $\\leq 10^{-10}$ & PASS","Q3R & order 3 coarse tensor recovery & $1.11\\times 10^{-16}$ & $\\leq 10^{-10}$ & PASS","QNEG & a perturbation of one coefficient violates fiber constancy & $0.125$ & $\\geq 10^{-1}$ & PASS","QSYM & symmetric interaction remains symmetric after exact lift & $0$ & $\\leq 10^{-10}$ & PASS","QHID & projected hidden tensor has zero coarse output & $0$ & $\\leq 10^{-10}$ & PASS","QTRN & exact quotient transitivity across two aggregation levels & $4.84\\times 10^{-16}$ & $\\leq 10^{-10}$ & PASS","QMLT & interaction defect identity across two levels & $9.46\\times 10^{-16}$ & $\\leq 10^{-10}$ & PASS","QFLD & fine vector field factors through the coarse field & $2.22\\times 10^{-16}$ & $\\leq 10^{-10}$ & PASS","QPRD & exact aggregate prediction commutation & $7.22\\times 10^{-16}$ & $\\leq 5\\times 10^{-10}$ & PASS","QCOC & exact commutation of the prediction and update cocycle & $3.89\\times 10^{-16}$ & $\\leq 5\\times 10^{-10}$ & PASS","QEVD & exact evidence aggregation at one step & $6.66\\times 10^{-16}$ & $\\leq 5\\times 10^{-10}$ & PASS","QMAS & fine nonlinear trajectory preserves total mass & $7.77\\times 10^{-16}$ & $\\leq 5\\times 10^{-10}$ & PASS","QPOS & fine nonlinear trajectory preserves nonnegativity & $0$ & $\\leq 5\\times 10^{-12}$ & PASS","QBAY & a likelihood constant within each fiber gives exact Bayesian refinement & $2.78\\times 10^{-16}$ & $\\leq 10^{-10}$ & PASS","QBEV & a likelihood constant within each fiber preserves evidence & $2.22\\times 10^{-16}$ & $\\leq 10^{-10}$ & PASS","QSCL & Bayesian update is invariant under global likelihood scale & $1.53\\times 10^{-16}$ & $\\leq 10^{-10}$ & PASS","QFDB & controlled dynamical perturbation respects the flow certificate & $0$ & $\\leq 2\\times 10^{-9}$ & PASS","QFSL & flow discrepancy has first order perturbation scaling & $3.37\\times 10^{-3}$ & $\\leq 0.08$ & PASS","QLPS & perturbed likelihood remains strictly positive & $0.8$ & $\\geq 10^{-12}$ & PASS","QOBB & variation of the likelihood within a fiber respects the Bayesian bound & $0$ & $\\leq 10^{-12}$ & PASS","QBSC & the Bayesian perturbation sweep remains below the likelihood certificate & $0$ & $\\leq 10^{-12}$ & PASS","QBSL & Bayesian discrepancy has first order scaling for small fiber mismatch & $4.63\\times 10^{-5}$ & $\\leq 0.08$ & PASS","QNMB & symmetric normalization bound uses the larger positive mass & $0$ & $\\leq 10^{-12}$ & PASS","QPST & recursive posterior certificate dominates every joint perturbation run & $0$ & $\\leq 10^{-10}$ & PASS","QPSL & joint posterior discrepancy has first order scaling for small perturbations & $1.23\\times 10^{-4}$ & $\\leq 0.12$ & PASS","QLEV & the evidence certificate at each step dominates the accumulated discrepancy & $0$ & $\\leq 10^{-10}$ & PASS","QPAR & hyperedge energy satisfies the scalar Parseval identity & $8.88\\times 10^{-16}$ & $\\leq 10^{-10}$ & PASS","QBST & the residual from retaining the largest energy hyperedges decreases to zero & $0$ & $\\leq 10^{-10}$ & PASS"]}