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bigoplus","display":"\\bigoplus_i","kind":"use","occurrences":[36]},{"id":"symbol-7d0f5e3799ea2757","tex":"\\bigoplus_{j\\in\\mathbb Z/N\\mathbb Z}","aliases":"⊕_j∈ Z/N Z bigoplus in mathbb mathbb","display":"\\bigoplus_{j\\in\\mathbb Z/N\\mathbb Z}","kind":"use","occurrences":[639]},{"id":"symbol-9b591bcc37a61369","tex":"\\bigoplus_\\alpha","aliases":"⊕_α bigoplus alpha","display":"\\bigoplus_\\alpha","kind":"use","occurrences":[772,882]},{"id":"symbol-90b882d77bb3725c","tex":"\\bigoplus_{\\alpha\\in I}","aliases":"⊕_α∈ I bigoplus alpha in","display":"\\bigoplus_{\\alpha\\in I}","kind":"use","occurrences":[393,764,768,780,1136,1137]},{"id":"symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"use","occurrences":[891]},{"id":"symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","occurrences":[388,1089]},{"id":"symbol-0c12df6de0a7dc07","tex":"\\bigcap_m","aliases":"⋂_m bigcap","display":"\\bigcap_m","kind":"use","occurrences":[388]},{"id":"symbol-08bf84a1d20b2d0b","tex":"\\bigcap_r","aliases":"⋂_r bigcap","display":"\\bigcap_r","kind":"use","occurrences":[1089]},{"id":"symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","occurrences":[1033,2089]}],"contexts":["Let $V$ be a finite packet type, $E$ a finite edge type, and $R$ a semiring.\nA directed multigraph is a pair of endpoint functions\n\\[\n  \\src,\\tgt:E\\longrightarrow V.\n\\]\nParallel edges remain distinct elements of $E$.  Rows are targets and columns\nare sources.  Thus an edge $e:i\\to j$ is represented by a matrix supported at\nrow $j$, column $i$.","\\begin{definition}[Packet graph and scalar edge system]\n\\label{def:edge-system}\nFor $a,b\\in V$ and $r\\in R$, define\n\\[\n  \\operatorname{packetMatrix}(a,b;r)_{yx}\n  =\n  \\begin{cases}\n  r,&y=b\\text{ and }x=a,","\\begin{definition}[Packet graph and scalar edge system]\n\\label{def:edge-system}\nFor $a,b\\in V$ and $r\\in R$, define\n\\[\n  \\operatorname{packetMatrix}(a,b;r)_{yx}\n  =\n  \\begin{cases}\n  r,&y=b\\text{ and }x=a,\\\\\n  0,&\\text{otherwise}.\n  \\end{cases}\n\\]\nAn edge system is a coefficient function $c:E\\to R$.  Its elementary edge\noperator and total transfer operator are\n\\[\n  L_e=\\operatorname{packetMatrix}(\\src(e),\\tgt(e);c(e)),\n  \\qquad\n  T=\\sum_{e\\in E}L_e.\n\\]\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp}","0,&\\text{otherwise}.\n  \\end{cases}\n\\]\nAn edge system is a coefficient function $c:E\\to R$.  Its elementary edge\noperator and total transfer operator are\n\\[\n  L_e=\\operatorname{packetMatrix}(\\src(e),\\tgt(e);c(e)),\n  \\qquad\n  T=\\sum_{e\\in E}L_e.\n\\]\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp}","Words are written in execution order. In $L_fL_e$, the rightmost operator\n$L_e$ acts first. For a written word $(e_1,\\ldots,e_n)$, $e_1$ acts first\nand the operator product is $L_{e_n}\\cdots L_{e_1}$.","\\begin{lemma}[Declared support]\n\\label{lem:edge-support}\nFor every edge $e$, $(L_e)_{yx}=0$ unless\n$y=\\tgt(e)$ and $x=\\src(e)$.\n\\end{lemma}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp_eq_zero_off_support}","\\begin{theorem}[Typed interface annihilation]\n\\label{thm:edge-annihilation}\nIf $\\tgt(e)\\ne\\src(f)$, then\n\\[\n  L_fL_e=0.\n\\]\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp_mul_edgeOp_eq_zero}","\\begin{proof}\nFor a matrix entry $(y,x)$,\n\\[\n  (L_fL_e)_{yx}=\\sum_{z\\in V}(L_f)_{yz}(L_e)_{zx}.\n\\]\nA potentially nonzero summand would require simultaneously\n$z=\\src(f)$ and $z=\\tgt(e)$, contrary to the hypothesis.\n\\end{proof}","\\begin{theorem}[Collapse of a matched edge pair]\n\\label{thm:edge-composable}\nIf $\\tgt(e)=\\src(f)$, then\n\\[\nL_fL_e\n=\n\\operatorname{packetMatrix}\n\\bigl(\\src(e),\\tgt(f);c(f)c(e)\\bigr).\n\\]\nTogether with \\cref{thm:edge-annihilation}, this gives both cases for\nthe composition of two edge operators.\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp_mul_edgeOp_of_composable}","\\begin{proof}\nThe matrix-product sum has exactly one potentially nonzero intermediate\nindex, namely $\\tgt(e)=\\src(f)$.  At that index the coefficients occur in\noperator order as $c(f)c(e)$, and the outer source and target are\n$\\src(e)$ and $\\tgt(f)$.\n\\end{proof}","\\begin{boundary}[Current fiber scope]\nThe compiled Paper III model above has one scalar coordinate per packet.\nFor a family of modules $(X_i)_{i\\in V}$, an intrinsic edge map would instead\nhave type $\\ell_e:X_{\\src(e)}\\to X_{\\tgt(e)}$, and its totalization on\n$\\bigoplus_iX_i$ would be\n$\\iota_{\\tgt(e)}\\ell_e\\pi_{\\src(e)}$.  The same support proof applies, but\nthis generalization with dependent fibers is not attributed to the released\nPaper III modules for scalar edge systems.\n\\end{boundary}","\\begin{definition}[Composable word and execution operator]\n\\label{def:word-op}\nA finite word $w=(e_1,\\ldots,e_n)$ is composable when\n\\[\n  \\tgt(e_r)=\\src(e_{r+1})\n  \\qquad (1\\le r<n).\n\\]\nSet\n\\[\n  L_w=L_{e_n}\\cdots L_{e_1},\n  \\qquad L_{\\varnothing}=I.\n\\]\nThis definition records the written order of execution even though matrix\ncomposition is read right to left.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp}","\\begin{theorem}[First broken interface]\n\\label{thm:first-break}\nIf a word begins $(e,f,\\ldots)$ and $\\tgt(e)\\ne\\src(f)$, then its word\noperator is zero.\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp_eq_zero_of_first_break}","\\begin{theorem}[Noncomposable words vanish]\n\\label{thm:nonwalk-vanish}\nIf $w$ is not composable, then $L_w=0$.\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp_eq_zero_of_not_composable}","\\begin{definition}[Recursive word expansion]\n\\label{def:recursive-word-expansion}\nDefine\n\\[\n  \\cW_0=I,\n  \\qquad\n  \\cW_{n+1}=T\\cW_n.\n\\]\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion}","\\begin{theorem}[Transfer power equals recursive word expansion]\n\\label{thm:power-recursive-expansion}\nFor every $n\\in\\N$,\n\\[\n  T^n=\\cW_n.\n\\]\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_eq_wordExpansion}","\\begin{proof}\nThe zero case is the identity matrix.  If $T^n=\\cW_n$, then the\nleft-successor power convention gives\n$T^{n+1}=TT^n=T\\cW_n=\\cW_{n+1}$.\n\\end{proof}","\\begin{remark}\nThe formal module also defines an explicit sum over tuples\n$w:\\operatorname{Fin}(n)\\to E$ and a second sum in which noncomposable tuples\nare replaced by zero.  It proves that the recursive matrix, the explicit tuple sum, and the sum\nfiltered by composability agree. The separate declarations make the finite\nenumeration claim directly checkable.\n\\end{remark}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.explicitWordExpansion; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.walkExpansion; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion_eq_explicitWordExpansion; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.explicitWordExpansion_eq_walkExpansion}","For a nonempty composable word $w=(e_1,\\ldots,e_n)$, define its reverse\ncoefficient product\n\\[\n  c^\\leftarrow(w)=c(e_n)c(e_{n-1})\\cdots c(e_1).\n\\]\nThe reversal is essential over a noncommutative semiring.","\\begin{lemma}[One word is one endpoint block]\n\\label{lem:word-endpoint}\nIf $w=(e_1,\\ldots,e_n)$ is nonempty and composable, then\n\\[\n  L_w\n  =\n  \\operatorname{packetMatrix}\n  \\bigl(\\src(e_1),\\tgt(e_n);c^\\leftarrow(w)\\bigr).\n\\]\n\\end{lemma}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.reverseCoefficient; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.terminalTarget; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp_eq_packetMatrix_of_composable}","\\begin{proof}\nInduct on $n$.  The case $n=1$ is the definition of $L_e$.  Suppose the\nformula holds for the prefix $(e_1,\\ldots,e_n)$ and append $f$.  Composability\ngives $\\tgt(e_n)=\\src(f)$.  Multiplying the two packet matrices leaves one\nintermediate index and changes the coefficient to\n$c(f)c(e_n)\\cdots c(e_1)$, with source $\\src(e_1)$ and target $\\tgt(f)$.\n\\end{proof}","Write $\\Walk_n(i,j)$ for the finite set of composable edge tuples of length $n$\nwhose first source is $i$ and whose final target is $j$. For $n=0$, it has\none empty walk when $i=j$ and is empty otherwise. Assign the empty walk the\ncoefficient $1$.","\\begin{theorem}[Finite operator expansion from words to walks]\n\\label{thm:word-to-walk}\nFor every $n\\in\\N$,\n\\[\n  \\boxed{\n  T^n\n  =\n  \\sum_{w\\in E^n}L_w\n  =\n  \\sum_{w\\in E^n}\n  \\begin{cases}\n  L_w,&w\\text{ composable},","\\begin{theorem}[Finite operator expansion from words to walks]\n\\label{thm:word-to-walk}\nFor every $n\\in\\N$,\n\\[\n  \\boxed{\n  T^n\n  =\n  \\sum_{w\\in E^n}L_w\n  =\n  \\sum_{w\\in E^n}\n  \\begin{cases}\n  L_w,&w\\text{ composable},\\\\\n  0,&w\\text{ not composable}.\n  \\end{cases}\n  }\n\\]\nParallel edges occur as distinct summands.\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_eq_explicitWordExpansion; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_eq_walkExpansion}","\\begin{proof}\nInduction and finite distributivity give\n\\[\n\\left(\\sum_{e\\in E}L_e\\right)^n\n=\n\\sum_{(e_1,\\ldots,e_n)\\in E^n}L_{e_n}\\cdots L_{e_1}.\n\\]\nEvery noncomposable summand is zero by\n\\cref{thm:nonwalk-vanish}, so inserting the composability filter does not\nchange the sum.  In Lean, tuple concatenation and the successor sum are\nhandled by \\leanname{tupleWordOp_snoc} and\n\\leanname{explicitWordExpansion_succ}; the two bridge equalities then yield\nthe displayed theorem.\n\\end{proof}","\\begin{corollary}[Endpoint coefficient walk sum]\n\\label{cor:word-to-walk-entry}\nFor all $n\\in\\N$ and $i,j\\in V$,\n\\[\n  \\boxed{\n  (T^n)_{ji}\n  =\n  \\sum_{w\\in\\Walk_n(i,j)}c^\\leftarrow(w).\n  }\n\\]\n\\end{corollary}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Connects; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.endpointWalkCoefficientSum; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_apply_eq_sum_walks}","\\begin{proof}\nTake the $(j,i)$ entry in \\cref{thm:word-to-walk}.  By\n\\cref{lem:word-endpoint}, a surviving word contributes\n$c^\\leftarrow(w)$ exactly when its endpoints are $i$ and $j$, and contributes\nzero otherwise.  At $n=0$, the identity matrix gives the unique empty\ndiagonal walk.\n\\end{proof}","\\begin{corollary}[Reachability obstruction at fixed length]\n\\label{cor:fixed-reachability}\nIf $\\Walk_n(i,j)=\\varnothing$, then $(T^n)_{ji}=0$.\n\\end{corollary}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_apply_eq_zero_of_no_walk}","\\begin{corollary}[Orbit support obstruction]\n\\label{cor:orbit-reachability}\nLet $S\\subseteq V$, and let $x\\in R^V$ be supported on $S$.  If $j$ is not\nreachable by a directed path from any vertex of $S$, then\n\\[\n  (T^nx)_j=0\\qquad\\text{for every }n\\in\\N.\n\\]\nHence the orbit under $T$ of $x$ cannot generate a vector with a nonzero\ncoordinate at $j$.\n\\end{corollary}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Reachable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_mulVec_apply_eq_zero_of_unreachable_support}\n\\statusleanprior{Komorebi.PacketNaturalityCompletion.Reachability.algebraicOrbitHull_le_reachable_support; Komorebi.PacketNaturalityCompletion.Reachability.unreachable_nonzero_packet_obstruction}","\\begin{proof}\nExpand $(T^nx)_j=\\sum_i(T^n)_{ji}x_i$.  Terms with $i\\notin S$ vanish by\nthe support of $x$, and terms with $i\\in S$ vanish by\n\\cref{cor:fixed-reachability}.  The final claim follows because the\n$j$-coordinate also vanishes on every finite linear combination of orbit\nvectors.\n\\end{proof}","Let every chart $a$ carry a linear frame equivalence\n$F_a:M\\xrightarrow{\\sim}M$.  The formal Paper III module defines transport\nfrom $a$ to $b$ by\n\\[\nJ_{b\\leftarrow a}=F_b^{-1}F_a.\n\\]\nThis frame formula means \\emph{synthesize in chart \\(a\\), then\nanalyze in chart \\(b\\)}.","\\begin{theorem}[Exact frame transport is coherent]\n\\label{thm:exact-frame-transport}\nFor every $a,b,c$,\n\\[\nJ_{a\\leftarrow a}=\\id,\\qquad\nJ_{a\\leftarrow b}J_{b\\leftarrow a}=\\id,\\qquad\nJ_{c\\leftarrow b}J_{b\\leftarrow a}=J_{c\\leftarrow a},\n\\]\nand therefore\n\\[\nJ_{a\\leftarrow c}J_{c\\leftarrow b}J_{b\\leftarrow a}=\\id.\n\\]\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.Transport.exactTransport_self; Komorebi.PaperIII.Transport.exactTransport_roundTrip; Komorebi.PaperIII.Transport.exactTransport_cocycle; Komorebi.PaperIII.Transport.exactTransport_triangleLoop}","\\begin{proof}\nEvery identity is cancellation of adjacent $F_rF_r^{-1}$ factors.  The Lean\ndefinitions use \\leanname{LinearEquiv.trans}; the displayed formulas use the\nequivalent right-to-left function-composition notation fixed in this paper.\n\\end{proof}","\\begin{theorem}[Every finite canonical path telescopes]\n\\label{thm:exact-path-transport}\nLet $p=(a_0,a_1,\\ldots,a_n)$ be any finite vertex list and put\n\\[\nJ_p\n=J_{a_n\\leftarrow a_{n-1}}\\cdots J_{a_1\\leftarrow a_0},\n\\qquad\nJ_{(a_0)}=\\id.\n\\]\nThen\n\\[\nJ_p=J_{a_n\\leftarrow a_0}.\n\\]\nIn particular, if $a_n=a_0$, then $J_p=\\id$.\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.Transport.pathEndpoint; Komorebi.PaperIII.Transport.exactPathTransport; Komorebi.PaperIII.Transport.exactPathTransport_eq_endpoint; Komorebi.PaperIII.Transport.exactPathTransport_loop}","The graph model above resolves a fixed scalar matrix.  Coordinate transport\nis a separate construction.  Let $X$ be an $R$-module, and let\n\\[\n\\cP^a=(P^a_i)_{i\\in I_a}\n\\]\nbe a finite complete family of pairwise orthogonal linear idempotents:\n\\[\nP^a_iP^a_j=\\delta_{ij}P^a_i,\n\\qquad\n\\sum_{i\\in I_a}P^a_i=I_X.\n\\]\nDefine the exact packet coordinate module\n\\[\n\\Coord(\\cP^a)\n=\n\\left\\{(x_i)_{i\\in I_a}:P^a_ix_i=x_i\\right\\}.\n\\]\nIts analysis and synthesis maps are\n\\[\n\\cA_a(x)=(P^a_ix)_i,\n\\qquad\n\\cS_a((x_i)_i)=\\sum_ix_i.\n\\]\nCompleteness and orthogonality give\n\\[\n\\cS_a\\cA_a=\\id_X,\n\\qquad\n\\cA_a\\cS_a=\\id_{\\Coord(\\cP^a)}.\n\\]","ordinate transport\nis a separate construction.  Let $X$ be an $R$-module, and let\n\\[\n\\cP^a=(P^a_i)_{i\\in I_a}\n\\]\nbe a finite complete family of pairwise orthogonal linear idempotents:\n\\[\nP^a_iP^a_j=\\delta_{ij}P^a_i,\n\\qquad\n\\sum_{i\\in I_a}P^a_i=I_X.\n\\]\nDefine the exact packet coordinate module\n\\[\n\\Coord(\\cP^a)\n=\n\\left\\{(x_i)_{i\\in I_a}:P^a_ix_i=x_i\\right\\}.\n\\]\nIts analysis and synthesis maps are\n\\[\n\\cA_a(x)=(P^a_ix)_i,\n\\qquad\n\\cS_a((x_i)_i)=\\sum_ix_i.\n\\]\nCompleteness and orthogonality give\n\\[\n\\cS_a\\cA_a=\\id_X,\n\\qquad\n\\cA_a\\cS_a=\\id_{\\Coord(\\cP^a)}.\n\\]","i)_{i\\in I_a}\n\\]\nbe a finite complete family of pairwise orthogonal linear idempotents:\n\\[\nP^a_iP^a_j=\\delta_{ij}P^a_i,\n\\qquad\n\\sum_{i\\in I_a}P^a_i=I_X.\n\\]\nDefine the exact packet coordinate module\n\\[\n\\Coord(\\cP^a)\n=\n\\left\\{(x_i)_{i\\in I_a}:P^a_ix_i=x_i\\right\\}.\n\\]\nIts analysis and synthesis maps are\n\\[\n\\cA_a(x)=(P^a_ix)_i,\n\\qquad\n\\cS_a((x_i)_i)=\\sum_ix_i.\n\\]\nCompleteness and orthogonality give\n\\[\n\\cS_a\\cA_a=\\id_X,\n\\qquad\n\\cA_a\\cS_a=\\id_{\\Coord(\\cP^a)}.\n\\]","\\begin{definition}[Canonical exact coordinate transport]\n\\label{def:canonical-J}\nFor two exact packet systems on the same ambient module, define\n\\[\n  J^X_{b\\leftarrow a}\n  =\n  \\cA_b\\cS_a:\n  \\Coord(\\cP^a)\\longrightarrow\\Coord(\\cP^b).\n\\]\nThe arrow points from the source coordinate system $a$ to the target\ncoordinate system $b$.\n\\end{definition}\n\\statusleanprior{Komorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv}","\\begin{theorem}[Identity, inverse, and cocycle for canonical transport]\n\\label{thm:J-cocycle}\nFor any exact packet systems $a,b,c$ on $X$,\n\\[\nJ^X_{a\\leftarrow a}=\\id,\n\\qquad\n\\left(J^X_{b\\leftarrow a}\\right)^{-1}=J^X_{a\\leftarrow b},\n\\qquad\nJ^X_{c\\leftarrow b}J^X_{b\\leftarrow a}=J^X_{c\\leftarrow a}.\n\\]\n\\end{theorem}\n\\statusleanprior{Komorebi.PaperII.NestedRefinement.refineCoordinates_self; Komorebi.PaperII.NestedRefinement.coarsenCoordinates_refineCoordinates; Komorebi.PaperII.NestedRefinement.refineCoordinates_coarsenCoordinates; Komorebi.PaperII.NestedRefinement.refineCoordinates_comp; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv_trans}","\\begin{proof}\nInsert $\\cS_b\\cA_b=\\id_X$:\n\\[\nJ^X_{c\\leftarrow b}J^X_{b\\leftarrow a}\n=\\cA_c\\cS_b\\cA_b\\cS_a\n=\\cA_c\\cS_a\n=J^X_{c\\leftarrow a}.\n\\]\nThe identity statement is $\\cA_a\\cS_a=\\id$.  Setting $c=a$ in the cocycle\ngives $J^X_{a\\leftarrow b}J^X_{b\\leftarrow a}=\\id$; interchanging $a$ and $b$\ngives the other composition, so the two transports are inverses.\n\\end{proof}","If $F:X\\to Y$ is linear and $\\cP^a,\\cP^b$ are source packet systems while\n$\\cQ^r,\\cQ^s$ are target packet systems, put\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_rF\\cS^X_a.\n\\]","\\begin{theorem}[Canonical coordinate naturality]\n\\label{thm:canonical-naturality}\nThe exact coordinate representation preserves typed composition and satisfies\n\\[\nJ^Y_{s\\leftarrow r}\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}J^X_{b\\leftarrow a}.\n\\]\n\\end{theorem}\n\\statusleanprior{Komorebi.PaperII.NestedRefinement.coordinateOperator_comp; Komorebi.PaperII.NestedRefinement.coordinateOperator_naturality}","\\begin{proof}\nBoth sides reduce to $\\cA^Y_sF\\cS^X_a$ after inserting the appropriate\nsynthesis--analysis identity.  The composition law is the same calculation\nwith the middle ambient module inserted.\n\\end{proof}","\\begin{corollary}[Canonical triangle holonomy is trivial]\n\\label{cor:J-flat}\nFor every triple $a,b,c$,\n\\[\nJ^X_{a\\leftarrow c}J^X_{c\\leftarrow b}J^X_{b\\leftarrow a}=\\id.\n\\]\n\\end{corollary}\n\\statusleanprior{Komorebi.PaperII.NestedRefinement.refineCoordinates_comp; Komorebi.PaperII.NestedRefinement.refineCoordinates_self}","\\begin{boundary}[Refinement is not reduction]\nAlthough Paper II uses the name \\emph{refineCoordinates}, $J$ is an\nequivalence between two complete coordinate presentations.  It neither\ntruncates packets nor projects to a smaller state space.  Any lossy\nrestriction, interpolation, or learned identification is extra data and\ncannot inherit \\cref{thm:J-cocycle} without proof.\n\\end{boundary}","The compiled Paper III API assumes only normalization at each chart.  All\nchart transports act on one module $M$, so chart labels distinguish routes\nbut not Lean types.","\\begin{definition}[Noncanonical connection normalized at each chart]\n\\label{def:minimal-connection}\nA minimal connection is a family\n\\[\n\\Gamma_{b\\leftarrow a}\\in\\operatorname{Aut}_R(M)\n\\qquad(a,b\\in B)\n\\]\nwith only\n\\[\n\\Gamma_{a\\leftarrow a}=\\id_M.\n\\]\nNeither inversion on reversed edges nor the cocycle law is a field of the\nstructure.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.Transport.Connection}","At a triple $(a,b,c)$, the compiled cocycle predicate is\n\\[\n\\Gamma_{c\\leftarrow b}\\Gamma_{b\\leftarrow a}\n=\\Gamma_{c\\leftarrow a}.\n\\]\nFlatness means this equality holds for every triple.  The formal\nmultiplicative relative defect and triangle holonomy are\n\\[\nD^\\times_{c,b,a}\n=\n\\bigl(\\Gamma_{c\\leftarrow b}\\Gamma_{b\\leftarrow a}\\bigr)\n\\Gamma_{c\\leftarrow a}^{-1},\n\\qquad\nH_{c,b,a}\n=\n\\Gamma_{a\\leftarrow c}\n\\Gamma_{c\\leftarrow b}\n\\Gamma_{b\\leftarrow a}.\n\\]\nThe defect is based at the target $c$.  The holonomy is based at $a$.\n\\statusleanthree{Komorebi.PaperIII.Transport.Connection.CocycleAt; Komorebi.PaperIII.Transport.Connection.Flat; Komorebi.PaperIII.Transport.Connection.cocycleDefect; Komorebi.PaperIII.Transport.C","At a triple $(a,b,c)$, the compiled cocycle predicate is\n\\[\n\\Gamma_{c\\leftarrow b}\\Gamma_{b\\leftarrow a}\n=\\Gamma_{c\\leftarrow a}.\n\\]\nFlatness means this equality holds for every triple.  The formal\nmultiplicative relative defect and triangle holonomy are\n\\[\nD^\\times_{c,b,a}\n=\n\\bigl(\\Gamma_{c\\leftarrow b}\\Gamma_{b\\leftarrow a}\\bigr)\n\\Gamma_{c\\leftarrow a}^{-1},\n\\qquad\nH_{c,b,a}\n=\n\\Gamma_{a\\leftarrow c}\n\\Gamma_{c\\leftarrow b}\n\\Gamma_{b\\leftarrow a}.\n\\]\nThe defect is based at the target $c$.  The holonomy is based at $a$.\n\\statusleanthree{Komorebi.PaperIII.Transport.Connection.CocycleAt; Komorebi.PaperIII.Transport.Connection.Flat; Komorebi.PaperIII.Transport.Connection.cocycleDefect; Komorebi.PaperIII.Transport.Connection.triangleHolonomy}","b\\leftarrow a}\n=\\Gamma_{c\\leftarrow a}.\n\\]\nFlatness means this equality holds for every triple.  The formal\nmultiplicative relative defect and triangle holonomy are\n\\[\nD^\\times_{c,b,a}\n=\n\\bigl(\\Gamma_{c\\leftarrow b}\\Gamma_{b\\leftarrow a}\\bigr)\n\\Gamma_{c\\leftarrow a}^{-1},\n\\qquad\nH_{c,b,a}\n=\n\\Gamma_{a\\leftarrow c}\n\\Gamma_{c\\leftarrow b}\n\\Gamma_{b\\leftarrow a}.\n\\]\nThe defect is based at the target $c$.  The holonomy is based at $a$.\n\\statusleanthree{Komorebi.PaperIII.Transport.Connection.CocycleAt; Komorebi.PaperIII.Transport.Connection.Flat; Komorebi.PaperIII.Transport.Connection.cocycleDefect; Komorebi.PaperIII.Transport.Connection.triangleHolonomy}","rrow a}.\n\\]\nFlatness means this equality holds for every triple.  The formal\nmultiplicative relative defect and triangle holonomy are\n\\[\nD^\\times_{c,b,a}\n=\n\\bigl(\\Gamma_{c\\leftarrow b}\\Gamma_{b\\leftarrow a}\\bigr)\n\\Gamma_{c\\leftarrow a}^{-1},\n\\qquad\nH_{c,b,a}\n=\n\\Gamma_{a\\leftarrow c}\n\\Gamma_{c\\leftarrow b}\n\\Gamma_{b\\leftarrow a}.\n\\]\nThe defect is based at the target $c$.  The holonomy is based at $a$.\n\\statusleanthree{Komorebi.PaperIII.Transport.Connection.CocycleAt; Komorebi.PaperIII.Transport.Connection.Flat; Komorebi.PaperIII.Transport.Connection.cocycleDefect; Komorebi.PaperIII.Transport.Connection.triangleHolonomy}","\\begin{theorem}[Checked flatness consequences]\n\\label{thm:minimal-flatness}\nFor the minimal connection API, flatness implies\n\\[\nD^\\times_{c,b,a}=\\id_M,\n\\qquad\nH_{c,b,a}=\\id_M\n\\]\nfor every $a,b,c$.\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.Transport.Connection.cocycleDefect_eq_refl_of_flat; Komorebi.PaperIII.Transport.Connection.triangleHolonomy_eq_refl_of_flat}","\\begin{proof}\nFlatness replaces the two-edge route in $D^\\times$ by the direct route,\nwhich cancels its inverse.  For $H$, flatness first collapses $a\\to b\\to c$\nto $a\\to c$, then collapses $a\\to c\\to a$ to the normalized transport\n$a\\to a$.\n\\end{proof}","\\begin{proposition}[The minimal API admits nontrivial holonomy]\n\\label{prop:twisted-loop}\nThere is a connection on three points over $\\mathbb Q$ for which transport\n$0\\to1$ is multiplication by $2$, every other transport between distinct\nendpoints is the identity, and\n\\[\nH_{2,1,0}(1)=2.\n\\]\nIn particular $H_{2,1,0}\\ne\\id$.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.Transport.twistedThreePointConnection; Komorebi.PaperIII.Transport.twisted_triangleHolonomy_apply_one; Komorebi.PaperIII.Transport.twisted_triangleHolonomy_ne_refl}","\\begin{boundary}[Why the minimal defect and loop are distinct]\nWithout reverse-edge inversion,\n$\\Gamma_{a\\leftarrow c}$ need not equal\n$\\Gamma_{c\\leftarrow a}^{-1}$.  Therefore the multiplicative relative defect\n$D^\\times_{c,b,a}$ and triangle holonomy $H_{c,b,a}$ are different\ncomparisons.  The formal module proves that global flatness trivializes both;\nit does not identify them for an arbitrary connection.\n\\end{boundary}","Let $R$ be a commutative ring, let $B$ be a type of charts or refinement\nstates, and let each $C_a$ be an $R$-module.  The chart type is not assumed\nfinite.  The following definition is deliberately not derived from analysis\nand synthesis.","\\begin{definition}[Normalized reversible connection]\n\\label{def:connection}\nA normalized reversible connection is a family of linear equivalences\n\\[\n\\Gamma_{b\\leftarrow a}:C_a\\xrightarrow{\\;\\sim\\;}C_b\n\\qquad(a,b\\in B)\n\\]\nsuch that\n\\[\n\\Gamma_{a\\leftarrow a}=\\id,\n\\qquad\n\\Gamma_{a\\leftarrow b}=\\Gamma_{b\\leftarrow a}^{-1}.\n\\]\nNo cocycle equation is included in the structure.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection}","\\begin{definition}[Calibration and flatness]\n\\label{def:calibration-flatness}\nWhen the $C_a$ are exact packet coordinates of one ambient module, the\nconnection is \\emph{canonically calibrated} if\n\\[\n\\Gamma_{b\\leftarrow a}=J_{b\\leftarrow a}\n\\quad\\text{for every }a,b.\n\\]\nIt is \\emph{flat} if\n\\[\n\\Gamma_{c\\leftarrow b}\\Gamma_{b\\leftarrow a}\n=\n\\Gamma_{c\\leftarrow a}\n\\quad\\text{for every }a,b,c.\n\\]\nThe canonical connection is flat.  A general flat connection is proved in\n\\cref{thm:flat-gauge-canonical} to be related to it by a gauge change.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.CocycleAt; Komorebi.PaperIII.Transport.ReversibleConnection.Flat; Komorebi.PaperIII.Transport.ReversibleConnection.canonicalConnection; Komorebi.PaperIII.Transport.Rev","\\begin{definition}[Calibration and flatness]\n\\label{def:calibration-flatness}\nWhen the $C_a$ are exact packet coordinates of one ambient module, the\nconnection is \\emph{canonically calibrated} if\n\\[\n\\Gamma_{b\\leftarrow a}=J_{b\\leftarrow a}\n\\quad\\text{for every }a,b.\n\\]\nIt is \\emph{flat} if\n\\[\n\\Gamma_{c\\leftarrow b}\\Gamma_{b\\leftarrow a}\n=\n\\Gamma_{c\\leftarrow a}\n\\quad\\text{for every }a,b,c.\n\\]\nThe canonical connection is flat.  A general flat connection is proved in\n\\cref{thm:flat-gauge-canonical} to be related to it by a gauge change.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.CocycleAt; Komorebi.PaperIII.Transport.ReversibleConnection.Flat; Komorebi.PaperIII.Transport.ReversibleConnection.canonicalConnection; Komorebi.PaperIII.Transport.ReversibleConnection.canonicalConnection_flat}","\\begin{remark}[A connection on a chart graph]\nOne may specify $\\Gamma_e$ only on the oriented edges of a chart graph,\ntogether with inverse maps on reversed edges.  Path transport and loop\nholonomy below still make sense.  A direct cocycle defect for $(a,b,c)$,\nhowever, requires a declared direct transport from $a$ to $c$.  The pairwise\nAPI in \\cref{def:connection} keeps this comparison total.\n\\end{remark}","\\begin{definition}[Cocycle defect]\n\\label{def:cocycle-defect}\nFor $a,b,c\\in B$, define the additive cocycle defect\n\\[\n\\Delta_{c,b,a}\n=\n\\Gamma_{c\\leftarrow b}\\Gamma_{b\\leftarrow a}\n-\\Gamma_{c\\leftarrow a}\n\\in\\Hom_R(C_a,C_c).\n\\]\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect}","\\begin{definition}[Triangle holonomy]\n\\label{def:triangle-holonomy}\nThe triangle holonomy based at $a$ is\n\\[\nH_{c,b,a}\n=\n\\Gamma_{a\\leftarrow c}\n\\Gamma_{c\\leftarrow b}\n\\Gamma_{b\\leftarrow a}\n\\in\\operatorname{Aut}_R(C_a).\n\\]\nIt traverses $a\\to b\\to c\\to a$.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.triangleHolonomy; Komorebi.PaperIII.Transport.ReversibleConnection.triangleObstruction}","Holonomy is attached to the chosen connection $\\Gamma$, never to an\nunspecified coordinate change.","\\begin{theorem}[Identity between defect and holonomy]\n\\label{thm:defect-holonomy}\nFor every triple $(a,b,c)$,\n\\[\n\\boxed{\nH_{c,b,a}-\\id_{C_a}\n=\n\\Gamma_{a\\leftarrow c}\\Delta_{c,b,a}.\n}\n\\]\nConsequently,\n\\[\n\\Delta_{c,b,a}=0\n\\quad\\Longleftrightarrow\\quad\nH_{c,b,a}=\\id_{C_a}.\n\\]\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.triangleHolonomy_sub_id; Komorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect_eq_zero_iff_triangleHolonomy_eq_refl}","\\begin{proof}\nUsing reversibility,\n\\[\n\\begin{aligned}\n\\Gamma_{a\\leftarrow c}\\Delta_{c,b,a}\n&=\n\\Gamma_{a\\leftarrow c}\n\\Gamma_{c\\leftarrow b}\n\\Gamma_{b\\leftarrow a}\n-\n\\Gamma_{a\\leftarrow c}\\Gamma_{c\\leftarrow a}\\\\\n&=H_{c,b,a}-\\id_{C_a}.\n\\end{aligned}\n\\]\nLeft composition by the equivalence $\\Gamma_{a\\leftarrow c}$ is injective,\nwhich gives the equivalence of the two vanishing statements.\n\\end{proof}","&=H_{c,b,a}-\\id_{C_a}.\n\\end{aligned}\n\\]\nLeft composition by the equivalence $\\Gamma_{a\\leftarrow c}$ is injective,\nwhich gives the equivalence of the two vanishing statements.\n\\end{proof}","\\begin{corollary}[Canonical transport has zero connection defect]\n\\label{cor:canonical-zero-defect}\nIf $\\Gamma_{b\\leftarrow a}=J_{b\\leftarrow a}$, then\n\\[\n\\Delta_{c,b,a}=0,\n\\qquad\nH_{c,b,a}=\\id\n\\]\nfor all triples.\n\\end{corollary}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.canonicalTransport_zero_connectionDefect; Komorebi.PaperIII.Transport.ReversibleConnection.canonicalTransport_triangleHolonomy_eq_refl}","\\begin{definition}[Connection transport along a path]\n\\label{def:path-transport}\nFor a vertex sequence\n$p=(a_0,a_1,\\ldots,a_n)$, define\n\\[\n\\Gamma_p\n=\n\\Gamma_{a_n\\leftarrow a_{n-1}}\\cdots\n\\Gamma_{a_1\\leftarrow a_0},\n\\qquad\n\\Gamma_{(a_0)}=\\id_{C_{a_0}}.\n\\]\nIf $a_n=a_0$, then $\\Gamma_p$ is the holonomy of the loop $p$.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport}","\\begin{proposition}[Flat pair transport collapses to endpoints]\n\\label{prop:flat-endpoints}\nIf $\\Gamma$ is flat, then every path satisfies\n\\[\n\\Gamma_p=\\Gamma_{a_n\\leftarrow a_0}.\n\\]\nIn particular, every loop has identity holonomy.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.pathTransport_eq_direct_of_flat; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport_eq_direct_of_flat; Komorebi.PaperIII.Transport.ReversibleConnection.indexedLoopTransport_eq_refl_of_flat}","\\begin{proof}\nInduct on the path length.  Appending $a_{n+1}$ left-multiplies the\ninductive expression by $\\Gamma_{a_{n+1}\\leftarrow a_n}$, and flatness\nreplaces the two factors by\n$\\Gamma_{a_{n+1}\\leftarrow a_0}$.  For a loop, normalization gives\n$\\Gamma_{a_0\\leftarrow a_0}=\\id$.\n\\end{proof}","\\begin{definition}[Gauge change]\n\\label{def:gauge-change}\nA gauge is a family of automorphisms $G_a\\in\\operatorname{Aut}_R(C_a)$.\nIt acts on a connection by\n\\[\n\\Gamma^G_{b\\leftarrow a}\n=\nG_b\\Gamma_{b\\leftarrow a}G_a^{-1}.\n\\]\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge}","\\begin{theorem}[Gauge covariance]\n\\label{thm:gauge-covariance}\nThe defect and triangle holonomy transform as\n\\[\n\\Delta^G_{c,b,a}\n=G_c\\Delta_{c,b,a}G_a^{-1},\n\\qquad\nH^G_{c,b,a}\n=G_aH_{c,b,a}G_a^{-1}.\n\\]\nThus flatness and triviality of triangle holonomy are gauge invariant.\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.gauge_cocycleDefect; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_triangleHolonomy; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_flat_iff; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_triangleHolonomy_eq_refl_iff}","\\begin{proof}\nSubstitute the definition of $\\Gamma^G$.  Adjacent factors\n$G_b^{-1}G_b$ and $G_c^{-1}G_c$ cancel.  In the defect, both the composite\nand direct terms acquire the same outer factors $G_c$ and $G_a^{-1}$.  In\nthe loop, all inner factors cancel and only conjugation at the base remains.\n\\end{proof}","\\begin{theorem}[Flat connections are gauges of the canonical pair transport]\n\\label{thm:flat-gauge-canonical}\nSuppose the $C_a$ are exact packet coordinate modules for one ambient module,\nso the canonical $J_{b\\leftarrow a}$ is available.  If $\\Gamma$ is a flat\nnormalized reversible pair connection, choose $o\\in B$ and set\n\\[\nG_a=\\Gamma_{a\\leftarrow o}J_{o\\leftarrow a}.\n\\]\nThen every pair transport satisfies\n\\[\n\\boxed{\n\\Gamma_{b\\leftarrow a}\n=G_bJ_{b\\leftarrow a}G_a^{-1}.\n}\n\\]\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.flatGauge; Komorebi.PaperIII.Transport.ReversibleConnection.flat_isGaugeOfCanonical}","}[Flat connections are gauges of the canonical pair transport]\n\\label{thm:flat-gauge-canonical}\nSuppose the $C_a$ are exact packet coordinate modules for one ambient module,\nso the canonical $J_{b\\leftarrow a}$ is available.  If $\\Gamma$ is a flat\nnormalized reversible pair connection, choose $o\\in B$ and set\n\\[\nG_a=\\Gamma_{a\\leftarrow o}J_{o\\leftarrow a}.\n\\]\nThen every pair transport satisfies\n\\[\n\\boxed{\n\\Gamma_{b\\leftarrow a}\n=G_bJ_{b\\leftarrow a}G_a^{-1}.\n}\n\\]\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.Transport.ReversibleConnection.flatGauge; Komorebi.PaperIII.Transport.ReversibleConnection.flat_isGaugeOfCanonical}","\\begin{proof}\nReversibility gives\n$G_a^{-1}=J_{a\\leftarrow o}\\Gamma_{o\\leftarrow a}$.  Therefore\n\\[\n\\begin{aligned}\nG_bJ_{b\\leftarrow a}G_a^{-1}\n&=\n\\Gamma_{b\\leftarrow o}\nJ_{o\\leftarrow b}J_{b\\leftarrow a}\nJ_{a\\leftarrow o}\n\\Gamma_{o\\leftarrow a}","\\begin{proof}\nReversibility gives\n$G_a^{-1}=J_{a\\leftarrow o}\\Gamma_{o\\leftarrow a}$.  Therefore\n\\[\n\\begin{aligned}\nG_bJ_{b\\leftarrow a}G_a^{-1}\n&=\n\\Gamma_{b\\leftarrow o}\nJ_{o\\leftarrow b}J_{b\\leftarrow a}\nJ_{a\\leftarrow o}\n\\Gamma_{o\\leftarrow a}\\\\\n&=\n\\Gamma_{b\\leftarrow o}\\Gamma_{o\\leftarrow a}\\\\\n&=\n\\Gamma_{b\\leftarrow a}.\n\\end{aligned}\n\\]\nThe first reduction uses the canonical cocycle twice; the second uses\nflatness of $\\Gamma$.\n\\end{proof}","&=\n\\Gamma_{b\\leftarrow a}.\n\\end{aligned}\n\\]\nThe first reduction uses the canonical cocycle twice; the second uses\nflatness of $\\Gamma$.\n\\end{proof}","\\begin{boundary}[Flat does not mean calibrated]\nThe theorem produces a gauge relating $\\Gamma$ to $J$.  It does not prove\n$\\Gamma=J$ before that gauge is chosen.  Conversely, nontrivial triangle\nholonomy cannot be removed by a gauge, because conjugation cannot turn a\nnonidentity automorphism into the identity.\n\\end{boundary}","Let $\\mathsf{Candidate}$, $\\mathsf{Packet}$, and $\\mathsf{Residual}$ be\narbitrary types, with a distinguished\n$0\\in\\mathsf{Residual}$, and let\n\\[\nr:\\mathsf{Candidate}\\longrightarrow\n\\mathsf{Packet}\\longrightarrow\\mathsf{Residual}\n\\]\nbe a residual function.","\\begin{definition}[Solution and obstruction packet for a candidate]\n\\label{def:generic-obstruction}\nA candidate $x$ is a solution when\n\\[\n\\operatorname{IsSolution}_r(x)\n\\quad\\Longleftrightarrow\\quad\n\\forall p,\\ r(x,p)=0.\n\\]\nAn obstruction packet for $x$ is a pair consisting of a named packet $p$\nand a certificate $r(x,p)\\ne0$.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPacket}","\\begin{theorem}[Exact logical force of a local obstruction]\n\\label{thm:generic-obstruction}\nEvery obstruction packet for $x$ proves that $x$ is not a solution, and\n\\[\n\\neg\\operatorname{IsSolution}_r(x)\n\\quad\\Longleftrightarrow\\quad\n\\text{there exists an obstruction packet for }x.\n\\]\nThe reverse implication from failure to a packet witness is classical and\ndoes not require the packet type to be finite.\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.NonlinearObstructions.ObstructionPacket.not_isSolution; Komorebi.PaperIII.NonlinearObstructions.obstructionPacket_of_not_isSolution; Komorebi.PaperIII.NonlinearObstructions.not_isSolution_iff_nonempty_obstructionPacket}","\\begin{boundary}[Candidate-specific means candidate-specific]\nThe theorem rules out the named $x$.  It neither constructs a solution nor\nshows that every candidate is obstructed.  It provides no minimality,\nstability, computability, or classification theorem for the packet witness.\n\\end{boundary}","For a reversible connection, specialize the generic residual at a fixed\ntriangle by taking a packet label to be a pair $(\\mu,\\nu)$ and the residual\nto be the corresponding block of $H-\\id$.  A nonzero block, together with its\nlabel and a proof of nonvanishing, gives an obstruction packet certificate\nin the sense of \\cref{def:generic-obstruction}.  The reconstruction theorem\nbelow supplies the further step: vanishing of all these local residuals is\nequivalent to the cocycle equation at that triangle.","Fix a triangle $(a,b,c)$ and resolve the base fiber $C_a$ by a finite complete\nfamily of idempotents $(\\Pi^a_\\mu)_{\\mu\\in I_a}$.  Orthogonality may also hold,\nas it does for the exact range coordinate projectors below, but the compiled\nreconstruction theorem does not require it.  In exact range coordinates,\n$\\Pi^a_\\mu$ is simply the projector that retains component $\\mu$ and zeros\nthe other components.","\\begin{definition}[Holonomy obstruction and packet residuals]\n\\label{def:obstruction-packet}\nDefine\n\\[\n\\Ob_{c,b,a}=H_{c,b,a}-\\id_{C_a},\n\\qquad\n\\Ob^{\\mu\\nu}_{c,b,a}\n=\n\\Pi^a_\\mu\\Ob_{c,b,a}\\Pi^a_\\nu.\n\\]\nThe map $\\Ob^{\\mu\\nu}_{c,b,a}$ is the obstruction block, or packet residual,\nfrom input packet $\\nu$ to output packet $\\mu$.  If it is nonzero, then\n$(\\mu,\\nu)$ and that nonvanishing proof form an obstruction packet\ncertificate.  Diagonal blocks detect holonomy within one packet.  Off-diagonal\nblocks detect packet mixing induced by holonomy.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi.PaperIII.HolonomyBlocks.PacketResolution.block}","\\begin{theorem}[Exact reconstruction and detection by obstruction blocks]\n\\label{thm:obstruction-detection}\nThe total obstruction has the exact finite resolution\n\\[\n\\boxed{\n\\Ob_{c,b,a}\n=\n\\sum_{\\mu,\\nu\\in I_a}\n\\Ob^{\\mu\\nu}_{c,b,a}.\n}\n\\]\nMoreover, the following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\Delta_{c,b,a}=0$;\n\\item $H_{c,b,a}=\\id$;\n\\item $\\Ob_{c,b,a}=0$;\n\\item $\\Ob^{\\mu\\nu}_{c,b,a}=0$ for every $\\mu,\\nu$.\n\\end{enumerate}\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.HolonomyBlocks.obstruction_sum_blocks; Komorebi.PaperIII.HolonomyBlocks.obstructionBlocks_zero_iff; Komorebi.PaperIII.HolonomyBlocks.obstructionDetection_equivalences}","onstruction and detection by obstruction blocks]\n\\label{thm:obstruction-detection}\nThe total obstruction has the exact finite resolution\n\\[\n\\boxed{\n\\Ob_{c,b,a}\n=\n\\sum_{\\mu,\\nu\\in I_a}\n\\Ob^{\\mu\\nu}_{c,b,a}.\n}\n\\]\nMoreover, the following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\Delta_{c,b,a}=0$;\n\\item $H_{c,b,a}=\\id$;\n\\item $\\Ob_{c,b,a}=0$;\n\\item $\\Ob^{\\mu\\nu}_{c,b,a}=0$ for every $\\mu,\\nu$.\n\\end{enumerate}\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.HolonomyBlocks.obstruction_sum_blocks; Komorebi.PaperIII.HolonomyBlocks.obstructionBlocks_zero_iff; Komorebi.PaperIII.HolonomyBlocks.obstructionDetection_equivalences}","\\begin{proof}\nInsert completeness on both sides:\n\\[\n\\Ob\n=\n\\left(\\sum_\\mu\\Pi^a_\\mu\\right)\n\\Ob\n\\left(\\sum_\\nu\\Pi^a_\\nu\\right)\n=\n\\sum_{\\mu,\\nu}\\Pi^a_\\mu\\Ob\\Pi^a_\\nu.\n\\]\nThus (iv) implies (iii), and (iii) implies (iv) by composition with the\nprojectors.  Conditions (ii) and (iii) are equivalent by the definition of\n$\\Ob$ and additive cancellation.\nConditions (i) and (ii) are equivalent by\n\\cref{thm:defect-holonomy}.\n\\end{proof}","\\begin{corollary}[Witness supported in one packet]\n\\label{cor:packet-witness}\nIf $\\Ob^{\\mu\\nu}_{c,b,a}\\ne0$, then there is a vector $x$ supported in packet\n$\\nu$ such that\n\\[\n\\Pi^a_\\mu\\Ob_{c,b,a}x\\ne0.\n\\]\nSuch an $x$ is an exact finite witness of nonflatness.\n\\end{corollary}\n\\statusleanthree{Komorebi.PaperIII.HolonomyBlocks.exists_packetWitness_of_obstructionBlock_ne_zero}","\\begin{proof}\nClassically choose $y$ on which the nonzero linear map\n$\\Pi^a_\\mu\\Ob\\Pi^a_\\nu$ is nonzero and set $x=\\Pi^a_\\nu y$.\nIdempotence gives $\\Pi^a_\\nu x=x$, and the required output is the chosen\nnonzero value.\n\\end{proof}","Vanishing or nonvanishing of the total triangle obstruction is gauge invariant\nby \\cref{thm:gauge-covariance}, but a fixed packet pattern need not be.  A\ngeneral $G_a$ can mix the projectors $\\Pi^a_\\mu$.","\\begin{proposition}[Gauge covariance when packet projectors are preserved]\n\\label{prop:packet-gauge}\nIf $G_a\\Pi^a_\\mu=\\Pi^a_\\mu G_a$ for every $\\mu$, then\n\\[\n\\left(\\Ob^G\\right)^{\\mu\\nu}\n=\nG_a\\Ob^{\\mu\\nu}G_a^{-1}.\n\\]\nConsequently the zero/nonzero status of each obstruction block is preserved\nby gauges that preserve the packet projectors.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.HolonomyBlocks.obstructionBlock_gauge_of_commutes; Komorebi.PaperIII.HolonomyBlocks.obstructionBlock_gauge_eq_zero_iff}","\\begin{proof}\nUse $\\Ob^G=G_a\\Ob G_a^{-1}$ and commute $G_a$ and $G_a^{-1}$ through the\ntwo packet projectors.  Composition by an equivalence is injective in both\ndirections.\n\\end{proof}","The passage from words to walks also applies to products of different matrices.\nLet $V$\nbe one finite packet type, let $R$ be a semiring, and let\n$B_1,\\ldots,B_m\\in R^{V\\times V}$ be a finite sequence of possibly different\npacket matrices.  Rows are targets and columns are sources as before.","\\begin{proposition}[Packet path expansion for different matrices]\n\\label{prop:heterogeneous-path}\nFor $i,j\\in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.he","\\begin{proposition}[Packet path expansion for different matrices]\n\\label{prop:heterogeneous-path}\nFor $i,j\\in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.heterogeneousPa","\\begin{proposition}[Packet path expansion for different matrices]\n\\label{prop:heterogeneous-path}\nFor $i,j\\in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.heterogeneousPacketPathExpan","\\begin{proposition}[Packet path expansion for different matrices]\n\\label{prop:heterogeneous-path}\nFor $i,j\\in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.heterogeneousPacketPathExpansion; Komore","\\begin{proposition}[Packet path expansion for different matrices]\n\\label{prop:heterogeneous-path}\nFor $i,j\\in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.heterogeneousPacketPathExpansion; Komorebi.PaperIII.EdgeOperators.SupportedStep; Komorebi.PaperIII.EdgeOperators.SupportedPath; Komorebi.PaperIII.EdgeOperators.het","\\begin{proposition}[Packet path expansion for different matrices]\n\\label{prop:heterogeneous-path}\nFor $i,j\\in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.heterogeneousPacketPathExpansion; Komorebi.PaperIII.EdgeOperators.SupportedStep; Komorebi.PaperIII.EdgeOperators.SupportedPath; Komorebi.PaperIII.EdgeOperators.heterogeneousPr","\\begin{proposition}[Packet path expansion for different matrices]\n\\label{prop:heterogeneous-path}\nFor $i,j\\in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.heterogeneousPacketPathExpansion; Komorebi.PaperIII.EdgeOperators.SupportedStep; Komorebi.PaperIII.EdgeOperators.SupportedPath; Komorebi.PaperIII.EdgeOperators.heterogeneousProduct_apply_eq_zero_of_no_supportedPath}","pansion for different matrices]\n\\label{prop:heterogeneous-path}\nFor $i,j\\in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.heterogeneousPacketPathExpansion; Komorebi.PaperIII.EdgeOperators.SupportedStep; Komorebi.PaperIII.EdgeOperators.SupportedPath; Komorebi.PaperIII.EdgeOperators.heterogeneousProduct_apply_eq_zero_of_no_supportedPath}","rices]\n\\label{prop:heterogeneous-path}\nFor $i,j\\in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.heterogeneousPacketPathExpansion; Komorebi.PaperIII.EdgeOperators.SupportedStep; Komorebi.PaperIII.EdgeOperators.SupportedPath; Komorebi.PaperIII.EdgeOperators.heterogeneousProduct_apply_eq_zero_of_no_supportedPath}","in V$ and $m\\geq1$, set $i_0=i$ and $i_m=j$.  Then\n\\[\n\\boxed{\n(B_m\\cdots B_1)_{ji}\n=\n\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}\n  B_m[i_m,i_{m-1}]\\cdots B_1[i_1,i_0].\n}\n\\]\nWhen $m=1$, the sum is over the unique empty tuple and its summand is\n$B_1[j,i]$.\nFor $m=0$, the product is the identity and the entry is the Kronecker delta\n$\\delta_{ji}$.  If every $B_r[q,p]$ vanishes outside a declared relation\n$\\mathcal R_r(p,q)$, then the endpoint entry is zero whenever there is no\npacket path compatible with the successive relations.\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.EdgeOperators.heterogeneousProduct; Komorebi.PaperIII.EdgeOperators.heterogeneousPathSum; Komorebi.PaperIII.EdgeOperators.heterogeneousPacketPathExpansion; Komorebi.PaperIII.EdgeOperators.SupportedStep; Komorebi.PaperIII.EdgeOperators.SupportedPath; Komorebi.PaperIII.EdgeOperators.heterogeneousProduct_apply_eq_zero_of_no_supportedPath}","\\begin{boundary}[Changing packet types along a chain]\nThe compiled theorems\n\\leanname{heterogeneousPacketPathExpansion_two} and\n\\leanname{heterogeneousPacketPathExpansion_three} allow different packet\ntypes at the successive stages of two- and three-factor products.  The\narbitrary-length theorem above permits different factors but uses one common\nfinite packet type $V$.  A fully dependent arbitrary-length chain of changing\nmodules and packet types remains outside the version 1.0.1 contract.\n\\end{boundary}","\\begin{remark}[Checked specialization]\nWhen every $B_r$ is one repeated endomorphism $T$, the recursive\nspecialization to packet powers is inherited from Paper I.\n\\end{remark}\n\\statusleanprior{Komorebi.PacketNaturalityCompletion.Packets.transferBlock_pow_eq_packetPathSum}","For a loop represented on one common finite scalar packet coordinate type,\nthe proposition resolves its successive transport matrices into packet\nwalks.  If $R$ is a ring, as in the reversible connection API, then\n$(H-I)_{ji}=H_{ji}-\\delta_{ji}$.  In particular, off-diagonal entries are\nunchanged.  Thus a nonzero off-diagonal holonomy obstruction in this model\nwith a common index type requires a transport word along the closed chart\nloop whose packet path runs from its input packet to its output packet.  This\nconclusion requires declared block supports for the connection matrices.\nAn arbitrary connection need not respect the transfer graph from\n\\cref{sec:edge-operators}.","\\begin{example}[A scalar triangle]\nLet $B=\\{0,1,2\\}$ and $C_0=C_1=C_2=\\bbk$ for a field $\\bbk$.  Choose\n$u,v,w\\in\\bbk^\\times$ and set\n\\[\n\\Gamma_{1\\leftarrow0}(x)=ux,\\qquad\n\\Gamma_{2\\leftarrow1}(x)=vx,\\qquad\n\\Gamma_{2\\leftarrow0}(x)=wx,\n\\]\nwith reverse transports given by inverse scalars.  Then\n\\[\n\\Delta_{2,1,0}(x)=(vu-w)x,\n\\]\nand the triangle holonomy based at $0$ is\n\\[\nH_{2,1,0}(x)=w^{-1}vux.\n\\]\nTherefore\n\\[\nH_{2,1,0}-\\id\n=w^{-1}\\Delta_{2,1,0}.\n\\]\nThe connection is flat on this triangle exactly when $vu=w$.  If the three\ncoordinate systems are literally the same and canonical transport is used,\nthen $u=v=w=1$ and the obstruction vanishes.  Choosing arbitrary nonzero\n$u,v,w$ is a connection choice, not a consequence of changing exact packet","\\begin{example}[A scalar triangle]\nLet $B=\\{0,1,2\\}$ and $C_0=C_1=C_2=\\bbk$ for a field $\\bbk$.  Choose\n$u,v,w\\in\\bbk^\\times$ and set\n\\[\n\\Gamma_{1\\leftarrow0}(x)=ux,\\qquad\n\\Gamma_{2\\leftarrow1}(x)=vx,\\qquad\n\\Gamma_{2\\leftarrow0}(x)=wx,\n\\]\nwith reverse transports given by inverse scalars.  Then\n\\[\n\\Delta_{2,1,0}(x)=(vu-w)x,\n\\]\nand the triangle holonomy based at $0$ is\n\\[\nH_{2,1,0}(x)=w^{-1}vux.\n\\]\nTherefore\n\\[\nH_{2,1,0}-\\id\n=w^{-1}\\Delta_{2,1,0}.\n\\]\nThe connection is flat on this triangle exactly when $vu=w$.  If the three\ncoordinate systems are literally the same and canonical transport is used,\nthen $u=v=w=1$ and the obstruction vanishes.  Choosing arbitrary nonzero\n$u,v,w$ is a connection choice, not a consequence of changing exact packet\ncoordinates.\n\\end{examp","\\begin{example}[A scalar triangle]\nLet $B=\\{0,1,2\\}$ and $C_0=C_1=C_2=\\bbk$ for a field $\\bbk$.  Choose\n$u,v,w\\in\\bbk^\\times$ and set\n\\[\n\\Gamma_{1\\leftarrow0}(x)=ux,\\qquad\n\\Gamma_{2\\leftarrow1}(x)=vx,\\qquad\n\\Gamma_{2\\leftarrow0}(x)=wx,\n\\]\nwith reverse transports given by inverse scalars.  Then\n\\[\n\\Delta_{2,1,0}(x)=(vu-w)x,\n\\]\nand the triangle holonomy based at $0$ is\n\\[\nH_{2,1,0}(x)=w^{-1}vux.\n\\]\nTherefore\n\\[\nH_{2,1,0}-\\id\n=w^{-1}\\Delta_{2,1,0}.\n\\]\nThe connection is flat on this triangle exactly when $vu=w$.  If the three\ncoordinate systems are literally the same and canonical transport is used,\nthen $u=v=w=1$ and the obstruction vanishes.  Choosing arbitrary nonzero\n$u,v,w$ is a connection choice, not a consequence of changing exact packet\ncoordinates.\n\\end{example}","le}[A scalar triangle]\nLet $B=\\{0,1,2\\}$ and $C_0=C_1=C_2=\\bbk$ for a field $\\bbk$.  Choose\n$u,v,w\\in\\bbk^\\times$ and set\n\\[\n\\Gamma_{1\\leftarrow0}(x)=ux,\\qquad\n\\Gamma_{2\\leftarrow1}(x)=vx,\\qquad\n\\Gamma_{2\\leftarrow0}(x)=wx,\n\\]\nwith reverse transports given by inverse scalars.  Then\n\\[\n\\Delta_{2,1,0}(x)=(vu-w)x,\n\\]\nand the triangle holonomy based at $0$ is\n\\[\nH_{2,1,0}(x)=w^{-1}vux.\n\\]\nTherefore\n\\[\nH_{2,1,0}-\\id\n=w^{-1}\\Delta_{2,1,0}.\n\\]\nThe connection is flat on this triangle exactly when $vu=w$.  If the three\ncoordinate systems are literally the same and canonical transport is used,\nthen $u=v=w=1$ and the obstruction vanishes.  Choosing arbitrary nonzero\n$u,v,w$ is a connection choice, not a consequence of changing exact packet\ncoordinates.\n\\end{example}","v,w\\in\\bbk^\\times$ and set\n\\[\n\\Gamma_{1\\leftarrow0}(x)=ux,\\qquad\n\\Gamma_{2\\leftarrow1}(x)=vx,\\qquad\n\\Gamma_{2\\leftarrow0}(x)=wx,\n\\]\nwith reverse transports given by inverse scalars.  Then\n\\[\n\\Delta_{2,1,0}(x)=(vu-w)x,\n\\]\nand the triangle holonomy based at $0$ is\n\\[\nH_{2,1,0}(x)=w^{-1}vux.\n\\]\nTherefore\n\\[\nH_{2,1,0}-\\id\n=w^{-1}\\Delta_{2,1,0}.\n\\]\nThe connection is flat on this triangle exactly when $vu=w$.  If the three\ncoordinate systems are literally the same and canonical transport is used,\nthen $u=v=w=1$ and the obstruction vanishes.  Choosing arbitrary nonzero\n$u,v,w$ is a connection choice, not a consequence of changing exact packet\ncoordinates.\n\\end{example}","amma_{2\\leftarrow0}(x)=wx,\n\\]\nwith reverse transports given by inverse scalars.  Then\n\\[\n\\Delta_{2,1,0}(x)=(vu-w)x,\n\\]\nand the triangle holonomy based at $0$ is\n\\[\nH_{2,1,0}(x)=w^{-1}vux.\n\\]\nTherefore\n\\[\nH_{2,1,0}-\\id\n=w^{-1}\\Delta_{2,1,0}.\n\\]\nThe connection is flat on this triangle exactly when $vu=w$.  If the three\ncoordinate systems are literally the same and canonical transport is used,\nthen $u=v=w=1$ and the obstruction vanishes.  Choosing arbitrary nonzero\n$u,v,w$ is a connection choice, not a consequence of changing exact packet\ncoordinates.\n\\end{example}","se scalars.  Then\n\\[\n\\Delta_{2,1,0}(x)=(vu-w)x,\n\\]\nand the triangle holonomy based at $0$ is\n\\[\nH_{2,1,0}(x)=w^{-1}vux.\n\\]\nTherefore\n\\[\nH_{2,1,0}-\\id\n=w^{-1}\\Delta_{2,1,0}.\n\\]\nThe connection is flat on this triangle exactly when $vu=w$.  If the three\ncoordinate systems are literally the same and canonical transport is used,\nthen $u=v=w=1$ and the obstruction vanishes.  Choosing arbitrary nonzero\n$u,v,w$ is a connection choice, not a consequence of changing exact packet\ncoordinates.\n\\end{example}","\\begin{example}[A transfer walk with two steps]\nLet a packet graph have edges $e:0\\to1$ and $f:1\\to2$, with no other edges.\nThen\n\\[\nT=L_e+L_f,\\qquad\n(T^2)_{20}=c(f)c(e).\n\\]\nThe products $L_eL_e$, $L_eL_f$, and $L_fL_f$ vanish by typed mismatch.\nOnly $L_fL_e$, corresponding to the walk $0\\to1\\to2$, contributes to the\n$(2,0)$ entry.  This example also fixes the noncommutative coefficient order.\n\\end{example}","\\begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is","\\begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence","\\begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exac","\\begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representati","\\begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreser","\\begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarr","\\begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\ci","\\begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{pro","\\begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{proposition}\n\\statusleanthre","begin{proposition}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{proposition}\n\\statusleanthree{Komor…","n}[Exact naturality for arbitrary functions]\n\\label{prop:set-naturality}\nLet $\\cA^X_a:X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.Tra…",":X\\xrightarrow{\\sim}X_a$ and\n$\\cA^X_b:X\\xrightarrow{\\sim}X_b$ be exact coordinate equivalences for $X$,\nwith inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.Transport.exactCoordinateMap; Komorebi.PaperIII.Transport.exactCoordinateMap_comp; Komor…","th inverses $\\cS^X_a$ and $\\cS^X_b$.  Likewise, let\n$\\cA^Y_r:Y\\xrightarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.Transport.exactCoordinateMap; Komorebi.PaperIII.Transport.exactCoordinateMap_comp; Komorebi.PaperIII.Transport.exactCoordinateMap_naturality}","tarrow{\\sim}Y_r$ and\n$\\cA^Y_s:Y\\xrightarrow{\\sim}Y_s$ be exact coordinate equivalences for $Y$,\nwith inverses $\\cS^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.Transport.exactCoordinateMap; Komorebi.PaperIII.Transport.exactCoordinateMap_comp; Komorebi.PaperIII.Transport.exactCoordinateMap_naturality}","^Y_r$ and $\\cS^Y_s$.  Let $F:X\\to Y$ be any function, with\nno linearity or differentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.Transport.exactCoordinateMap; Komorebi.PaperIII.Transport.exactCoordinateMap_comp; Komorebi.PaperIII.Transport.exactCoordinateMap_naturality}","fferentiability hypothesis.  Set\n\\[\nJ^X_{b\\leftarrow a}=\\cA^X_b\\circ\\cS^X_a,\n\\qquad\nJ^Y_{s\\leftarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.Transport.exactCoordinateMap; Komorebi.PaperIII.Transport.exactCoordinateMap_comp; Komorebi.PaperIII.Transport.exactCoordinateMap_naturality}","tarrow r}=\\cA^Y_s\\circ\\cS^Y_r,\n\\]\nand define\n\\[\n\\widehat F_{r\\leftarrow a}\n=\n\\cA^Y_r\\circ F\\circ\\cS^X_a.\n\\]\nThen\n\\[\nJ^Y_{s\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}\n=\n\\widehat F_{s\\leftarrow b}\\circ J^X_{b\\leftarrow a}.\n\\]\nIf $\\cA^Z_t:Z\\xrightarrow{\\sim}Z_t$ is another exact coordinate equivalence\nand $G:Y\\to Z$ is any function, then exact coordinate representation also\npreserves composition:\n\\[\n\\widehat{G\\circ F}_{t\\leftarrow a}\n=\n\\widehat G_{t\\leftarrow r}\\circ\\widehat F_{r\\leftarrow a}.\n\\]\n\\end{proposition}\n\\statusleanthree{Komorebi.PaperIII.Transport.exactCoordinateMap; Komorebi.PaperIII.Transport.exactCoordinateMap_comp; Komorebi.PaperIII.Transport.exactCoordinateMap_naturality}","\\begin{proof}\nBoth sides of the naturality equation reduce to\n$\\cA^Y_s\\circ F\\circ\\cS^X_a$ using\n$\\cS\\circ\\cA=\\id$ in $X$ and $Y$.  For composition, the middle\n$\\cS^Y\\circ\\cA^Y$ cancels in the same way.\n\\end{proof}","\\begin{boundary}[Where nonlinear defects can enter]\nA nonlinear \\emph{coordinate} defect cannot be blamed on the exact canonical\n$J$ of \\cref{def:canonical-J}; \\cref{prop:set-naturality} makes that defect\nidentically zero.  Nonzero defects require extra structure such as a\nnoncanonical connection, truncation, approximation, restricted domains, or\nfailure of an operation to respect a declared packet model.\n\\end{boundary}","\\begin{definition}[Finite grammar for operation trees]\n\\label{def:compiled-operation-tree}\nFor types of input labels $I$, constants $K$, unary symbols $U$, and binary\nsymbols $B$, the released formal grammar has four constructors:\n\\[\n\\begin{aligned}\nt ::={}&\\operatorname{input}(i)\n\\mid\\operatorname{constant}(k)\n\\mid\\operatorname{unary}(u,t)","\\begin{definition}[Finite grammar for operation trees]\n\\label{def:compiled-operation-tree}\nFor types of input labels $I$, constants $K$, unary symbols $U$, and binary\nsymbols $B$, the released formal grammar has four constructors:\n\\[\n\\begin{aligned}\nt ::={}&\\operatorname{input}(i)\n\\mid\\operatorname{constant}(k)\n\\mid\\operatorname{unary}(u,t)\\\\\n&\\mid\\operatorname{binary}(b,t_1,t_2).\n\\end{aligned}\n\\]\nGiven an arbitrary value type $Y$, evaluation takes completely explicit maps\n\\[\nI\\to Y,\\qquad K\\to Y,\\qquad U\\to Y\\to Y,\\qquad\nB\\to Y\\to Y\\to Y.\n\\]\nNo algebraic, continuity, or differentiability law is imposed on these\nsemantics.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.NonlinearObstructions.OperationTree; Komorebi.PaperIII.NonlinearObstructions.OperationTree.eval}","&\\mid\\operatorname{binary}(b,t_1,t_2).\n\\end{aligned}\n\\]\nGiven an arbitrary value type $Y$, evaluation takes completely explicit maps\n\\[\nI\\to Y,\\qquad K\\to Y,\\qquad U\\to Y\\to Y,\\qquad\nB\\to Y\\to Y\\to Y.\n\\]\nNo algebraic, continuity, or differentiability law is imposed on these\nsemantics.\n\\end{definition}\n\\statusleanthree{Komorebi.PaperIII.NonlinearObstructions.OperationTree; Komorebi.PaperIII.NonlinearObstructions.OperationTree.eval}","When $I$ has decidable equality, let $\\operatorname{inputs}(t)$ be the finite\nset of input labels actually occurring in $t$.","\\begin{theorem}[Input locality for operation trees]\n\\label{thm:tree-locality}\nLet $\\eta_1,\\eta_2:I\\to Y$ be two input environments.  If\n\\[\n\\eta_1(i)=\\eta_2(i)\n\\qquad\\text{for every }i\\in\\operatorname{inputs}(t),\n\\]\nthen the two evaluations of $t$ are equal under the same constant, unary, and\nbinary semantics.  In particular, if\n$\\operatorname{inputs}(t)=\\varnothing$, evaluation is independent of the\ninput environment.\n\\end{theorem}\n\\statusleanthree{Komorebi.PaperIII.NonlinearObstructions.OperationTree.eval_congr_on_inputs; Komorebi.PaperIII.NonlinearObstructions.OperationTree.eval_eq_of_inputs_eq_empty}","\\begin{proof}\nStructural induction on $t$.  The input case is the agreement hypothesis;\nthe constant case is reflexive; the unary case applies the induction\nhypothesis to its child; and the binary case applies it independently to the\nleft and right subtrees, whose input sets form a union.\n\\end{proof}","\\begin{definition}[Finitary colored operation signature]\n\\label{def:signature}\nA finitary colored signature consists of:\n\\begin{itemize}\n\\item a finite type $V$ of colors;\n\\item a type $\\Sigma$ of operation symbols;\n\\item an arity $r(\\sigma)\\in\\N$ for every $\\sigma\\in\\Sigma$;\n\\item input colors\n$\\operatorname{in}_\\sigma:\\operatorname{Fin}(r(\\sigma))\\to V$;\n\\item an output color $\\operatorname{out}(\\sigma)\\in V$.\n\\end{itemize}\nA realization assigns a space $X_v$ to each color and a map\n\\[\n\\Phi_\\sigma:\n\\prod_{q\\in\\operatorname{Fin}(r(\\sigma))}X_{\\operatorname{in}_\\sigma(q)}\n\\longrightarrow X_{\\operatorname{out}(\\sigma)}.\n\\]\n\\end{definition}\n\\statusroadmap{The intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.}","el{def:signature}\nA finitary colored signature consists of:\n\\begin{itemize}\n\\item a finite type $V$ of colors;\n\\item a type $\\Sigma$ of operation symbols;\n\\item an arity $r(\\sigma)\\in\\N$ for every $\\sigma\\in\\Sigma$;\n\\item input colors\n$\\operatorname{in}_\\sigma:\\operatorname{Fin}(r(\\sigma))\\to V$;\n\\item an output color $\\operatorname{out}(\\sigma)\\in V$.\n\\end{itemize}\nA realization assigns a space $X_v$ to each color and a map\n\\[\n\\Phi_\\sigma:\n\\prod_{q\\in\\operatorname{Fin}(r(\\sigma))}X_{\\operatorname{in}_\\sigma(q)}\n\\longrightarrow X_{\\operatorname{out}(\\sigma)}.\n\\]\n\\end{definition}\n\\statusroadmap{The intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.}","lored signature consists of:\n\\begin{itemize}\n\\item a finite type $V$ of colors;\n\\item a type $\\Sigma$ of operation symbols;\n\\item an arity $r(\\sigma)\\in\\N$ for every $\\sigma\\in\\Sigma$;\n\\item input colors\n$\\operatorname{in}_\\sigma:\\operatorname{Fin}(r(\\sigma))\\to V$;\n\\item an output color $\\operatorname{out}(\\sigma)\\in V$.\n\\end{itemize}\nA realization assigns a space $X_v$ to each color and a map\n\\[\n\\Phi_\\sigma:\n\\prod_{q\\in\\operatorname{Fin}(r(\\sigma))}X_{\\operatorname{in}_\\sigma(q)}\n\\longrightarrow X_{\\operatorname{out}(\\sigma)}.\n\\]\n\\end{definition}\n\\statusroadmap{The intended Lean target is a dependent inductive typed tree; no such declaration is part of the version 1.0.1 completion contract.}","\\begin{definition}[Typed operation tree and evaluation]\n\\label{def:operation-tree}\nA typed operation tree is recursively either a leaf of declared color or a\nnode labeled by $\\sigma$, with one correctly colored child for each input\nslot of $\\sigma$.  Evaluation is recursive:\n\\[\n\\ev(\\operatorname{node}(\\sigma;t_1,\\ldots,t_{r(\\sigma)}))\n=\n\\Phi_\\sigma(\\ev(t_1),\\ldots,\\ev(t_{r(\\sigma)})).\n\\]\n\\end{definition}\n\\statusroadmap{Candidate API: \\leanname{Komorebi.PaperIII.NonlinearObstructions.ColoredOperationTree} and \\leanname{ColoredOperationTree.eval}.  This is stronger than the compiled uncolored unary--binary grammar.}","Suppose every $X_v$ has a finite complete packet family\n$(P^v_i)_{i\\in I_v}$.  A packet labeling assigns an admissible packet index\nto each edge or node of a typed tree.  For multilinear $\\Phi_\\sigma$, inserting\n\\[\n\\id_{X_v}=\\sum_{i\\in I_v}P^v_i\n\\]\nat every input and output yields a finite sum over trees with packet labels.\nThis is the branching analogue of inserting intermediate projectors in\n\\cref{prop:heterogeneous-path}.","For coordinate models of the source and target colors, an additive target\ncoordinate space, and a chosen noncanonical connection, a pointwise node\ndefect can be defined by\n\\[\n\\mathfrak D^\\Gamma_{b,a}(\\Phi)\n=\n\\Gamma^Y_{b\\leftarrow a}\\circ\\widehat\\Phi_a\n-\n\\widehat\\Phi_b\\circ\n\\left(\\prod_q\\Gamma^{X_q}_{b\\leftarrow a}\\right).\n\\]\nThis expression compares two \\emph{chosen} routes.  With canonical exact\ntransport in every slot it vanishes by \\cref{prop:set-naturality}.","\\addlinespace\nTwisted three-point loop\n& Lean-backed III\n& \\leanname{twistedThreePointConnection}; \\leanname{twisted_triangleHolonomy_ne_refl}\n& Concrete rational connection; the selected loop sends \\(1\\) to \\(2\\).","\\medskip\\noindent The notational change from \\(\\cA,\\cS\\) to \\(\\Acal,\\Scal\\) is typographical: both denote analysis and synthesis for the declared packet system. A connection remains additional data. Smooth canonical transport is constructed from a moving projector family, whereas the algebraic exact coordinate transport in Part~I compares fixed exact descriptions of one ambient object.\n\\section{Introduction}\\label{sec:introduction}","A finite family of pairwise annihilating idempotents gives exact coordinates.  If\n\\[\nP_\\alpha P_\\beta=0\n\\quad\\text{for }\\alpha\\ne\\beta,\n\\qquad\n\\sum_\\alpha P_\\alpha=I,\n\\]\nthen every vector is the sum of its packet components.  Linear operators become block matrices in these coordinates.  A nonlinear map cannot be reconstructed from its diagonal restrictions, since mixed interactions between packet components are usually present.  It can still be represented exactly if its input is the complete packet tuple.","A linear correction map\n\\[\nD:U\\longrightarrow V\n\\]\nplaces the obstruction to solving $Du=-r$ in the canonical quotient $\\coker D$.  This quotient has a universal property and behaves naturally under commuting squares.  Mapping cones extend the same construction to cochain complexes.  In a Hilbert space with closed range, the quotient is represented by the orthogonal projection onto $\\Ker D^*$.  The Moore Penrose inverse then gives the least norm correction.  Relevant background appears in \\cite{Weibel1994,Penrose1955}.","The first question is representation.  Given a nonlinear map $F$, can its packet form be written without approximation?  The answer is yes.  Analysis and synthesis are inverse coordinate maps, so conjugation preserves the full equation.","The third question is correction.  Suppose an approximate solution leaves a defect $r$.  A declared linearized correction map $D$ removes the part of $r$ lying in $\\im D$.  The quotient class\n\\[\n[r]\\in\\coker D\n\\]\nis the part that no allowed first order correction can remove.  This quotient class is the obstruction.  Its vanishing must be supplemented by a nonlinear theorem.","These four layers fit into the sequence\n\\[\n\\begin{gathered}\n\\text{exact coordinates}\n\\longrightarrow\n\\text{packet interactions}\n\\longrightarrow\n\\text{correction classes}\n\\\\\n\\longrightarrow\n\\text{global gluing}.\n\\end{gathered}\n\\]\nEach arrow needs hypotheses.  Exact coordinates need a complete packet system.  Differential interactions need regularity.  A cokernel obstruction needs a specified correction map.  Holonomy needs invertible local transports.","\\subsection*{Standing conventions}\nAll packet systems are finite unless a statement says otherwise.  In a normed setting every packet projector and every coarsening operator is bounded.  Differentiability of operator valued maps is taken in operator norm.  A Banach space isomorphism is a bounded linear bijection with bounded inverse.  For a decreasing filtration $(F^mX)$, completeness means that the canonical map\n\\[\nX\\longrightarrow\\varprojlim_m X/F^mX\n\\]\nis an isomorphism and separatedness means $\\bigcap_mF^mX=0$.  Statements about congruence classes in a filtered nonlinear problem also assume that the nonlinear map preserves filtration congruences at the stated depth.  These assumptions are repeated where they are essential to a proof.","ng conventions}\nAll packet systems are finite unless a statement says otherwise.  In a normed setting every packet projector and every coarsening operator is bounded.  Differentiability of operator valued maps is taken in operator norm.  A Banach space isomorphism is a bounded linear bijection with bounded inverse.  For a decreasing filtration $(F^mX)$, completeness means that the canonical map\n\\[\nX\\longrightarrow\\varprojlim_m X/F^mX\n\\]\nis an isomorphism and separatedness means $\\bigcap_mF^mX=0$.  Statements about congruence classes in a filtered nonlinear problem also assume that the nonlinear map preserves filtration congruences at the stated depth.  These assumptions are repeated where they are essential to a proof.","a normed setting every packet projector and every coarsening operator is bounded.  Differentiability of operator valued maps is taken in operator norm.  A Banach space isomorphism is a bounded linear bijection with bounded inverse.  For a decreasing filtration $(F^mX)$, completeness means that the canonical map\n\\[\nX\\longrightarrow\\varprojlim_m X/F^mX\n\\]\nis an isomorphism and separatedness means $\\bigcap_mF^mX=0$.  Statements about congruence classes in a filtered nonlinear problem also assume that the nonlinear map preserves filtration congruences at the stated depth.  These assumptions are repeated where they are essential to a proof.","Let $X$ be a module over a unital ring.  A finite packet system on $X$ is a family\n\\[\n\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\subseteq\\End(X)\n\\]\nof nonzero idempotents such that\n\\[\nP_\\alpha P_\\beta=0\n\\quad\\text{when }\\alpha\\ne\\beta,\n\\qquad\n\\sum_{\\alpha\\in I}P_\\alpha=I_X.\n\\]\nWrite\n\\[\nX_\\alpha=P_\\alpha X,\n\\qquad\nX^{\\Pcal}=\\bigoplus_{\\alpha\\in I}X_\\alpha.\n\\]\nThe analysis and synthesis maps are\n\\[\n\\Acal_{\\Pcal}x=(P_\\alpha x)_\\alpha,\n\\qquad\n\\Scal_{\\Pcal}(x_\\alpha)_\\alpha=\\sum_\\alpha x_\\alpha.\n\\]\nThey are inverse isomorphisms.","\\begin{definition}[Nonlinear packet coordinate map]\\label{def:nonlinear_packet_map}\nLet $\\Pcal$ be a packet system on $X$ and let $\\Qcal$ be a packet system on $Y$.  For any map $F:X\\to Y$, define\n\\[\nF_{\\Qcal\\leftarrow\\Pcal}\n=\n\\Acal_{\\Qcal}F\\Scal_{\\Pcal}.\n\\]\nThus\n\\[\nF_{\\Qcal\\leftarrow\\Pcal}((x_\\alpha)_\\alpha)\n=\n\\left(\nQ_\\gamma F\\left(\\sum_\\alpha x_\\alpha\\right)\n\\right)_\\gamma.\n\\]\n\\end{definition}","\\begin{theorem}[Exact nonlinear coordinate theorem]\\label{thm:exact_nonlinear_coordinates}\nFor every map $F:X\\to Y$,\n\\[\n\\Scal_{\\Qcal}F_{\\Qcal\\leftarrow\\Pcal}\\Acal_{\\Pcal}=F.\n\\]\nIf $G:Y\\to Z$ and $\\Rcal$ is a packet system on $Z$, then\n\\[\n(G\\circ F)_{\\Rcal\\leftarrow\\Pcal}\n=\nG_{\\Rcal\\leftarrow\\Qcal}\n\\circ\nF_{\\Qcal\\leftarrow\\Pcal}.\n\\]\nNo linearity, continuity, or differentiability assumption is needed.\n\\end{theorem}","\\begin{proof}\nThe identities $\\Scal_{\\Pcal}\\Acal_{\\Pcal}=I_X$ and $\\Scal_{\\Qcal}\\Acal_{\\Qcal}=I_Y$ give\n\\[\n\\Scal_{\\Qcal}F_{\\Qcal\\leftarrow\\Pcal}\\Acal_{\\Pcal}\n=\n\\Scal_{\\Qcal}\\Acal_{\\Qcal}F\\Scal_{\\Pcal}\\Acal_{\\Pcal}\n=F.\n\\]\nFor composition,\n\\[\nG_{\\Rcal\\leftarrow\\Qcal}F_{\\Qcal\\leftarrow\\Pcal}\n=\n\\Acal_{\\Rcal}G\\Scal_{\\Qcal}\\Acal_{\\Qcal}F\\Scal_{\\Pcal}\n=\n\\Acal_{\\Rcal}(G\\circ F)\\Scal_{\\Pcal}.\n\\]\n\\end{proof}","\\begin{corollary}[Dynamical equivalence]\\label{cor:dynamical_equivalence}\nSuppose $X=Y$ and the same packet system is used on both sides.  Packet coordinates preserve fixed points, periodic points, injectivity, surjectivity, and invertibility.  In particular,\n\\[\nF(x)=x\n\\quad\\Longleftrightarrow\\quad\nF_{\\Pcal\\leftarrow\\Pcal}(\\Acal_{\\Pcal}x)=\\Acal_{\\Pcal}x.\n\\]\n\\end{corollary}","\\begin{proof}\nThe packet map is conjugate to $F$ through the isomorphism $\\Acal_{\\Pcal}$.  Every listed property is invariant under conjugacy.\n\\end{proof}","\\begin{proposition}[Change of packet coordinates]\\label{prop:change_packet_coordinates}\nLet $\\Pcal$ and $\\Pcal'$ be packet systems on $X$.  Let $\\Qcal$ and $\\Qcal'$ be packet systems on $Y$.  For any two packet systems $\\Ucal$ and $\\Vcal$ on the same space, write\n\\[\nJ_{\\Ucal\\leftarrow\\Vcal}\n=\n\\Acal_{\\Ucal}\\Scal_{\\Vcal}.\n\\]\nThen\n\\[\nF_{\\Qcal'\\leftarrow\\Pcal'}\n=\nJ_{\\Qcal'\\leftarrow\\Qcal}\nF_{\\Qcal\\leftarrow\\Pcal}\nJ_{\\Pcal\\leftarrow\\Pcal'}.\n\\]\n\\end{proposition}","\\begin{proof}\nInsert $\\Scal_{\\Qcal}\\Acal_{\\Qcal}=I_Y$ and $\\Scal_{\\Pcal}\\Acal_{\\Pcal}=I_X$ into the definition of $F_{\\Qcal'\\leftarrow\\Pcal'}$.\n\\end{proof}","\\begin{example}[A mixed quadratic map]\\label{ex:mixed_quadratic}\nLet $X=Y=\\bbk^2$ with coordinate packets.  For\n\\[\nF(x_1,x_2)=(x_1x_2,x_1^2+x_2),\n\\]\nthe first output packet depends on both input packets.  The second output packet contains a quadratic contribution from the first input packet and a linear contribution from the second.  Packet coordinates record these interactions exactly.  Replacing $F$ by the two diagonal restrictions would lose the product $x_1x_2$.\n\\end{example}","egin{example}[A mixed quadratic map]\\label{ex:mixed_quadratic}\nLet $X=Y=\\bbk^2$ with coordinate packets.  For\n\\[\nF(x_1,x_2)=(x_1x_2,x_1^2+x_2),\n\\]\nthe first output packet depends on both input packets.  The second output packet contains a quadratic contribution from the first input packet and a linear contribution from the second.  Packet coordinates record these interactions exactly.  Replacing $F$ by the two diagonal restrictions would lose the product $x_1x_2$.\n\\end{example}","c}\nLet $X=Y=\\bbk^2$ with coordinate packets.  For\n\\[\nF(x_1,x_2)=(x_1x_2,x_1^2+x_2),\n\\]\nthe first output packet depends on both input packets.  The second output packet contains a quadratic contribution from the first input packet and a linear contribution from the second.  Packet coordinates record these interactions exactly.  Replacing $F$ by the two diagonal restrictions would lose the product $x_1x_2$.\n\\end{example}","Let $X$ and $Y$ be Banach spaces over the same field $\\mathbb R$ or $\\mathbb C$ and let the packet projectors on both spaces be bounded. In the complex case, derivatives are complex Fr\\'echet derivatives and the Taylor integral below has a real parameter. Assume that $F:X\\to Y$ is $m$ times continuously Fr\\'echet differentiable. The same local statements hold on open domains containing the points and segments in question. For $1\\le r\\le m$, the derivative\n\\[\nD^rF(x):X^r\\longrightarrow Y\n\\]\nis a continuous symmetric multilinear map.","s over the same field $\\mathbb R$ or $\\mathbb C$ and let the packet projectors on both spaces be bounded. In the complex case, derivatives are complex Fr\\'echet derivatives and the Taylor integral below has a real parameter. Assume that $F:X\\to Y$ is $m$ times continuously Fr\\'echet differentiable. The same local statements hold on open domains containing the points and segments in question. For $1\\le r\\le m$, the derivative\n\\[\nD^rF(x):X^r\\longrightarrow Y\n\\]\nis a continuous symmetric multilinear map.","R$ or $\\mathbb C$ and let the packet projectors on both spaces be bounded. In the complex case, derivatives are complex Fr\\'echet derivatives and the Taylor integral below has a real parameter. Assume that $F:X\\to Y$ is $m$ times continuously Fr\\'echet differentiable. The same local statements hold on open domains containing the points and segments in question. For $1\\le r\\le m$, the derivative\n\\[\nD^rF(x):X^r\\longrightarrow Y\n\\]\nis a continuous symmetric multilinear map.","The range $X_\\alpha$ is closed because\n$X_\\alpha=\\ker(I-P_\\alpha)$. It is therefore Banach with its inherited norm. The same holds for every $Y_\\gamma$. Give the finite product of ranges the maximum norm. Packet analysis and synthesis are bounded, since they are formed from finitely many bounded projectors and a finite sum. Their algebraic inverse identities from \\cref{sec:nonlinear_coordinates} make them bounded linear isomorphisms. For a nonempty input family,\n$\\|\\Acal_{\\Pcal}\\|\\le\\max_\\alpha\\|P_\\alpha\\|$ and\n$\\|\\Scal_{\\Pcal}\\|\\le |I|$. An empty complete family can occur only on the zero space. Its coordinate product is the zero vector space with norm zero, and the coordinate identities hold directly.","It is therefore Banach with its inherited norm. The same holds for every $Y_\\gamma$. Give the finite product of ranges the maximum norm. Packet analysis and synthesis are bounded, since they are formed from finitely many bounded projectors and a finite sum. Their algebraic inverse identities from \\cref{sec:nonlinear_coordinates} make them bounded linear isomorphisms. For a nonempty input family,\n$\\|\\Acal_{\\Pcal}\\|\\le\\max_\\alpha\\|P_\\alpha\\|$ and\n$\\|\\Scal_{\\Pcal}\\|\\le |I|$. An empty complete family can occur only on the zero space. Its coordinate product is the zero vector space with norm zero, and the coordinate identities hold directly.","e same holds for every $Y_\\gamma$. Give the finite product of ranges the maximum norm. Packet analysis and synthesis are bounded, since they are formed from finitely many bounded projectors and a finite sum. Their algebraic inverse identities from \\cref{sec:nonlinear_coordinates} make them bounded linear isomorphisms. For a nonempty input family,\n$\\|\\Acal_{\\Pcal}\\|\\le\\max_\\alpha\\|P_\\alpha\\|$ and\n$\\|\\Scal_{\\Pcal}\\|\\le |I|$. An empty complete family can occur only on the zero space. Its coordinate product is the zero vector space with norm zero, and the coordinate identities hold directly.","\\begin{definition}[Packet jet]\\label{def:packet_jet}\nFor $1\\le r\\le m$, define\n\\[\nJ^{r}_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}F(x)\n=\nQ_\\gamma D^rF(x)\n\\big|_{X_{\\alpha_1}\\times\\cdots\\times X_{\\alpha_r}}.\n\\]\nThis is a continuous multilinear map\n\\[\nX_{\\alpha_1}\\times\\cdots\\times X_{\\alpha_r}\n\\longrightarrow\nY_\\gamma.\n\\]\n\\end{definition}","\\begin{theorem}[Exact reconstruction of derivatives]\\label{thm:jet_reconstruction}\nFor $h_1,\\ldots,h_r\\in X$,\n\\[\nD^rF(x)[h_1,\\ldots,h_r]\n=\n\\sum_{\\gamma}\n\\sum_{\\alpha_1,\\ldots,\\alpha_r}\nJ^{r}_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}F(x)\n[P_{\\alpha_1}h_1,\\ldots,P_{\\alpha_r}h_r].\n\\]\n\\end{theorem}","\\begin{proof}\nInsert $I_Y=\\sum_\\gamma Q_\\gamma$ on the output and $I_X=\\sum_\\alpha P_\\alpha$ in every input slot.  Multilinearity expands the finite sums.\n\\end{proof}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.96\\linewidth]{packet_jet_worked_example.pdf}\n\\caption{Second packet jets for $F(x,y)=(x+xy,y+x^2)$ at the origin. Coordinate axis projectors give $DF(0)=I$. The first output has mixed second derivatives equal to one, while the second has a repeated first input derivative equal to two. Each grid records the ordered pair of input packets. The mixed entries retain an interaction that diagonal restrictions would lose.}\n\\label{fig:packet-jet-example}\n\\end{figure}\n\\clearpage","\\begin{proposition}[Jet norm bound]\\label{prop:jet_norm_bound}\nGive every packet range its inherited subspace norm. The restricted jet satisfies the sharper estimate\n\\[\n\\left\\|J^r_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}F(x)\\right\\|\n\\le \\|Q_\\gamma\\|\\,\\|D^rF(x)\\|.\n\\]\nIts ambient extension inserts $P_{\\alpha_j}$ in the $j$th slot and has norm at most\n$\\|Q_\\gamma\\|\\,\\|D^rF(x)\\|\\prod_j\\|P_{\\alpha_j}\\|$.\nConsequently the operator norm of the restricted packet jet also satisfies\n\\[\n\\left\\|J^{r}_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}F(x)\\right\\|\n\\le\n\\|Q_\\gamma\\|\\,\\|D^rF(x)\\|\n\\prod_{j=1}^r\\|P_{\\alpha_j}\\|.\n\\]\nWhen all packets are orthogonal projections on Hilbert spaces, every nonzero projector has norm one and the bound reduces to the norm of the full derivative.\n\\end{proposition}","cket range its inherited subspace norm. The restricted jet satisfies the sharper estimate\n\\[\n\\left\\|J^r_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}F(x)\\right\\|\n\\le \\|Q_\\gamma\\|\\,\\|D^rF(x)\\|.\n\\]\nIts ambient extension inserts $P_{\\alpha_j}$ in the $j$th slot and has norm at most\n$\\|Q_\\gamma\\|\\,\\|D^rF(x)\\|\\prod_j\\|P_{\\alpha_j}\\|$.\nConsequently the operator norm of the restricted packet jet also satisfies\n\\[\n\\left\\|J^{r}_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}F(x)\\right\\|\n\\le\n\\|Q_\\gamma\\|\\,\\|D^rF(x)\\|\n\\prod_{j=1}^r\\|P_{\\alpha_j}\\|.\n\\]\nWhen all packets are orthogonal projections on Hilbert spaces, every nonzero projector has norm one and the bound reduces to the norm of the full derivative.\n\\end{proposition}","\\begin{proof}\nFor $h_j\\in X_{\\alpha_j}$, the inherited norm gives\n\\[\n\\|Q_\\gamma D^rF(x)[h_1,\\ldots,h_r]\\|\n\\le \\|Q_\\gamma\\|\\,\\|D^rF(x)\\|\\prod_j\\|h_j\\|.\n\\]\nTaking the multilinear operator norm proves the sharper restricted estimate. For ambient inputs, replace $h_j$ by $P_{\\alpha_j}h_j$ and apply the operator norm bound to each projector. Every nonzero idempotent has norm at least one, since it fixes a nonzero vector in its range. The displayed product bound therefore also follows for the restricted jet. If a zero packet is retained as a redundant index, multilinearity makes any jet with that input or output packet zero, so both estimates remain valid.\n\\end{proof}","\\begin{corollary}[Packet Taylor formula]\\label{cor:packet_taylor}\nAssume $F$ is $m+1$ times continuously Fr\\'echet differentiable on an open set containing the segment $x+th$ for $0\\le t\\le1$.  Then\n\\[\n\\begin{aligned}\nQ_\\gamma F(x+h)\n={}&Q_\\gamma F(x)\n+\n\\sum_{r=1}^{m}\\frac1{r!}\n\\sum_{\\alpha_1,\\ldots,\\alpha_r}\nJ^{r}_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}F(x)\n[P_{\\alpha_1}h,\\ldots,P_{\\alpha_r}h]","\\begin{corollary}[Packet Taylor formula]\\label{cor:packet_taylor}\nAssume $F$ is $m+1$ times continuously Fr\\'echet differentiable on an open set containing the segment $x+th$ for $0\\le t\\le1$.  Then\n\\[\n\\begin{aligned}\nQ_\\gamma F(x+h)\n={}&Q_\\gamma F(x)\n+\n\\sum_{r=1}^{m}\\frac1{r!}\n\\sum_{\\alpha_1,\\ldots,\\alpha_r}\nJ^{r}_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}F(x)\n[P_{\\alpha_1}h,\\ldots,P_{\\alpha_r}h]\\\\\n&+Q_\\gamma R_{m+1}(x,h),\n\\end{aligned}\n\\]\nwhere the Bochner integral remainder is\n\\[\nR_{m+1}(x,h)\n=\n\\frac1{m!}\n\\int_0^1(1-t)^m\nD^{m+1}F(x+th)[h,\\ldots,h]\\,dt.\n\\]\n\\end{corollary}","&+Q_\\gamma R_{m+1}(x,h),\n\\end{aligned}\n\\]\nwhere the Bochner integral remainder is\n\\[\nR_{m+1}(x,h)\n=\n\\frac1{m!}\n\\int_0^1(1-t)^m\nD^{m+1}F(x+th)[h,\\ldots,h]\\,dt.\n\\]\n\\end{corollary}","\\begin{proof}\nSet $g(t)=F(x+th)$ on the open preimage of the domain under $t\\mapsto x+th$. It contains $[0,1]$, and repeated differentiation gives\n$g^{(r)}(t)=D^rF(x+th)[h,\\ldots,h]$. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put\n\\[\nT_j=\\frac1{j!}\\int_0^1(1-t)^j g^{(j+1)}(t)\\,dt\n\\qquad(0\\le j\\le m).\n\\]\nThe fundamental theorem of calculus gives $T_0=g(1)-g(0)$. For $j\\ge1$, integration by parts gives\n$T_j=T_{j-1}-g^{(j)}(0)/j!$. Iterating this relation proves the Taylor identity with remainder $T_m=R_{m+1}(x,h)$ and its exact factorial normalization.","$g(t)=F(x+th)$ on the open preimage of the domain under $t\\mapsto x+th$. It contains $[0,1]$, and repeated differentiation gives\n$g^{(r)}(t)=D^rF(x+th)[h,\\ldots,h]$. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put\n\\[\nT_j=\\frac1{j!}\\int_0^1(1-t)^j g^{(j+1)}(t)\\,dt\n\\qquad(0\\le j\\le m).\n\\]\nThe fundamental theorem of calculus gives $T_0=g(1)-g(0)$. For $j\\ge1$, integration by parts gives\n$T_j=T_{j-1}-g^{(j)}(0)/j!$. Iterating this relation proves the Taylor identity with remainder $T_m=R_{m+1}(x,h)$ and its exact factorial normalization.","open preimage of the domain under $t\\mapsto x+th$. It contains $[0,1]$, and repeated differentiation gives\n$g^{(r)}(t)=D^rF(x+th)[h,\\ldots,h]$. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put\n\\[\nT_j=\\frac1{j!}\\int_0^1(1-t)^j g^{(j+1)}(t)\\,dt\n\\qquad(0\\le j\\le m).\n\\]\nThe fundamental theorem of calculus gives $T_0=g(1)-g(0)$. For $j\\ge1$, integration by parts gives\n$T_j=T_{j-1}-g^{(j)}(0)/j!$. Iterating this relation proves the Taylor identity with remainder $T_m=R_{m+1}(x,h)$ and its exact factorial normalization.","t\\mapsto x+th$. It contains $[0,1]$, and repeated differentiation gives\n$g^{(r)}(t)=D^rF(x+th)[h,\\ldots,h]$. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put\n\\[\nT_j=\\frac1{j!}\\int_0^1(1-t)^j g^{(j+1)}(t)\\,dt\n\\qquad(0\\le j\\le m).\n\\]\nThe fundamental theorem of calculus gives $T_0=g(1)-g(0)$. For $j\\ge1$, integration by parts gives\n$T_j=T_{j-1}-g^{(j)}(0)/j!$. Iterating this relation proves the Taylor identity with remainder $T_m=R_{m+1}(x,h)$ and its exact factorial normalization.","h,\\ldots,h]$. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put\n\\[\nT_j=\\frac1{j!}\\int_0^1(1-t)^j g^{(j+1)}(t)\\,dt\n\\qquad(0\\le j\\le m).\n\\]\nThe fundamental theorem of calculus gives $T_0=g(1)-g(0)$. For $j\\ge1$, integration by parts gives\n$T_j=T_{j-1}-g^{(j)}(0)/j!$. Iterating this relation proves the Taylor identity with remainder $T_m=R_{m+1}(x,h)$ and its exact factorial normalization.","Apply $Q_\\gamma$ to the identity and use \\cref{thm:jet_reconstruction}. Pairwise annihilation and idempotence retain exactly the output packet $\\gamma$. Boundedness of $Q_\\gamma$ allows it to pass through the Bochner integral. In particular, if\n$\\|D^{m+1}F(x+th)\\|\\le M$ along the segment, then\n\\[\n\\|Q_\\gamma R_{m+1}(x,h)\\|\n\\le \\frac{\\|Q_\\gamma\\|M\\|h\\|^{m+1}}{(m+1)!}.\n\\]\nFor $m=0$ the Taylor sum is empty. For $h=0$, multilinearity makes every positive-order term and the remainder zero.\n\\end{proof}","mma$ to the identity and use \\cref{thm:jet_reconstruction}. Pairwise annihilation and idempotence retain exactly the output packet $\\gamma$. Boundedness of $Q_\\gamma$ allows it to pass through the Bochner integral. In particular, if\n$\\|D^{m+1}F(x+th)\\|\\le M$ along the segment, then\n\\[\n\\|Q_\\gamma R_{m+1}(x,h)\\|\n\\le \\frac{\\|Q_\\gamma\\|M\\|h\\|^{m+1}}{(m+1)!}.\n\\]\nFor $m=0$ the Taylor sum is empty. For $h=0$, multilinearity makes every positive-order term and the remainder zero.\n\\end{proof}","The relation to derivatives of the complete coordinate representation is also literal. For an open domain $U\\subseteq X$, put\n$\\widehat F=\\Acal_{\\Qcal}\\circ F\\circ\\Scal_{\\Pcal}$ on the open coordinate set $\\Acal_{\\Pcal}(U)$. Boundedness and linearity of analysis and synthesis imply, for $1\\le r\\le m$,\n\\[\nD^r\\widehat F(u)[v_1,\\ldots,v_r]\n=\\Acal_{\\Qcal}\\bigl(D^rF(\\Scal_{\\Pcal}u)\n[\\Scal_{\\Pcal}v_1,\\ldots,\\Scal_{\\Pcal}v_r]\\bigr).\n\\]\nIndeed, the first derivative follows from the chain rule. Repeating it adds no higher derivatives of the fixed linear coordinate maps. If each $v_j$ is supported in one coordinate slot $\\alpha_j$, synthesis includes that slot into $X$. Its $\\gamma$ output component is exactly the range jet defined above, including when a chosen direction is zero.","on the open coordinate set $\\Acal_{\\Pcal}(U)$. Boundedness and linearity of analysis and synthesis imply, for $1\\le r\\le m$,\n\\[\nD^r\\widehat F(u)[v_1,\\ldots,v_r]\n=\\Acal_{\\Qcal}\\bigl(D^rF(\\Scal_{\\Pcal}u)\n[\\Scal_{\\Pcal}v_1,\\ldots,\\Scal_{\\Pcal}v_r]\\bigr).\n\\]\nIndeed, the first derivative follows from the chain rule. Repeating it adds no higher derivatives of the fixed linear coordinate maps. If each $v_j$ is supported in one coordinate slot $\\alpha_j$, synthesis includes that slot into $X$. Its $\\gamma$ output component is exactly the range jet defined above, including when a chosen direction is zero.","(U)$. Boundedness and linearity of analysis and synthesis imply, for $1\\le r\\le m$,\n\\[\nD^r\\widehat F(u)[v_1,\\ldots,v_r]\n=\\Acal_{\\Qcal}\\bigl(D^rF(\\Scal_{\\Pcal}u)\n[\\Scal_{\\Pcal}v_1,\\ldots,\\Scal_{\\Pcal}v_r]\\bigr).\n\\]\nIndeed, the first derivative follows from the chain rule. Repeating it adds no higher derivatives of the fixed linear coordinate maps. If each $v_j$ is supported in one coordinate slot $\\alpha_j$, synthesis includes that slot into $X$. Its $\\gamma$ output component is exactly the range jet defined above, including when a chosen direction is zero.","d synthesis imply, for $1\\le r\\le m$,\n\\[\nD^r\\widehat F(u)[v_1,\\ldots,v_r]\n=\\Acal_{\\Qcal}\\bigl(D^rF(\\Scal_{\\Pcal}u)\n[\\Scal_{\\Pcal}v_1,\\ldots,\\Scal_{\\Pcal}v_r]\\bigr).\n\\]\nIndeed, the first derivative follows from the chain rule. Repeating it adds no higher derivatives of the fixed linear coordinate maps. If each $v_j$ is supported in one coordinate slot $\\alpha_j$, synthesis includes that slot into $X$. Its $\\gamma$ output component is exactly the range jet defined above, including when a chosen direction is zero.","is imply, for $1\\le r\\le m$,\n\\[\nD^r\\widehat F(u)[v_1,\\ldots,v_r]\n=\\Acal_{\\Qcal}\\bigl(D^rF(\\Scal_{\\Pcal}u)\n[\\Scal_{\\Pcal}v_1,\\ldots,\\Scal_{\\Pcal}v_r]\\bigr).\n\\]\nIndeed, the first derivative follows from the chain rule. Repeating it adds no higher derivatives of the fixed linear coordinate maps. If each $v_j$ is supported in one coordinate slot $\\alpha_j$, synthesis includes that slot into $X$. Its $\\gamma$ output component is exactly the range jet defined above, including when a chosen direction is zero.","At a fixed base point, the nonzero packet jets form a directed hypergraph.  An order $r$ interaction has $r$ input packet labels and one output packet label.  The support therefore records interactions relative to the chosen packet system.","For a block $B=\\{i_1<\\cdots<i_s\\}$ and an intermediate packet $\\gamma$, write\n\\[\nG_{\\gamma,B}(x\\mid h)\n=\nJ^s_{\\gamma\\mid\\alpha_{i_1},\\ldots,\\alpha_{i_s}}G(x)\n[h_{i_1},\\ldots,h_{i_s}].\n\\]","\\begin{theorem}[Packet Fa\\`a di Bruno formula]\\label{thm:packet_faa}\nLet $n\\ge1$, let $X,Y,Z$ be Banach spaces over the same real or complex field, and let $G:X\\to Y$ and $F:Y\\to Z$ be $n$ times continuously Fr\\'echet differentiable. Let $\\Pcal$, $\\Qcal$, and $\\Rcal$ be packet systems on $X$, $Y$, and $Z$.  Fix input packets $\\alpha_1,\\ldots,\\alpha_n$, an output packet $\\delta$, and directions $h_i\\in X_{\\alpha_i}$.  Then\n\\[\n\\begin{aligned}\n&J^{n}_{\\delta\\mid\\alpha_1,\\ldots,\\alpha_n}(F\\circ G)(x)\n[h_1,\\ldots,h_n]","di Bruno formula]\\label{thm:packet_faa}\nLet $n\\ge1$, let $X,Y,Z$ be Banach spaces over the same real or complex field, and let $G:X\\to Y$ and $F:Y\\to Z$ be $n$ times continuously Fr\\'echet differentiable. Let $\\Pcal$, $\\Qcal$, and $\\Rcal$ be packet systems on $X$, $Y$, and $Z$.  Fix input packets $\\alpha_1,\\ldots,\\alpha_n$, an output packet $\\delta$, and directions $h_i\\in X_{\\alpha_i}$.  Then\n\\[\n\\begin{aligned}\n&J^{n}_{\\delta\\mid\\alpha_1,\\ldots,\\alpha_n}(F\\circ G)(x)\n[h_1,\\ldots,h_n]\\\\\n&=\n\\sum_{\\pi=\\{B_1,\\ldots,B_k\\}}\n\\sum_{\\gamma_1,\\ldots,\\gamma_k}\nJ^{k}_{\\delta\\mid\\gamma_1,\\ldots,\\gamma_k}F(Gx)\n\\big[\nG_{\\gamma_1,B_1}(x\\mid h),\\ldots,\nG_{\\gamma_k,B_k}(x\\mid h)\n\\big].\n\\end{aligned}\n\\]\nThe first sum runs over all set partitions of $\\{1,\\ldots,n\\}$.\n\\end{theorem}","&=\n\\sum_{\\pi=\\{B_1,\\ldots,B_k\\}}\n\\sum_{\\gamma_1,\\ldots,\\gamma_k}\nJ^{k}_{\\delta\\mid\\gamma_1,\\ldots,\\gamma_k}F(Gx)\n\\big[\nG_{\\gamma_1,B_1}(x\\mid h),\\ldots,\nG_{\\gamma_k,B_k}(x\\mid h)\n\\big].\n\\end{aligned}\n\\]\nThe first sum runs over all set partitions of $\\{1,\\ldots,n\\}$.\n\\end{theorem}","\\begin{proof}\nFirst derive the unprojected partition formula. At order one it is the ordinary chain rule. Suppose it holds at order $n$ and the maps are $C^{n+1}$. Differentiating a summand in direction $h_{n+1}$ has two kinds of contribution. A derivative of the outer map creates the new singleton block $\\{n+1\\}$. A derivative of an inner block jet adjoins $n+1$ to that block. Every partition of $\\{1,\\ldots,n+1\\}$ arises exactly once: if $\\{n+1\\}$ is a block, remove that block. Otherwise remove $n+1$ from the unique block that contains it. The chain and product rules for bounded multilinear evaluation justify the differentiation. This proves the unprojected formula by induction.","begin{proof}\nFirst derive the unprojected partition formula. At order one it is the ordinary chain rule. Suppose it holds at order $n$ and the maps are $C^{n+1}$. Differentiating a summand in direction $h_{n+1}$ has two kinds of contribution. A derivative of the outer map creates the new singleton block $\\{n+1\\}$. A derivative of an inner block jet adjoins $n+1$ to that block. Every partition of $\\{1,\\ldots,n+1\\}$ arises exactly once: if $\\{n+1\\}$ is a block, remove that block. Otherwise remove $n+1$ from the unique block that contains it. The chain and product rules for bounded multilinear evaluation justify the differentiation. This proves the unprojected formula by induction.","artition formula. At order one it is the ordinary chain rule. Suppose it holds at order $n$ and the maps are $C^{n+1}$. Differentiating a summand in direction $h_{n+1}$ has two kinds of contribution. A derivative of the outer map creates the new singleton block $\\{n+1\\}$. A derivative of an inner block jet adjoins $n+1$ to that block. Every partition of $\\{1,\\ldots,n+1\\}$ arises exactly once: if $\\{n+1\\}$ is a block, remove that block. Otherwise remove $n+1$ from the unique block that contains it. The chain and product rules for bounded multilinear evaluation justify the differentiation. This proves the unprojected formula by induction.","le. Suppose it holds at order $n$ and the maps are $C^{n+1}$. Differentiating a summand in direction $h_{n+1}$ has two kinds of contribution. A derivative of the outer map creates the new singleton block $\\{n+1\\}$. A derivative of an inner block jet adjoins $n+1$ to that block. Every partition of $\\{1,\\ldots,n+1\\}$ arises exactly once: if $\\{n+1\\}$ is a block, remove that block. Otherwise remove $n+1$ from the unique block that contains it. The chain and product rules for bounded multilinear evaluation justify the differentiation. This proves the unprojected formula by induction.","Insert $I_Y=\\sum_\\gamma Q_\\gamma$ in each argument of the outer derivative and expand the finite sums by multilinearity. Apply $R_\\delta$ to the output. The directions already lie in the declared input ranges, and each intermediate projection lands in its literal output range. The resulting summands are exactly the displayed packet jets. List directions within a block in increasing index order and blocks in increasing least-element order. Symmetry of the higher derivative makes the final value independent of the block ordering.\n\\end{proof}","\\begin{proposition}[Number of labeled composition terms]\\label{prop:faa_term_count}\nFor $n\\ge1$, if $Y$ has $M$ packets, then the number of packet labeled terms at order $n$ before zero jets are removed is\n\\[\nN_{n,M}\n=\n\\sum_{k=1}^{n}S(n,k)M^k,\n\\]\nwhere $S(n,k)$ is a Stirling number of the second kind.  Thus $N_{n,M}$ is the $n$th Touchard polynomial evaluated at $M$.\n\\end{proposition}","\\begin{proof}\nA set partition with $k$ blocks receives one independent intermediate packet label on each block.  It therefore gives $M^k$ labeled terms.  There are $S(n,k)$ such partitions.\n\\end{proof}","\\begin{corollary}[Structural pruning]\\label{cor:structural_pruning}\nFor a map $H$ with fixed input and output packet systems, let $\\supp_r H(x)$ be the set of external profiles $(\\delta;\\alpha_1,\\ldots,\\alpha_r)$ for which the restricted jet $J^r_{\\delta\\mid\\alpha_1,\\ldots,\\alpha_r}H(x)$ is nonzero. A profile belongs to the admissible composition set if it admits a partition $\\pi=\\{B_1,\\ldots,B_k\\}$ and intermediate labels $\\gamma_1,\\ldots,\\gamma_k$ such that the outer jet of $F$ and every inner block jet of $G$ in \\cref{thm:packet_faa} are nonzero. Then $\\supp_n(F\\circ G)(x)$ is contained in this admissible set. In particular, a summand vanishes whenever one of its constituent packet jets vanishes.\n\\end{corollary}","pruning]\\label{cor:structural_pruning}\nFor a map $H$ with fixed input and output packet systems, let $\\supp_r H(x)$ be the set of external profiles $(\\delta;\\alpha_1,\\ldots,\\alpha_r)$ for which the restricted jet $J^r_{\\delta\\mid\\alpha_1,\\ldots,\\alpha_r}H(x)$ is nonzero. A profile belongs to the admissible composition set if it admits a partition $\\pi=\\{B_1,\\ldots,B_k\\}$ and intermediate labels $\\gamma_1,\\ldots,\\gamma_k$ such that the outer jet of $F$ and every inner block jet of $G$ in \\cref{thm:packet_faa} are nonzero. Then $\\supp_n(F\\circ G)(x)$ is contained in this admissible set. In particular, a summand vanishes whenever one of its constituent packet jets vanishes.\n\\end{corollary}","with fixed input and output packet systems, let $\\supp_r H(x)$ be the set of external profiles $(\\delta;\\alpha_1,\\ldots,\\alpha_r)$ for which the restricted jet $J^r_{\\delta\\mid\\alpha_1,\\ldots,\\alpha_r}H(x)$ is nonzero. A profile belongs to the admissible composition set if it admits a partition $\\pi=\\{B_1,\\ldots,B_k\\}$ and intermediate labels $\\gamma_1,\\ldots,\\gamma_k$ such that the outer jet of $F$ and every inner block jet of $G$ in \\cref{thm:packet_faa} are nonzero. Then $\\supp_n(F\\circ G)(x)$ is contained in this admissible set. In particular, a summand vanishes whenever one of its constituent packet jets vanishes.\n\\end{corollary}","t systems, let $\\supp_r H(x)$ be the set of external profiles $(\\delta;\\alpha_1,\\ldots,\\alpha_r)$ for which the restricted jet $J^r_{\\delta\\mid\\alpha_1,\\ldots,\\alpha_r}H(x)$ is nonzero. A profile belongs to the admissible composition set if it admits a partition $\\pi=\\{B_1,\\ldots,B_k\\}$ and intermediate labels $\\gamma_1,\\ldots,\\gamma_k$ such that the outer jet of $F$ and every inner block jet of $G$ in \\cref{thm:packet_faa} are nonzero. Then $\\supp_n(F\\circ G)(x)$ is contained in this admissible set. In particular, a summand vanishes whenever one of its constituent packet jets vanishes.\n\\end{corollary}","ernal profiles $(\\delta;\\alpha_1,\\ldots,\\alpha_r)$ for which the restricted jet $J^r_{\\delta\\mid\\alpha_1,\\ldots,\\alpha_r}H(x)$ is nonzero. A profile belongs to the admissible composition set if it admits a partition $\\pi=\\{B_1,\\ldots,B_k\\}$ and intermediate labels $\\gamma_1,\\ldots,\\gamma_k$ such that the outer jet of $F$ and every inner block jet of $G$ in \\cref{thm:packet_faa} are nonzero. Then $\\supp_n(F\\circ G)(x)$ is contained in this admissible set. In particular, a summand vanishes whenever one of its constituent packet jets vanishes.\n\\end{corollary}","\\begin{example}[The second order composition law]\\label{ex:second_order_composition}\nAt order two, the partition formula has two shapes.  For packet directions $h_1$ and $h_2$,\n\\[\n\\begin{aligned}\nR_\\delta D^2(F\\circ G)(x)[h_1,h_2]\n={}&\n\\sum_{\\gamma_1,\\gamma_2}\nJ^2_{\\delta\\mid\\gamma_1,\\gamma_2}F(Gx)\n[J^1_{\\gamma_1\\mid\\alpha_1}G(x)h_1,J^1_{\\gamma_2\\mid\\alpha_2}G(x)h_2]","\\begin{example}[The second order composition law]\\label{ex:second_order_composition}\nAt order two, the partition formula has two shapes.  For packet directions $h_1$ and $h_2$,\n\\[\n\\begin{aligned}\nR_\\delta D^2(F\\circ G)(x)[h_1,h_2]\n={}&\n\\sum_{\\gamma_1,\\gamma_2}\nJ^2_{\\delta\\mid\\gamma_1,\\gamma_2}F(Gx)\n[J^1_{\\gamma_1\\mid\\alpha_1}G(x)h_1,J^1_{\\gamma_2\\mid\\alpha_2}G(x)h_2]\\\\\n&+\n\\sum_\\gamma\nJ^1_{\\delta\\mid\\gamma}F(Gx)\n[J^2_{\\gamma\\mid\\alpha_1,\\alpha_2}G(x)[h_1,h_2]].\n\\end{aligned}\n\\]\nThe first line records two first order changes entering a quadratic interaction of $F$.  The second records a genuine second order interaction created inside $G$ before $F$ is applied.\n\\end{example}","&+\n\\sum_\\gamma\nJ^1_{\\delta\\mid\\gamma}F(Gx)\n[J^2_{\\gamma\\mid\\alpha_1,\\alpha_2}G(x)[h_1,h_2]].\n\\end{aligned}\n\\]\nThe first line records two first order changes entering a quadratic interaction of $F$.  The second records a genuine second order interaction created inside $G$ before $F$ is applied.\n\\end{example}","Let $X$ be a Banach space.  Fix $s\\in X$ and continuous multilinear maps\n\\[\nB_r:X^r\\longrightarrow X,\n\\qquad\n2\\le r\\le d.\n\\]\nSet\n\\[\na=\\|s\\|,\n\\qquad\nb_r=\\|B_r\\|,\n\\qquad\n\\phi(t)=a+\\sum_{r=2}^{d}b_rt^r.\n\\]\nAn ordered rooted tree has leaves labeled by $s$ and every internal vertex of arity $r$ labeled by $B_r$.  Its value is obtained by evaluating from the leaves toward the root.","\\begin{theorem}[Polynomial tree expansion]\\label{thm:polynomial_tree_expansion}\nSuppose there is $R>0$ such that\n\\[\n\\phi(R)\\le R\n\\]\nand\n\\[\nq=\n\\sum_{r=2}^{d}r b_rR^{r-1}<1.\n\\]\nThen the map\n\\[\n\\Phi(x)=s+\\sum_{r=2}^{d}B_r(x,\\ldots,x)\n\\]\nhas a unique fixed point in the closed ball $\\overline B_R(0)$.  The sum of the values of all finite ordered rooted trees converges absolutely to that fixed point.\n\\end{theorem}","\\begin{proof}\nFor $x,y\\in\\overline B_R(0)$, multilinearity and telescoping in the $r$ input positions give\n\\[\n\\|B_r(x,\\ldots,x)-B_r(y,\\ldots,y)\\|\n\\le\nr b_rR^{r-1}\\|x-y\\|.\n\\]\nHence $\\Phi$ is $q$ Lipschitz on the ball.  The inequality $\\phi(R)\\le R$ shows that $\\Phi$ preserves the ball.  The contraction mapping theorem gives a unique fixed point $u$ in that ball.","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  T","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative c","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bou","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\p","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tr","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolu","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its","Let $\\mathfrak T_m$ be the finite set of ordered rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","rooted trees of height less than $m$.  A leaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","eaf has value $s$.  If the root of $\\tau$ has arity $r$ and ordered subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","red subtrees $\\tau_1,\\ldots,\\tau_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","_r$, define\n\\[\nB_\\tau(s)=B_r(B_{\\tau_1}(s),\\ldots,B_{\\tau_r}(s)).\n\\]\nInduction on $m$ shows that the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","the Picard iterate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","terate $u_0=0$ and $u_{m+1}=\\Phi(u_m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","m)$ satisfies\n\\[\nu_m=\\sum_{\\tau\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","u\\in\\mathfrak T_m}B_\\tau(s).\n\\]\nDefine a nonnegative scalar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","alar weight by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","by assigning weight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","ight $a$ to a leaf and applying the scalar map $(t_1,\\ldots,t_r)\\mapsto b_rt_1\\cdots t_r$ at a vertex of arity $r$.  A second induction gives\n\\[\n\\sum_{\\tau\\in\\mathfrak T_m}\\|B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","B_\\tau(s)\\|\n\\le\n\\sum_{\\tau\\in\\mathfrak T_m}w(\\tau)\n=t_m,\n\\]\nwhere $t_0=0$ and $t_{m+1}=\\phi(t_m)$.  The sequence $(t_m)$ is increasing because $\\phi$ has nonnegative coefficients.  It is bounded by $R$ because $t_0\\le R$ and $\\phi(R)\\le R$.  The sum of the norms of all tree values is therefore finite.  The tree series converges absolutely.  Its partial sums are the Picard iterates, so its sum is $u$.\n\\end{proof}","\\begin{corollary}[Quadratic Catalan expansion]\\label{cor:quadratic_tree_expansion}\nLet $B:X\\times X\\to X$ be bounded and consider\n\\[\nu=s+B(u,u).\n\\]\nPut $a=\\|s\\|$ and $b=\\|B\\|$.  If $4ab<1$, then\n\\[\nu=\n\\sum_{n\\ge1}\n\\sum_{\\substack{\\tau\\text{ full ordered binary tree}","\\begin{corollary}[Quadratic Catalan expansion]\\label{cor:quadratic_tree_expansion}\nLet $B:X\\times X\\to X$ be bounded and consider\n\\[\nu=s+B(u,u).\n\\]\nPut $a=\\|s\\|$ and $b=\\|B\\|$.  If $4ab<1$, then\n\\[\nu=\n\\sum_{n\\ge1}\n\\sum_{\\substack{\\tau\\text{ full ordered binary tree}\\\\\\text{with }n\\text{ leaves}}}\nB_\\tau(s)\n\\]\nconverges absolutely.  It also satisfies\n\\[\n\\|u\\|\n\\le\n\\frac{1-\\sqrt{1-4ab}}{2b}\n\\]\nwhen $b>0$.\n\\end{corollary}","text{with }n\\text{ leaves}}}\nB_\\tau(s)\n\\]\nconverges absolutely.  It also satisfies\n\\[\n\\|u\\|\n\\le\n\\frac{1-\\sqrt{1-4ab}}{2b}\n\\]\nwhen $b>0$.\n\\end{corollary}","\\begin{proof}\nIf $b=0$, the equation is $u=s$ and the tree series contains only its one leaf.  If $a=0$, every tree value is zero.  Choose any $R$ with $0<R<1/(2b)$ and apply \\cref{thm:polynomial_tree_expansion} to obtain uniqueness of the zero fixed point in that ball.","Assume now that $a>0$ and $b>0$.  Take\n\\[\nR=\n\\frac{1-\\sqrt{1-4ab}}{2b}.\n\\]\nThen $a+bR^2=R$ and $2bR<1$.  Apply \\cref{thm:polynomial_tree_expansion}.  The number of full ordered binary trees with $n$ leaves is the Catalan number $C_{n-1}$, which gives the displayed majorant.\n\\end{proof}","\\begin{theorem}[Packet labeled tree reconstruction]\\label{thm:packet_tree_reconstruction}\nLet $X$ carry a packet system $\\Pcal$.  At each internal vertex of a tree, insert the output identity and one input identity for every child.  Summing over all internal packet labels reconstructs the original tree value exactly.\n\\end{theorem}","\\begin{proof}\nAt a vertex labeled by an $r$ multilinear map $B_r$, insert\n\\[\nI_X=\\sum_\\beta P_\\beta\n\\]\non the output and in every input slot.  Multilinearity gives an exact finite sum of packet blocks at that vertex.  Apply this identity successively from the leaves toward the root.  Since the tree and the packet set are finite, all sums are finite and their total reconstructs the original tree value.\n\\end{proof}","The tree theorem separates combinatorial growth from analytic convergence.  The number of trees may grow rapidly, while multilinear bounds control their values. The scalar map $\\phi$ controls the total weight of the expansion.  Packet pruning changes the combinatorics but not the proof.  It removes trees whose values are already zero before any norm estimate is used.","\\begin{example}[Catalan parity packets]\\label{ex:catalan_parity}\nOver a field of characteristic zero, the formal equation\n\\[\nu=z+u^2\n\\]\nhas the branch\n\\[\nu(z)=\\frac{1-\\sqrt{1-4z}}2.\n\\]\nWrite $u=u_0+u_1$, where $u_0$ contains the even powers of $z$ and $u_1$ contains the odd powers.  Exact packet projection gives\n\\[\nu_0=u_0^2+u_1^2,\n\\qquad\nu_1=z+2u_0u_1.\n\\]\nThe two equations remain coupled.  Their sum is the original fixed point equation.\n\\end{example}","Let $G$ be a finite abelian group and let\n\\[\nA=\\bigoplus_{g\\in G}A_g\n\\]\nbe a $G$ graded module.","\\begin{definition}[Grade respecting operation]\\label{def:grade_respecting}\nAn $r$ multilinear map $\\mu:A^r\\to A$ respects the grading when\n\\[\n\\mu(A_{g_1},\\ldots,A_{g_r})\n\\subseteq\nA_{g_1+\\cdots+g_r}.\n\\]\n\\end{definition}","\\begin{theorem}[Graded packet convolution]\\label{thm:graded_convolution}\nIf $a_j=\\sum_{g\\in G}a_{j,g}$ and $\\mu$ respects the grading, then\n\\[\nP_k\\mu(a_1,\\ldots,a_r)\n=\n\\sum_{g_1+\\cdots+g_r=k}\n\\mu(a_{1,g_1},\\ldots,a_{r,g_r}).\n\\]\n\\end{theorem}","\\begin{proof}\nExpand every input into its homogeneous components.  Each term lands in the component whose label is the sum of its input labels.  Projection onto $A_k$ retains exactly the displayed terms.\n\\end{proof}","\\begin{corollary}[Cyclic product formula]\\label{cor:cyclic_product}\nFor a cyclically graded algebra\n\\[\nA=\\bigoplus_{j\\in\\mathbb Z/N\\mathbb Z}A_j,\n\\qquad\nA_iA_j\\subseteq A_{i+j},\n\\]\none has\n\\[\nP_k(ab)=\n\\sum_{i+j=k}a_ib_j.\n\\]\n\\end{corollary}","\\begin{theorem}[Packet Maurer Cartan equations]\\label{thm:packet_mc}\nLet\n\\[\n\\Lcal=\\bigoplus_{g\\in G}\\Lcal_g\n\\]\nbe a differential graded Lie algebra over a ring in which two is invertible.  Assume\n\\[\nd\\Lcal_g\\subseteq\\Lcal_g,\n\\qquad\n[\\Lcal_g,\\Lcal_h]\\subseteq\\Lcal_{g+h}.\n\\]\nFor a degree one element $\\eta=\\sum_g\\eta_g$, the Maurer Cartan equation\n\\[\nd\\eta+\\frac12[\\eta,\\eta]=0\n\\]\nis equivalent to\n\\[\nd\\eta_k\n+\n\\frac12\n\\sum_{g+h=k}[\\eta_g,\\eta_h]\n=0\n\\]\nfor every $k\\in G$.\n\\end{theorem}","t_mc}\nLet\n\\[\n\\Lcal=\\bigoplus_{g\\in G}\\Lcal_g\n\\]\nbe a differential graded Lie algebra over a ring in which two is invertible.  Assume\n\\[\nd\\Lcal_g\\subseteq\\Lcal_g,\n\\qquad\n[\\Lcal_g,\\Lcal_h]\\subseteq\\Lcal_{g+h}.\n\\]\nFor a degree one element $\\eta=\\sum_g\\eta_g$, the Maurer Cartan equation\n\\[\nd\\eta+\\frac12[\\eta,\\eta]=0\n\\]\nis equivalent to\n\\[\nd\\eta_k\n+\n\\frac12\n\\sum_{g+h=k}[\\eta_g,\\eta_h]\n=0\n\\]\nfor every $k\\in G$.\n\\end{theorem}","\\begin{proof}\nThe differential preserves the group label and the bracket adds labels.  Apply the projector onto $\\Lcal_k$ and use exact reconstruction.\n\\end{proof}","\\begin{proposition}[Packet equations for $L_\\infty$ operations]\\label{prop:packet_l_infinity}\nLet $\\Lcal$ be complete and carry operations $\\ell_r$ that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges.  Assume the Maurer Cartan series\n\\[\n\\sum_{r\\ge1}\\frac1{r!}\\ell_r(\\eta,\\ldots,\\eta)\n\\]\nconverges and the required factorials are invertible.  Then its packet of label $k$ is\n\\[\n\\sum_{r\\ge1}\\frac1{r!}\n\\sum_{g_1+\\cdots+g_r=k}\n\\ell_r(\\eta_{g_1},\\ldots,\\eta_{g_r}).\n\\]\nThe full Maurer Cartan equation holds if and only if every packet equation vanishes.\n\\end{proposition}","ions]\\label{prop:packet_l_infinity}\nLet $\\Lcal$ be complete and carry operations $\\ell_r$ that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges.  Assume the Maurer Cartan series\n\\[\n\\sum_{r\\ge1}\\frac1{r!}\\ell_r(\\eta,\\ldots,\\eta)\n\\]\nconverges and the required factorials are invertible.  Then its packet of label $k$ is\n\\[\n\\sum_{r\\ge1}\\frac1{r!}\n\\sum_{g_1+\\cdots+g_r=k}\n\\ell_r(\\eta_{g_1},\\ldots,\\eta_{g_r}).\n\\]\nThe full Maurer Cartan equation holds if and only if every packet equation vanishes.\n\\end{proposition}","bel{prop:packet_l_infinity}\nLet $\\Lcal$ be complete and carry operations $\\ell_r$ that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges.  Assume the Maurer Cartan series\n\\[\n\\sum_{r\\ge1}\\frac1{r!}\\ell_r(\\eta,\\ldots,\\eta)\n\\]\nconverges and the required factorials are invertible.  Then its packet of label $k$ is\n\\[\n\\sum_{r\\ge1}\\frac1{r!}\n\\sum_{g_1+\\cdots+g_r=k}\n\\ell_r(\\eta_{g_1},\\ldots,\\eta_{g_r}).\n\\]\nThe full Maurer Cartan equation holds if and only if every packet equation vanishes.\n\\end{proposition}","\\begin{proof}\nApply \\cref{thm:graded_convolution} to each operation $\\ell_r$.  Continuity of the packet projectors permits the convergent sums to be projected term by term.\n\\end{proof}","\\begin{definition}[Divided square]\\label{def:divided_square}\nA divided square on degree one elements is a map $q$ satisfying\n\\[\nq(x+y)=q(x)+q(y)+[x,y],\n\\qquad\nq(ax)=a^2q(x),\n\\]\nwhenever the expressions are defined.\n\\end{definition}","\\begin{proposition}[Divided square packet equations]\\label{prop:divided_square_packets}\nAssume $q(\\Lcal_g)\\subseteq\\Lcal_{2g}$.  Choose an ordering of the finite label set.  The equation\n\\[\nd\\eta+q(\\eta)=0\n\\]\nis equivalent to\n\\[\nd\\eta_k\n+\n\\sum_{2g=k}q(\\eta_g)\n+\n\\sum_{\\substack{g<h","\\begin{proposition}[Divided square packet equations]\\label{prop:divided_square_packets}\nAssume $q(\\Lcal_g)\\subseteq\\Lcal_{2g}$.  Choose an ordering of the finite label set.  The equation\n\\[\nd\\eta+q(\\eta)=0\n\\]\nis equivalent to\n\\[\nd\\eta_k\n+\n\\sum_{2g=k}q(\\eta_g)\n+\n\\sum_{\\substack{g<h\\\\g+h=k}}[\\eta_g,\\eta_h]\n=0\n\\]\nfor every $k$.\n\\end{proposition}","g+h=k}}[\\eta_g,\\eta_h]\n=0\n\\]\nfor every $k$.\n\\end{proposition}","\\begin{proof}\nRepeated use of the polarization identity for $q$ gives\n\\[\nq\\left(\\sum_g\\eta_g\\right)\n=\n\\sum_gq(\\eta_g)\n+\n\\sum_{g<h}[\\eta_g,\\eta_h].\n\\]\nProject onto the component of label $k$.\n\\end{proof}","\\begin{example}[Matrix difference grading]\\label{ex:matrix_difference_grading}\nLet $A=M_N(\\bbk)$ and assign the matrix unit $E_{rs}$ the label $r-s$ modulo $N$.  Since\n\\[\nE_{rs}E_{uv}=\\delta_{su}E_{rv},\n\\]\nthe output label is\n\\[\n(r-s)+(u-v)=r-v\n\\]\nwhenever the product is nonzero.  Matrix multiplication therefore obeys the cyclic convolution law exactly.  The same grading resolves commutators and polynomial matrix equations.\n\\end{example}","Let\n\\[\nF_f:X_f\\longrightarrow Y_f,\n\\qquad\nF_c:X_c\\longrightarrow Y_c\n\\]\nbe fine and coarse maps.  Let\n\\[\nC_X:X_f\\longrightarrow X_c,\n\\qquad\nC_Y:Y_f\\longrightarrow Y_c\n\\]\nbe linear coarsening maps.  Whenever a norm, a Lipschitz constant, or a Fr\\'echet derivative is used, the spaces are Banach spaces and the coarsening maps are bounded.","\\begin{definition}[Nonlinear refinement defect]\\label{def:nonlinear_defect}\nThe pointwise defect of $F$ is\n\\[\n\\Delta_F(x)\n=\nC_YF_f(x)-F_c(C_Xx).\n\\]\nThe two models are exactly compatible when $\\Delta_F$ vanishes everywhere.\n\\end{definition}","The defect is a vector in $Y_c$.  It determines an obstruction class after a correction map into $Y_c$ has been chosen and the defect has been passed to the corresponding cokernel.  Norm estimates measure the size of the defect.  The quotient determines whether an allowed linear correction can remove it.","\\begin{theorem}[Exact defect under composition]\\label{thm:defect_composition_identity}\nLet $G_f:Y_f\\to Z_f$ and $G_c:Y_c\\to Z_c$, with coarsening $C_Z:Z_f\\to Z_c$.  Then\n\\[\n\\Delta_{G\\circ F}(x)\n=\n\\Delta_G(F_f(x))\n+\nG_c\\big(F_c(C_Xx)+\\Delta_F(x)\\big)\n-\nG_c\\big(F_c(C_Xx)\\big).\n\\]\nIf $G_c$ is $L$ Lipschitz on the relevant set, then\n\\[\n\\|\\Delta_{G\\circ F}(x)\\|\n\\le\n\\|\\Delta_G(F_f(x))\\|\n+\nL\\|\\Delta_F(x)\\|.\n\\]\n\\end{theorem}","\\begin{proof}\nAdd and subtract $G_c(C_YF_f(x))$.  Use\n\\[\nC_YF_f(x)=F_c(C_Xx)+\\Delta_F(x).\n\\]\nThe norm estimate follows from the Lipschitz bound.\n\\end{proof}","\\begin{corollary}[Linear product rule]\\label{cor:linear_defect_product}\nIf all maps in \\cref{thm:defect_composition_identity} are linear, then\n\\[\n\\Delta_{GF}\n=\n\\Delta_G F_f+G_c\\Delta_F.\n\\]\n\\end{corollary}","\\begin{proof}\nLinearity turns the last difference in \\cref{thm:defect_composition_identity} into $G_c\\Delta_F(x)$.\n\\end{proof}","\\begin{theorem}[Iterated orbit defect]\\label{thm:iterated_orbit_defect}\nLet $F_f:X_f\\to X_f$ and $F_c:X_c\\to X_c$, with coarsening $C:X_f\\to X_c$.  Suppose $F_c$ is $q$ Lipschitz.  For a fine orbit starting at $x$, assume\n\\[\n\\|\\Delta_F(F_f^j x)\\|\\le\\varepsilon_j.\n\\]\nThen\n\\[\n\\left\\|CF_f^n(x)-F_c^n(Cx)\\right\\|\n\\le\n\\sum_{j=0}^{n-1}q^{n-1-j}\\varepsilon_j.\n\\]\nIf $0\\le q<1$ and every $\\varepsilon_j\\le\\varepsilon$, then\n\\[\n\\left\\|CF_f^n(x)-F_c^n(Cx)\\right\\|\n\\le\n\\varepsilon\\frac{1-q^n}{1-q}.\n\\]\n\\end{theorem}","rated orbit defect]\\label{thm:iterated_orbit_defect}\nLet $F_f:X_f\\to X_f$ and $F_c:X_c\\to X_c$, with coarsening $C:X_f\\to X_c$.  Suppose $F_c$ is $q$ Lipschitz.  For a fine orbit starting at $x$, assume\n\\[\n\\|\\Delta_F(F_f^j x)\\|\\le\\varepsilon_j.\n\\]\nThen\n\\[\n\\left\\|CF_f^n(x)-F_c^n(Cx)\\right\\|\n\\le\n\\sum_{j=0}^{n-1}q^{n-1-j}\\varepsilon_j.\n\\]\nIf $0\\le q<1$ and every $\\varepsilon_j\\le\\varepsilon$, then\n\\[\n\\left\\|CF_f^n(x)-F_c^n(Cx)\\right\\|\n\\le\n\\varepsilon\\frac{1-q^n}{1-q}.\n\\]\n\\end{theorem}","\\begin{proof}\nLet\n\\[\ne_n=\\left\\|CF_f^n(x)-F_c^n(Cx)\\right\\|.\n\\]\nThe definition of the defect and the Lipschitz property give\n\\[\ne_{n+1}\\le\\varepsilon_n+qe_n.\n\\]\nIterating this recurrence proves the first estimate.  The second is the geometric sum.\n\\end{proof}","\\begin{theorem}[Fixed point stability]\\label{thm:fixed_point_stability}\nAssume $F_c$ is a contraction with constant $q<1$.  Let $x_f$ and $x_c$ be fixed points of $F_f$ and $F_c$.  If\n\\[\n\\|\\Delta_F(x_f)\\|\\le\\varepsilon,\n\\]\nthen\n\\[\n\\|Cx_f-x_c\\|\n\\le\n\\frac{\\varepsilon}{1-q}.\n\\]\nWhen the defect at $x_f$ is zero, the fixed points are compatible.\n\\end{theorem}","\\begin{proof}\nUsing the fixed point equations,\n\\[\n\\begin{aligned}\n\\|Cx_f-x_c\\|\n&\\le\n\\|CF_f(x_f)-F_c(Cx_f)\\|\n+\n\\|F_c(Cx_f)-F_c(x_c)\\|\\\\\n&\\le\n\\varepsilon+q\\|Cx_f-x_c\\|.\n\\end{aligned}\n\\]\nRearrange.\n\\end{proof}","\\begin{corollary}[Several refinement levels]\\label{cor:multilevel_fixed_points}\nLet $C_\\ell:X_{\\ell+1}\\to X_\\ell$ for $0\\le\\ell<m$.  Suppose $x_\\ell$ is a fixed point at level $\\ell$, the coarse map at level $\\ell$ has contraction constant $q_\\ell<1$, and the adjacent defect at $x_{\\ell+1}$ is at most $\\varepsilon_\\ell$.  Write\n\\[\nC_{a,b}=C_aC_{a+1}\\cdots C_{b-1}\n\\quad\\text{for }a<b,\n\\qquad\nC_{a,a}=I.\n\\]\nThen\n\\[\n\\|C_{0,m}x_m-x_0\\|\n\\le\n\\sum_{\\ell=0}^{m-1}\n\\|C_{0,\\ell}\\|\n\\frac{\\varepsilon_\\ell}{1-q_\\ell}.\n\\]\n\\end{corollary}","y}[Several refinement levels]\\label{cor:multilevel_fixed_points}\nLet $C_\\ell:X_{\\ell+1}\\to X_\\ell$ for $0\\le\\ell<m$.  Suppose $x_\\ell$ is a fixed point at level $\\ell$, the coarse map at level $\\ell$ has contraction constant $q_\\ell<1$, and the adjacent defect at $x_{\\ell+1}$ is at most $\\varepsilon_\\ell$.  Write\n\\[\nC_{a,b}=C_aC_{a+1}\\cdots C_{b-1}\n\\quad\\text{for }a<b,\n\\qquad\nC_{a,a}=I.\n\\]\nThen\n\\[\n\\|C_{0,m}x_m-x_0\\|\n\\le\n\\sum_{\\ell=0}^{m-1}\n\\|C_{0,\\ell}\\|\n\\frac{\\varepsilon_\\ell}{1-q_\\ell}.\n\\]\n\\end{corollary}","\\begin{proof}\nInsert the intermediate vectors $C_{0,\\ell}x_\\ell$ and telescope.  Apply \\cref{thm:fixed_point_stability} to every adjacent pair.\n\\end{proof}","\\begin{proposition}[Derivative defects]\\label{prop:derivative_defects}\nAssume $F_f$ and $F_c$ are $r$ times Fr\\'echet differentiable and the coarsening maps are bounded and linear.  Then\n\\[\nD\\Delta_F(x)\n=\nC_YDF_f(x)-DF_c(C_Xx)C_X.\n\\]\nFor $r\\ge2$,\n\\[\nD^r\\Delta_F(x)[h_1,\\ldots,h_r]\n=\nC_YD^rF_f(x)[h_1,\\ldots,h_r]\n-\nD^rF_c(C_Xx)[C_Xh_1,\\ldots,C_Xh_r].\n\\]\n\\end{proposition}","\\begin{proof}\nDifferentiate the defining identity for $\\Delta_F$.  The coarsening maps are linear, so their higher derivatives vanish.\n\\end{proof}","\\begin{example}[Sharp affine bound]\\label{ex:sharp_affine_bound}\nLet $F_c(y)=qy+b$ with $0<q<1$.  Choose a fine fixed point whose coarsening has constant defect $\\varepsilon$ in the direction of the scalar state.  The fixed point mismatch is exactly $\\varepsilon/(1-q)$.  Thus the constant in \\cref{thm:fixed_point_stability} cannot be improved without more information.\n\\end{example}","\\begin{definition}[Linear correction datum]\\label{def:linear_correction_datum}\nA linear correction datum consists of a homomorphism\n\\[\nD:U\\longrightarrow V\n\\]\nand a defect $r\\in V$.  A correction is an element $u\\in U$ satisfying\n\\[\nDu=-r.\n\\]\nThe obstruction class is\n\\[\n\\Ob_D(r)=[r]\\in\\coker D=V/\\im D.\n\\]\n\\end{definition}","\\begin{theorem}[Cokernel obstruction criterion]\\label{thm:cokernel_obstruction}\nA linear correction exists if and only if\n\\[\n\\Ob_D(r)=0.\n\\]\nThe quotient map\n\\[\nq_D:V\\longrightarrow\\coker D\n\\]\nhas the following universal property.  If $\\varphi:V\\to W$ is linear and $\\varphi D=0$, then there is a unique linear map\n\\[\n\\overline\\varphi:\\coker D\\longrightarrow W\n\\]\nwith $\\varphi=\\overline\\varphi q_D$.\n\\end{theorem}","\\begin{proof}\nThe class of $r$ is zero exactly when $r\\in\\im D$, which is equivalent to solvability of $Du=-r$.  If $\\varphi D=0$, then $\\varphi$ vanishes on $\\im D$.  It therefore descends uniquely to the quotient.\n\\end{proof}","The universal property explains why the cokernel is the canonical obstruction object relative to $D$.  Any linear obstruction test that vanishes on correctable defects factors through it.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.96\\linewidth]{correction_cokernel_plane.pdf}\n\\caption{A specified correction range in $\\mathbb R^2$. For $L(a,b)=(a,0)$, the image is the horizontal axis. The defect $(2,3)$ splits into $(2,0)$ in that image and the remaining representative $(0,3)$. Subtracting the image component leaves the same cokernel class. The functional $(x,y)\\mapsto y$ vanishes on the correction range and detects the value three. The correction equation $Lu=-(2,3)$ therefore has no solution.}\n\\label{fig:correction-cokernel-example}\n\\end{figure}\n\\clearpage","_plane.pdf}\n\\caption{A specified correction range in $\\mathbb R^2$. For $L(a,b)=(a,0)$, the image is the horizontal axis. The defect $(2,3)$ splits into $(2,0)$ in that image and the remaining representative $(0,3)$. Subtracting the image component leaves the same cokernel class. The functional $(x,y)\\mapsto y$ vanishes on the correction range and detects the value three. The correction equation $Lu=-(2,3)$ therefore has no solution.}\n\\label{fig:correction-cokernel-example}\n\\end{figure}\n\\clearpage","\\begin{theorem}[Naturality of obstruction classes]\\label{thm:obstruction_naturality}\nSuppose\n\\[\n\\begin{array}{ccc}\nU & \\xrightarrow{D} & V\\\\\n\\mathllap{a}\\downarrow & & \\downarrow\\mathrlap{b}\\\\\nU' & \\xrightarrow{D'} & V'\n\\end{array}\n\\]\nis a commuting square.  Then there is a unique induced map\n\\[\n\\overline b:\\coker D\\longrightarrow\\coker D'\n\\]\nsatisfying\n\\[\n\\overline b\\,\\Ob_D(r)=\\Ob_{D'}(br).\n\\]\n\\end{theorem}","U' & \\xrightarrow{D'} & V'\n\\end{array}\n\\]\nis a commuting square.  Then there is a unique induced map\n\\[\n\\overline b:\\coker D\\longrightarrow\\coker D'\n\\]\nsatisfying\n\\[\n\\overline b\\,\\Ob_D(r)=\\Ob_{D'}(br).\n\\]\n\\end{theorem}","\\begin{proof}\nThe relation $bD=D'a$ shows that the quotient map $q_{D'}b$ vanishes on $\\im D$.  Apply the universal property in \\cref{thm:cokernel_obstruction}.\n\\end{proof}","\\begin{theorem}[Packet decomposition of an obstruction]\\label{thm:packet_obstruction_decomposition}\nSuppose\n\\[\nU=\\bigoplus_{\\alpha\\in I}U_\\alpha,\n\\qquad\nV=\\bigoplus_{\\alpha\\in I}V_\\alpha,\n\\]\nand $D(U_\\alpha)\\subseteq V_\\alpha$ for every $\\alpha$.  Let\n\\[\nD_\\alpha:U_\\alpha\\longrightarrow V_\\alpha\n\\]\nbe the restriction.  Then\n\\[\n\\coker D\n\\cong\n\\bigoplus_{\\alpha\\in I}\\coker D_\\alpha.\n\\]\nUnder this isomorphism, the obstruction class of $r=\\sum_\\alpha r_\\alpha$ is the tuple\n\\[\n\\big([r_\\alpha]\\big)_\\alpha.\n\\]\nHence $r$ is correctable if and only if every packet obstruction vanishes.\n\\end{theorem}","of an obstruction]\\label{thm:packet_obstruction_decomposition}\nSuppose\n\\[\nU=\\bigoplus_{\\alpha\\in I}U_\\alpha,\n\\qquad\nV=\\bigoplus_{\\alpha\\in I}V_\\alpha,\n\\]\nand $D(U_\\alpha)\\subseteq V_\\alpha$ for every $\\alpha$.  Let\n\\[\nD_\\alpha:U_\\alpha\\longrightarrow V_\\alpha\n\\]\nbe the restriction.  Then\n\\[\n\\coker D\n\\cong\n\\bigoplus_{\\alpha\\in I}\\coker D_\\alpha.\n\\]\nUnder this isomorphism, the obstruction class of $r=\\sum_\\alpha r_\\alpha$ is the tuple\n\\[\n\\big([r_\\alpha]\\big)_\\alpha.\n\\]\nHence $r$ is correctable if and only if every packet obstruction vanishes.\n\\end{theorem}","truction_decomposition}\nSuppose\n\\[\nU=\\bigoplus_{\\alpha\\in I}U_\\alpha,\n\\qquad\nV=\\bigoplus_{\\alpha\\in I}V_\\alpha,\n\\]\nand $D(U_\\alpha)\\subseteq V_\\alpha$ for every $\\alpha$.  Let\n\\[\nD_\\alpha:U_\\alpha\\longrightarrow V_\\alpha\n\\]\nbe the restriction.  Then\n\\[\n\\coker D\n\\cong\n\\bigoplus_{\\alpha\\in I}\\coker D_\\alpha.\n\\]\nUnder this isomorphism, the obstruction class of $r=\\sum_\\alpha r_\\alpha$ is the tuple\n\\[\n\\big([r_\\alpha]\\big)_\\alpha.\n\\]\nHence $r$ is correctable if and only if every packet obstruction vanishes.\n\\end{theorem}","goplus_{\\alpha\\in I}U_\\alpha,\n\\qquad\nV=\\bigoplus_{\\alpha\\in I}V_\\alpha,\n\\]\nand $D(U_\\alpha)\\subseteq V_\\alpha$ for every $\\alpha$.  Let\n\\[\nD_\\alpha:U_\\alpha\\longrightarrow V_\\alpha\n\\]\nbe the restriction.  Then\n\\[\n\\coker D\n\\cong\n\\bigoplus_{\\alpha\\in I}\\coker D_\\alpha.\n\\]\nUnder this isomorphism, the obstruction class of $r=\\sum_\\alpha r_\\alpha$ is the tuple\n\\[\n\\big([r_\\alpha]\\big)_\\alpha.\n\\]\nHence $r$ is correctable if and only if every packet obstruction vanishes.\n\\end{theorem}","\\begin{proof}\nThe grading assumption gives\n\\[\n\\im D=\\bigoplus_\\alpha\\im D_\\alpha.\n\\]\nTaking the quotient of the direct sum by this direct sum of images gives the displayed isomorphism.\n\\end{proof}","\\begin{proposition}[Projectors on the obstruction quotient]\\label{prop:obstruction_quotient_projectors}\nLet $(Q_\\alpha)_{\\alpha\\in I}$ be a complete family of pairwise annihilating idempotents on $V$.  Assume\n\\[\nQ_\\alpha(\\im D)\\subseteq\\im D\n\\]\nfor every $\\alpha$.  Then each $Q_\\alpha$ induces an idempotent $\\overline Q_\\alpha$ on $\\coker D$, and\n\\[\n\\coker D\n=\n\\bigoplus_{\\alpha\\in I}\\overline Q_\\alpha(\\coker D).\n\\]\nFor every defect $r\\in V$,\n\\[\n\\overline Q_\\alpha\\Ob_D(r)\n=\n[Q_\\alpha r].\n\\]\nNo packet decomposition of the correction space $U$ is required.\n\\end{proposition}","obstruction quotient]\\label{prop:obstruction_quotient_projectors}\nLet $(Q_\\alpha)_{\\alpha\\in I}$ be a complete family of pairwise annihilating idempotents on $V$.  Assume\n\\[\nQ_\\alpha(\\im D)\\subseteq\\im D\n\\]\nfor every $\\alpha$.  Then each $Q_\\alpha$ induces an idempotent $\\overline Q_\\alpha$ on $\\coker D$, and\n\\[\n\\coker D\n=\n\\bigoplus_{\\alpha\\in I}\\overline Q_\\alpha(\\coker D).\n\\]\nFor every defect $r\\in V$,\n\\[\n\\overline Q_\\alpha\\Ob_D(r)\n=\n[Q_\\alpha r].\n\\]\nNo packet decomposition of the correction space $U$ is required.\n\\end{proposition}","n quotient]\\label{prop:obstruction_quotient_projectors}\nLet $(Q_\\alpha)_{\\alpha\\in I}$ be a complete family of pairwise annihilating idempotents on $V$.  Assume\n\\[\nQ_\\alpha(\\im D)\\subseteq\\im D\n\\]\nfor every $\\alpha$.  Then each $Q_\\alpha$ induces an idempotent $\\overline Q_\\alpha$ on $\\coker D$, and\n\\[\n\\coker D\n=\n\\bigoplus_{\\alpha\\in I}\\overline Q_\\alpha(\\coker D).\n\\]\nFor every defect $r\\in V$,\n\\[\n\\overline Q_\\alpha\\Ob_D(r)\n=\n[Q_\\alpha r].\n\\]\nNo packet decomposition of the correction space $U$ is required.\n\\end{proposition}","lete family of pairwise annihilating idempotents on $V$.  Assume\n\\[\nQ_\\alpha(\\im D)\\subseteq\\im D\n\\]\nfor every $\\alpha$.  Then each $Q_\\alpha$ induces an idempotent $\\overline Q_\\alpha$ on $\\coker D$, and\n\\[\n\\coker D\n=\n\\bigoplus_{\\alpha\\in I}\\overline Q_\\alpha(\\coker D).\n\\]\nFor every defect $r\\in V$,\n\\[\n\\overline Q_\\alpha\\Ob_D(r)\n=\n[Q_\\alpha r].\n\\]\nNo packet decomposition of the correction space $U$ is required.\n\\end{proposition}","\\begin{proof}\nIf $v-v'\\in\\im D$, then\n\\[\nQ_\\alpha(v-v')\\in\\im D.\n\\]\nHence $[v]\\mapsto[Q_\\alpha v]$ is well defined on the quotient.  The idempotent, annihilation, and reconstruction identities descend from $V$ to $V/\\im D$.  The formula for an obstruction class follows from the definition.\n\\end{proof}","\\begin{theorem}[Relative obstruction exact sequence]\\label{thm:relative_obstruction_sequence}\nSuppose the following diagram of modules commutes and has exact rows:\n\\[\n\\begin{array}{ccccccccc}\n0&\\longrightarrow&U_0&\\xrightarrow{i_U}&U_f&\\xrightarrow{p_U}&U_c&\\longrightarrow&0\\\\[4pt]\n&&\\mathllap{D_0}\\downarrow&&\\mathllap{D_f}\\downarrow&&\\mathllap{D_c}\\downarrow&&\\\\[4pt]\n0&\\longrightarrow&V_0&\\xrightarrow{i_V}&V_f&\\xrightarrow{p_V}&V_c&\\longrightarrow&0.\n\\end{array}\n\\]\nThere is a natural exact sequence\n\\[\n\\begin{aligned}\n0&\\longrightarrow\\Ker D_0\n\\longrightarrow\\Ker D_f\n\\longrightarrow\\Ker D_c\n\\xrightarrow{\\delta}\\coker D_0","[4pt]\n0&\\longrightarrow&V_0&\\xrightarrow{i_V}&V_f&\\xrightarrow{p_V}&V_c&\\longrightarrow&0.\n\\end{array}\n\\]\nThere is a natural exact sequence\n\\[\n\\begin{aligned}\n0&\\longrightarrow\\Ker D_0\n\\longrightarrow\\Ker D_f\n\\longrightarrow\\Ker D_c\n\\xrightarrow{\\delta}\\coker D_0\\\\\n&\\longrightarrow\\coker D_f\n\\longrightarrow\\coker D_c\n\\longrightarrow0.\n\\end{aligned}\n\\]\nFor $u_c\\in\\Ker D_c$, choose $u_f\\in U_f$ with $p_Uu_f=u_c$.  There is a unique $v_0\\in V_0$ such that\n\\[\nD_fu_f=i_Vv_0,\n\\]\nand the connecting map is\n\\[\n\\delta(u_c)=[v_0].\n\\]\n\\end{theorem}","&\\longrightarrow\\coker D_f\n\\longrightarrow\\coker D_c\n\\longrightarrow0.\n\\end{aligned}\n\\]\nFor $u_c\\in\\Ker D_c$, choose $u_f\\in U_f$ with $p_Uu_f=u_c$.  There is a unique $v_0\\in V_0$ such that\n\\[\nD_fu_f=i_Vv_0,\n\\]\nand the connecting map is\n\\[\n\\delta(u_c)=[v_0].\n\\]\n\\end{theorem}","\\begin{proof}\nSince $D_cu_c=0$, commutativity gives\n\\[\np_VD_fu_f=D_cp_Uu_f=0.\n\\]\nExactness of the lower row gives a unique $v_0$ with $D_fu_f=i_Vv_0$.  A different lift has the form $u_f+i_Uu_0$.  The corresponding element of $V_0$ changes by $D_0u_0$, so $[v_0]$ is independent of the lift.","The map $\\Ker D_0\\to\\Ker D_f$ is injective because $i_U$ is injective.  If $u_f\\in\\Ker D_f$ maps to zero in $U_c$, then $u_f=i_Uu_0$ for a unique $u_0$.  The identity\n\\[\ni_VD_0u_0=D_fi_Uu_0=0\n\\]\nand injectivity of $i_V$ show that $u_0\\in\\Ker D_0$.  This proves exactness at $\\Ker D_f$.","If $u_c$ comes from an element of $\\Ker D_f$, one may choose that element as its lift and obtain $\\delta(u_c)=0$.  Conversely, if $\\delta(u_c)=0$, then $v_0=D_0u_0$ for some $u_0$.  Replacing $u_f$ by $u_f-i_Uu_0$ gives an element of $\\Ker D_f$ that maps to $u_c$.  This proves exactness at $\\Ker D_c$.","The class $[v_0]\\in\\coker D_0$ maps to zero in $\\coker D_f$ whenever it equals $\\delta(u_c)$, since the chosen lift satisfies $i_Vv_0=D_fu_f$.  Conversely, if $i_Vv_0=D_fu_f$, then $u_c=p_Uu_f$ lies in $\\Ker D_c$ and $\\delta(u_c)=[v_0]$.  This proves exactness at $\\coker D_0$.","If a class in $\\coker D_f$ comes from $\\coker D_0$, its image in $\\coker D_c$ is zero.  Conversely, suppose $[v_f]$ maps to zero in $\\coker D_c$.  Choose $u_c$ with $D_cu_c=p_Vv_f$ and lift it to $u_f\\in U_f$.  Then\n\\[\np_V(v_f-D_fu_f)=0,\n\\]\nso $v_f-D_fu_f=i_Vv_0$ for some $v_0\\in V_0$.  Thus $[v_f]$ comes from $[v_0]$.  Surjectivity at the last term follows from surjectivity of $p_V$.\n\\end{proof}","\\begin{corollary}[Coarse vanishing and fine vanishing]\\label{cor:coarse_fine_obstruction}\nLet $[r_f]\\in\\coker D_f$.  If its image in $\\coker D_c$ is zero, then $[r_f]$ comes from $\\coker D_0$.  Coarse vanishing therefore means that the fine obstruction is supported in information removed by coarsening.  It does not imply $[r_f]=0$.\n\\end{corollary}","\\begin{proof}\nThis is exactness of the sequence in \\cref{thm:relative_obstruction_sequence} at $\\coker D_f$.\n\\end{proof}","\\begin{example}[A fine obstruction erased by coarsening]\\label{ex:coarse_cancellation}\nTake all three correction spaces to be zero.  Let\n\\[\nV_f=\\bbk^2,\n\\qquad\nV_c=\\bbk,\n\\qquad\np_V(x,y)=x+y,\n\\]\nand let\n\\[\nV_0=\\Ker p_V=\\Span\\{(1,-1)\\}.\n\\]\nThe fine defect $(1,-1)$ is a nonzero class in $\\coker D_f=V_f$, while its coarse image is zero.  The class comes from $\\coker D_0=V_0$, exactly as \\cref{cor:coarse_fine_obstruction} predicts.\n\\end{example}","\\begin{definition}[Mapping cone]\\label{def:mapping_cone}\nLet $D:C^\\bullet\\to E^\\bullet$ be a cochain map.  Its mapping cone is\n\\[\n\\Cone(D)^n=E^n\\oplus C^{n+1}\n\\]\nwith differential\n\\[\nd_{\\Cone}(e,c)=(d_Ee+Dc,-d_Cc).\n\\]\n\\end{definition}","\\begin{theorem}[Cohomological obstruction through the cone]\\label{thm:cone_obstruction}\nLet $e\\in E^n$ satisfy $d_Ee=0$.  The class of $(e,0)$ in\n\\[\nH^n(\\Cone(D))\n\\]\nvanishes if and only if the class $[e]\\in H^n(E)$ lies in the image of\n\\[\nH^n(D):H^n(C)\\longrightarrow H^n(E).\n\\]\nWhen $C$ and $E$ are concentrated in degree zero,\n\\[\nH^0(\\Cone(D))\\cong\\coker D.\n\\]\n\\end{theorem}","\\begin{proof}\nThe class of $(e,0)$ vanishes exactly when there are $e'\\in E^{n-1}$ and $c\\in C^n$ with\n\\[\n(e,0)=d_{\\Cone}(e',c).\n\\]\nThis means\n\\[\ne=d_Ee'+Dc,\n\\qquad\nd_Cc=0.\n\\]\nEquivalently, $[e]=H^n(D)[c]$.  The degree zero statement follows directly from the definition of the cone differential.\n\\end{proof}","\\begin{corollary}[Packet obstruction complexes]\\label{cor:packet_obstruction_complexes}\nIf $C^\\bullet$ and $E^\\bullet$ split into finite packet subcomplexes and $D$ preserves those packets, then $\\Cone(D)$ splits into packet subcomplexes and\n\\[\nH^n(\\Cone(D))\n\\cong\n\\bigoplus_\\alpha H^n(\\Cone(D_\\alpha)).\n\\]\n\\end{corollary}","\\begin{example}[A two component obstruction]\\label{ex:two_component_obstruction}\nLet\n\\[\nD:\\bbk\\longrightarrow\\bbk^2,\n\\qquad\nD(t)=(t,t).\n\\]\nThe cokernel is one dimensional.  The defect $(a,b)$ is correctable exactly when $a=b$.  The obstruction can be represented by $a-b$, but this scalar representative depends on a choice of isomorphism from the cokernel to $\\bbk$.  The quotient class itself is canonical.\n\\end{example}","\\begin{theorem}[Orthogonal obstruction projector]\\label{thm:orthogonal_obstruction}\nLet $D:H_1\\to H_2$ be a bounded operator between Hilbert spaces with closed range.  Let $D^\\dagger$ be its Moore Penrose inverse.  Then\n\\[\nP_{\\mathrm{ob}}=I_{H_2}-DD^\\dagger\n\\]\nis the orthogonal projector onto\n\\[\n(\\im D)^\\perp=\\Ker D^*.\n\\]\nFor $r\\in H_2$, the following are equivalent:\n\\begin{enumerate}\n\\item $Du=-r$ is solvable\n\\item $[r]=0$ in $\\coker D$\n\\item $P_{\\mathrm{ob}}r=0$.\n\\end{enumerate}\nWhen these conditions hold, $-D^\\dagger r$ is the least norm correction.  For every $r$,\n\\[\n\\dist(r,\\im D)=\\|P_{\\mathrm{ob}}r\\|.\n\\]\n\\end{theorem}","gonal obstruction projector]\\label{thm:orthogonal_obstruction}\nLet $D:H_1\\to H_2$ be a bounded operator between Hilbert spaces with closed range.  Let $D^\\dagger$ be its Moore Penrose inverse.  Then\n\\[\nP_{\\mathrm{ob}}=I_{H_2}-DD^\\dagger\n\\]\nis the orthogonal projector onto\n\\[\n(\\im D)^\\perp=\\Ker D^*.\n\\]\nFor $r\\in H_2$, the following are equivalent:\n\\begin{enumerate}\n\\item $Du=-r$ is solvable\n\\item $[r]=0$ in $\\coker D$\n\\item $P_{\\mathrm{ob}}r=0$.\n\\end{enumerate}\nWhen these conditions hold, $-D^\\dagger r$ is the least norm correction.  For every $r$,\n\\[\n\\dist(r,\\im D)=\\|P_{\\mathrm{ob}}r\\|.\n\\]\n\\end{theorem}","uction projector]\\label{thm:orthogonal_obstruction}\nLet $D:H_1\\to H_2$ be a bounded operator between Hilbert spaces with closed range.  Let $D^\\dagger$ be its Moore Penrose inverse.  Then\n\\[\nP_{\\mathrm{ob}}=I_{H_2}-DD^\\dagger\n\\]\nis the orthogonal projector onto\n\\[\n(\\im D)^\\perp=\\Ker D^*.\n\\]\nFor $r\\in H_2$, the following are equivalent:\n\\begin{enumerate}\n\\item $Du=-r$ is solvable\n\\item $[r]=0$ in $\\coker D$\n\\item $P_{\\mathrm{ob}}r=0$.\n\\end{enumerate}\nWhen these conditions hold, $-D^\\dagger r$ is the least norm correction.  For every $r$,\n\\[\n\\dist(r,\\im D)=\\|P_{\\mathrm{ob}}r\\|.\n\\]\n\\end{theorem}","\\label{thm:orthogonal_obstruction}\nLet $D:H_1\\to H_2$ be a bounded operator between Hilbert spaces with closed range.  Let $D^\\dagger$ be its Moore Penrose inverse.  Then\n\\[\nP_{\\mathrm{ob}}=I_{H_2}-DD^\\dagger\n\\]\nis the orthogonal projector onto\n\\[\n(\\im D)^\\perp=\\Ker D^*.\n\\]\nFor $r\\in H_2$, the following are equivalent:\n\\begin{enumerate}\n\\item $Du=-r$ is solvable\n\\item $[r]=0$ in $\\coker D$\n\\item $P_{\\mathrm{ob}}r=0$.\n\\end{enumerate}\nWhen these conditions hold, $-D^\\dagger r$ is the least norm correction.  For every $r$,\n\\[\n\\dist(r,\\im D)=\\|P_{\\mathrm{ob}}r\\|.\n\\]\n\\end{theorem}","operator between Hilbert spaces with closed range.  Let $D^\\dagger$ be its Moore Penrose inverse.  Then\n\\[\nP_{\\mathrm{ob}}=I_{H_2}-DD^\\dagger\n\\]\nis the orthogonal projector onto\n\\[\n(\\im D)^\\perp=\\Ker D^*.\n\\]\nFor $r\\in H_2$, the following are equivalent:\n\\begin{enumerate}\n\\item $Du=-r$ is solvable\n\\item $[r]=0$ in $\\coker D$\n\\item $P_{\\mathrm{ob}}r=0$.\n\\end{enumerate}\nWhen these conditions hold, $-D^\\dagger r$ is the least norm correction.  For every $r$,\n\\[\n\\dist(r,\\im D)=\\|P_{\\mathrm{ob}}r\\|.\n\\]\n\\end{theorem}","$D^\\dagger$ be its Moore Penrose inverse.  Then\n\\[\nP_{\\mathrm{ob}}=I_{H_2}-DD^\\dagger\n\\]\nis the orthogonal projector onto\n\\[\n(\\im D)^\\perp=\\Ker D^*.\n\\]\nFor $r\\in H_2$, the following are equivalent:\n\\begin{enumerate}\n\\item $Du=-r$ is solvable\n\\item $[r]=0$ in $\\coker D$\n\\item $P_{\\mathrm{ob}}r=0$.\n\\end{enumerate}\nWhen these conditions hold, $-D^\\dagger r$ is the least norm correction.  For every $r$,\n\\[\n\\dist(r,\\im D)=\\|P_{\\mathrm{ob}}r\\|.\n\\]\n\\end{theorem}","agger$ be its Moore Penrose inverse.  Then\n\\[\nP_{\\mathrm{ob}}=I_{H_2}-DD^\\dagger\n\\]\nis the orthogonal projector onto\n\\[\n(\\im D)^\\perp=\\Ker D^*.\n\\]\nFor $r\\in H_2$, the following are equivalent:\n\\begin{enumerate}\n\\item $Du=-r$ is solvable\n\\item $[r]=0$ in $\\coker D$\n\\item $P_{\\mathrm{ob}}r=0$.\n\\end{enumerate}\nWhen these conditions hold, $-D^\\dagger r$ is the least norm correction.  For every $r$,\n\\[\n\\dist(r,\\im D)=\\|P_{\\mathrm{ob}}r\\|.\n\\]\n\\end{theorem}","\\begin{proof}\nClosed range gives the orthogonal decomposition\n\\[\nH_2=\\im D\\oplus\\Ker D^*.\n\\]\nThe Moore Penrose identities show that $DD^\\dagger$ is the orthogonal projector onto $\\im D$.  The stated equivalences and distance formula follow from this decomposition.  The least norm property is the standard characterization of $D^\\dagger r$.\n\\end{proof}","\\begin{theorem}[Right inverse correction theorem]\\label{thm:right_inverse_correction}\nLet $D:X\\to Y$ be a bounded linear map between Banach spaces.  Assume $Z=\\im D$ is closed and there is a bounded linear right inverse\n\\[\nR:Z\\longrightarrow X,\n\\qquad\nDR=I_Z.\n\\]\nPut $X_R=R(Z)$.  Let\n\\[\nN:\\overline B_\\rho(0)\\cap X_R\\longrightarrow Z\n\\]\nbe $L$ Lipschitz.  Let $y\\in Z$ and put\n\\[\nq=\\|R\\|L.\n\\]\nAssume $q<1$ and\n\\[\n\\|R(y-N(0))\\|\n\\le\n(1-q)\\rho.\n\\]\nThen the equation\n\\[\nDx+N(x)=y\n\\]\nhas a unique solution in $\\overline B_\\rho(0)\\cap X_R$.\n\\end{theorem}","begin{theorem}[Right inverse correction theorem]\\label{thm:right_inverse_correction}\nLet $D:X\\to Y$ be a bounded linear map between Banach spaces.  Assume $Z=\\im D$ is closed and there is a bounded linear right inverse\n\\[\nR:Z\\longrightarrow X,\n\\qquad\nDR=I_Z.\n\\]\nPut $X_R=R(Z)$.  Let\n\\[\nN:\\overline B_\\rho(0)\\cap X_R\\longrightarrow Z\n\\]\nbe $L$ Lipschitz.  Let $y\\in Z$ and put\n\\[\nq=\\|R\\|L.\n\\]\nAssume $q<1$ and\n\\[\n\\|R(y-N(0))\\|\n\\le\n(1-q)\\rho.\n\\]\nThen the equation\n\\[\nDx+N(x)=y\n\\]\nhas a unique solution in $\\overline B_\\rho(0)\\cap X_R$.\n\\end{theorem}","em}[Right inverse correction theorem]\\label{thm:right_inverse_correction}\nLet $D:X\\to Y$ be a bounded linear map between Banach spaces.  Assume $Z=\\im D$ is closed and there is a bounded linear right inverse\n\\[\nR:Z\\longrightarrow X,\n\\qquad\nDR=I_Z.\n\\]\nPut $X_R=R(Z)$.  Let\n\\[\nN:\\overline B_\\rho(0)\\cap X_R\\longrightarrow Z\n\\]\nbe $L$ Lipschitz.  Let $y\\in Z$ and put\n\\[\nq=\\|R\\|L.\n\\]\nAssume $q<1$ and\n\\[\n\\|R(y-N(0))\\|\n\\le\n(1-q)\\rho.\n\\]\nThen the equation\n\\[\nDx+N(x)=y\n\\]\nhas a unique solution in $\\overline B_\\rho(0)\\cap X_R$.\n\\end{theorem}","_inverse_correction}\nLet $D:X\\to Y$ be a bounded linear map between Banach spaces.  Assume $Z=\\im D$ is closed and there is a bounded linear right inverse\n\\[\nR:Z\\longrightarrow X,\n\\qquad\nDR=I_Z.\n\\]\nPut $X_R=R(Z)$.  Let\n\\[\nN:\\overline B_\\rho(0)\\cap X_R\\longrightarrow Z\n\\]\nbe $L$ Lipschitz.  Let $y\\in Z$ and put\n\\[\nq=\\|R\\|L.\n\\]\nAssume $q<1$ and\n\\[\n\\|R(y-N(0))\\|\n\\le\n(1-q)\\rho.\n\\]\nThen the equation\n\\[\nDx+N(x)=y\n\\]\nhas a unique solution in $\\overline B_\\rho(0)\\cap X_R$.\n\\end{theorem}","bounded linear map between Banach spaces.  Assume $Z=\\im D$ is closed and there is a bounded linear right inverse\n\\[\nR:Z\\longrightarrow X,\n\\qquad\nDR=I_Z.\n\\]\nPut $X_R=R(Z)$.  Let\n\\[\nN:\\overline B_\\rho(0)\\cap X_R\\longrightarrow Z\n\\]\nbe $L$ Lipschitz.  Let $y\\in Z$ and put\n\\[\nq=\\|R\\|L.\n\\]\nAssume $q<1$ and\n\\[\n\\|R(y-N(0))\\|\n\\le\n(1-q)\\rho.\n\\]\nThen the equation\n\\[\nDx+N(x)=y\n\\]\nhas a unique solution in $\\overline B_\\rho(0)\\cap X_R$.\n\\end{theorem}","\\begin{proof}\nThe space $X_R$ is closed.  Indeed, if $Rz_n\\to x$, then $z_n=DRz_n\\to Dx$, so $x=R(Dx)$.  Define\n\\[\n\\Psi(x)=R(y-N(x)).\n\\]\nThis map takes $X_R$ to itself.  For $x\\in\\overline B_\\rho(0)\\cap X_R$,\n\\[\n\\begin{aligned}\n\\|\\Psi(x)\\|\n&\\le\n\\|R(y-N(0))\\|\n+\n\\|R\\|\\,\\|N(x)-N(0)\\|","\\begin{proof}\nThe space $X_R$ is closed.  Indeed, if $Rz_n\\to x$, then $z_n=DRz_n\\to Dx$, so $x=R(Dx)$.  Define\n\\[\n\\Psi(x)=R(y-N(x)).\n\\]\nThis map takes $X_R$ to itself.  For $x\\in\\overline B_\\rho(0)\\cap X_R$,\n\\[\n\\begin{aligned}\n\\|\\Psi(x)\\|\n&\\le\n\\|R(y-N(0))\\|\n+\n\\|R\\|\\,\\|N(x)-N(0)\\|\\\\\n&\\le\n(1-q)\\rho+q\\rho\n=\\rho.\n\\end{aligned}\n\\]\nAlso,\n\\[\n\\|\\Psi(x)-\\Psi(x')\\|\n\\le q\\|x-x'\\|.\n\\]\nThe contraction mapping theorem gives a unique fixed point in the stated set.  Applying $D$ to the fixed point identity gives the equation.  Conversely, a solution in $X_R$ equals $R(Dx)$ and is therefore a fixed point of $\\Psi$.\n\\end{proof}","&\\le\n(1-q)\\rho+q\\rho\n=\\rho.\n\\end{aligned}\n\\]\nAlso,\n\\[\n\\|\\Psi(x)-\\Psi(x')\\|\n\\le q\\|x-x'\\|.\n\\]\nThe contraction mapping theorem gives a unique fixed point in the stated set.  Applying $D$ to the fixed point identity gives the equation.  Conversely, a solution in $X_R$ equals $R(Dx)$ and is therefore a fixed point of $\\Psi$.\n\\end{proof}","\\begin{corollary}[Stability with respect to the target]\\label{cor:correction_stability}\nUnder the hypotheses of \\cref{thm:right_inverse_correction}, let $x$ and $x'$ be the fixed points associated with targets $y$ and $y'$.  If both fixed point maps preserve the same ball, then\n\\[\n\\|x-x'\\|\n\\le\n\\frac{\\|R\\|}{1-q}\\|y-y'\\|.\n\\]\n\\end{corollary}","\\begin{proof}\nSubtract the two fixed point equations and use the Lipschitz bound for $N$:\n\\[\n\\|x-x'\\|\n\\le\n\\|R\\|\\|y-y'\\|+q\\|x-x'\\|.\n\\]\n\\end{proof}","\\begin{theorem}[Local Lyapunov Schmidt reduction]\\label{thm:lyapunov_schmidt}\nLet $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points","\\begin{theorem}[Local Lyapunov Schmidt reduction]\\label{thm:lyapunov_schmidt}\nLet $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[","\\begin{theorem}[Local Lyapunov Schmidt reduction]\\label{thm:lyapunov_schmidt}\nLet $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\","\\begin{theorem}[Local Lyapunov Schmidt reduction]\\label{thm:lyapunov_schmidt}\nLet $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the b","\\begin{theorem}[Local Lyapunov Schmidt reduction]\\label{thm:lyapunov_schmidt}\nLet $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,","\\begin{theorem}[Local Lyapunov Schmidt reduction]\\label{thm:lyapunov_schmidt}\nLet $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,","\\begin{theorem}[Local Lyapunov Schmidt reduction]\\label{thm:lyapunov_schmidt}\nLet $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","hmidt reduction]\\label{thm:lyapunov_schmidt}\nLet $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","ion]\\label{thm:lyapunov_schmidt}\nLet $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","et $\\mathcal U\\subseteq X$ be an open neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","en neighborhood of zero and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","ro and let $F:\\mathcal U\\to Y$ be a $C^r$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","$ map between Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","en Banach spaces with $r\\ge1$, $F(0)=0$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","$, and $L=DF(0)$.  Assume there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","me there are closed decompositions\n\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","\\[\nX=K\\oplus X_1,\n\\qquad\nY=R\\oplus O,\n\\]\nwhere $K=\\Ker L$, $R=\\im L$, and\n\\[\nL_1=L|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","|_{X_1}:X_1\\longrightarrow R\n\\]\nis a Banach space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","space isomorphism.  Let $\\pi_R$ and $\\pi_O$ be the two projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","projections on $Y$.  Then there are neighborhoods $U_K\\subseteq K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","K$ and $U_1\\subseteq X_1$ of zero and a unique $C^r$ map\n\\[\n\\psi:U_K\\longrightarrow U_1\n\\]\nwith $\\psi(0)=0$ such that\n\\[\n\\pi_RF(k+\\psi(k))=0\n\\]\nfor every $k\\in U_K$.  After the neighborhoods are made smaller if needed, the solutions of $F(x)=0$ near zero are exactly the points\n\\[\nx=k+\\psi(k)\n\\]\nfor which the reduced obstruction map\n\\[\n\\kappa(k)=\\pi_OF(k+\\psi(k))\n\\]\nvanishes.  At the base point,\n\\[\nD\\psi(0)=0,\n\\qquad\nD\\kappa(0)=0.\n\\]\n\\end{theorem}","\\begin{proof}\nDefine\n\\[\nG(k,x_1)=\\pi_RF(k+x_1).\n\\]\nThe derivative of $G$ in the $X_1$ variable at $(0,0)$ is $L_1$, which is invertible.  The implicit function theorem gives the unique map $\\psi$.  Every nearby $x$ has a unique decomposition $x=k+x_1$.  The range equation forces $x_1=\\psi(k)$, and the remaining equation is precisely $\\kappa(k)=0$.","Differentiate $G(k,\\psi(k))=0$ at zero.  Since $L$ vanishes on $K$, one obtains $L_1D\\psi(0)=0$, hence $D\\psi(0)=0$.  Finally,\n\\[\nD\\kappa(0)=\\pi_OL|_K=0.\n\\]\n\\end{proof}","\\begin{corollary}[Quadratic reduced obstruction]\\label{cor:quadratic_reduced_obstruction}\nAssume the hypotheses of \\cref{thm:lyapunov_schmidt} with $r\\ge2$.  For $k_1,k_2\\in K$,\n\\[\nD^2\\kappa(0)[k_1,k_2]\n=\n\\pi_O D^2F(0)[k_1,k_2].\n\\]\n\\end{corollary}","\\begin{proof}\nDifferentiate $\\kappa(k)=\\pi_OF(k+\\psi(k))$ twice.  The terms containing $D\\psi(0)$ vanish.  The term containing $D^2\\psi(0)$ is multiplied by $\\pi_OL$, which is zero because $L(X)=R$.\n\\end{proof}","\\begin{example}[A purely quadratic obstruction]\\label{ex:pure_quadratic_obstruction}\nLet\n\\[\nF(x,y)=(y,x^2)\n\\]\non $\\mathbb R^2$.  At the origin,\n\\[\nL(h,k)=(k,0).\n\\]\nThe kernel is the $x$ axis, the range is the first output axis, and the cokernel is the second output axis.  The range equation forces $y=0$.  The reduced obstruction map is\n\\[\n\\kappa(x)=x^2.\n\\]\nThe linear obstruction vanishes for every kernel direction, but the nonlinear equation still has only the solution $x=0$ near the origin.\n\\end{example}","truction}\nLet\n\\[\nF(x,y)=(y,x^2)\n\\]\non $\\mathbb R^2$.  At the origin,\n\\[\nL(h,k)=(k,0).\n\\]\nThe kernel is the $x$ axis, the range is the first output axis, and the cokernel is the second output axis.  The range equation forces $y=0$.  The reduced obstruction map is\n\\[\n\\kappa(x)=x^2.\n\\]\nThe linear obstruction vanishes for every kernel direction, but the nonlinear equation still has only the solution $x=0$ near the origin.\n\\end{example}","\\begin{remark}\\label{rem:higher_obstruction}\nThe reduced map $\\kappa$ begins at quadratic order.  This is why vanishing of the first order cokernel class need not settle a nonlinear problem.  Higher terms can produce a genuine obstruction on the kernel directions.\n\\end{remark}","Let $X$ and $Y$ be modules with decreasing filtrations\n\\[\nX=F^0X\\supseteq F^1X\\supseteq\\cdots,\n\\qquad\nY=F^0Y\\supseteq F^1Y\\supseteq\\cdots.\n\\]\nAssume both are separated and complete for their filtration topologies.  Let $\\Phi:X\\to Y$ be continuous and filtration congruence preserving in the sense that\n\\[\nh\\in F^mX\n\\quad\\Longrightarrow\\quad\n\\Phi(x+h)-\\Phi(x)\\in F^mY\n\\]\nfor every $x\\in X$ and every $m$.  This condition makes the statement $\\Phi(x)\\in F^mY$ depend only on the class of $x$ modulo $F^mX$.","g filtrations\n\\[\nX=F^0X\\supseteq F^1X\\supseteq\\cdots,\n\\qquad\nY=F^0Y\\supseteq F^1Y\\supseteq\\cdots.\n\\]\nAssume both are separated and complete for their filtration topologies.  Let $\\Phi:X\\to Y$ be continuous and filtration congruence preserving in the sense that\n\\[\nh\\in F^mX\n\\quad\\Longrightarrow\\quad\n\\Phi(x+h)-\\Phi(x)\\in F^mY\n\\]\nfor every $x\\in X$ and every $m$.  This condition makes the statement $\\Phi(x)\\in F^mY$ depend only on the class of $x$ modulo $F^mX$.","\\cdots,\n\\qquad\nY=F^0Y\\supseteq F^1Y\\supseteq\\cdots.\n\\]\nAssume both are separated and complete for their filtration topologies.  Let $\\Phi:X\\to Y$ be continuous and filtration congruence preserving in the sense that\n\\[\nh\\in F^mX\n\\quad\\Longrightarrow\\quad\n\\Phi(x+h)-\\Phi(x)\\in F^mY\n\\]\nfor every $x\\in X$ and every $m$.  This condition makes the statement $\\Phi(x)\\in F^mY$ depend only on the class of $x$ modulo $F^mX$.","uad\nY=F^0Y\\supseteq F^1Y\\supseteq\\cdots.\n\\]\nAssume both are separated and complete for their filtration topologies.  Let $\\Phi:X\\to Y$ be continuous and filtration congruence preserving in the sense that\n\\[\nh\\in F^mX\n\\quad\\Longrightarrow\\quad\n\\Phi(x+h)-\\Phi(x)\\in F^mY\n\\]\nfor every $x\\in X$ and every $m$.  This condition makes the statement $\\Phi(x)\\in F^mY$ depend only on the class of $x$ modulo $F^mX$.","Suppose $x_m$ satisfies\n\\[\n\\Phi(x_m)\\in F^mY.\n\\]\nAssume that corrections $h\\in F^mX$ obey a first order congruence\n\\[\n\\Phi(x_m+h)\n\\equiv\n\\Phi(x_m)+L_mh\n\\pmod{F^{m+1}Y},\n\\]\nwhere $L_m:F^mX\\to F^mY$ is a module homomorphism satisfying\n\\[\nL_m(F^{m+1}X)\\subseteq F^{m+1}Y\n\\]\nand therefore induces\n\\[\n\\overline L_m:\nF^mX/F^{m+1}X\n\\longrightarrow\nF^mY/F^{m+1}Y.\n\\]","At depth $m$, only the associated graded correction is visible. The cokernel below has finite length whenever the target graded piece has finite length, even if the completed problem is infinite. Vanishing of the depth obstruction decides whether the current approximation can be improved by one level. When the graded pieces are finite, the branching count below applies. Finite length alone does not imply finite cardinality.","\\begin{definition}[Depth obstruction]\\label{def:depth_obstruction}\nThe depth $m$ obstruction of $x_m$ is\n\\[\n\\Ob_m(x_m)\n=\n[\\Phi(x_m)]\n\\in\n\\coker\\overline L_m,\n\\]\nwhere $\\Phi(x_m)$ is first viewed in $F^mY/F^{m+1}Y$.\n\\end{definition}","\\begin{theorem}[One step lifting criterion]\\label{thm:one_step_lifting}\nThe approximation $x_m$ has a correction $h_m\\in F^mX$ such that\n\\[\n\\Phi(x_m+h_m)\\in F^{m+1}Y\n\\]\nif and only if\n\\[\n\\Ob_m(x_m)=0.\n\\]\nWhen a correction exists, the set of its classes in\n\\[\nF^mX/F^{m+1}X\n\\]\nis a coset of $\\Ker\\overline L_m$.  If the graded pieces are vector spaces, this coset is an affine space.\n\\end{theorem}","\\begin{proof}\nThe next depth condition is equivalent to\n\\[\n\\overline L_m[h_m]\n=-[\\Phi(x_m)]\n\\]\nin $F^mY/F^{m+1}Y$.  This is solvable exactly when the displayed defect has zero cokernel class.  If $h_m$ and $h_m'$ are two solutions, their difference lies in $\\Ker\\overline L_m$.  Adding an element of the kernel to one solution gives another, so the solution set is a coset of the kernel.\n\\end{proof}","\\begin{corollary}[Finite branching count]\\label{cor:finite_branch_count}\nSuppose the graded pieces are finite vector spaces over a finite field.  If the depth obstruction vanishes, then the number of correction classes at depth $m$ is\n\\[\n|\\Ker\\overline L_m|.\n\\]\nIf the obstruction does not vanish, the number is zero.\n\\end{corollary}","\\begin{lemma}[Well defined lift classes]\\label{lem:well_defined_lift_classes}\nUnder the filtration congruence assumption, the condition\n\\[\n\\Phi(x)\\in F^mY\n\\]\ndepends only on the class of $x$ in $X/F^mX$.  Fix a representative $x_m$ of a depth $m$ class.  The map\n\\[\n[h]\\longmapsto[x_m+h]_{m+1}\n\\]\nidentifies correction classes $[h]\\in F^mX/F^{m+1}X$ with the classes at depth $m+1$ that reduce to $[x_m]_m$.  Under this identification, the solutions of the associated graded correction equation in \\cref{thm:one_step_lifting} are exactly the children whose defect lies in $F^{m+1}Y$.  A different representative of the same parent gives a translated parametrization of the same set of children.\n\\end{lemma}","the class of $x$ in $X/F^mX$.  Fix a representative $x_m$ of a depth $m$ class.  The map\n\\[\n[h]\\longmapsto[x_m+h]_{m+1}\n\\]\nidentifies correction classes $[h]\\in F^mX/F^{m+1}X$ with the classes at depth $m+1$ that reduce to $[x_m]_m$.  Under this identification, the solutions of the associated graded correction equation in \\cref{thm:one_step_lifting} are exactly the children whose defect lies in $F^{m+1}Y$.  A different representative of the same parent gives a translated parametrization of the same set of children.\n\\end{lemma}","\\begin{proof}\nIf $x'-x\\in F^mX$, then filtration congruence preservation gives\n\\[\n\\Phi(x')-\\Phi(x)\\in F^mY.\n\\]\nThus one defect lies in $F^mY$ exactly when the other does.  If $h-h'\\in F^{m+1}X$, then $x_m+h$ and $x_m+h'$ define the same depth $m+1$ class.  Conversely, two depth $m+1$ classes reducing to $[x_m]_m$ differ from $x_m$ by elements of $F^mX$, and they are equal precisely when those elements agree modulo $F^{m+1}X$.  The first order congruence shows that such a child has defect in $F^{m+1}Y$ exactly when its correction class solves the associated graded equation.  Replacing $x_m$ by another representative translates the correction coordinate by an element of $F^mX/F^{m+1}X$ and leaves the child classes themselv","\\begin{proof}\nIf $x'-x\\in F^mX$, then filtration congruence preservation gives\n\\[\n\\Phi(x')-\\Phi(x)\\in F^mY.\n\\]\nThus one defect lies in $F^mY$ exactly when the other does.  If $h-h'\\in F^{m+1}X$, then $x_m+h$ and $x_m+h'$ define the same depth $m+1$ class.  Conversely, two depth $m+1$ classes reducing to $[x_m]_m$ differ from $x_m$ by elements of $F^mX$, and they are equal precisely when those elements agree modulo $F^{m+1}X$.  The first order congruence shows that such a child has defect in $F^{m+1}Y$ exactly when its correction class solves the associated graded equation.  Replacing $x_m$ by another representative translates the correction coordinate by an element of $F^mX/F^{m+1}X$ and leaves the child classes themselves unchanged.\n\\end{proof}","'-x\\in F^mX$, then filtration congruence preservation gives\n\\[\n\\Phi(x')-\\Phi(x)\\in F^mY.\n\\]\nThus one defect lies in $F^mY$ exactly when the other does.  If $h-h'\\in F^{m+1}X$, then $x_m+h$ and $x_m+h'$ define the same depth $m+1$ class.  Conversely, two depth $m+1$ classes reducing to $[x_m]_m$ differ from $x_m$ by elements of $F^mX$, and they are equal precisely when those elements agree modulo $F^{m+1}X$.  The first order congruence shows that such a child has defect in $F^{m+1}Y$ exactly when its correction class solves the associated graded equation.  Replacing $x_m$ by another representative translates the correction coordinate by an element of $F^mX/F^{m+1}X$ and leaves the child classes themselves unchanged.\n\\end{proof}",")\\in F^mY.\n\\]\nThus one defect lies in $F^mY$ exactly when the other does.  If $h-h'\\in F^{m+1}X$, then $x_m+h$ and $x_m+h'$ define the same depth $m+1$ class.  Conversely, two depth $m+1$ classes reducing to $[x_m]_m$ differ from $x_m$ by elements of $F^mX$, and they are equal precisely when those elements agree modulo $F^{m+1}X$.  The first order congruence shows that such a child has defect in $F^{m+1}Y$ exactly when its correction class solves the associated graded equation.  Replacing $x_m$ by another representative translates the correction coordinate by an element of $F^mX/F^{m+1}X$ and leaves the child classes themselves unchanged.\n\\end{proof}","$, then $x_m+h$ and $x_m+h'$ define the same depth $m+1$ class.  Conversely, two depth $m+1$ classes reducing to $[x_m]_m$ differ from $x_m$ by elements of $F^mX$, and they are equal precisely when those elements agree modulo $F^{m+1}X$.  The first order congruence shows that such a child has defect in $F^{m+1}Y$ exactly when its correction class solves the associated graded equation.  Replacing $x_m$ by another representative translates the correction coordinate by an element of $F^mX/F^{m+1}X$ and leaves the child classes themselves unchanged.\n\\end{proof}","$m+1$ classes reducing to $[x_m]_m$ differ from $x_m$ by elements of $F^mX$, and they are equal precisely when those elements agree modulo $F^{m+1}X$.  The first order congruence shows that such a child has defect in $F^{m+1}Y$ exactly when its correction class solves the associated graded equation.  Replacing $x_m$ by another representative translates the correction coordinate by an element of $F^mX/F^{m+1}X$ and leaves the child classes themselves unchanged.\n\\end{proof}","\\begin{definition}[Lift tree]\\label{def:lift_tree}\nFix an initial approximation.  A vertex at depth $m$ is a class $[x]_m\\in X/F^mX$ with $\\Phi(x)\\in F^mY$ that reduces to the fixed initial class.  A depth $m+1$ vertex is joined to its reduction in $X/F^mX$.  By \\cref{lem:well_defined_lift_classes}, this rooted graph is independent of the chosen representatives.  It is called the lift tree.\n\\end{definition}","\\begin{theorem}[Finitely branching lift tree]\\label{thm:finitely_branching_lift_tree}\nAssume every level of the lift tree is nonempty and every vertex has only finitely many children.  Then the tree contains an infinite branch.  If $X$ and $Y$ are separated and complete and $\\Phi$ is continuous, that branch determines an exact solution of $\\Phi(x)=0$.\n\\end{theorem}","Choose representatives recursively.  Once $x_m$ represents the selected depth $m$ vertex, choose a representative $x_{m+1}$ of its selected child with\n\\[\nx_{m+1}-x_m\\in F^mX.\n\\]\nThe resulting sequence is Cauchy for the filtration topology.  Each representative satisfies $\\Phi(x_m)\\in F^mY$.  The conclusion now follows from \\cref{thm:complete_filtered_lifting}.\n\\end{proof}","\\begin{corollary}[Unique lift branch]\\label{cor:unique_lift_branch}\nSuppose an initial approximation is fixed, every depth obstruction vanishes along its descendants, and every map $\\overline L_m$ has zero kernel.  Then at most one compatible lift branch extends the initial approximation.\n\\end{corollary}","\\begin{proof}\nBy \\cref{thm:one_step_lifting}, the correction classes at each depth form a coset of $\\Ker\\overline L_m$.  A zero kernel leaves at most one correction class, so induction gives at most one branch.\n\\end{proof}","\\begin{theorem}[Complete filtered lifting]\\label{thm:complete_filtered_lifting}\nAssume there is a compatible sequence\n\\[\nx_{m+1}\\equiv x_m\\pmod{F^mX}\n\\]\nwith\n\\[\n\\Phi(x_m)\\in F^mY\n\\]\nfor every $m$.  Then $x_m$ converges to an element $x_\\infty\\in X$ and\n\\[\n\\Phi(x_\\infty)=0.\n\\]\nIn particular, if the obstruction vanishes at every depth and compatible corrections are chosen, an exact solution exists.\n\\end{theorem}","\\begin{proof}\nFor every fixed $r$, the congruences imply\n\\[\nx_n-x_m\\in F^rX\n\\]\nwhenever $n\\ge m\\ge r$.  Thus $(x_m)$ is Cauchy for the filtration topology.  Completeness gives a limit $x_\\infty$.","Fix $r$.  For every $m\\ge r$, one has $\\Phi(x_m)\\in F^mY\\subseteq F^rY$.  The submodule $F^rY$ is open and therefore closed in the filtration topology.  Continuity of $\\Phi$ gives $\\Phi(x_m)\\to\\Phi(x_\\infty)$, so $\\Phi(x_\\infty)\\in F^rY$.  Since this holds for every $r$ and $Y$ is separated,\n\\[\n\\Phi(x_\\infty)\\in\\bigcap_rF^rY=0.\n\\]\n\\end{proof}","\\begin{theorem}[Naturality of filtered obstructions]\\label{thm:filtered_obstruction_naturality}\nSuppose a morphism of filtered lifting problems sends $x_m$ to $x_m'$, intertwines $L_m$ with $L_m'$, and sends $\\Phi(x_m)$ to $\\Phi'(x_m')$ on the associated graded pieces.  Then the induced cokernel map sends\n\\[\n\\Ob_m(x_m)\n\\quad\\text{to}\\quad\n\\Ob_m'(x_m').\n\\]\n\\end{theorem}","\\begin{proof}\nApply \\cref{thm:obstruction_naturality} to the induced commuting square on the depth $m$ graded pieces.\n\\end{proof}","\\begin{example}[Three lifting outcomes]\\label{ex:three_lifting_outcomes}\nThe scalar equation $x^2-2=0$ at $x=3$ modulo $7$ has invertible derivative.  It has one correction modulo $49$ and gives $x=10$ modulo $49$.","For $x^2-3=0$ at $x=0$ modulo $3$, the derivative is zero and the defect class is nonzero.  No correction exists.","For $x^2=0$ at $x=0$ modulo $3$, the derivative and defect both vanish.  Every correction digit lies in the kernel, so the lift branches.\n\\end{example}","\\begin{proposition}[Maurer Cartan lifting obstruction]\\label{prop:maurer_cartan_lifting}\nLet $\\Lcal$ be a complete filtered differential graded Lie algebra over a ring in which two is invertible.  Assume\n\\[\ndF^i\\Lcal\\subseteq F^i\\Lcal,\n\\qquad\n[F^i\\Lcal,F^j\\Lcal]\\subseteq F^{i+j}\\Lcal.\n\\]\nFor a degree one element $\\eta$, put\n\\[\n\\Omega(\\eta)=d\\eta+\\frac12[\\eta,\\eta].\n\\]\nSuppose $\\Omega(\\eta)\\in F^m\\Lcal^2$ with $m\\ge1$.  For every $h\\in F^m\\Lcal^1$,\n\\[\n\\Omega(\\eta+h)\n\\equiv\n\\Omega(\\eta)+d_\\eta h\n\\pmod{F^{m+1}\\Lcal^2},\n\\qquad\nd_\\eta=d+[\\eta,\\mathord\\cdot].\n\\]\nThe obstruction to improving $\\eta$ by one depth is therefore the class of $\\Omega(\\eta)$ in the cokernel of\n\\[\n\\gr^m(d_\\eta):\n\\gr^m\\Lcal^1\n\\longrightarrow\n\\gr^m\\Lcal^2.\n\\]\n\\end{proposition}","tion}[Maurer Cartan lifting obstruction]\\label{prop:maurer_cartan_lifting}\nLet $\\Lcal$ be a complete filtered differential graded Lie algebra over a ring in which two is invertible.  Assume\n\\[\ndF^i\\Lcal\\subseteq F^i\\Lcal,\n\\qquad\n[F^i\\Lcal,F^j\\Lcal]\\subseteq F^{i+j}\\Lcal.\n\\]\nFor a degree one element $\\eta$, put\n\\[\n\\Omega(\\eta)=d\\eta+\\frac12[\\eta,\\eta].\n\\]\nSuppose $\\Omega(\\eta)\\in F^m\\Lcal^2$ with $m\\ge1$.  For every $h\\in F^m\\Lcal^1$,\n\\[\n\\Omega(\\eta+h)\n\\equiv\n\\Omega(\\eta)+d_\\eta h\n\\pmod{F^{m+1}\\Lcal^2},\n\\qquad\nd_\\eta=d+[\\eta,\\mathord\\cdot].\n\\]\nThe obstruction to improving $\\eta$ by one depth is therefore the class of $\\Omega(\\eta)$ in the cokernel of\n\\[\n\\gr^m(d_\\eta):\n\\gr^m\\Lcal^1\n\\longrightarrow\n\\gr^m\\Lcal^2.\n\\]\n\\end{proposition}","lifting obstruction]\\label{prop:maurer_cartan_lifting}\nLet $\\Lcal$ be a complete filtered differential graded Lie algebra over a ring in which two is invertible.  Assume\n\\[\ndF^i\\Lcal\\subseteq F^i\\Lcal,\n\\qquad\n[F^i\\Lcal,F^j\\Lcal]\\subseteq F^{i+j}\\Lcal.\n\\]\nFor a degree one element $\\eta$, put\n\\[\n\\Omega(\\eta)=d\\eta+\\frac12[\\eta,\\eta].\n\\]\nSuppose $\\Omega(\\eta)\\in F^m\\Lcal^2$ with $m\\ge1$.  For every $h\\in F^m\\Lcal^1$,\n\\[\n\\Omega(\\eta+h)\n\\equiv\n\\Omega(\\eta)+d_\\eta h\n\\pmod{F^{m+1}\\Lcal^2},\n\\qquad\nd_\\eta=d+[\\eta,\\mathord\\cdot].\n\\]\nThe obstruction to improving $\\eta$ by one depth is therefore the class of $\\Omega(\\eta)$ in the cokernel of\n\\[\n\\gr^m(d_\\eta):\n\\gr^m\\Lcal^1\n\\longrightarrow\n\\gr^m\\Lcal^2.\n\\]\n\\end{proposition}","\\label{prop:maurer_cartan_lifting}\nLet $\\Lcal$ be a complete filtered differential graded Lie algebra over a ring in which two is invertible.  Assume\n\\[\ndF^i\\Lcal\\subseteq F^i\\Lcal,\n\\qquad\n[F^i\\Lcal,F^j\\Lcal]\\subseteq F^{i+j}\\Lcal.\n\\]\nFor a degree one element $\\eta$, put\n\\[\n\\Omega(\\eta)=d\\eta+\\frac12[\\eta,\\eta].\n\\]\nSuppose $\\Omega(\\eta)\\in F^m\\Lcal^2$ with $m\\ge1$.  For every $h\\in F^m\\Lcal^1$,\n\\[\n\\Omega(\\eta+h)\n\\equiv\n\\Omega(\\eta)+d_\\eta h\n\\pmod{F^{m+1}\\Lcal^2},\n\\qquad\nd_\\eta=d+[\\eta,\\mathord\\cdot].\n\\]\nThe obstruction to improving $\\eta$ by one depth is therefore the class of $\\Omega(\\eta)$ in the cokernel of\n\\[\n\\gr^m(d_\\eta):\n\\gr^m\\Lcal^1\n\\longrightarrow\n\\gr^m\\Lcal^2.\n\\]\n\\end{proposition}","e.  Assume\n\\[\ndF^i\\Lcal\\subseteq F^i\\Lcal,\n\\qquad\n[F^i\\Lcal,F^j\\Lcal]\\subseteq F^{i+j}\\Lcal.\n\\]\nFor a degree one element $\\eta$, put\n\\[\n\\Omega(\\eta)=d\\eta+\\frac12[\\eta,\\eta].\n\\]\nSuppose $\\Omega(\\eta)\\in F^m\\Lcal^2$ with $m\\ge1$.  For every $h\\in F^m\\Lcal^1$,\n\\[\n\\Omega(\\eta+h)\n\\equiv\n\\Omega(\\eta)+d_\\eta h\n\\pmod{F^{m+1}\\Lcal^2},\n\\qquad\nd_\\eta=d+[\\eta,\\mathord\\cdot].\n\\]\nThe obstruction to improving $\\eta$ by one depth is therefore the class of $\\Omega(\\eta)$ in the cokernel of\n\\[\n\\gr^m(d_\\eta):\n\\gr^m\\Lcal^1\n\\longrightarrow\n\\gr^m\\Lcal^2.\n\\]\n\\end{proposition}","uad\n[F^i\\Lcal,F^j\\Lcal]\\subseteq F^{i+j}\\Lcal.\n\\]\nFor a degree one element $\\eta$, put\n\\[\n\\Omega(\\eta)=d\\eta+\\frac12[\\eta,\\eta].\n\\]\nSuppose $\\Omega(\\eta)\\in F^m\\Lcal^2$ with $m\\ge1$.  For every $h\\in F^m\\Lcal^1$,\n\\[\n\\Omega(\\eta+h)\n\\equiv\n\\Omega(\\eta)+d_\\eta h\n\\pmod{F^{m+1}\\Lcal^2},\n\\qquad\nd_\\eta=d+[\\eta,\\mathord\\cdot].\n\\]\nThe obstruction to improving $\\eta$ by one depth is therefore the class of $\\Omega(\\eta)$ in the cokernel of\n\\[\n\\gr^m(d_\\eta):\n\\gr^m\\Lcal^1\n\\longrightarrow\n\\gr^m\\Lcal^2.\n\\]\n\\end{proposition}","{i+j}\\Lcal.\n\\]\nFor a degree one element $\\eta$, put\n\\[\n\\Omega(\\eta)=d\\eta+\\frac12[\\eta,\\eta].\n\\]\nSuppose $\\Omega(\\eta)\\in F^m\\Lcal^2$ with $m\\ge1$.  For every $h\\in F^m\\Lcal^1$,\n\\[\n\\Omega(\\eta+h)\n\\equiv\n\\Omega(\\eta)+d_\\eta h\n\\pmod{F^{m+1}\\Lcal^2},\n\\qquad\nd_\\eta=d+[\\eta,\\mathord\\cdot].\n\\]\nThe obstruction to improving $\\eta$ by one depth is therefore the class of $\\Omega(\\eta)$ in the cokernel of\n\\[\n\\gr^m(d_\\eta):\n\\gr^m\\Lcal^1\n\\longrightarrow\n\\gr^m\\Lcal^2.\n\\]\n\\end{proposition}","\\begin{proof}\nDirect expansion gives\n\\[\n\\Omega(\\eta+h)\n=\n\\Omega(\\eta)+dh+[\\eta,h]+\\frac12[h,h].\n\\]\nThe filtration rule places $[h,h]$ in $F^{2m}\\Lcal^2$, which is contained in $F^{m+1}\\Lcal^2$ because $m\\ge1$.  The congruence follows.  The last statement is \\cref{thm:one_step_lifting} applied to the induced graded correction map.\n\\end{proof}","\\begin{standardfact}[Spectral sequence of a finite filtered complex]\\label{fact:filtered_spectral_sequence}\nLet $C^\\bullet$ be a cochain complex with a finite decreasing filtration by subcomplexes\n\\[\nC^\\bullet=F^0C^\\bullet\\supseteq F^1C^\\bullet\\supseteq\\cdots\\supseteq F^{N+1}C^\\bullet=0.\n\\]\nThere is a natural spectral sequence with\n\\[\nE_0^{p,q}=\\gr^pC^{p+q},\n\\qquad\nE_1^{p,q}=H^{p+q}(\\gr^pC^\\bullet),\n\\]\nwhich converges to the associated graded object of $H^{p+q}(C^\\bullet)$ with its induced filtration.  A filtered cochain map induces a morphism of spectral sequences.\n\\end{standardfact}","d complex]\\label{fact:filtered_spectral_sequence}\nLet $C^\\bullet$ be a cochain complex with a finite decreasing filtration by subcomplexes\n\\[\nC^\\bullet=F^0C^\\bullet\\supseteq F^1C^\\bullet\\supseteq\\cdots\\supseteq F^{N+1}C^\\bullet=0.\n\\]\nThere is a natural spectral sequence with\n\\[\nE_0^{p,q}=\\gr^pC^{p+q},\n\\qquad\nE_1^{p,q}=H^{p+q}(\\gr^pC^\\bullet),\n\\]\nwhich converges to the associated graded object of $H^{p+q}(C^\\bullet)$ with its induced filtration.  A filtered cochain map induces a morphism of spectral sequences.\n\\end{standardfact}","This is the standard construction for a filtered complex, in the form given in \\cite{Weibel1994,McCleary2001}. When $C^\\bullet$ is a correction or deformation complex, the groups on its pages can be read as graded obstruction candidates. For an arbitrary filtered complex, they are cohomology classes of the associated graded problem.","\\begin{proposition}[Fine obstruction survival]\\label{prop:fine_obstruction_survival}\nA class on the first page contributes to the total obstruction only if it is represented by a permanent cycle, so it is neither killed nor identified as a boundary on any later page.  If the spectral sequence degenerates at the first page, then $H^n(C^\\bullet)=0$ exactly when $E_1^{p,n-p}=0$ for every $p$.\n\\end{proposition}","\\begin{proof}\nConvergence identifies the last page with the associated graded object of the total cohomology.  A first page class contributes to that object exactly when it determines a nonzero class on the last page.  If the sequence degenerates at the first page, the associated graded pieces of $H^n(C^\\bullet)$ are the groups $E_1^{p,n-p}$.  A finite filtered module is zero exactly when all its associated graded pieces are zero.\n\\end{proof}","\\begin{corollary}[Direct packet complexes]\\label{cor:direct_packet_no_cancellation}\nSuppose\n\\[\nC^\\bullet=\\bigoplus_{\\alpha\\in I}C_\\alpha^\\bullet\n\\]\nis a finite direct sum of subcomplexes.  Then\n\\[\nH^n(C^\\bullet)\n\\cong\n\\bigoplus_{\\alpha\\in I}H^n(C_\\alpha^\\bullet).\n\\]\nA total cohomology class vanishes if and only if every packet class vanishes.\n\\end{corollary}","\\begin{example}[A fine class killed at the next page]\\label{ex:spectral_cancellation}\nLet $C^0=\\bbk x$ and $C^1=\\bbk y$, with differential $d(x)=y$.  Give $C^0$ the filtration\n\\[\nF^0C^0=\\bbk x,\n\\qquad\nF^1C^0=0,\n\\]\nand give $C^1$ the filtration\n\\[\nF^0C^1=F^1C^1=\\bbk y,\n\\qquad\nF^2C^1=0.\n\\]\nThe associated graded differential is zero in the relevant positions, so both $x$ and $y$ give nonzero classes on the next page.  The next differential sends the class of $x$ to the class of $y$.  The total complex is acyclic.  Thus nonzero fine obstruction classes can cancel through a later refinement differential.\n\\end{example}","x:spectral_cancellation}\nLet $C^0=\\bbk x$ and $C^1=\\bbk y$, with differential $d(x)=y$.  Give $C^0$ the filtration\n\\[\nF^0C^0=\\bbk x,\n\\qquad\nF^1C^0=0,\n\\]\nand give $C^1$ the filtration\n\\[\nF^0C^1=F^1C^1=\\bbk y,\n\\qquad\nF^2C^1=0.\n\\]\nThe associated graded differential is zero in the relevant positions, so both $x$ and $y$ give nonzero classes on the next page.  The next differential sends the class of $x$ to the class of $y$.  The total complex is acyclic.  Thus nonzero fine obstruction classes can cancel through a later refinement differential.\n\\end{example}","ion}\nLet $C^0=\\bbk x$ and $C^1=\\bbk y$, with differential $d(x)=y$.  Give $C^0$ the filtration\n\\[\nF^0C^0=\\bbk x,\n\\qquad\nF^1C^0=0,\n\\]\nand give $C^1$ the filtration\n\\[\nF^0C^1=F^1C^1=\\bbk y,\n\\qquad\nF^2C^1=0.\n\\]\nThe associated graded differential is zero in the relevant positions, so both $x$ and $y$ give nonzero classes on the next page.  The next differential sends the class of $x$ to the class of $y$.  The total complex is acyclic.  Thus nonzero fine obstruction classes can cancel through a later refinement differential.\n\\end{example}","Let $U,V,Y_P,Y_Q$ be Banach spaces. On an open neighborhood of $(u_0,v_0)\\in U\\times V$, split the variables and equations as\n\\[\nx=(u,v),\n\\qquad\nF(u,v)=\n\\begin{pmatrix}\nF_P(u,v)","Let $U,V,Y_P,Y_Q$ be Banach spaces. On an open neighborhood of $(u_0,v_0)\\in U\\times V$, split the variables and equations as\n\\[\nx=(u,v),\n\\qquad\nF(u,v)=\n\\begin{pmatrix}\nF_P(u,v)\\\\\nF_Q(u,v)\n\\end{pmatrix}.\n\\]\nAssume $F$ takes values in $Y_P\\times Y_Q$, is twice continuously Fr\\'echet differentiable near $(u_0,v_0)$, satisfies $F_Q(u_0,v_0)=0$, and\n\\[\nA=D_vF_Q(u_0,v_0)\n\\]\nis a Banach space isomorphism from $V$ to $Y_Q$. The implicit function theorem gives a local map $v=\\varphi(u)$, with $\\varphi(u_0)=v_0$, satisfying\n\\[\nF_Q(u,\\varphi(u))=0.\n\\]\nDefine the reduced equation\n\\[\nR(u)=F_P(u,\\varphi(u)).\n\\]","F_Q(u,v)\n\\end{pmatrix}.\n\\]\nAssume $F$ takes values in $Y_P\\times Y_Q$, is twice continuously Fr\\'echet differentiable near $(u_0,v_0)$, satisfies $F_Q(u_0,v_0)=0$, and\n\\[\nA=D_vF_Q(u_0,v_0)\n\\]\nis a Banach space isomorphism from $V$ to $Y_Q$. The implicit function theorem gives a local map $v=\\varphi(u)$, with $\\varphi(u_0)=v_0$, satisfying\n\\[\nF_Q(u,\\varphi(u))=0.\n\\]\nDefine the reduced equation\n\\[\nR(u)=F_P(u,\\varphi(u)).\n\\]","\\begin{theorem}[Exact local elimination]\\label{thm:exact_local_elimination}\nAfter the neighborhoods are chosen small enough, the equation\n\\[\nF(u,v)=0\n\\]\nis equivalent to\n\\[\nv=\\varphi(u)\n\\qquad\\text{and}\\qquad\nR(u)=0.\n\\]\nThus eliminating the hidden variable loses no local solutions.\n\\end{theorem}","\\begin{proof}\nThe hidden equation $F_Q(u,v)=0$ has the unique local solution $v=\\varphi(u)$ by the implicit function theorem.  After this identity is imposed, the remaining component of $F$ is exactly $R(u)$.\n\\end{proof}","\\begin{theorem}[Nonlinear Schur derivative]\\label{thm:schur_derivative}\nAt $(u_0,v_0)$,\n\\[\nDR\n=\nD_uF_P\n-\nD_vF_P A^{-1}D_uF_Q.\n\\]\n\\end{theorem}","\\begin{proof}\nDifferentiate $F_Q(u,\\varphi(u))=0$:\n\\[\nD_uF_Q+A D\\varphi=0.\n\\]\nThus\n\\[\nD\\varphi=-A^{-1}D_uF_Q.\n\\]\nSubstitute this into\n\\[\nDR=D_uF_P+D_vF_PD\\varphi.\n\\]\n\\end{proof}","\\begin{theorem}[Second Schur derivative]\\label{thm:schur_second_derivative}\nFor retained directions $h,k$, set\n\\[\n\\widehat h=(h,D\\varphi h),\n\\qquad\n\\widehat k=(k,D\\varphi k).\n\\]\nThen\n\\[\nD^2\\varphi[h,k]\n=\n-A^{-1}D^2F_Q[\\widehat h,\\widehat k]\n\\]\nand\n\\[\nD^2R[h,k]\n=\nD^2F_P[\\widehat h,\\widehat k]\n-\nD_vF_P A^{-1}D^2F_Q[\\widehat h,\\widehat k].\n\\]\nAll derivatives on the right are evaluated at $(u_0,v_0)$.\n\\end{theorem}","\\begin{proof}\nDifferentiate the identity $F_Q(u,\\varphi(u))=0$ twice.  The chain rule gives\n\\[\nD^2F_Q[\\widehat h,\\widehat k]\n+A D^2\\varphi[h,k]\n=0.\n\\]\nThis proves the first formula.  Differentiate $R(u)=F_P(u,\\varphi(u))$ twice and substitute the first formula.\n\\end{proof}","\\begin{corollary}[Packet jets of the reduced equation]\\label{cor:reduced_packet_jets}\nIf the retained and hidden spaces carry packet systems, then the first and second packet jets of $R$ are obtained by projecting the formulas in \\cref{thm:schur_derivative,thm:schur_second_derivative}.  Hidden packet feedback is carried by the factors involving $A^{-1}$.\n\\end{corollary}","\\begin{example}[Two retained variables and one hidden variable]\\label{ex:finite_schur}\nSuppose the derivative of $F$ at a solution has block form\n\\[\nDF=\n\\begin{pmatrix}\nA&B","\\begin{example}[Two retained variables and one hidden variable]\\label{ex:finite_schur}\nSuppose the derivative of $F$ at a solution has block form\n\\[\nDF=\n\\begin{pmatrix}\nA&B\\\\\nC&D\n\\end{pmatrix},\n\\]\nwhere $D$ is invertible.  Eliminating the hidden variable replaces the retained derivative $A$ by\n\\[\nA-BD^{-1}C.\n\\]\nThe feedback term is exact.  Keeping only $A$ would ignore the path from retained variables into the hidden variable and back.\n\\end{example}","C&D\n\\end{pmatrix},\n\\]\nwhere $D$ is invertible.  Eliminating the hidden variable replaces the retained derivative $A$ by\n\\[\nA-BD^{-1}C.\n\\]\nThe feedback term is exact.  Keeping only $A$ would ignore the path from retained variables into the hidden variable and back.\n\\end{example}","Let $\\Gamma=(V,E)$ be a finite connected graph and fix a vector space $X$.  At every vertex take one labeled copy of $X$.  Every oriented edge $u\\to v$ carries an invertible transport\n\\[\nT_{vu}\\in\\operatorname{GL}(X),\n\\qquad\nT_{uv}=T_{vu}^{-1}.\n\\]\nEquivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber.  The matrices below are then the transports in those frames.  For a path $\\gamma=(v_0,\\ldots,v_r)$, set\n\\[\nT_\\gamma\n=\nT_{v_rv_{r-1}}\\cdots T_{v_1v_0}.\n\\]\nFor a loop, this product is its holonomy. In the based loop group, $\\gamma_2\\gamma_1$ means first traverse $\\gamma_1$ and then $\\gamma_2$, so $T_{\\gamma_2\\gamma_1}=T_{\\gamma_2}T_{\\gamma_1}$.  Standard graph t","Let $\\Gamma=(V,E)$ be a finite connected graph and fix a vector space $X$.  At every vertex take one labeled copy of $X$.  Every oriented edge $u\\to v$ carries an invertible transport\n\\[\nT_{vu}\\in\\operatorname{GL}(X),\n\\qquad\nT_{uv}=T_{vu}^{-1}.\n\\]\nEquivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber.  The matrices below are then the transports in those frames.  For a path $\\gamma=(v_0,\\ldots,v_r)$, set\n\\[\nT_\\gamma\n=\nT_{v_rv_{r-1}}\\cdots T_{v_1v_0}.\n\\]\nFor a loop, this product is its holonomy. In the based loop group, $\\gamma_2\\gamma_1$ means first traverse $\\gamma_1$ and then $\\gamma_2$, so $T_{\\gamma_2\\gamma_1}=T_{\\gamma_2}T_{\\gamma_1}$.  Standard graph terminology may be found in \\cite{GrossTucker1987}.","e connected graph and fix a vector space $X$.  At every vertex take one labeled copy of $X$.  Every oriented edge $u\\to v$ carries an invertible transport\n\\[\nT_{vu}\\in\\operatorname{GL}(X),\n\\qquad\nT_{uv}=T_{vu}^{-1}.\n\\]\nEquivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber.  The matrices below are then the transports in those frames.  For a path $\\gamma=(v_0,\\ldots,v_r)$, set\n\\[\nT_\\gamma\n=\nT_{v_rv_{r-1}}\\cdots T_{v_1v_0}.\n\\]\nFor a loop, this product is its holonomy. In the based loop group, $\\gamma_2\\gamma_1$ means first traverse $\\gamma_1$ and then $\\gamma_2$, so $T_{\\gamma_2\\gamma_1}=T_{\\gamma_2}T_{\\gamma_1}$.  Standard graph terminology may be found in \\cite{GrossTucker1987}.","or space $X$.  At every vertex take one labeled copy of $X$.  Every oriented edge $u\\to v$ carries an invertible transport\n\\[\nT_{vu}\\in\\operatorname{GL}(X),\n\\qquad\nT_{uv}=T_{vu}^{-1}.\n\\]\nEquivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber.  The matrices below are then the transports in those frames.  For a path $\\gamma=(v_0,\\ldots,v_r)$, set\n\\[\nT_\\gamma\n=\nT_{v_rv_{r-1}}\\cdots T_{v_1v_0}.\n\\]\nFor a loop, this product is its holonomy. In the based loop group, $\\gamma_2\\gamma_1$ means first traverse $\\gamma_1$ and then $\\gamma_2$, so $T_{\\gamma_2\\gamma_1}=T_{\\gamma_2}T_{\\gamma_1}$.  Standard graph terminology may be found in \\cite{GrossTucker1987}.","nsport\n\\[\nT_{vu}\\in\\operatorname{GL}(X),\n\\qquad\nT_{uv}=T_{vu}^{-1}.\n\\]\nEquivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber.  The matrices below are then the transports in those frames.  For a path $\\gamma=(v_0,\\ldots,v_r)$, set\n\\[\nT_\\gamma\n=\nT_{v_rv_{r-1}}\\cdots T_{v_1v_0}.\n\\]\nFor a loop, this product is its holonomy. In the based loop group, $\\gamma_2\\gamma_1$ means first traverse $\\gamma_1$ and then $\\gamma_2$, so $T_{\\gamma_2\\gamma_1}=T_{\\gamma_2}T_{\\gamma_1}$.  Standard graph terminology may be found in \\cite{GrossTucker1987}.","\\qquad\nT_{uv}=T_{vu}^{-1}.\n\\]\nEquivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber.  The matrices below are then the transports in those frames.  For a path $\\gamma=(v_0,\\ldots,v_r)$, set\n\\[\nT_\\gamma\n=\nT_{v_rv_{r-1}}\\cdots T_{v_1v_0}.\n\\]\nFor a loop, this product is its holonomy. In the based loop group, $\\gamma_2\\gamma_1$ means first traverse $\\gamma_1$ and then $\\gamma_2$, so $T_{\\gamma_2\\gamma_1}=T_{\\gamma_2}T_{\\gamma_1}$.  Standard graph terminology may be found in \\cite{GrossTucker1987}.","}^{-1}.\n\\]\nEquivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber.  The matrices below are then the transports in those frames.  For a path $\\gamma=(v_0,\\ldots,v_r)$, set\n\\[\nT_\\gamma\n=\nT_{v_rv_{r-1}}\\cdots T_{v_1v_0}.\n\\]\nFor a loop, this product is its holonomy. In the based loop group, $\\gamma_2\\gamma_1$ means first traverse $\\gamma_1$ and then $\\gamma_2$, so $T_{\\gamma_2\\gamma_1}=T_{\\gamma_2}T_{\\gamma_1}$.  Standard graph terminology may be found in \\cite{GrossTucker1987}.","valently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber.  The matrices below are then the transports in those frames.  For a path $\\gamma=(v_0,\\ldots,v_r)$, set\n\\[\nT_\\gamma\n=\nT_{v_rv_{r-1}}\\cdots T_{v_1v_0}.\n\\]\nFor a loop, this product is its holonomy. In the based loop group, $\\gamma_2\\gamma_1$ means first traverse $\\gamma_1$ and then $\\gamma_2$, so $T_{\\gamma_2\\gamma_1}=T_{\\gamma_2}T_{\\gamma_1}$.  Standard graph terminology may be found in \\cite{GrossTucker1987}.","\\begin{definition}[Gauge change]\\label{def:gauge_change}\nA gauge is a family $G_v\\in\\operatorname{GL}(X)$.  It changes the edge transports to\n\\[\nT'_{vu}=G_vT_{vu}G_u^{-1}.\n\\]\n\\end{definition}","\\begin{proposition}[Gauge covariance]\\label{prop:gauge_covariance}\nFor a loop $\\gamma$ based at $v_0$,\n\\[\n\\Hol'(\\gamma)\n=\nG_{v_0}\\Hol(\\gamma)G_{v_0}^{-1}.\n\\]\nThus the conjugacy class of each loop holonomy is gauge invariant.\n\\end{proposition}","\\begin{theorem}[Spanning tree normalization]\\label{thm:tree_normalization}\nChoose a root and a spanning tree $\\Tcal$ of $\\Gamma$.  There is a gauge in which every tree edge transport is the identity.  In that gauge, the transport on each edge outside the tree is the holonomy of its fundamental cycle, with the orientation used to define that cycle.\n\\end{theorem}","\\begin{corollary}[Number of independent cycle tests]\\label{cor:cycle_number}\nFor a connected graph with $|V|$ vertices and $|E|$ unoriented edges, a spanning tree leaves\n\\[\n|E|-|V|+1\n\\]\nfundamental cycle holonomies.\n\\end{corollary}","\\begin{proof}\nA spanning tree has $|V|-1$ edges.  Every remaining edge creates one fundamental cycle.\n\\end{proof}","Let $A$ be a transport system on copies of a fixed vector space $V_0$, and let $B$ be a transport system on copies of a fixed vector space $W_0$.  Fix a root $r$ and an isomorphism\n\\[\nU_r:V_0\\longrightarrow W_0.\n\\]","\\begin{theorem}[Fundamental holonomy intertwining criterion]\\label{thm:holonomy_intertwining}\nThe following conditions are equivalent.\n\\begin{enumerate}\n\\item There is a family of isomorphisms $U_v:V_0\\to W_0$ with the prescribed root value $U_r$ and satisfying\n\\[\nU_vA_{vu}=B_{vu}U_u\n\\]\nfor every oriented edge $u\\to v$.\n\\item For every fundamental loop $\\gamma$ based at $r$,\n\\[\nU_r\\Hol_A(\\gamma)=\\Hol_B(\\gamma)U_r.\n\\]\n\\item The same identity holds for every loop based at $r$.\n\\end{enumerate}\nWhen these conditions hold, the family $(U_v)$ is uniquely determined by $U_r$.\n\\end{theorem}","omy_intertwining}\nThe following conditions are equivalent.\n\\begin{enumerate}\n\\item There is a family of isomorphisms $U_v:V_0\\to W_0$ with the prescribed root value $U_r$ and satisfying\n\\[\nU_vA_{vu}=B_{vu}U_u\n\\]\nfor every oriented edge $u\\to v$.\n\\item For every fundamental loop $\\gamma$ based at $r$,\n\\[\nU_r\\Hol_A(\\gamma)=\\Hol_B(\\gamma)U_r.\n\\]\n\\item The same identity holds for every loop based at $r$.\n\\end{enumerate}\nWhen these conditions hold, the family $(U_v)$ is uniquely determined by $U_r$.\n\\end{theorem}","begin{enumerate}\n\\item There is a family of isomorphisms $U_v:V_0\\to W_0$ with the prescribed root value $U_r$ and satisfying\n\\[\nU_vA_{vu}=B_{vu}U_u\n\\]\nfor every oriented edge $u\\to v$.\n\\item For every fundamental loop $\\gamma$ based at $r$,\n\\[\nU_r\\Hol_A(\\gamma)=\\Hol_B(\\gamma)U_r.\n\\]\n\\item The same identity holds for every loop based at $r$.\n\\end{enumerate}\nWhen these conditions hold, the family $(U_v)$ is uniquely determined by $U_r$.\n\\end{theorem}","family of isomorphisms $U_v:V_0\\to W_0$ with the prescribed root value $U_r$ and satisfying\n\\[\nU_vA_{vu}=B_{vu}U_u\n\\]\nfor every oriented edge $u\\to v$.\n\\item For every fundamental loop $\\gamma$ based at $r$,\n\\[\nU_r\\Hol_A(\\gamma)=\\Hol_B(\\gamma)U_r.\n\\]\n\\item The same identity holds for every loop based at $r$.\n\\end{enumerate}\nWhen these conditions hold, the family $(U_v)$ is uniquely determined by $U_r$.\n\\end{theorem}","Conversely, choose a spanning tree.  Starting from $U_r$, propagate $U_v$ along the unique tree path by the rule\n\\[\nU_v=B_{vu}U_uA_{vu}^{-1}\n\\]\non every tree edge.  This is the only possible extension satisfying the tree edge equations.  An edge outside the tree closes one fundamental loop.  Its intertwining equation holds exactly when $U_r$ intertwines the two corresponding fundamental holonomies.  The second condition therefore supplies every remaining edge equation.  Uniqueness follows from propagation along the tree.\n\\end{proof}","\\begin{theorem}[Gauge classification of transports]\\label{thm:gauge_classification}\nLet $G\\subseteq\\operatorname{GL}(X)$ be the chosen structure group and fix a root $r$.  Restrict both edge transports and gauge values to $G$.  Gauge equivalence classes of $G$ valued transport systems are naturally identified with\n\\[\n\\Hom(\\pi_1(\\Gamma,r),G)/G,\n\\]\nwhere $G$ acts by conjugation.  After a spanning tree and orientations of the remaining edges are chosen, this set is identified with simultaneous conjugacy classes of\n\\[\nb_1=|E|-|V|+1\n\\]\ntuples in $G$.\n\\end{theorem}","$ be the chosen structure group and fix a root $r$.  Restrict both edge transports and gauge values to $G$.  Gauge equivalence classes of $G$ valued transport systems are naturally identified with\n\\[\n\\Hom(\\pi_1(\\Gamma,r),G)/G,\n\\]\nwhere $G$ acts by conjugation.  After a spanning tree and orientations of the remaining edges are chosen, this set is identified with simultaneous conjugacy classes of\n\\[\nb_1=|E|-|V|+1\n\\]\ntuples in $G$.\n\\end{theorem}","up and fix a root $r$.  Restrict both edge transports and gauge values to $G$.  Gauge equivalence classes of $G$ valued transport systems are naturally identified with\n\\[\n\\Hom(\\pi_1(\\Gamma,r),G)/G,\n\\]\nwhere $G$ acts by conjugation.  After a spanning tree and orientations of the remaining edges are chosen, this set is identified with simultaneous conjugacy classes of\n\\[\nb_1=|E|-|V|+1\n\\]\ntuples in $G$.\n\\end{theorem}","\\begin{proof}\nA transport system assigns a group element to every oriented edge, with inversion under reversal.  The product along a based loop gives a homomorphism from the free group $\\pi_1(\\Gamma,r)$ to $G$.  A gauge change conjugates this homomorphism by the gauge value at the root.  Conversely, normalize a transport system on a spanning tree.  The remaining edge values prescribe the images of a free basis of $\\pi_1(\\Gamma,r)$, and every homomorphism arises in this way.  A further gauge preserving the tree normalization is constant at all vertices and acts by simultaneous conjugation.  The number of non tree edges is $b_1$.\n\\end{proof}","ansport system assigns a group element to every oriented edge, with inversion under reversal.  The product along a based loop gives a homomorphism from the free group $\\pi_1(\\Gamma,r)$ to $G$.  A gauge change conjugates this homomorphism by the gauge value at the root.  Conversely, normalize a transport system on a spanning tree.  The remaining edge values prescribe the images of a free basis of $\\pi_1(\\Gamma,r)$, and every homomorphism arises in this way.  A further gauge preserving the tree normalization is constant at all vertices and acts by simultaneous conjugation.  The number of non tree edges is $b_1$.\n\\end{proof}","jugates this homomorphism by the gauge value at the root.  Conversely, normalize a transport system on a spanning tree.  The remaining edge values prescribe the images of a free basis of $\\pi_1(\\Gamma,r)$, and every homomorphism arises in this way.  A further gauge preserving the tree normalization is constant at all vertices and acts by simultaneous conjugation.  The number of non tree edges is $b_1$.\n\\end{proof}","\\begin{corollary}[Abelian cohomology class]\\label{cor:abelian_holonomy_class}\nWhen the structure group $A$ is abelian, gauge classes of transports are naturally\n\\[\nH^1(\\Gamma,A)\n\\cong\n\\Hom(H_1(\\Gamma,\\mathbb Z),A).\n\\]\nAfter a basis of $H_1(\\Gamma,\\mathbb Z)$ is chosen, this group is identified with $A^{b_1}$.\n\\end{corollary}","\\begin{proof}\nAn edge assignment is an $A$ valued one cochain.  A gauge change multiplies it by a coboundary.  A graph has no two cells, so every one cochain is a cocycle.  The usual computation of the first cohomology of a connected graph gives $A^{b_1}$.\n\\end{proof}","Finite graph transport has a continuous counterpart.  Let $H$ be a Hilbert space and let\n\\[\n\\Pcal(t)=(P_\\alpha(t))_{\\alpha\\in I}\n\\]\nbe a complete orthogonal packet system with finitely many labels. Assume that it is continuously differentiable in operator norm for $t\\in[0,T]$, where $T\\ge0$. Here every $P_\\alpha(t)$ is a bounded self adjoint projection, so $P_\\alpha(t)^*=P_\\alpha(t)=P_\\alpha(t)^2$.","Zero projections are permitted in this section and contribute zero to all generator sums. When $T=0$, every partition product is the empty product $I$. Completeness and orthogonality give\n\\[\nP_\\alpha(t)P_\\beta(t)=0\n\\quad\\text{for }\\alpha\\ne\\beta,\n\\qquad\n\\sum_\\alpha P_\\alpha(t)=I.\n\\]\nDefine the canonical generator\n\\[\nK(t)=\\sum_\\alpha \\dot P_\\alpha(t)P_\\alpha(t).\n\\]","\\begingroup\n\\setlength{\\abovedisplayskip}{6pt plus 2pt minus 2pt}\n\\setlength{\\belowdisplayskip}{6pt plus 2pt minus 2pt}\n\\begin{lemma}[Complete packet commutator identity]\\label{lem:packet_commutator}\nFor every packet label $\\beta$,\n\\[\n\\dot P_\\beta(t)=[K(t),P_\\beta(t)].\n\\]\nThe generator is also skew adjoint:\n\\[\nK(t)^*=-K(t).\n\\]\n\\end{lemma}","\\begin{proof}\nDifferentiating $P_\\alpha^2=P_\\alpha$ gives\n\\[\nP_\\alpha\\dot P_\\alpha P_\\alpha=0.\n\\]\nDifferentiating $P_\\beta P_\\alpha=0$ for $\\alpha\\ne\\beta$ gives\n\\[\nP_\\beta\\dot P_\\alpha=-\\dot P_\\beta P_\\alpha.\n\\]\nHence\n\\[\nK P_\\beta=\\dot P_\\beta P_\\beta\n\\]\nand\n\\[\nP_\\beta K=-\\dot P_\\beta(I-P_\\beta).\n\\]\nTheir difference is $\\dot P_\\beta$.  Also,\n\\[\nK+K^*\n=\n\\sum_\\alpha(\\dot P_\\alpha P_\\alpha+P_\\alpha\\dot P_\\alpha)\n=\n\\sum_\\alpha\\dot P_\\alpha\n=0.\n\\]\n\\end{proof}\n\\endgroup","\\begin{theorem}[Canonical packet transport]\\label{thm:kato_transport}\nLet $U(t)$ solve\n\\[\n\\dot U(t)=K(t)U(t),\n\\qquad\nU(0)=I.\n\\]\nThen $U(t)$ is unitary and transports every packet exactly:\n\\[\nP_\\alpha(t)U(t)=U(t)P_\\alpha(0)\n\\]\nfor all $\\alpha$.\n\\end{theorem}","\\begin{proof}\nThe continuous bounded generator has an evolution inverse. Indeed, solve $\\dot V=-VK$ with $V(0)=I$. Differentiation gives $(VU)^{\\prime}=0$, hence $VU=I$. The product $W=UV$ satisfies $\\dot W=KW-WK$ and $W(0)=I$; uniqueness of this bounded linear equation gives $UV=I$. Thus $U$ is invertible.","Skew adjointness of $K(t)$ gives\n\\[\n\\frac{d}{dt}(U(t)^*U(t))=0,\n\\]\nso $U(t)$ is unitary.  By \\cref{lem:packet_commutator},\n\\[\n\\frac{d}{dt}\\big(U(t)^{-1}P_\\alpha(t)U(t)\\big)\n=\nU(t)^{-1}\\big(\\dot P_\\alpha-[K,P_\\alpha]\\big)U(t)\n=0.\n\\]\nThe expression equals $P_\\alpha(0)$ at time zero.\n\\end{proof}","\\begin{corollary}[One moving subspace]\\label{cor:one_moving_subspace}\nFor one orthogonal projection $P(t)$ and its complement, the canonical generator is\n\\[\nK(t)=[\\dot P(t),P(t)].\n\\]\nThe transport in \\cref{thm:kato_transport} maps $\\im P(0)$ unitarily onto $\\im P(t)$.\n\\end{corollary}","\\begin{proof}\nApply the complete packet formula to $P(t)$ and $I-P(t)$, then simplify the sum.\n\\end{proof}","\\begin{definition}[Direct rotation of nearby packet systems]\\label{def:packet_direct_rotation}\nLet $(P_\\alpha)_{\\alpha\\in I}$ and $(Q_\\alpha)_{\\alpha\\in I}$ be complete orthogonal packet systems on the same Hilbert space.  Put\n\\[\nS(\\Qcal,\\Pcal)=\\sum_{\\alpha\\in I}Q_\\alpha P_\\alpha.\n\\]\nWhen $S(\\Qcal,\\Pcal)$ is invertible, define\n\\[\n\\Ucal(\\Qcal,\\Pcal)\n=\nS(\\Qcal,\\Pcal)\n\\big(S(\\Qcal,\\Pcal)^*S(\\Qcal,\\Pcal)\\big)^{-1/2}.\n\\]\n\\end{definition}","\\begin{proposition}[Exact direct packet rotation]\\label{prop:packet_direct_rotation}\nIf\n\\[\n\\max\\bigl(\\{0\\}\\cup\\{\\|P_\\alpha-Q_\\alpha\\|:\\alpha\\in I\\}\\bigr)<1,\n\\]\nthen $S(\\Qcal,\\Pcal)$ is invertible.  The operator $\\Ucal(\\Qcal,\\Pcal)$ is unitary and satisfies\n\\[\nQ_\\alpha\\Ucal(\\Qcal,\\Pcal)\n=\n\\Ucal(\\Qcal,\\Pcal)P_\\alpha\n\\]\nfor every $\\alpha$.\n\\end{proposition}","\\begin{proof}\nThe identities\n\\[\nQ_\\alpha S(\\Qcal,\\Pcal)\n=\nS(\\Qcal,\\Pcal)P_\\alpha\n=\nQ_\\alpha P_\\alpha\n\\]\nfollow from pairwise orthogonality.  Also\n\\[\nS^*S\n=\n\\sum_\\alpha P_\\alpha Q_\\alpha P_\\alpha,\n\\qquad\nSS^*\n=\n\\sum_\\alpha Q_\\alpha P_\\alpha Q_\\alpha.\n\\]\nPut $\\delta=\\max(\\{0\\}\\cup\\{\\|P_\\alpha-Q_\\alpha\\|:\\alpha\\in I\\})$.  On $P_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}",")\n=\nS(\\Qcal,\\Pcal)P_\\alpha\n=\nQ_\\alpha P_\\alpha\n\\]\nfollow from pairwise orthogonality.  Also\n\\[\nS^*S\n=\n\\sum_\\alpha P_\\alpha Q_\\alpha P_\\alpha,\n\\qquad\nSS^*\n=\n\\sum_\\alpha Q_\\alpha P_\\alpha Q_\\alpha.\n\\]\nPut $\\delta=\\max(\\{0\\}\\cup\\{\\|P_\\alpha-Q_\\alpha\\|:\\alpha\\in I\\})$.  On $P_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","a P_\\alpha\n\\]\nfollow from pairwise orthogonality.  Also\n\\[\nS^*S\n=\n\\sum_\\alpha P_\\alpha Q_\\alpha P_\\alpha,\n\\qquad\nSS^*\n=\n\\sum_\\alpha Q_\\alpha P_\\alpha Q_\\alpha.\n\\]\nPut $\\delta=\\max(\\{0\\}\\cup\\{\\|P_\\alpha-Q_\\alpha\\|:\\alpha\\in I\\})$.  On $P_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","m pairwise orthogonality.  Also\n\\[\nS^*S\n=\n\\sum_\\alpha P_\\alpha Q_\\alpha P_\\alpha,\n\\qquad\nSS^*\n=\n\\sum_\\alpha Q_\\alpha P_\\alpha Q_\\alpha.\n\\]\nPut $\\delta=\\max(\\{0\\}\\cup\\{\\|P_\\alpha-Q_\\alpha\\|:\\alpha\\in I\\})$.  On $P_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","\\sum_\\alpha P_\\alpha Q_\\alpha P_\\alpha,\n\\qquad\nSS^*\n=\n\\sum_\\alpha Q_\\alpha P_\\alpha Q_\\alpha.\n\\]\nPut $\\delta=\\max(\\{0\\}\\cup\\{\\|P_\\alpha-Q_\\alpha\\|:\\alpha\\in I\\})$.  On $P_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","a P_\\alpha Q_\\alpha P_\\alpha,\n\\qquad\nSS^*\n=\n\\sum_\\alpha Q_\\alpha P_\\alpha Q_\\alpha.\n\\]\nPut $\\delta=\\max(\\{0\\}\\cup\\{\\|P_\\alpha-Q_\\alpha\\|:\\alpha\\in I\\})$.  On $P_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","*\n=\n\\sum_\\alpha Q_\\alpha P_\\alpha Q_\\alpha.\n\\]\nPut $\\delta=\\max(\\{0\\}\\cup\\{\\|P_\\alpha-Q_\\alpha\\|:\\alpha\\in I\\})$.  On $P_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","$\\delta=\\max(\\{0\\}\\cup\\{\\|P_\\alpha-Q_\\alpha\\|:\\alpha\\in I\\})$.  On $P_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","a\\|:\\alpha\\in I\\})$.  On $P_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","_\\alpha H$,\n\\[\nP_\\alpha Q_\\alpha P_\\alpha\n=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","=\nP_\\alpha-P_\\alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","alpha(P_\\alpha-Q_\\alpha)P_\\alpha,\n\\]\nso this block is bounded below by $1-\\delta>0$.  The same argument on $Q_\\alpha H$ applies to $Q_\\alpha P_\\alpha Q_\\alpha$.  Thus both $S^*S$ and $SS^*$ are invertible.  It follows that $S$ is injective and surjective.  The polar factor $\\Ucal=S(S^*S)^{-1/2}$ is unitary.  Since $S^*S$ commutes with every $P_\\alpha$, the intertwining identity for $S$ passes to $\\Ucal$.\n\\end{proof}","\\begin{lemma}[One step direct rotation estimate]\\label{lem:one_step_direct_rotation}\nAssume $\\Pcal(t)$ is twice continuously differentiable in operator norm on $[0,T]$.  There are constants $h_0>0$ and $C>0$ such that, whenever $t,t+h\\in[0,T]$ and $|h|\\le h_0$,\n\\[\n\\left\\|\n\\Ucal(\\Pcal(t+h),\\Pcal(t))\n-\nI\n-\nhK(t)\n\\right\\|\n\\le\nCh^2.\n\\]\nThe constants can be chosen uniformly in $t$.\n\\end{lemma}","\\begin{proof}\nTaylor's theorem in the Banach space of bounded operators gives\n\\[\nP_\\alpha(t+h)\n=\nP_\\alpha(t)+h\\dot P_\\alpha(t)+E_{\\alpha}(t,h),\n\\qquad\n\\|E_{\\alpha}(t,h)\\|\\le C_1h^2,\n\\]\nuniformly in $t$ and $\\alpha$.  Hence\n\\[\nS(\\Pcal(t+h),\\Pcal(t))\n=\nI+hK(t)+E(t,h),\n\\qquad\n\\|E(t,h)\\|\\le C_2h^2.\n\\]\nSince $K(t)^*=-K(t)$,\n\\[\nS^*S=I+E_2(t,h),\n\\qquad\n\\|E_2(t,h)\\|\\le C_3h^2.\n\\]\nFor small $h$, the spectrum of $S^*S$ lies in a fixed compact subinterval of $(0,\\infty)$.  The functional calculus for $z^{-1/2}$ gives\n\\[\n(S^*S)^{-1/2}=I+E_3(t,h),\n\\qquad\n\\|E_3(t,h)\\|\\le C_4h^2.\n\\]\nMultiplying the last two expansions proves the estimate.\n\\end{proof}","proof}\nTaylor's theorem in the Banach space of bounded operators gives\n\\[\nP_\\alpha(t+h)\n=\nP_\\alpha(t)+h\\dot P_\\alpha(t)+E_{\\alpha}(t,h),\n\\qquad\n\\|E_{\\alpha}(t,h)\\|\\le C_1h^2,\n\\]\nuniformly in $t$ and $\\alpha$.  Hence\n\\[\nS(\\Pcal(t+h),\\Pcal(t))\n=\nI+hK(t)+E(t,h),\n\\qquad\n\\|E(t,h)\\|\\le C_2h^2.\n\\]\nSince $K(t)^*=-K(t)$,\n\\[\nS^*S=I+E_2(t,h),\n\\qquad\n\\|E_2(t,h)\\|\\le C_3h^2.\n\\]\nFor small $h$, the spectrum of $S^*S$ lies in a fixed compact subinterval of $(0,\\infty)$.  The functional calculus for $z^{-1/2}$ gives\n\\[\n(S^*S)^{-1/2}=I+E_3(t,h),\n\\qquad\n\\|E_3(t,h)\\|\\le C_4h^2.\n\\]\nMultiplying the last two expansions proves the estimate.\n\\end{proof}","bounded operators gives\n\\[\nP_\\alpha(t+h)\n=\nP_\\alpha(t)+h\\dot P_\\alpha(t)+E_{\\alpha}(t,h),\n\\qquad\n\\|E_{\\alpha}(t,h)\\|\\le C_1h^2,\n\\]\nuniformly in $t$ and $\\alpha$.  Hence\n\\[\nS(\\Pcal(t+h),\\Pcal(t))\n=\nI+hK(t)+E(t,h),\n\\qquad\n\\|E(t,h)\\|\\le C_2h^2.\n\\]\nSince $K(t)^*=-K(t)$,\n\\[\nS^*S=I+E_2(t,h),\n\\qquad\n\\|E_2(t,h)\\|\\le C_3h^2.\n\\]\nFor small $h$, the spectrum of $S^*S$ lies in a fixed compact subinterval of $(0,\\infty)$.  The functional calculus for $z^{-1/2}$ gives\n\\[\n(S^*S)^{-1/2}=I+E_3(t,h),\n\\qquad\n\\|E_3(t,h)\\|\\le C_4h^2.\n\\]\nMultiplying the last two expansions proves the estimate.\n\\end{proof}","P_\\alpha(t)+h\\dot P_\\alpha(t)+E_{\\alpha}(t,h),\n\\qquad\n\\|E_{\\alpha}(t,h)\\|\\le C_1h^2,\n\\]\nuniformly in $t$ and $\\alpha$.  Hence\n\\[\nS(\\Pcal(t+h),\\Pcal(t))\n=\nI+hK(t)+E(t,h),\n\\qquad\n\\|E(t,h)\\|\\le C_2h^2.\n\\]\nSince $K(t)^*=-K(t)$,\n\\[\nS^*S=I+E_2(t,h),\n\\qquad\n\\|E_2(t,h)\\|\\le C_3h^2.\n\\]\nFor small $h$, the spectrum of $S^*S$ lies in a fixed compact subinterval of $(0,\\infty)$.  The functional calculus for $z^{-1/2}$ gives\n\\[\n(S^*S)^{-1/2}=I+E_3(t,h),\n\\qquad\n\\|E_3(t,h)\\|\\le C_4h^2.\n\\]\nMultiplying the last two expansions proves the estimate.\n\\end{proof}","P_\\alpha(t)+E_{\\alpha}(t,h),\n\\qquad\n\\|E_{\\alpha}(t,h)\\|\\le C_1h^2,\n\\]\nuniformly in $t$ and $\\alpha$.  Hence\n\\[\nS(\\Pcal(t+h),\\Pcal(t))\n=\nI+hK(t)+E(t,h),\n\\qquad\n\\|E(t,h)\\|\\le C_2h^2.\n\\]\nSince $K(t)^*=-K(t)$,\n\\[\nS^*S=I+E_2(t,h),\n\\qquad\n\\|E_2(t,h)\\|\\le C_3h^2.\n\\]\nFor small $h$, the spectrum of $S^*S$ lies in a fixed compact subinterval of $(0,\\infty)$.  The functional calculus for $z^{-1/2}$ gives\n\\[\n(S^*S)^{-1/2}=I+E_3(t,h),\n\\qquad\n\\|E_3(t,h)\\|\\le C_4h^2.\n\\]\nMultiplying the last two expansions proves the estimate.\n\\end{proof}","\\begin{theorem}[Direct rotations converge to packet transport]\\label{thm:direct_rotation_convergence}\nAssume the complete packet family $\\Pcal(t)$ is twice continuously differentiable in operator norm on $[0,T]$.  For a partition\n\\[\n0=t_0<t_1<\\cdots<t_N=T\n\\]\nwith sufficiently small mesh, define\n\\[\nV_\\pi\n=\n\\Ucal(\\Pcal(t_N),\\Pcal(t_{N-1}))\n\\cdots\n\\Ucal(\\Pcal(t_1),\\Pcal(t_0)).\n\\]\nThen\n\\[\nV_\\pi\\longrightarrow U(T)\n\\]\nin operator norm, where $U$ is the canonical transport from \\cref{thm:kato_transport}.  On a fixed compact interval the error is bounded by a constant times the mesh.\n\\end{theorem}","to packet transport]\\label{thm:direct_rotation_convergence}\nAssume the complete packet family $\\Pcal(t)$ is twice continuously differentiable in operator norm on $[0,T]$.  For a partition\n\\[\n0=t_0<t_1<\\cdots<t_N=T\n\\]\nwith sufficiently small mesh, define\n\\[\nV_\\pi\n=\n\\Ucal(\\Pcal(t_N),\\Pcal(t_{N-1}))\n\\cdots\n\\Ucal(\\Pcal(t_1),\\Pcal(t_0)).\n\\]\nThen\n\\[\nV_\\pi\\longrightarrow U(T)\n\\]\nin operator norm, where $U$ is the canonical transport from \\cref{thm:kato_transport}.  On a fixed compact interval the error is bounded by a constant times the mesh.\n\\end{theorem}","\\begin{proof}\nLet $U(t,s)$ be the exact propagator generated by $K$.  Since $K$ is continuously differentiable, the integral equation for $U(t+h,t)$ gives a uniform estimate\n\\[\n\\|U(t+h,t)-I-hK(t)\\|\n\\le\nC_1h^2.\n\\]\nBy \\cref{lem:one_step_direct_rotation}, the direct rotation $V_j$ from $t_j$ to $t_{j+1}$ therefore satisfies\n\\[\n\\|V_j-U(t_{j+1},t_j)\\|\n\\le\nC_2h_j^2,\n\\qquad\nh_j=t_{j+1}-t_j.\n\\]\nBoth $V_j$ and the exact propagator are unitary.  A telescoping product comparison gives\n\\[\n\\|V_\\pi-U(T,0)\\|\n\\le\n\\sum_j\\|V_j-U(t_{j+1},t_j)\\|\n\\le\nC_2\\sum_jh_j^2\n\\le\nC_2T|\\pi|.\n\\]\nThis proves norm convergence and the stated first order global bound.\n\\end{proof}","s continuously differentiable, the integral equation for $U(t+h,t)$ gives a uniform estimate\n\\[\n\\|U(t+h,t)-I-hK(t)\\|\n\\le\nC_1h^2.\n\\]\nBy \\cref{lem:one_step_direct_rotation}, the direct rotation $V_j$ from $t_j$ to $t_{j+1}$ therefore satisfies\n\\[\n\\|V_j-U(t_{j+1},t_j)\\|\n\\le\nC_2h_j^2,\n\\qquad\nh_j=t_{j+1}-t_j.\n\\]\nBoth $V_j$ and the exact propagator are unitary.  A telescoping product comparison gives\n\\[\n\\|V_\\pi-U(T,0)\\|\n\\le\n\\sum_j\\|V_j-U(t_{j+1},t_j)\\|\n\\le\nC_2\\sum_jh_j^2\n\\le\nC_2T|\\pi|.\n\\]\nThis proves norm convergence and the stated first order global bound.\n\\end{proof}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.96\\linewidth]{rotating_projector_transport.pdf}\n\\caption{A rotating rank one packet and its orthogonal complement. With $u(\\theta)=(\\cos\\theta,\\sin\\theta)$ and $P(\\theta)=u(\\theta)u(\\theta)^{\\mathsf T}$, the canonical generator is $[P',P]=\\bigl(\\begin{smallmatrix}0&-1","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.96\\linewidth]{rotating_projector_transport.pdf}\n\\caption{A rotating rank one packet and its orthogonal complement. With $u(\\theta)=(\\cos\\theta,\\sin\\theta)$ and $P(\\theta)=u(\\theta)u(\\theta)^{\\mathsf T}$, the canonical generator is $[P',P]=\\bigl(\\begin{smallmatrix}0&-1\\\\1&0\\end{smallmatrix}\\bigr)$. For a positive step $\\delta<\\pi/2$, the polar unitary is the usual rotation $R(\\delta)$. On each displayed mesh, the direct rotation product equals $R(\\pi/3)$ exactly. This special planar example illustrates the construction. The general theorem supplies convergence for families whose step products need not already equal the continuous transport.}\n\\label{fig:rotating-projector-example}\n\\end{figure}\n\\clearpage","1&0\\end{smallmatrix}\\bigr)$. For a positive step $\\delta<\\pi/2$, the polar unitary is the usual rotation $R(\\delta)$. On each displayed mesh, the direct rotation product equals $R(\\pi/3)$ exactly. This special planar example illustrates the construction. The general theorem supplies convergence for families whose step products need not already equal the continuous transport.}\n\\label{fig:rotating-projector-example}\n\\end{figure}\n\\clearpage","\\begin{theorem}[Convergence of sampled transports]\\label{thm:sampled_transport_convergence}\nAssume $K$ is Lipschitz on $[0,T]$.  For a partition\n\\[\n0=t_0<t_1<\\cdots<t_N=T,\n\\]\ndefine\n\\[\nU_\\pi\n=\n\\exp\\big((t_N-t_{N-1})K(t_{N-1})\\big)\n\\cdots\n\\exp\\big((t_1-t_0)K(t_0)\\big).\n\\]\nAs the mesh $|\\pi|$ tends to zero,\n\\[\nU_\\pi\\longrightarrow U(T)\n\\]\nin operator norm.  There is a constant $C$, depending only on $T$, the supremum norm of $K$, and its Lipschitz constant, such that\n\\[\n\\|U_\\pi-U(T)\\|\n\\le C|\\pi|.\n\\]\n\\end{theorem}","egin{theorem}[Convergence of sampled transports]\\label{thm:sampled_transport_convergence}\nAssume $K$ is Lipschitz on $[0,T]$.  For a partition\n\\[\n0=t_0<t_1<\\cdots<t_N=T,\n\\]\ndefine\n\\[\nU_\\pi\n=\n\\exp\\big((t_N-t_{N-1})K(t_{N-1})\\big)\n\\cdots\n\\exp\\big((t_1-t_0)K(t_0)\\big).\n\\]\nAs the mesh $|\\pi|$ tends to zero,\n\\[\nU_\\pi\\longrightarrow U(T)\n\\]\nin operator norm.  There is a constant $C$, depending only on $T$, the supremum norm of $K$, and its Lipschitz constant, such that\n\\[\n\\|U_\\pi-U(T)\\|\n\\le C|\\pi|.\n\\]\n\\end{theorem}","of sampled transports]\\label{thm:sampled_transport_convergence}\nAssume $K$ is Lipschitz on $[0,T]$.  For a partition\n\\[\n0=t_0<t_1<\\cdots<t_N=T,\n\\]\ndefine\n\\[\nU_\\pi\n=\n\\exp\\big((t_N-t_{N-1})K(t_{N-1})\\big)\n\\cdots\n\\exp\\big((t_1-t_0)K(t_0)\\big).\n\\]\nAs the mesh $|\\pi|$ tends to zero,\n\\[\nU_\\pi\\longrightarrow U(T)\n\\]\nin operator norm.  There is a constant $C$, depending only on $T$, the supremum norm of $K$, and its Lipschitz constant, such that\n\\[\n\\|U_\\pi-U(T)\\|\n\\le C|\\pi|.\n\\]\n\\end{theorem}","nsport_convergence}\nAssume $K$ is Lipschitz on $[0,T]$.  For a partition\n\\[\n0=t_0<t_1<\\cdots<t_N=T,\n\\]\ndefine\n\\[\nU_\\pi\n=\n\\exp\\big((t_N-t_{N-1})K(t_{N-1})\\big)\n\\cdots\n\\exp\\big((t_1-t_0)K(t_0)\\big).\n\\]\nAs the mesh $|\\pi|$ tends to zero,\n\\[\nU_\\pi\\longrightarrow U(T)\n\\]\nin operator norm.  There is a constant $C$, depending only on $T$, the supremum norm of $K$, and its Lipschitz constant, such that\n\\[\n\\|U_\\pi-U(T)\\|\n\\le C|\\pi|.\n\\]\n\\end{theorem}","\\begin{proof}\nOn each interval, compare the exact propagator with the frozen exponential.  Variation of constants and the Lipschitz bound give a local error bounded by a constant times the square of the interval length.  Every factor is unitary, so multiplication does not amplify norms.  Summing the local errors gives a bound by a constant times\n\\[\n\\sum_j(t_{j+1}-t_j)^2\n\\le\nT|\\pi|.\n\\]\n\\end{proof}","\\Needspace{10\\baselineskip}\n\\begin{corollary}[Continuous holonomy from graph products]\\label{cor:continuous_holonomy}\nAssume the complete packet family is twice continuously differentiable in operator norm on $[0,T]$. If $P_\\alpha(0)=P_\\alpha(T)$ for every $\\alpha$, then $U(T)$ preserves every initial packet.  Its restriction to each packet is the holonomy of the closed packet path.  The products in \\cref{thm:direct_rotation_convergence,thm:sampled_transport_convergence} converge to these holonomies.  The direct rotation product has the added property that every factor maps each sampled packet exactly to its successor.\n\\end{corollary}","\\begin{example}[A rotating line]\\label{ex:rotating_line}\nIn $\\mathbb R^2$, let\n\\[\nv(t)=(\\cos t,\\sin t)\n\\]\nand let $P(t)$ be the orthogonal projection onto the line spanned by $v(t)$.  The canonical transport is the ordinary rotation matrix\n\\[\nU(t)=\n\\begin{pmatrix}\n\\cos t&-\\sin t","\\begin{example}[A rotating line]\\label{ex:rotating_line}\nIn $\\mathbb R^2$, let\n\\[\nv(t)=(\\cos t,\\sin t)\n\\]\nand let $P(t)$ be the orthogonal projection onto the line spanned by $v(t)$.  The canonical transport is the ordinary rotation matrix\n\\[\nU(t)=\n\\begin{pmatrix}\n\\cos t&-\\sin t\\\\\n\\sin t&\\cos t\n\\end{pmatrix}.\n\\]\nIt carries the initial line and its orthogonal complement to the two moving packets.  Products of short rotations converge to the same transport.\n\\end{example}","\\begin{standardfact}[Curvature of a projected connection]\\label{fact:projected_curvature}\nLet $P$ be a smooth orthogonal projector valued map on a parameter manifold.  The connection\n\\[\n\\nabla=P\\,d\n\\]\non the range bundle has curvature\n\\[\n\\Fcal=P(dP\\wedge dP)P.\n\\]\n\\end{standardfact}","\\begin{proof}\nFor a section $s$ satisfying $Ps=s$,\n\\[\n\\nabla^2s=P\\,d(P\\,ds)=P(dP)\\wedge ds.\n\\]\nSince $ds=(dP)s+Pds$ and $P(dP)P=0$, the surviving term is\n\\[\nP(dP\\wedge dP)Ps.\\qedhere\n\\]\n\\end{proof}","\\vpage{2}{An atlas of the objects}{ORIENTATION / KEEP THE TYPES VISIBLE}{Traditional paper: Definition 2.1 (p. 2), Definition 5.5 (p. 11), Definition 7.4 (p. 16)}\nRows are targets and columns are sources throughout. If $E_{ji}$ is a matrix unit, then $E_{ji}e_i=e_j$. The notation $e_i$ means a basis vector. A named graph edge is written $a,b,c,d$ or $e,f$ when its role is explicit.","midrule\n$\\src,\\tgt:E\\to V$ & The ordered endpoints of each distinct edge, including parallel edges. & \\pagerefV{3} to \\pagerefV{4}","$L_e=c(e)E_{\\tgt(e),\\src(e)}$ & One operator supported on a packet. & \\pagerefV{3}","$L_w=L_{e_n}\\cdots L_{e_1}$ & Execute the written word from left to right. & \\pagerefV{4} to \\pagerefV{5}","$T=\\sum_eL_e$ & Add all edge operators before taking powers. & \\pagerefV{4} to \\pagerefV{7}","$F_a$ & A synthesis frame from chart $a$ into the ambient space. & \\pagerefV{9}","$\\mathcal A_a,\\mathcal S_a$ & Inverse analysis and synthesis maps, which may be maps of sets. & \\pagerefV{10}","$J_{b\\leftarrow a}$ & Canonical coordinate transport, forced by the exact charts. & \\pagerefV{9} to \\pagerefV{10}","$\\Gamma_{b\\leftarrow a}$ & Additional connection data, with reversibility stated explicitly. & \\pagerefV{11} to \\pagerefV{12}","$\\Delta_{c,b,a}$ & The difference between transport through two steps and direct transport, a map from $C_a$ to $C_c$. & \\pagerefV{12}","$H_{c,b,a}$ & Loop transport based at $a$, an automorphism of $C_a$. & \\pagerefV{13}","$G_a$ & An invertible gauge change at each chart. & \\pagerefV{14} to \\pagerefV{15}","$\\Pi_\\mu\\Ob\\Pi_\\nu$ & Block from input to output of $\\Ob=H-I$. & \\pagerefV{16} to \\pagerefV{18}","$r(x,p)$ & A residual at packet $p$ for the named candidate $x$. & \\pagerefV{19}","$t,\\operatorname{inputs}(t)$ & A finite operation tree and its syntactically used input labels. & \\pagerefV{20}","\\takeaway{A $C_a\\to C_c$ defect and a $C_a\\to C_a$ holonomy are not the same typed map. Returning the defect to the base fiber requires a specified inverse transport.}","\\vpage{3}{Read the product in execution order}{MODEL A / ONE EDGE IS ONE MATRIX ENTRY}{Traditional paper: Lemma 2.2 (p. 2), Theorem 2.3 (p. 2), Theorem 2.4 (p. 3)}\nWork over $\\ZZ$ with packets $0,1,2$. Let $e:0\\to1$ have coefficient $2$, and let $f:1\\to2$ have coefficient $3$. The elementary operators are\n\\[\nL_e=2E_{10},\\qquad L_f=3E_{21}.\n\\]\nThe location of the nonzero entry records the type of the transfer, not an arbitrary matrix convention.","\\vfig[.94]{edge_orientation}\n\\platecaption{Each grid is an actual $3\\times3$ matrix. The rightmost matrix factor acts first. In $L_fL_e$, packet $1$ is the shared intermediate index. In $L_eL_f$ the interface is broken.}","On a vector $(x_0,x_1,x_2)^\\mathsf T$, the first step produces $(0,2x_0,0)^\\mathsf T$, and the second produces $(0,0,6x_0)^\\mathsf T$. Thus\n\\[\nL_fL_e=6E_{20},\\qquad L_eL_f=0.\n\\]\nFor general edges, a summand in the entrywise product can survive only when the same intermediate packet is both $\\tgt(e)$ and $\\src(f)$:\n\\[\n(L_fL_e)_{yx}=\\sum_z(L_f)_{yz}(L_e)_{zx}.\n\\]\nThis is why a broken interface annihilates the entire product, even if both edge coefficients are nonzero.","\\takeaway{Write the traversed edges first. Then reverse their order when writing the operator product: $e$ followed by $f$ is $L_fL_e$.}","\\vpage{4}{All sixteen words, only five walks}{MODEL A / PARALLEL EDGES REMAIN DISTINCT}{Traditional paper: Theorem 3.8 (p. 6), Corollary 3.9 (p. 6)}\nAdd four edges on three packets: $a:0\\to1$ with coefficient $2$, $b:0\\to1$ with coefficient $-1$, $c:1\\to2$ with coefficient $3$, and $d:2\\to0$ with coefficient $1$. The parallel edges $a$ and $b$ are separate labels even though their source and target agree.","The surviving words are $ac,bc,cd,da,db$. Their coefficients are $6,-3,3,2,-1$, respectively. Their endpoints determine where these numbers enter the matrix:\n\\[\nT=\\mat{0&0&1","The surviving words are $ac,bc,cd,da,db$. Their coefficients are $6,-3,3,2,-1$, respectively. Their endpoints determine where these numbers enter the matrix:\n\\[\nT=\\mat{0&0&1\\\\1&0&0\\\\0&3&0},\\qquad\nT^2=\\mat{0&3&0\\\\0&0&1\\\\3&0&0}.\n\\]\nAt row $2$, column $0$, the two parallel choices give $6-3=3$. Deduplicating the edge labels before summing would alter the operator. On the other hand, adding the parallel coefficients in the already assembled matrix is legitimate: both act in the same supported matrix position.","3&0&0}.\n\\]\nAt row $2$, column $0$, the two parallel choices give $6-3=3$. Deduplicating the edge labels before summing would alter the operator. On the other hand, adding the parallel coefficients in the already assembled matrix is legitimate: both act in the same supported matrix position.","\\vpage{5}{Coefficients need not commute}{MODEL A / A MATRIX RING AS THE COEFFICIENT SEMIRING}{Traditional paper: Lemma 3.7 (p. 5), Section 9 (p. 20)}\nThe edge theorem allows a noncommutative coefficient semiring. Use $R=M_2(\\QQ)$ and let the two coefficients themselves be matrices:\n\\[\nA=\\mat{1&1","\\vpage{5}{Coefficients need not commute}{MODEL A / A MATRIX RING AS THE COEFFICIENT SEMIRING}{Traditional paper: Lemma 3.7 (p. 5), Section 9 (p. 20)}\nThe edge theorem allows a noncommutative coefficient semiring. Use $R=M_2(\\QQ)$ and let the two coefficients themselves be matrices:\n\\[\nA=\\mat{1&1\\\\0&1},\\qquad B=\\mat{1&0\\\\1&1}.\n\\]\nAttach $A$ to $0\\to1$ and $B$ to $1\\to2$. The single surviving endpoint coefficient is $BA$, not $AB$.","1&1}.\n\\]\nAttach $A$ to $0\\to1$ and $B$ to $1\\to2$. The single surviving endpoint coefficient is $BA$, not $AB$.","\\vfig[.96]{ordered_matrix_coefficients}\n\\platecaption{All entries are exact integers. The two products differ on the diagonal and act differently on the same input vector. This example is within the scalar model with semiring coefficients by taking the scalar ring to be $M_2(\\QQ)$.}","The distinction is visible before any abstraction:\n\\[\nBA\\mat{1\\\\0}=\\mat{1\\\\1},\\qquad\nAB\\mat{1\\\\0}=\\mat{2\\\\1}.\n\\]\nFor a written walk $(e_1,\\ldots,e_n)$, its coefficient must therefore be\n\\[\nc^\\leftarrow(w)=c(e_n)c(e_{n-1})\\cdots c(e_1).\n\\]\nA commutative example can hide an orientation mistake because reversing the coefficients changes nothing. This one cannot.","1}.\n\\]\nFor a written walk $(e_1,\\ldots,e_n)$, its coefficient must therefore be\n\\[\nc^\\leftarrow(w)=c(e_n)c(e_{n-1})\\cdots c(e_1).\n\\]\nA commutative example can hide an orientation mistake because reversing the coefficients changes nothing. This one cannot.","\\head{One ring is not a dependent family of fibers}\nThe $3\\times3$ matrix over $M_2(\\QQ)$ can be displayed as an ordinary $6\\times6$ matrix with $2\\times2$ blocks. This is a concrete representation of the example over one ring. It does not establish an arbitrary family of differently typed edge fibers, which is outside the finite formal release's scalar interface for edge systems.","\\vpage{6}{A reachable packet can receive zero}{MODEL A / CANCELLATION IS NOT A BROKEN INTERFACE}{Traditional paper: Corollary 3.9 (p. 6), Corollary 3.10 (p. 6)}\nConsider a diamond: two routes of length two from packet $0$ to packet $3$. Three edge coefficients are $1$ and the lower final coefficient is $-1$. Both routes are correctly typed. Neither is annihilated by an interface mismatch.","\\vfig[.96]{walk_cancellation}\n\\platecaption{The dashed edge carries the coefficient $-1$. It is an actual edge, not an uncertain or missing one. Both paths exist. The unsigned count is two, while their exact operator contributions sum to zero.}","The endpoint identity gives\n\\[\n(T^2)_{30}=(1)(1)+(-1)(1)=0.\n\\]\nThus the paper's reachability implication has a definite direction:\n\\[\n\\text{no compatible walk}\\ \\Longrightarrow\\ (T^n)_{ji}=0.\n\\]\nIts converse is false. A zero entry may come from absent paths, zero coefficients, zero divisors in the coefficient ring, or cancellation among nonzero terms. The diamond exhibits the last possibility over $\\ZZ$, where no zero divisor is needed.","e endpoint identity gives\n\\[\n(T^2)_{30}=(1)(1)+(-1)(1)=0.\n\\]\nThus the paper's reachability implication has a definite direction:\n\\[\n\\text{no compatible walk}\\ \\Longrightarrow\\ (T^n)_{ji}=0.\n\\]\nIts converse is false. A zero entry may come from absent paths, zero coefficients, zero divisors in the coefficient ring, or cancellation among nonzero terms. The diamond exhibits the last possibility over $\\ZZ$, where no zero divisor is needed.","\\vpage{7}{Reachability is only an upper bound}{MODEL A / AN ORBIT THAT STAYS IN TWO DIMENSIONS}{Traditional paper: Corollary 3.11 (p. 6)}\nKeep the diamond and add an isolated packet $4$. Start with the single seed $e_0\\in\\QQ^5$. Direct multiplication gives\n\\[\nTe_0=e_1+e_2,\\qquad T^2e_0=0.\n\\]\nEvery later orbit vector is zero. The orbit hull generated by $T$ is therefore exactly the span of the first two displayed columns.","A vector belongs to this hull precisely when it has the form\n\\[\n(a,b,b,0,0)^\\mathsf T,\\qquad a,b\\in\\QQ.\n\\]\nPacket $4$ is excluded because it is unreachable. Packet $3$ is also absent, but for a different reason: the two contributions cancel. Even the reachable packets $1$ and $2$ remain linked. Powers of the single total operator do not separate them.","\\head{The admissible operator grammar matters}\nAllowing each individual edge operator is a different generation problem. For instance, applying the edge $0\\to1$ and then $1\\to3$ individually produces $e_3$. It does not follow that $e_3$ belongs to the orbit of the summed operator $T$. The paper's theorem on orbit support concerns the explicitly stated operator action.","\\vpage{8}{Paths through changing packet lists}{MODEL B / RECTANGULAR PRODUCTS WITH TWO OR THREE STEPS}{Traditional paper: Proposition 8.1 (p. 19)}\nHere the packet lists have sizes $2,3,2,1$ in succession. Let the actual maps be\n\\[\n\\QQ^2\\xrightarrow{A}\\QQ^3\\xrightarrow{B}\\QQ^2\\xrightarrow{C}\\QQ.\n\\]\nUnlike a power of one matrix, each step has its own entries and its own source and target list.","\\vfig[.98]{rectangular_paths}\n\\platecaption{The lower board expands one final row into all six choices of intermediate indices for each input packet. The entries are $C_{0j}B_{jk}A_{ki}$, with $k\\in\\{0,1,2\\}$ and $j\\in\\{0,1\\}$. Zero terms remain visible.}","The intermediate and total products are\n\\[\nBA=\\mat{-1&5\\\\2&4},\\qquad CBA=\\mat{-4&6}.\n\\]\nFor input packet $0$, the six contributions sum to $2-4+0+0-2+0=-4$. For input packet $1$, they sum to $0-2+12+0-1-3=6$. These are literal matrix entries, not probabilities or path counts.","2&4},\\qquad CBA=\\mat{-4&6}.\n\\]\nFor input packet $0$, the six contributions sum to $2-4+0+0-2+0=-4$. For input packet $1$, they sum to $0-2+12+0-1-3=6$. These are literal matrix entries, not probabilities or path counts.","\\vpage{9}{One vector, three exact frames}{MODEL C / CANONICAL TRANSPORT HAS NO HOLONOMY}{Traditional paper: Theorem 4.1 (p. 7), Theorem 4.2 (p. 7)}\nLet $F_a$ synthesize chart coordinates into one ambient $\\QQ^2$. Choose\n\\[\nF_0=I,\\qquad F_1=\\mat{1&1","\\vpage{9}{One vector, three exact frames}{MODEL C / CANONICAL TRANSPORT HAS NO HOLONOMY}{Traditional paper: Theorem 4.1 (p. 7), Theorem 4.2 (p. 7)}\nLet $F_a$ synthesize chart coordinates into one ambient $\\QQ^2$. Choose\n\\[\nF_0=I,\\qquad F_1=\\mat{1&1\\\\0&1},\\qquad\nF_2=\\mat{2&0\\\\1&1}.\n\\]\nThe same ambient vector $x=(3,2)^\\mathsf T$ has chart coordinates $u_a=F_a^{-1}x$.","1&1}.\n\\]\nThe same ambient vector $x=(3,2)^\\mathsf T$ has chart coordinates $u_a=F_a^{-1}x$.","\\vfig[.98]{frame_coordinates}\n\\platecaption{Each panel is a coordinate plane, not a different physical location. The numerical arrows differ because the frames differ. Multiplying each coordinate vector by its corresponding $F_a$ recovers the same $x$.}","Transport from chart $a$ to chart $b$ is forced:\n\\[\nJ_{b\\leftarrow a}=F_b^{-1}F_a.\n\\]\nThe route $0\\to1\\to2$ agrees with the direct route because the middle frame cancels:\n\\[\n(F_2^{-1}F_1)(F_1^{-1}F_0)=F_2^{-1}F_0.\n\\]\nReturning to chart $0$ cancels every remaining frame. The final coordinate vector is $(3,2)^\\mathsf T$, not a sheared or rotated variant.","\\vpage{10}{Nonlinear charts still cancel exactly}{MODEL C / NATURALITY DOES NOT REQUIRE DERIVATIVES}{Traditional paper: Proposition 10.1 (p. 20)}\nExact coordinate equivalences can be nonlinear. On $\\QQ^2$, use\n\\[\n\\mathcal A(x,y)=(x,y+x^2),\\qquad\n\\mathcal S(u,v)=(u,v-u^2).\n\\]\nBoth composites are the identity by direct substitution. The real drawing below is a geometric view of these same polynomial formulas.","Now represent the nonlinear operation $F(x,y)=(x+y,xy)$ in the new chart. Set $w=u+v-u^2$. Its correct coordinate formula is\n\\[\n\\widehat F_b(u,v)=\\mathcal A F\\mathcal S(u,v)\n=\\bigl(w,\\ u(v-u^2)+w^2\\bigr).\n\\]\nFor $(x,y)=(1,2)$, applying $F$ first gives $(3,2)$ and then $(3,11)$ in the new chart. Changing the input first gives $(u,v)=(1,3)$, and $\\widehat F_b(1,3)=(3,11)$ as well.","\\vpage{11}{Reversibility is a real hypothesis}{MODEL D / THE MINIMAL AND REVERSIBLE INTERFACES}{Traditional paper: Definition 5.1 (p. 9), Definition 5.5 (p. 11)}\nChoose the forward scalar transports $\\Gamma_{1\\leftarrow0}=2$, $\\Gamma_{2\\leftarrow1}=3$, and $\\Gamma_{2\\leftarrow0}=5$. Their multiplicative comparison between transport through two steps and direct transport is $D^\\times=6/5$. A loop also needs a declared return map $\\Gamma_{0\\leftarrow2}$.","l hypothesis}{MODEL D / THE MINIMAL AND REVERSIBLE INTERFACES}{Traditional paper: Definition 5.1 (p. 9), Definition 5.5 (p. 11)}\nChoose the forward scalar transports $\\Gamma_{1\\leftarrow0}=2$, $\\Gamma_{2\\leftarrow1}=3$, and $\\Gamma_{2\\leftarrow0}=5$. Their multiplicative comparison between transport through two steps and direct transport is $D^\\times=6/5$. A loop also needs a declared return map $\\Gamma_{0\\leftarrow2}$.","In the first row, choose the return scalar $1/6$. Then\n\\[\nH=(1/6)\\cdot3\\cdot2=1,\n\\qquad \\Gamma_{2\\leftarrow1}\\Gamma_{1\\leftarrow0}-\\Gamma_{2\\leftarrow0}=1.\n\\]\nThis triangle loop is the identity even though its cocycle equation fails. The return is not the inverse of the direct transport: $(1/6)5\\ne1$. Thus one cannot transfer the reversible equivalence between defect and holonomy to the minimal interface.","In the reversible row, the return is $1/5$. Now the loop is $6/5$, and the additive defect of one is converted into the loop obstruction $1/5$ by that inverse return map.","\\takeaway{Self normalization means $\\Gamma_{a\\leftarrow a}=I$. Reversible additionally means $\\Gamma_{a\\leftarrow b}=\\Gamma_{b\\leftarrow a}^{-1}$. These are different assumptions.}","\\vpage{12}{Measure the mismatch at its base}{MODEL D / AN EXACT REVERSIBLE TRIANGLE}{Traditional paper: Theorem 5.10 (p. 13), Section 9 (p. 20)}\nRetain the reversible scalar model with forward values $2,3,5$ and all reverse values given by inverses. Two routes from chart $0$ to chart $2$ multiply the input by $6$ and $5$, respectively. The additive discrepancy equals the original input.","The defect lives at the target of the forward routes:\n\\[\n\\Delta_{2,1,0}=3\\cdot2-5=1.\n\\]\nThe holonomy lives back at chart $0$:\n\\[\nH_{2,1,0}=5^{-1}\\cdot3\\cdot2=\\frac65,\n\\qquad H-I=\\frac15=\\Gamma_{0\\leftarrow2}\\Delta.\n\\]\nFor input $5$, the direct output and the output after two steps are $25$ and $30$. Returning by $1/5$ gives $5$ and $6$. The target discrepancy of $5$ has become a base discrepancy of $1$. The types and locations of these differences matter even in an example in one dimension.","defect lives at the target of the forward routes:\n\\[\n\\Delta_{2,1,0}=3\\cdot2-5=1.\n\\]\nThe holonomy lives back at chart $0$:\n\\[\nH_{2,1,0}=5^{-1}\\cdot3\\cdot2=\\frac65,\n\\qquad H-I=\\frac15=\\Gamma_{0\\leftarrow2}\\Delta.\n\\]\nFor input $5$, the direct output and the output after two steps are $25$ and $30$. Returning by $1/5$ gives $5$ and $6$. The target discrepancy of $5$ has become a base discrepancy of $1$. The types and locations of these differences matter even in an example in one dimension.","Because the return map is invertible, it cannot turn a nonzero defect into zero. This is the mechanism behind $\\Delta=0\\Longleftrightarrow H=I$ in the reversible interface.","\\vpage{13}{The identity can hide in the spectrum}{MODEL E / A SHEAR WITH NONTRIVIAL HOLONOMY}{Traditional paper: Definition 5.9 (p. 12), Definition 7.4 (p. 16)}\nPut $\\Gamma_{1\\leftarrow0}=H$, $\\Gamma_{2\\leftarrow1}=I$, and $\\Gamma_{2\\leftarrow0}=I$, with reverse maps given by inverses, where\n\\[\nH=\\mat{1&1","\\vpage{13}{The identity can hide in the spectrum}{MODEL E / A SHEAR WITH NONTRIVIAL HOLONOMY}{Traditional paper: Definition 5.9 (p. 12), Definition 7.4 (p. 16)}\nPut $\\Gamma_{1\\leftarrow0}=H$, $\\Gamma_{2\\leftarrow1}=I$, and $\\Gamma_{2\\leftarrow0}=I$, with reverse maps given by inverses, where\n\\[\nH=\\mat{1&1\\\\0&1}.\n\\]\nThe loop $0\\to1\\to2\\to0$ then has exactly this shear as its holonomy.","0&1}.\n\\]\nThe loop $0\\to1\\to2\\to0$ then has exactly this shear as its holonomy.","\\vfig[.94]{shear_holonomy}\n\\platecaption{The dashed square is the original set. The solid parallelogram is its exact image. The second coordinate basis vector moves to $(1,1)^\\mathsf T$. Its nonzero displacement is the first basis vector.}","The usual scalar invariants can be misleading:\n\\[\n\\operatorname{tr}H=2,\\qquad \\det H=1,\\qquad\n\\chi_H(t)=(t-1)^2.\n\\]\nThese are also the invariants of the $2\\times2$ identity matrix. Nevertheless,\n\\[\n\\Ob=H-I=\\mat{0&1","The usual scalar invariants can be misleading:\n\\[\n\\operatorname{tr}H=2,\\qquad \\det H=1,\\qquad\n\\chi_H(t)=(t-1)^2.\n\\]\nThese are also the invariants of the $2\\times2$ identity matrix. Nevertheless,\n\\[\n\\Ob=H-I=\\mat{0&1\\\\0&0}\\ne0,\n\\qquad \\Ob e_1=e_0.\n\\]\nA complete obstruction test must therefore retain the operator or a certificate strong enough to detect its nonzero action. Trace, determinant, and eigenvalue lists alone do not distinguish this loop from the identity.","\\head{What the picture certifies, and what it does not}\nThe plotted corners are the images of the four corners of the unit square under the displayed matrix. The equation $\\Ob e_1=e_0$ is the exact witness. No smooth connection, curvature tensor, or continuum limit is used in this model. Its holonomy is the finite algebraic loop product. The traditional paper treats smooth transport separately under additional hypotheses.","\\takeaway{Flatness is the equality $H=I$, not agreement of a short list of spectral summaries. A single explicit moved vector can disprove that equality.}","\\vpage{14}{Gauge changes the picture, not flatness}{MODEL E / CONJUGATION AND ITS LIMITS}{Traditional paper: Theorem 6.2 (p. 14)}\nAt the base chart, change coordinates by $D=\\operatorname{diag}(2,1)$. Transform every connection edge by the prescribed endpoint gauges. The loop transforms by conjugation, giving\n\\[\nH^D=DHD^{-1}=\\mat{1&2","\\vpage{14}{Gauge changes the picture, not flatness}{MODEL E / CONJUGATION AND ITS LIMITS}{Traditional paper: Theorem 6.2 (p. 14)}\nAt the base chart, change coordinates by $D=\\operatorname{diag}(2,1)$. Transform every connection edge by the prescribed endpoint gauges. The loop transforms by conjugation, giving\n\\[\nH^D=DHD^{-1}=\\mat{1&2\\\\0&1}.\n\\]\nThe shear looks stronger in these fixed numerical axes, but the verdict on flatness is unchanged by gauge.","The rank of the obstruction is one on both sides. Trace, determinant, and characteristic polynomial are preserved under this conjugation. Most importantly,\n\\[\nD(H-I)D^{-1}=0\\quad\\Longleftrightarrow\\quad H-I=0.\n\\]\nAn invertible coordinate change cannot turn a nonidentity loop into the identity. The fact that a particular residual norm changes does not contradict the invariance of exact vanishing.","\\head{The base chart determines the outer gauge}\nIn a triangle product, adjacent intermediate gauges cancel:\n\\[\n(G_a\\Gamma_{a\\leftarrow c}G_c^{-1})\n(G_c\\Gamma_{c\\leftarrow b}G_b^{-1})\n(G_b\\Gamma_{b\\leftarrow a}G_a^{-1})\n=G_aHG_a^{-1}.\n\\]\nOnly the base gauge survives around the loop. For the additive defect, the surviving outer gauges instead belong to its different source and target fibers.","\\vpage{15}{Flat is not the same as calibrated}{MODEL D / A FLAT CONNECTION WITH NONIDENTITY EDGES}{Traditional paper: Theorem 6.3 (p. 14)}\nMake all three canonical charts literally identical, so $J_{b\\leftarrow a}=1$ over $\\QQ$. Choose scalar gauges $g_0=1$, $g_1=2$, and $g_2=3$, and define\n\\[\n\\Gamma_{b\\leftarrow a}=g_b g_a^{-1}.\n\\]\nThe edge $0\\to1$ is multiplication by two, although its canonical counterpart is the identity.","\\vfig[.97]{flat_not_calibrated}\n\\platecaption{The rulers display the chosen gauge scales: the marks are $g_a n$ for integer $n$. They visualize the supplied scalar factors, not an intrinsic metric. Relative transport is the ratio of the target and source scales.}","The route through two steps $0\\to1\\to2$ has multiplier $(3/2)2=3$, equal to the direct route $0\\to2$. More generally,\n\\[\n(g_cg_b^{-1})(g_bg_a^{-1})=g_cg_a^{-1}\n\\]\nfor every triple. Thus this connection is flat and reversible, but it is not calibrated to the original $J$, whose values are identities.","\\head{Recovering the gauge from one root}\nChoose a base chart $o$. The theorem constructs\n\\[\nG_a=\\Gamma_{a\\leftarrow o}J_{o\\leftarrow a},\n\\qquad\n\\Gamma_{b\\leftarrow a}=G_bJ_{b\\leftarrow a}G_a^{-1}.\n\\]\nWith $o=0$ and $J=1$, the recovered $G_a$ are exactly $1,2,3$. The statement concerns the complete pairwise interface and its cocycle. It is not a topological claim about an arbitrary graph with missing direct comparisons.","\\vpage{16}{Resolve the entire obstruction}{MODEL F / PACKET BLOCKS FROM INPUT TO OUTPUT}{Traditional paper: Theorem 7.5 (p. 17), Corollary 7.6 (p. 18)}\nUse three coordinate packets with projectors $P_0,P_1,P_2$, and set\n\\[\nH=I+2E_{01}+3E_{12},\\qquad \\Ob=H-I.\n\\]\nHere $H$ is invertible, so it can be the loop product of a reversible connection on three charts. Its entries outside the diagonal encode two effects that mix packets.","\\vfig[.95]{all_obstruction_blocks}\n\\platecaption{The board on the right contains all nine actual embedded blocks $P_\\mu\\Ob P_\\nu$. Row labels are output packets and column labels are input packets. Only the addresses $(0,1)$ and $(1,2)$ carry nonzero maps.}","Completeness on both sides supplies the reconstruction:\n\\[\n\\Ob=(P_0+P_1+P_2)\\Ob(P_0+P_1+P_2)\n=\\sum_{\\mu,\\nu}P_\\mu\\Ob P_\\nu.\n\\]\nFor $x=(4,5,6)^\\mathsf T$, the input in packet $1$ contributes $10e_0$ and the input in packet $2$ contributes $18e_1$. Their sum is the full displacement $\\Ob x$.","\\head{A block becomes a witness when it acts}\nThe nonzero block $2E_{01}$ has the explicit source vector $e_1$, supported entirely in packet $1$, and $P_0\\Ob e_1=2e_0\\ne0$. Similarly, $P_1\\Ob e_2=3e_1\\ne0$. The general theorem supplies such a witness classically when a nonzero block is known. The example displays it concretely.","\\vpage{17}{Why diagonal checks are insufficient}{MODEL E / COMPLETENESS, TWO SIDES, AND OBLIQUE PACKETS}{Traditional paper: Theorem 7.5 (p. 17)}\nReturn to $\\Ob=E_{01}$. With the two standard coordinate projectors, both diagonal blocks vanish:\n\\[\nP_0\\Ob P_0=P_1\\Ob P_1=0.\n\\]\nThe obstruction nevertheless survives in the block outside the diagonal $P_0\\Ob P_1=E_{01}$.","There is a second failure mode. Supplying only $P_0$ gives a family of idempotents but not a complete one. The single tested block is zero, while the omitted packet carries the missing input. The implication ``all tested blocks vanish, therefore the operator vanishes'' requires the completeness hypothesis.","\\head{Exact reconstruction does not require right angles}\nFor the displayed $Q_0,Q_1$,\n\\[\nQ_0+Q_1=I,\\qquad Q_i^2=Q_i,\\qquad Q_0Q_1=Q_1Q_0=0,\n\\]\nbut $Q_0^\\mathsf T\\ne Q_0$. Expanding both copies of $I$ still proves\n\\[\n\\sum_{i,j}Q_i\\Ob Q_j=\\Ob.\n\\]\nThe block reconstruction identity itself uses the condition that the projectors sum to the identity. Idempotence is additionally used when constructing a witness that remains supported in one packet after projection. The release does not require pairwise annihilation for its general reconstruction statement.","\\vpage{18}{A gauge can move every packet label}{MODEL E / FIXED VERSUS TRANSPORTED PROJECTORS}{Traditional paper: Proposition 7.7 (p. 18)}\nThe diagonal gauge on V14 commuted with each standard projector. Now use the gauge that mixes packets\n\\[\nG=\\mat{1&0\\\\1&1},\\qquad \\Ob=E_{01}.\n\\]\nThe total obstruction transforms by conjugation, but its pattern in the \\emph{fixed} standard packets changes.","Direct multiplication gives\n\\[\nG\\Ob G^{-1}=\\mat{-1&1\\\\-1&1}.\n\\]\nFor example, the original $(0,0)$ block vanishes, while the new $(0,0)$ block relative to the old coordinate projectors does not. This is why the theorem about preservation of each block's zero status requires $G P_i=P_iG$.","-1&1}.\n\\]\nFor example, the original $(0,0)$ block vanishes, while the new $(0,0)$ block relative to the old coordinate projectors does not. This is why the theorem about preservation of each block's zero status requires $G P_i=P_iG$.","\\head{Transporting the packet frame as well}\nDefine $Q_i=GP_iG^{-1}$. Then the identities cancel exactly:\n\\[\nQ_i(G\\Ob G^{-1})Q_j=G(P_i\\Ob P_j)G^{-1}.\n\\]\nRelative to these transported packets, the corresponding labeled blocks again have matching vanishing status. These are the same oblique projectors displayed on V17. One must distinguish moving the operator alone in a fixed resolution from moving the entire resolved description.","\\vpage{19}{An obstruction rejects one candidate}{MODEL G / THE QUANTIFIERS ARE THE CONTENT}{Traditional paper: Theorem 7.2 (p. 15)}\nLet the candidates be rational numbers and the packet labels be $0,1$. Define\n\\[\nr(x,0)=x^2-1,\\qquad r(x,1)=x-1.\n\\]\nA candidate is a solution when both residuals vanish. The five candidate columns below are a display of this explicit residual function, not a search over an unspecified class.","\\vfig[.97]{candidate_residual_table}\n\\platecaption{A shaded cell is a nonzero exact residual. At $x=-1$, the first equation passes but the second supplies an obstruction. At $x=1$, both pass. The rejected candidate does not obstruct the successful one.}","The statement proved by the generic certificate has the form\n\\[\n\\neg\\bigl(\\forall p,\\ r(x,p)=0\\bigr)\n\\quad\\Longleftrightarrow\\quad\n\\exists p,\\ r(x,p)\\ne0.\n\\]\nThe same candidate $x$ appears on both sides. A certificate consists of the packet label and its nonvanishing proof. The general converse uses classical reasoning and does not itself provide an algorithm to find the label.","\\head{What would be required to rule out all solutions?}\nOne would need the stronger assertion $\\forall x\\,\\exists p,\\ r(x,p)\\ne0$. The value $x=1$ disproves that assertion in this example. A witness for $x=-1$ supplies no evidence for it. Likewise, finding an obstruction block for one triangle of one chosen connection does not classify every possible connection.","\\vpage{20}{A tree reads its named leaves}{MODEL H / ACTUAL SYNTAX, ARBITRARY SEMANTICS}{Traditional paper: Definition 11.1 (p. 22), Theorem 11.2 (p. 22)}\nUse the grammar's constant, input, unary, and binary constructors to build\n\\[\nt=(x^2+3)-y.\n\\]\nHere ``square,'' ``add,'' and ``subtract'' are explicitly supplied meanings of operation symbols. At $x=2,y=5$, every node value can be evaluated exactly.","The input set is $\\operatorname{inputs}(t)=\\{x,y\\}$. If two environments agree at $x$ and $y$, structural induction proves that the two evaluations agree: input leaves agree, constants agree, and every parent applies the same supplied operation to equal child values.","\\head{Syntactic locality is not minimal dependence}\nThe tree $x-x$ has input set $\\{x\\}$, although its value is always zero under ordinary subtraction. Locality provides a sufficient agreement condition. It does not compute the smallest set of semantically relevant variables. Nor does it prove that two different trees are equivalent.","The semantics may act on arbitrary values. For example, a binary concatenation symbol applied to strings ``ab'' and ``cd'' gives ``abcd.'' No linearity or differentiability is required. Conversely, swapping the ordered inputs of the subtraction root above changes the value from $2$ to $-2$.","\\vpage{21}{Where a nonlinear defect can enter}{MODEL H / EXACT COORDINATES VERSUS A CHOSEN RULE}{Traditional paper: Proposition 10.1 (p. 20), Formalization target 11.7 (p. 24), Theorem 20.2 (p. 43)}\nTake the unary operation $f(x)=x^2$ over $\\QQ$. Suppose both input and output transport are chosen as multiplication by two, while the same numerical formula $f$ is reused at the target. The two routes give\n\\[\n2f(x)=2x^2,\\qquad f(2x)=4x^2.\n\\]\nTheir difference is $-2x^2$, which is generally nonzero.","1}{Where a nonlinear defect can enter}{MODEL H / EXACT COORDINATES VERSUS A CHOSEN RULE}{Traditional paper: Proposition 10.1 (p. 20), Formalization target 11.7 (p. 24), Theorem 20.2 (p. 43)}\nTake the unary operation $f(x)=x^2$ over $\\QQ$. Suppose both input and output transport are chosen as multiplication by two, while the same numerical formula $f$ is reused at the target. The two routes give\n\\[\n2f(x)=2x^2,\\qquad f(2x)=4x^2.\n\\]\nTheir difference is $-2x^2$, which is generally nonzero.","COORDINATES VERSUS A CHOSEN RULE}{Traditional paper: Proposition 10.1 (p. 20), Formalization target 11.7 (p. 24), Theorem 20.2 (p. 43)}\nTake the unary operation $f(x)=x^2$ over $\\QQ$. Suppose both input and output transport are chosen as multiplication by two, while the same numerical formula $f$ is reused at the target. The two routes give\n\\[\n2f(x)=2x^2,\\qquad f(2x)=4x^2.\n\\]\nTheir difference is $-2x^2$, which is generally nonzero.","\\vfig[.96]{node_defect_not_coordinate_defect}\n\\platecaption{Solid: $2x^2$. Dashed: $4x^2$. The curves are a real visualization of exact polynomial formulas. The table contains exact evaluations over the rationals. The mismatch is caused by the specified combination of transport and formula.}","For a canonical coordinate change with input and output chart $v=2x$, the target formula is not still $v^2$. It is\n\\[\n\\widehat f_b(v)=2f(v/2)=\\frac{v^2}{2}.\n\\]\nConsequently $\\widehat f_b(2x)=2x^2$, and the exact coordinate defect vanishes. This is the same distinction illustrated in the example in two dimensions on V10.","\\vpage{22}{Mixed terms are not optional}{FINITE EXAMPLES / WHY MULTILINEARITY MATTERS}{Traditional paper: Formalization target 11.6 (p. 23), Theorem 18.3 (p. 39)}\nA bilinear node distributes over two independently resolved inputs:\n\\[\n(a+b)(c+d)=ac+ad+bc+bd.\n\\]\nThe four terms correspond to all ordered choices of one input packet from each slot. Keeping only matching packet labels loses the mixed terms.","\\vfig[.95]{bilinear_vs_nonlinear}\n\\platecaption{The rectangle has width $2+3$ and height $5+7$, with differently scaled horizontal and vertical display units. Each cell is labeled by its actual product. The lighter diagonal pair contributes only $31$ of the total $60$.}","For a unary square, inserting $a+b$ does not give $a^2+b^2$. The extra $2ab$ arises because the operation is nonlinear in its single argument. It can be analyzed by representing the square using a binary multiplication node with a repeated input, but that is an additional choice of syntax and semantics.","\\vpage{23}{From loop data to a packet witness}{CAPSTONE / A COMPLETE FINITE DIAGNOSIS}{Traditional paper: Theorem 5.10 (p. 13), Theorem 6.2 (p. 14), Corollary 7.6 (p. 18)}\nSupply a reversible connection on three copies of $\\QQ^2$ by choosing\n\\[\n\\Gamma_{1\\leftarrow0}=A,\\qquad\\Gamma_{2\\leftarrow1}=B,\n\\qquad\\Gamma_{2\\leftarrow0}=C=I,\n\\]\nand declaring every reverse edge to be the inverse. This time both nontrivial forward matrices are retained.","\\vfig[.97]{worked_holonomy_diagnosis}\n\\platecaption{The vector row starts with the source witness $e_0$ and follows it through $A$, then $B$, then $C^{-1}$. Returning to the base produces $(1,1)^\\mathsf T$, whose displacement from the input is $e_1$.}","The calculation gives\n\\[\n\\Delta=BA-C=\\mat{0&1\\\\1&1},\\qquad H=C^{-1}BA=\\mat{1&1\\\\1&2}.\n\\]\nChoose the standard packet projectors. The $(1,0)$ block is nonzero, and the explicit candidate vector satisfies $P_1(H-I)e_0=e_1$. This proves nonflatness of the supplied connection at the supplied triangle.","1&2}.\n\\]\nChoose the standard packet projectors. The $(1,0)$ block is nonzero, and the explicit candidate vector satisfies $P_1(H-I)e_0=e_1$. This proves nonflatness of the supplied connection at the supplied triangle.","\\head{Repairing data is not the same as changing gauge}\nReplacing the direct map $C=I$ by $C=BA$, and updating its reverse, makes the new loop equal to $I$. That changes the connection. It cannot be a gauge of the original nonflat connection, because gauge preserves whether holonomy is the identity. For this complete example on three charts all cocycles then hold. Changing one edge in a larger network would require checking the other affected loops as well.","\\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\\sffamily\\small,line cap=round,line join=round]\n\\node[] at (0.97500,4.30000) {$H$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (0.00000,3.10000) rectangle ++(0.65000,0.65000);\n\\node[] at (0.32500,3.42500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (0.65000,3.10000) rectangle ++(0.65000,0.65000);\n\\node[] at (0.97500,3.42500) {$2$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (1.30000,3.10000) rectangle ++(0.65000,0.65000);\n\\node[] at (1.62500,3.42500) {$0$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.00000,2.45000) rectangle ++(","\\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\\sffamily\\small,line cap=round,line join=round]\n\\node[] at (0.97500,4.30000) {$H$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (0.00000,3.10000) rectangle ++(0.65000,0.65000);\n\\node[] at 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at (0.32500,2.12500) {$0$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.65000,1.80000) rectangle ++(0.65000,0.65000);\n\\node[] at (0.97500,2.12500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (1.30000,1.80000) rectangle ++(","ll=black!0,draw=black!40,line width=.35pt,] (0.00000,2.45000) rectangle ++(0.65000,0.65000);\n\\node[] at (0.32500,2.77500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (0.65000,2.45000) rectangle ++(0.65000,0.65000);\n\\node[] at (0.97500,2.77500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (1.30000,2.45000) rectangle ++(0.65000,0.65000);\n\\node[] at (1.62500,2.77500) {$3$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.00000,1.80000) rectangle ++(0.65000,0.65000);\n\\node[] at (0.32500,2.12500) {$0$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.65000,1.80000) rectangle ++(0.65000,0.65000);\n\\node[] at (0.97500,2.12500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (1.30000,1.80000) rectangle ++(0.65000,0.65000);\n\\node[] at (1.62500,2.12500) {$1$};\n\\node[] at (4.47500,4.30000) {$\\Ob=H-I$};\n\\draw[fill=black!0,draw=black!40,line widt","ll=black!12,draw=black!40,line width=.35pt,] (0.65000,2.45000) rectangle ++(0.65000,0.65000);\n\\node[] at (0.97500,2.77500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (1.30000,2.45000) rectangle ++(0.65000,0.65000);\n\\node[] at (1.62500,2.77500) {$3$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.00000,1.80000) rectangle ++(0.65000,0.65000);\n\\node[] at (0.32500,2.12500) {$0$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.65000,1.80000) rectangle ++(0.65000,0.65000);\n\\node[] at (0.97500,2.12500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (1.30000,1.80000) rectangle ++(0.65000,0.65000);\n\\node[] at (1.62500,2.12500) {$1$};\n\\node[] at (4.47500,4.30000) {$\\Ob=H-I$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (3.50000,3.10000) rectangle ++(0.65000,0.65000);\n\\node[] at (3.82500,3.42500) {$0$};\n\\draw[fill=black!12,draw=black!40,line wid","ill=black!12,draw=black!40,line width=.35pt,] (1.30000,2.45000) rectangle ++(0.65000,0.65000);\n\\node[] at (1.62500,2.77500) {$3$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.00000,1.80000) rectangle ++(0.65000,0.65000);\n\\node[] at (0.32500,2.12500) {$0$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.65000,1.80000) rectangle ++(0.65000,0.65000);\n\\node[] at (0.97500,2.12500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (1.30000,1.80000) rectangle ++(0.65000,0.65000);\n\\node[] at (1.62500,2.12500) {$1$};\n\\node[] at (4.47500,4.30000) {$\\Ob=H-I$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (3.50000,3.10000) rectangle ++(0.65000,0.65000);\n\\node[] at (3.82500,3.42500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (4.15000,3.10000) rectangle ++(0.65000,0.65000);\n\\node[] at 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(4.67500,1.92500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.3","h=.35pt,] (0.85000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (1.27500,2.77500) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.00000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (0.42500,1.92500) {$0$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.85000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (1.27500,1.92500) {$0$};\n\\node[] at (5.10000,3.75000) {$G$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (4.25000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (4.67500,2.77500) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (5.10000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (5.52500,2.77500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (4.25000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (4.67500,1.92500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (5.10000,1.50000) rectangle ++","=.35pt,] (0.00000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (0.42500,1.92500) {$0$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.85000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (1.27500,1.92500) {$0$};\n\\node[] at (5.10000,3.75000) {$G$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (4.25000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (4.67500,2.77500) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (5.10000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (5.52500,2.77500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (4.25000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (4.67500,1.92500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (5.10000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (5.52500,1.92500) {$1$};\n\\node[] at (9.35000,3.75000) {$G\\Ob G^{-1}$};\n\\draw[fill=black!12,draw=black!40,lin","=.35pt,] (0.85000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (1.27500,1.92500) {$0$};\n\\node[] at (5.10000,3.75000) {$G$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (4.25000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (4.67500,2.77500) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (5.10000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (5.52500,2.77500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (4.25000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (4.67500,1.92500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (5.10000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (5.52500,1.92500) {$1$};\n\\node[] at (9.35000,3.75000) {$G\\Ob G^{-1}$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (8.50000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (8.92500,2.77500) {$-1$};\n\\draw[fill=black!12,draw=black!40,l","ll=black!12,draw=black!40,line width=.35pt,] (4.25000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (4.67500,2.77500) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (5.10000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (5.52500,2.77500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (4.25000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (4.67500,1.92500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (5.10000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (5.52500,1.92500) {$1$};\n\\node[] at (9.35000,3.75000) {$G\\Ob G^{-1}$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (8.50000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (8.92500,2.77500) {$-1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.35000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (9.77500,2.77500) {$1$};\n\\draw[fill=black!12,draw=black!40,l","ll=black!0,draw=black!40,line width=.35pt,] (5.10000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (5.52500,2.77500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (4.25000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (4.67500,1.92500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (5.10000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (5.52500,1.92500) {$1$};\n\\node[] at (9.35000,3.75000) {$G\\Ob G^{-1}$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (8.50000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (8.92500,2.77500) {$-1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.35000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (9.77500,2.77500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (8.50000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (8.92500,1.92500) {$-1$};\n\\draw[fill=black!12,draw=black!40,",".35pt,] (5.10000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (5.52500,2.77500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (4.25000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (4.67500,1.92500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (5.10000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (5.52500,1.92500) {$1$};\n\\node[] at (9.35000,3.75000) {$G\\Ob G^{-1}$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (8.50000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (8.92500,2.77500) {$-1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.35000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (9.77500,2.77500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (8.50000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (8.92500,1.92500) {$-1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.35000,1.50000) rectangle",".25000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (4.67500,1.92500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (5.10000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (5.52500,1.92500) {$1$};\n\\node[] at (9.35000,3.75000) {$G\\Ob G^{-1}$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (8.50000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (8.92500,2.77500) {$-1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.35000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (9.77500,2.77500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (8.50000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (8.92500,1.92500) {$-1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.35000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (9.77500,1.92500) {$1$};\n\\node[] at (0.85000,0.65000) {one nonzero entry};\n\\node[] at (5.10000,0.65000) {$","10000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (5.52500,1.92500) {$1$};\n\\node[] at (9.35000,3.75000) {$G\\Ob G^{-1}$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (8.50000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (8.92500,2.77500) {$-1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.35000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (9.77500,2.77500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (8.50000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (8.92500,1.92500) {$-1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.35000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (9.77500,1.92500) {$1$};\n\\node[] at (0.85000,0.65000) {one nonzero entry};\n\\node[] at (5.10000,0.65000) {$GP_0\\ne P_0G$};\n\\node[] at (9.35000,0.65000) {four nonzero entries};\n\\node[] at (5.50000,-0.20000) {fixed axes: different packet pattern /","=black!12,draw=black!40,line width=.35pt,] (8.50000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (8.92500,2.77500) {$-1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.35000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (9.77500,2.77500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (8.50000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (8.92500,1.92500) {$-1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.35000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (9.77500,1.92500) {$1$};\n\\node[] at (0.85000,0.65000) {one nonzero entry};\n\\node[] at (5.10000,0.65000) {$GP_0\\ne P_0G$};\n\\node[] at (9.35000,0.65000) {four nonzero entries};\n\\node[] at (5.50000,-0.20000) {fixed axes: different packet pattern / moving axes: conjugate labeled blocks};\n\\end{tikzpicture}","=black!12,draw=black!40,line width=.35pt,] (9.35000,2.35000) rectangle ++(0.85000,0.85000);\n\\node[] at (9.77500,2.77500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (8.50000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (8.92500,1.92500) {$-1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.35000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (9.77500,1.92500) {$1$};\n\\node[] at (0.85000,0.65000) {one nonzero entry};\n\\node[] at (5.10000,0.65000) {$GP_0\\ne P_0G$};\n\\node[] at (9.35000,0.65000) {four nonzero entries};\n\\node[] at (5.50000,-0.20000) {fixed axes: different packet pattern / moving axes: conjugate labeled blocks};\n\\end{tikzpicture}","000);\n\\node[] at (9.77500,2.77500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (8.50000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (8.92500,1.92500) {$-1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.35000,1.50000) rectangle ++(0.85000,0.85000);\n\\node[] at (9.77500,1.92500) {$1$};\n\\node[] at (0.85000,0.65000) {one nonzero entry};\n\\node[] at (5.10000,0.65000) {$GP_0\\ne P_0G$};\n\\node[] at (9.35000,0.65000) {four nonzero entries};\n\\node[] at (5.50000,-0.20000) {fixed axes: different packet pattern / moving axes: conjugate labeled blocks};\n\\end{tikzpicture}",".4143) -- (2.9100,1.5132) -- (3.0150,1.6167) -- (3.1200,1.7248) -- (3.2250,1.8375) -- (3.3300,1.9548) -- (3.4350,2.0767) -- (3.5400,2.2032) -- (3.6450,2.3343) -- (3.7500,2.4700) -- (3.8550,2.6103) -- (3.9600,2.7552) -- (4.0650,2.9047) -- (4.1700,3.0588) -- (4.2750,3.2175) -- (4.3800,3.3808) -- (4.4850,3.5487) -- (4.5900,3.7212) -- (4.6950,3.8983) -- (4.8000,4.0800);\n\\node[] at (3.80000,2.10000) {$2x^2$};\n\\node[] at (3.80000,4.20000) {$4x^2$};\n\\node[] at (5.50000,0.10000) {$x$};\n\\draw[black!25] (6.20000,0.00000) -- (6.20000,4.75000);\n\\node[anchor=east] at (8.75000,3.94000) {$x$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.00000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,3.94000) {$0$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.87000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,3.94000) {$1$};\n\\draw[fill=black!0,draw=black!40,line width","6167) -- (3.1200,1.7248) -- (3.2250,1.8375) -- (3.3300,1.9548) -- (3.4350,2.0767) -- (3.5400,2.2032) -- (3.6450,2.3343) -- (3.7500,2.4700) -- (3.8550,2.6103) -- (3.9600,2.7552) -- (4.0650,2.9047) -- (4.1700,3.0588) -- (4.2750,3.2175) -- (4.3800,3.3808) -- (4.4850,3.5487) -- (4.5900,3.7212) -- (4.6950,3.8983) -- (4.8000,4.0800);\n\\node[] at (3.80000,2.10000) {$2x^2$};\n\\node[] at (3.80000,4.20000) {$4x^2$};\n\\node[] at (5.50000,0.10000) {$x$};\n\\draw[black!25] (6.20000,0.00000) -- (6.20000,4.75000);\n\\node[anchor=east] at (8.75000,3.94000) {$x$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.00000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,3.94000) {$0$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.87000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,3.94000) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (10.74000,3.65000) rectangle +","375) -- (3.3300,1.9548) -- (3.4350,2.0767) -- (3.5400,2.2032) -- (3.6450,2.3343) -- (3.7500,2.4700) -- (3.8550,2.6103) -- (3.9600,2.7552) -- (4.0650,2.9047) -- (4.1700,3.0588) -- (4.2750,3.2175) -- (4.3800,3.3808) -- (4.4850,3.5487) -- (4.5900,3.7212) -- (4.6950,3.8983) -- (4.8000,4.0800);\n\\node[] at (3.80000,2.10000) {$2x^2$};\n\\node[] at (3.80000,4.20000) {$4x^2$};\n\\node[] at (5.50000,0.10000) {$x$};\n\\draw[black!25] (6.20000,0.00000) -- (6.20000,4.75000);\n\\node[anchor=east] at (8.75000,3.94000) {$x$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.00000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,3.94000) {$0$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.87000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,3.94000) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (10.74000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.1","(3.8550,2.6103) -- (3.9600,2.7552) -- (4.0650,2.9047) -- (4.1700,3.0588) -- (4.2750,3.2175) -- (4.3800,3.3808) -- (4.4850,3.5487) -- (4.5900,3.7212) -- (4.6950,3.8983) -- (4.8000,4.0800);\n\\node[] at (3.80000,2.10000) {$2x^2$};\n\\node[] at (3.80000,4.20000) {$4x^2$};\n\\node[] at (5.50000,0.10000) {$x$};\n\\draw[black!25] (6.20000,0.00000) -- (6.20000,4.75000);\n\\node[anchor=east] at (8.75000,3.94000) {$x$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.00000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,3.94000) {$0$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.87000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,3.94000) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (10.74000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,3.94000) {$2$};\n\\node[anchor=east] at (8.75000,3.09000) {$2f(x)$};\n\\draw[fill=black!10,draw=black!","900,3.7212) -- (4.6950,3.8983) -- (4.8000,4.0800);\n\\node[] at (3.80000,2.10000) {$2x^2$};\n\\node[] at (3.80000,4.20000) {$4x^2$};\n\\node[] at (5.50000,0.10000) {$x$};\n\\draw[black!25] (6.20000,0.00000) -- (6.20000,4.75000);\n\\node[anchor=east] at (8.75000,3.94000) {$x$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.00000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,3.94000) {$0$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.87000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,3.94000) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (10.74000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,3.94000) {$2$};\n\\node[anchor=east] at (8.75000,3.09000) {$2f(x)$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.00000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,3.09000) {$0$};\n\\draw[fill=black!10,draw=black","t (5.50000,0.10000) {$x$};\n\\draw[black!25] (6.20000,0.00000) -- (6.20000,4.75000);\n\\node[anchor=east] at (8.75000,3.94000) {$x$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.00000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,3.94000) {$0$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.87000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,3.94000) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (10.74000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,3.94000) {$2$};\n\\node[anchor=east] at (8.75000,3.09000) {$2f(x)$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.00000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,3.09000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,3.09000) {$2$};\n\\draw[fill=black!10,draw=blac","l=black!0,draw=black!40,line width=.35pt,] (9.00000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,3.94000) {$0$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.87000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,3.94000) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (10.74000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,3.94000) {$2$};\n\\node[anchor=east] at (8.75000,3.09000) {$2f(x)$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.00000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,3.09000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,3.09000) {$2$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,3.09000) {$8$};\n\\node[anchor=east] at (8.750","0000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,3.94000) {$0$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.87000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,3.94000) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (10.74000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,3.94000) {$2$};\n\\node[anchor=east] at (8.75000,3.09000) {$2f(x)$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.00000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,3.09000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,3.09000) {$2$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,3.09000) {$8$};\n\\node[anchor=east] at (8.75000,2.24000) {$f(2x)$};\n\\draw[fill=black!10,draw=bla","3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,3.94000) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (10.74000,3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,3.94000) {$2$};\n\\node[anchor=east] at (8.75000,3.09000) {$2f(x)$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.00000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,3.09000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,3.09000) {$2$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,3.09000) {$8$};\n\\node[anchor=east] at (8.75000,2.24000) {$f(2x)$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.00000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,2.24000) {$0$};\n\\draw[fill=black!10,draw=bla","3.65000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,3.94000) {$2$};\n\\node[anchor=east] at (8.75000,3.09000) {$2f(x)$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.00000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,3.09000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,3.09000) {$2$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,3.09000) {$8$};\n\\node[anchor=east] at (8.75000,2.24000) {$f(2x)$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.00000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,2.24000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,2.24000) {$4$};\n\\draw[fill=black!10,draw=bla","lack!10,draw=black!40,line width=.35pt,] (9.00000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,3.09000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,3.09000) {$2$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,3.09000) {$8$};\n\\node[anchor=east] at (8.75000,2.24000) {$f(2x)$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.00000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,2.24000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,2.24000) {$4$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,2.24000) {$16$};\n\\node[anchor=east] at (8.75","00,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,3.09000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,3.09000) {$2$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,3.09000) {$8$};\n\\node[anchor=east] at (8.75000,2.24000) {$f(2x)$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.00000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,2.24000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,2.24000) {$4$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,2.24000) {$16$};\n\\node[anchor=east] at (8.75000,1.39000) {$2f(x)-f(2x)$};\n\\draw[fill=black!10,d",".80000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,3.09000) {$2$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,3.09000) {$8$};\n\\node[anchor=east] at (8.75000,2.24000) {$f(2x)$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.00000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,2.24000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,2.24000) {$4$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,2.24000) {$16$};\n\\node[anchor=east] at (8.75000,1.39000) {$2f(x)-f(2x)$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.00000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,1.39000) {$0$};\n\\draw[fill=black!10,d","2.80000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,3.09000) {$8$};\n\\node[anchor=east] at (8.75000,2.24000) {$f(2x)$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.00000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,2.24000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,2.24000) {$4$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,2.24000) {$16$};\n\\node[anchor=east] at (8.75000,1.39000) {$2f(x)-f(2x)$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.00000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,1.39000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,1.39000) {$-2$};\n\\draw[fill=black!10,","lack!10,draw=black!40,line width=.35pt,] (9.00000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,2.24000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,2.24000) {$4$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,2.24000) {$16$};\n\\node[anchor=east] at (8.75000,1.39000) {$2f(x)-f(2x)$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.00000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,1.39000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,1.39000) {$-2$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,1.39000) {$-8$};\n\\node[] at (9.65000,","0,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,2.24000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,2.24000) {$4$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,2.24000) {$16$};\n\\node[anchor=east] at (8.75000,1.39000) {$2f(x)-f(2x)$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.00000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,1.39000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,1.39000) {$-2$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,1.39000) {$-8$};\n\\node[] at (9.65000,-0.05000) {correct chart formula: $f_b(v)=v^2/2$};\n\\end{t","rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,2.24000) {$4$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,1.95000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,2.24000) {$16$};\n\\node[anchor=east] at (8.75000,1.39000) {$2f(x)-f(2x)$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.00000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,1.39000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,1.39000) {$-2$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,1.39000) {$-8$};\n\\node[] at (9.65000,-0.05000) {correct chart formula: $f_b(v)=v^2/2$};\n\\end{tikzpicture}",") rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,2.24000) {$16$};\n\\node[anchor=east] at (8.75000,1.39000) {$2f(x)-f(2x)$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.00000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,1.39000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,1.39000) {$-2$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,1.39000) {$-8$};\n\\node[] at (9.65000,-0.05000) {correct chart formula: $f_b(v)=v^2/2$};\n\\end{tikzpicture}","ack!10,draw=black!40,line width=.35pt,] (9.00000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (9.43500,1.39000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,1.39000) {$-2$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,1.39000) {$-8$};\n\\node[] at (9.65000,-0.05000) {correct chart formula: $f_b(v)=v^2/2$};\n\\end{tikzpicture}","tangle ++(0.87000,0.58000);\n\\node[] at (9.43500,1.39000) {$0$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (10.30500,1.39000) {$-2$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,1.10000) rectangle ++(0.87000,0.58000);\n\\node[] at (11.17500,1.39000) {$-8$};\n\\node[] at (9.65000,-0.05000) {correct chart formula: $f_b(v)=v^2/2$};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\\sffamily\\small,line cap=round,line join=round]\n\\node[font=\\sffamily\\bfseries] at (2.10000,4.45000) {$(x,y)$ chart};\n\\node[font=\\sffamily\\bfseries] at (9.00000,4.45000) {$(u,v)=(x,y+x^2)$ chart};\n\\draw[->] (0.50000,0.30000) -- (4.00000,0.30000);\n\\draw[->] (0.50000,0.30000) -- (0.50000,3.80000);\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (0.50000,0.30000) rectangle ++(2.80000,2.80000);\n\\draw[black!40] (0.50000,0.30000) -- (3.30000,0.30000);\n\\draw[black!40] (0.50000,1.00000) -- (3.30000,1.00000);\n\\draw[black!40] (0.50000,1.70000) -- (3.30000,1.70000);\n\\draw[black!40] (0.50000,2.40000) -- 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width=.35pt,] (0.00000,3.65000) rectangle ++(4.00000,0.75000);\n\\node[] at (2.00000,4.02500) {connection data};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (4.00000,3.65000) rectangle ++(2.00000,0.75000);\n\\node[] at (5.00000,4.02500) {$\\Gamma_{0\\leftarrow2}$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (6.00000,3.65000) rectangle ++(1.80000,0.75000);\n\\node[] at (6.90000,4.02500) {$H$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (7.80000,3.65000) rectangle ++(1.80000,0.75000);\n\\node[] at (8.70000,4.02500) {$D^\\times$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.60000,3.65000) rectangle ++(3.10000,0.75000);\n\\node[] at (11.15000,4.02500) {$\\Gamma_{0\\leftarrow2}\\Gamma_{2\\leftarrow0}$};\n\\draw[fill=black!0,draw=black!40,","dth=.35pt,] (0.00000,3.65000) rectangle ++(4.00000,0.75000);\n\\node[] at (2.00000,4.02500) {connection data};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (4.00000,3.65000) rectangle ++(2.00000,0.75000);\n\\node[] at (5.00000,4.02500) {$\\Gamma_{0\\leftarrow2}$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (6.00000,3.65000) rectangle ++(1.80000,0.75000);\n\\node[] at (6.90000,4.02500) {$H$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (7.80000,3.65000) rectangle ++(1.80000,0.75000);\n\\node[] at (8.70000,4.02500) {$D^\\times$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.60000,3.65000) rectangle ++(3.10000,0.75000);\n\\node[] at (11.15000,4.02500) {$\\Gamma_{0\\leftarrow2}\\Gamma_{2\\leftarrow0}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.00000,2.70000) rectangle ++(4.00000,0.80000);\n\\node[] at (2.00000,3.10000) {self normalization only};\n\\draw[fill=bla","k!40,line width=.35pt,] (4.00000,3.65000) rectangle ++(2.00000,0.75000);\n\\node[] at (5.00000,4.02500) {$\\Gamma_{0\\leftarrow2}$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (6.00000,3.65000) rectangle ++(1.80000,0.75000);\n\\node[] at (6.90000,4.02500) {$H$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (7.80000,3.65000) rectangle ++(1.80000,0.75000);\n\\node[] at (8.70000,4.02500) {$D^\\times$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.60000,3.65000) rectangle ++(3.10000,0.75000);\n\\node[] at (11.15000,4.02500) {$\\Gamma_{0\\leftarrow2}\\Gamma_{2\\leftarrow0}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.00000,2.70000) rectangle ++(4.00000,0.80000);\n\\node[] at (2.00000,3.10000) {self normalization only};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (4.00000,2.70000) rectangle ++(2.00000,0.80000);\n\\node[] at (5.00000,3.10000) {$\\frac{1}{6}$};\n\\draw[fill=b","10,draw=black!40,line width=.35pt,] (6.00000,3.65000) rectangle ++(1.80000,0.75000);\n\\node[] at (6.90000,4.02500) {$H$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (7.80000,3.65000) rectangle ++(1.80000,0.75000);\n\\node[] at (8.70000,4.02500) {$D^\\times$};\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (9.60000,3.65000) rectangle ++(3.10000,0.75000);\n\\node[] at (11.15000,4.02500) {$\\Gamma_{0\\leftarrow2}\\Gamma_{2\\leftarrow0}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.00000,2.70000) rectangle ++(4.00000,0.80000);\n\\node[] at (2.00000,3.10000) {self normalization only};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (4.00000,2.70000) rectangle ++(2.00000,0.80000);\n\\node[] at (5.00000,3.10000) {$\\frac{1}{6}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (6.00000,2.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (6.90000,3.10000) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (7.",") rectangle ++(3.10000,0.75000);\n\\node[] at (11.15000,4.02500) {$\\Gamma_{0\\leftarrow2}\\Gamma_{2\\leftarrow0}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.00000,2.70000) rectangle ++(4.00000,0.80000);\n\\node[] at (2.00000,3.10000) {self normalization only};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (4.00000,2.70000) rectangle ++(2.00000,0.80000);\n\\node[] at (5.00000,3.10000) {$\\frac{1}{6}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (6.00000,2.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (6.90000,3.10000) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (7.80000,2.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (8.70000,3.10000) {$6/5$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.60000,2.70000) rectangle ++(3.10000,0.80000);\n\\node[] at (11.15000,3.10000) {$\\frac{5}{6}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.00000,1.70000) re","e width=.35pt,] (0.00000,2.70000) rectangle ++(4.00000,0.80000);\n\\node[] at (2.00000,3.10000) {self normalization only};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (4.00000,2.70000) rectangle ++(2.00000,0.80000);\n\\node[] at (5.00000,3.10000) {$\\frac{1}{6}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (6.00000,2.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (6.90000,3.10000) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (7.80000,2.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (8.70000,3.10000) {$6/5$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.60000,2.70000) rectangle ++(3.10000,0.80000);\n\\node[] at (11.15000,3.10000) {$\\frac{5}{6}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.00000,1.70000) rectangle ++(4.00000,0.80000);\n\\node[] at (2.00000,2.10000) {reversible};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (4.00000,1.70","!0,draw=black!40,line width=.35pt,] (4.00000,2.70000) rectangle ++(2.00000,0.80000);\n\\node[] at (5.00000,3.10000) {$\\frac{1}{6}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (6.00000,2.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (6.90000,3.10000) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (7.80000,2.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (8.70000,3.10000) {$6/5$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.60000,2.70000) rectangle ++(3.10000,0.80000);\n\\node[] at (11.15000,3.10000) {$\\frac{5}{6}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.00000,1.70000) rectangle ++(4.00000,0.80000);\n\\node[] at (2.00000,2.10000) {reversible};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (4.00000,1.70000) rectangle ++(2.00000,0.80000);\n\\node[] at (5.00000,2.10000) {$\\frac{1}{5}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (6.00","l=black!0,draw=black!40,line width=.35pt,] (6.00000,2.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (6.90000,3.10000) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (7.80000,2.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (8.70000,3.10000) {$6/5$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.60000,2.70000) rectangle ++(3.10000,0.80000);\n\\node[] at (11.15000,3.10000) {$\\frac{5}{6}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.00000,1.70000) rectangle ++(4.00000,0.80000);\n\\node[] at (2.00000,2.10000) {reversible};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (4.00000,1.70000) rectangle ++(2.00000,0.80000);\n\\node[] at (5.00000,2.10000) {$\\frac{1}{5}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (6.00000,1.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (6.90000,2.10000) {$\\frac{6}{5}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (7.800","black!40,line width=.35pt,] (9.60000,2.70000) rectangle ++(3.10000,0.80000);\n\\node[] at (11.15000,3.10000) {$\\frac{5}{6}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.00000,1.70000) rectangle ++(4.00000,0.80000);\n\\node[] at (2.00000,2.10000) {reversible};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (4.00000,1.70000) rectangle ++(2.00000,0.80000);\n\\node[] at (5.00000,2.10000) {$\\frac{1}{5}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (6.00000,1.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (6.90000,2.10000) {$\\frac{6}{5}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (7.80000,1.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (8.70000,2.10000) {$6/5$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.60000,1.70000) rectangle ++(3.10000,0.80000);\n\\node[] at (11.15000,2.10000) {$1$};\n\\node[] at (6.35000,0.50000) {$\\Gamma_{1\\leftarrow0}=2,\\quad\\Gamma_{2\\le","=black!40,line width=.35pt,] (0.00000,1.70000) rectangle ++(4.00000,0.80000);\n\\node[] at (2.00000,2.10000) {reversible};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (4.00000,1.70000) rectangle ++(2.00000,0.80000);\n\\node[] at (5.00000,2.10000) {$\\frac{1}{5}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (6.00000,1.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (6.90000,2.10000) {$\\frac{6}{5}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (7.80000,1.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (8.70000,2.10000) {$6/5$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.60000,1.70000) rectangle ++(3.10000,0.80000);\n\\node[] at (11.15000,2.10000) {$1$};\n\\node[] at (6.35000,0.50000) {$\\Gamma_{1\\leftarrow0}=2,\\quad\\Gamma_{2\\leftarrow1}=3,\\quad\\Gamma_{2\\leftarrow0}=5$};\n\\node[] at (6.35000,-0.20000) {The first row has H = identity at this triangle, but fails its cocycle.}","ack!40,line width=.35pt,] (4.00000,1.70000) rectangle ++(2.00000,0.80000);\n\\node[] at (5.00000,2.10000) {$\\frac{1}{5}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (6.00000,1.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (6.90000,2.10000) {$\\frac{6}{5}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (7.80000,1.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (8.70000,2.10000) {$6/5$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.60000,1.70000) rectangle ++(3.10000,0.80000);\n\\node[] at (11.15000,2.10000) {$1$};\n\\node[] at (6.35000,0.50000) {$\\Gamma_{1\\leftarrow0}=2,\\quad\\Gamma_{2\\leftarrow1}=3,\\quad\\Gamma_{2\\leftarrow0}=5$};\n\\node[] at (6.35000,-0.20000) {The first row has H = identity at this triangle, but fails its cocycle.};\n\\end{tikzpicture}","draw=black!40,line width=.35pt,] (6.00000,1.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (6.90000,2.10000) {$\\frac{6}{5}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (7.80000,1.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (8.70000,2.10000) {$6/5$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.60000,1.70000) rectangle ++(3.10000,0.80000);\n\\node[] at (11.15000,2.10000) {$1$};\n\\node[] at (6.35000,0.50000) {$\\Gamma_{1\\leftarrow0}=2,\\quad\\Gamma_{2\\leftarrow1}=3,\\quad\\Gamma_{2\\leftarrow0}=5$};\n\\node[] at (6.35000,-0.20000) {The first row has H = identity at this triangle, but fails its cocycle.};\n\\end{tikzpicture}","00000,1.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (6.90000,2.10000) {$\\frac{6}{5}$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (7.80000,1.70000) rectangle ++(1.80000,0.80000);\n\\node[] at (8.70000,2.10000) {$6/5$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (9.60000,1.70000) rectangle ++(3.10000,0.80000);\n\\node[] at (11.15000,2.10000) {$1$};\n\\node[] at (6.35000,0.50000) {$\\Gamma_{1\\leftarrow0}=2,\\quad\\Gamma_{2\\leftarrow1}=3,\\quad\\Gamma_{2\\leftarrow0}=5$};\n\\node[] at (6.35000,-0.20000) {The first row has H = identity at this triangle, but fails its cocycle.};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\\sffamily\\small,line cap=round,line join=round]\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (1.00000,1.00000) {$0$};\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (5.00000,1.00000) {$1$};\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (3.00000,4.00000) {$2$};\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (1,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$2$} (5,1);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (5,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$3$} (3,4);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,dashed] (1,1","\\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\\sffamily\\small,line cap=round,line join=round]\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (1.00000,1.00000) {$0$};\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (5.00000,1.00000) {$1$};\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (3.00000,4.00000) {$2$};\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (1,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$2$} (5,1);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (5,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$3$} (3,4);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,dashed] (1,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$5$} (3,4);\n\\node[] at (3.00000,4.800","\\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\\sffamily\\small,line cap=round,line join=round]\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (1.00000,1.00000) {$0$};\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (5.00000,1.00000) {$1$};\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (3.00000,4.00000) {$2$};\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (1,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$2$} (5,1);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (5,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$3$} (3,4);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,dashed] (1,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$5$} (3,4);\n\\node[] at (3.00000,4.80000) {reverse of direct edge: $1/5$};\n\\node[] at (3.00000,0.00000) {forward routes: 5 versu","ine join=round]\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (1.00000,1.00000) {$0$};\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (5.00000,1.00000) {$1$};\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (3.00000,4.00000) {$2$};\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (1,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$2$} (5,1);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (5,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$3$} (3,4);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,dashed] (1,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$5$} (3,4);\n\\node[] at (3.00000,4.80000) {reverse of direct edge: $1/5$};\n\\node[] at (3.00000,0.00000) {forward routes: 5 versus 6};\n\\draw[black!25] (6.00000,-0.35000) -- (6.00000,4.60000);\n\\node[font=\\sffamily\\bfseries] at (9.00000,4.70000) {loo","ill=white,minimum size=7mm,inner sep=1pt,] at (5.00000,1.00000) {$1$};\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (3.00000,4.00000) {$2$};\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (1,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$2$} (5,1);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (5,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$3$} (3,4);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,dashed] (1,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$5$} (3,4);\n\\node[] at (3.00000,4.80000) {reverse of direct edge: $1/5$};\n\\node[] at (3.00000,0.00000) {forward routes: 5 versus 6};\n\\draw[black!25] (6.00000,-0.35000) -- (6.00000,4.60000);\n\\node[font=\\sffamily\\bfseries] at (9.00000,4.70000) {loop input $1$ returns $6/5$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,3.10000) rectangle ++(1.00000,0.6600","] at (3.00000,4.00000) {$2$};\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (1,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$2$} (5,1);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (5,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$3$} (3,4);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,dashed] (1,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$5$} (3,4);\n\\node[] at (3.00000,4.80000) {reverse of direct edge: $1/5$};\n\\node[] at (3.00000,0.00000) {forward routes: 5 versus 6};\n\\draw[black!25] (6.00000,-0.35000) -- (6.00000,4.60000);\n\\node[font=\\sffamily\\bfseries] at (9.00000,4.70000) {loop input $1$ returns $6/5$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,3.10000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,3.43000) {$0$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,3.10000) rectangle ++(3.00000","4mm,shorten >=4mm,] (1,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$2$} (5,1);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (5,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$3$} (3,4);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,dashed] (1,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$5$} (3,4);\n\\node[] at (3.00000,4.80000) {reverse of direct edge: $1/5$};\n\\node[] at (3.00000,0.00000) {forward routes: 5 versus 6};\n\\draw[black!25] (6.00000,-0.35000) -- (6.00000,4.60000);\n\\node[font=\\sffamily\\bfseries] at (9.00000,4.70000) {loop input $1$ returns $6/5$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,3.10000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,3.43000) {$0$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,3.10000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,3.43000) {$1$};\n\\draw[fill=black!8,dr","ootnotesize] {$3$} (3,4);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,dashed] (1,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$5$} (3,4);\n\\node[] at (3.00000,4.80000) {reverse of direct edge: $1/5$};\n\\node[] at (3.00000,0.00000) {forward routes: 5 versus 6};\n\\draw[black!25] (6.00000,-0.35000) -- (6.00000,4.60000);\n\\node[font=\\sffamily\\bfseries] at (9.00000,4.70000) {loop input $1$ returns $6/5$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,3.10000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,3.43000) {$0$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,3.10000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,3.43000) {$1$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,2.32000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,2.65000) {$1$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,2.32000)","{$3$} (3,4);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,dashed] (1,1) -- node[sloped,above,font=\\sffamily\\footnotesize] {$5$} (3,4);\n\\node[] at (3.00000,4.80000) {reverse of direct edge: $1/5$};\n\\node[] at (3.00000,0.00000) {forward routes: 5 versus 6};\n\\draw[black!25] (6.00000,-0.35000) -- (6.00000,4.60000);\n\\node[font=\\sffamily\\bfseries] at (9.00000,4.70000) {loop input $1$ returns $6/5$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,3.10000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,3.43000) {$0$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,3.10000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,3.43000) {$1$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,2.32000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,2.65000) {$1$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,2.32000) rectangle ++(","3,4);\n\\node[] at (3.00000,4.80000) {reverse of direct edge: $1/5$};\n\\node[] at (3.00000,0.00000) {forward routes: 5 versus 6};\n\\draw[black!25] (6.00000,-0.35000) -- (6.00000,4.60000);\n\\node[font=\\sffamily\\bfseries] at (9.00000,4.70000) {loop input $1$ returns $6/5$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,3.10000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,3.43000) {$0$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,3.10000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,3.43000) {$1$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,2.32000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,2.65000) {$1$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,2.32000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,2.65000) {$\\frac{6}{5}$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,1.54000) rec","!25] (6.00000,-0.35000) -- (6.00000,4.60000);\n\\node[font=\\sffamily\\bfseries] at (9.00000,4.70000) {loop input $1$ returns $6/5$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,3.10000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,3.43000) {$0$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,3.10000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,3.43000) {$1$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,2.32000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,2.65000) {$1$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,2.32000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,2.65000) {$\\frac{6}{5}$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,1.54000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,1.87000) {$2$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,1.54000) rec","ill=black!8,draw=black!40,line width=.35pt,] (7.00000,3.10000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,3.43000) {$0$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,3.10000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,3.43000) {$1$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,2.32000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,2.65000) {$1$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,2.32000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,2.65000) {$\\frac{6}{5}$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,1.54000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,1.87000) {$2$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,1.54000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,1.87000) {$\\frac{36}{25}$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,","ll=black!16,draw=black!40,line width=.35pt,] (8.00000,3.10000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,3.43000) {$1$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,2.32000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,2.65000) {$1$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,2.32000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,2.65000) {$\\frac{6}{5}$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,1.54000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,1.87000) {$2$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,1.54000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,1.87000) {$\\frac{36}{25}$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,0.76000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,1.09000) {$3$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,0.76000) r","8,draw=black!40,line width=.35pt,] (7.00000,2.32000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,2.65000) {$1$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,2.32000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,2.65000) {$\\frac{6}{5}$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,1.54000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,1.87000) {$2$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,1.54000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,1.87000) {$\\frac{36}{25}$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,0.76000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,1.09000) {$3$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,0.76000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,1.09000) {$\\frac{216}{125}$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00","6,draw=black!40,line width=.35pt,] (8.00000,2.32000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,2.65000) {$\\frac{6}{5}$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,1.54000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,1.87000) {$2$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,1.54000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,1.87000) {$\\frac{36}{25}$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,0.76000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,1.09000) {$3$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,0.76000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,1.09000) {$\\frac{216}{125}$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,-0.02000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,0.31000) {$4$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,-0.0200","draw=black!40,line width=.35pt,] (7.00000,1.54000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,1.87000) {$2$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,1.54000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,1.87000) {$\\frac{36}{25}$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,0.76000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,1.09000) {$3$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,0.76000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,1.09000) {$\\frac{216}{125}$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,-0.02000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,0.31000) {$4$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,-0.02000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,0.31000) {$\\frac{1296}{625}$};\n\\node[] at (7.50000,4.08000) {loops};\n\\node[] at (9.","draw=black!40,line width=.35pt,] (8.00000,1.54000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,1.87000) {$\\frac{36}{25}$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,0.76000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,1.09000) {$3$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,0.76000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,1.09000) {$\\frac{216}{125}$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,-0.02000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,0.31000) {$4$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,-0.02000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,0.31000) {$\\frac{1296}{625}$};\n\\node[] at (7.50000,4.08000) {loops};\n\\node[] at (9.50000,4.08000) {returned value};\n\\end{tikzpicture}","w=black!40,line width=.35pt,] (7.00000,0.76000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,1.09000) {$3$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,0.76000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,1.09000) {$\\frac{216}{125}$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,-0.02000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,0.31000) {$4$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,-0.02000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,0.31000) {$\\frac{1296}{625}$};\n\\node[] at (7.50000,4.08000) {loops};\n\\node[] at (9.50000,4.08000) {returned value};\n\\end{tikzpicture}","=black!40,line width=.35pt,] (8.00000,0.76000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,1.09000) {$\\frac{216}{125}$};\n\\draw[fill=black!8,draw=black!40,line width=.35pt,] (7.00000,-0.02000) rectangle ++(1.00000,0.66000);\n\\node[] at (7.50000,0.31000) {$4$};\n\\draw[fill=black!16,draw=black!40,line width=.35pt,] (8.00000,-0.02000) rectangle ++(3.00000,0.66000);\n\\node[] at (9.50000,0.31000) {$\\frac{1296}{625}$};\n\\node[] at (7.50000,4.08000) {loops};\n\\node[] at (9.50000,4.08000) {returned value};\n\\end{tikzpicture}","00,0.40000) -- (2.00000,0.40000) -- (2.00000,1.80000) -- (0.60000,1.80000) -- cycle;\n\\draw[fill=black!10,line width=1pt] (0.60000,0.40000) -- (2.00000,0.40000) -- (3.40000,1.80000) -- (2.00000,1.80000) -- cycle;\n\\draw[->,line width=1.4pt] (0.60000,0.40000) -- (2.00000,1.80000);\n\\node[font=\\sffamily\\bfseries] at (2.00000,2.75000) {The unit square under H};\n\\node[anchor=west] at (2.12000,2.02000) {$(1,1)$};\n\\node[] at (6.95000,3.55000) {$H$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (6.20000,2.25000) rectangle ++(0.75000,0.75000);\n\\node[] at (6.57500,2.62500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (6.95000,2.25000) rectangle ++(0.75000,0.75000);\n\\node[] at (7.32500,2.62500) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (6.20000,1.50000) rectangle ++(0.75000,0.75000);\n\\node[] at (6.57500,1.87500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.3","0000,1.80000) -- (0.60000,1.80000) -- cycle;\n\\draw[fill=black!10,line width=1pt] (0.60000,0.40000) -- (2.00000,0.40000) -- (3.40000,1.80000) -- (2.00000,1.80000) -- cycle;\n\\draw[->,line width=1.4pt] (0.60000,0.40000) -- (2.00000,1.80000);\n\\node[font=\\sffamily\\bfseries] at (2.00000,2.75000) {The unit square under H};\n\\node[anchor=west] at (2.12000,2.02000) {$(1,1)$};\n\\node[] at (6.95000,3.55000) {$H$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (6.20000,2.25000) rectangle ++(0.75000,0.75000);\n\\node[] at (6.57500,2.62500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (6.95000,2.25000) rectangle ++(0.75000,0.75000);\n\\node[] at (7.32500,2.62500) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (6.20000,1.50000) rectangle ++(0.75000,0.75000);\n\\node[] at (6.57500,1.87500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (6.95000,1.50000) rectangle ++","0) -- (2.00000,1.80000) -- cycle;\n\\draw[->,line width=1.4pt] (0.60000,0.40000) -- (2.00000,1.80000);\n\\node[font=\\sffamily\\bfseries] at (2.00000,2.75000) {The unit square under H};\n\\node[anchor=west] at (2.12000,2.02000) {$(1,1)$};\n\\node[] at (6.95000,3.55000) {$H$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (6.20000,2.25000) rectangle ++(0.75000,0.75000);\n\\node[] at (6.57500,2.62500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (6.95000,2.25000) rectangle ++(0.75000,0.75000);\n\\node[] at (7.32500,2.62500) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (6.20000,1.50000) rectangle ++(0.75000,0.75000);\n\\node[] at (6.57500,1.87500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (6.95000,1.50000) rectangle ++(0.75000,0.75000);\n\\node[] at (7.32500,1.87500) {$1$};\n\\node[] at (10.45000,3.55000) {$H-I$};\n\\draw[fill=black!0,draw=black!40,line width=","00000,2.75000) {The unit square under H};\n\\node[anchor=west] at (2.12000,2.02000) {$(1,1)$};\n\\node[] at (6.95000,3.55000) {$H$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (6.20000,2.25000) rectangle ++(0.75000,0.75000);\n\\node[] at (6.57500,2.62500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (6.95000,2.25000) rectangle ++(0.75000,0.75000);\n\\node[] at (7.32500,2.62500) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (6.20000,1.50000) rectangle ++(0.75000,0.75000);\n\\node[] at (6.57500,1.87500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (6.95000,1.50000) rectangle ++(0.75000,0.75000);\n\\node[] at (7.32500,1.87500) {$1$};\n\\node[] at (10.45000,3.55000) 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zero};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\\sffamily\\small,line cap=round,line join=round]\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (0.70000,2.20000) {$0$};\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (3.40000,3.75000) {$1$};\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (3.40000,0.65000) {$2$};\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (6.10000,2.20000) {$3$};\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (0.7,2.2) -- node[sloped,above,font=\\sffamily\\footnotesize] {$1$} (3.4,3.75);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (0.7,2.2) -- node[sloped,above,font=","\\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\\sffamily\\small,line cap=round,line join=round]\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (0.70000,2.20000) {$0$};\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (3.40000,3.75000) 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<=4mm,shorten >=4mm,] (0.7,2.2) -- node[sloped,above,font=\\sffamily\\footnotesize] {$1$} (3.4,0.65);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (3.4,3.75) -- node[sloped,above,font=\\sffamily\\footnotesize] {$1$} (6.1,2.2);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,dashed] (3.4,0.65) -- node[sloped,above,","000) {$0$};\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (3.40000,3.75000) {$1$};\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (3.40000,0.65000) {$2$};\n\\node[circle,draw,fill=white,minimum size=7mm,inner sep=1pt,] at (6.10000,2.20000) {$3$};\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (0.7,2.2) -- node[sloped,above,font=\\sffamily\\footnotesize] {$1$} (3.4,3.75);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (0.7,2.2) -- node[sloped,above,font=\\sffamily\\footnotesize] {$1$} (3.4,0.65);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (3.4,3.75) -- 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(6.1,2.2);\n\\node[] at (3.40000,4.70000) {two walks of length two};\n\\draw[black!30] (7.00000,-0.10000) -- (7.00000,4.75000);\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (8.20000,3.40000) rectangle ++(3.00000,0.650","0000) {$3$};\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (0.7,2.2) -- node[sloped,above,font=\\sffamily\\footnotesize] {$1$} (3.4,3.75);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (0.7,2.2) -- node[sloped,above,font=\\sffamily\\footnotesize] {$1$} (3.4,0.65);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (3.4,3.75) -- node[sloped,above,font=\\sffamily\\footnotesize] {$1$} (6.1,2.2);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,dashed] (3.4,0.65) -- node[sloped,above,font=\\sffamily\\footnotesize] {$-1$} (6.1,2.2);\n\\node[] at (3.40000,4.70000) {two walks of length two};\n\\draw[black!30] (7.00000,-0.10000) -- (7.00000,4.75000);\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (8.20000,3.40000) rectangle ++(3.00000,0.65000);\n\\node[] at (9.70000,3.72500) {unsigned count: $2$};\n\\draw[fill=black!14,draw=black!40,line width=.35pt,] (8.20000,2.25000) rectang","3.75);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (0.7,2.2) -- node[sloped,above,font=\\sffamily\\footnotesize] {$1$} (3.4,0.65);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,] (3.4,3.75) -- node[sloped,above,font=\\sffamily\\footnotesize] {$1$} (6.1,2.2);\n\\draw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,dashed] (3.4,0.65) -- node[sloped,above,font=\\sffamily\\footnotesize] {$-1$} (6.1,2.2);\n\\node[] at (3.40000,4.70000) {two walks of length two};\n\\draw[black!30] (7.00000,-0.10000) -- (7.00000,4.75000);\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (8.20000,3.40000) rectangle ++(3.00000,0.65000);\n\\node[] at (9.70000,3.72500) {unsigned count: $2$};\n\\draw[fill=black!14,draw=black!40,line width=.35pt,] (8.20000,2.25000) rectangle ++(3.00000,0.65000);\n\\node[] at (9.70000,2.57500) {upper route: $+1$};\n\\draw[fill=black!14,draw=black!40,line width=.35pt,] (8.20000,1.100","aw[->,line width=.85pt,shorten <=4mm,shorten >=4mm,dashed] (3.4,0.65) -- node[sloped,above,font=\\sffamily\\footnotesize] {$-1$} (6.1,2.2);\n\\node[] at (3.40000,4.70000) {two walks of length two};\n\\draw[black!30] (7.00000,-0.10000) -- (7.00000,4.75000);\n\\draw[fill=black!10,draw=black!40,line width=.35pt,] (8.20000,3.40000) rectangle ++(3.00000,0.65000);\n\\node[] at (9.70000,3.72500) {unsigned count: $2$};\n\\draw[fill=black!14,draw=black!40,line width=.35pt,] (8.20000,2.25000) rectangle ++(3.00000,0.65000);\n\\node[] at (9.70000,2.57500) {upper route: $+1$};\n\\draw[fill=black!14,draw=black!40,line width=.35pt,] (8.20000,1.10000) rectangle ++(3.00000,0.65000);\n\\node[] at (9.70000,1.42500) {lower route: $-1$};\n\\node[] at (9.70000,0.20000) {$(T^2)_{30}=0$};\n\\end{tikzpicture}",".40000,4.70000) {two walks of length two};\n\\draw[black!30] (7.00000,-0.10000) -- 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(9.70000,0.20000) {$(T^2)_{30}=0$};\n\\end{tikzpicture}","00,0.65000);\n\\node[] at (9.70000,3.72500) {unsigned count: $2$};\n\\draw[fill=black!14,draw=black!40,line width=.35pt,] (8.20000,2.25000) rectangle ++(3.00000,0.65000);\n\\node[] at (9.70000,2.57500) {upper route: $+1$};\n\\draw[fill=black!14,draw=black!40,line width=.35pt,] (8.20000,1.10000) rectangle ++(3.00000,0.65000);\n\\node[] at (9.70000,1.42500) {lower route: $-1$};\n\\node[] at (9.70000,0.20000) {$(T^2)_{30}=0$};\n\\end{tikzpicture}","\\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\\sffamily\\small,line cap=round,line join=round]\n\\node[] at (0.67000,4.35000) {$A$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (0.00000,3.13000) rectangle ++(0.67000,0.67000);\n\\node[] at (0.33500,3.46500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (0.67000,3.13000) rectangle ++(0.67000,0.67000);\n\\node[] at (1.00500,3.46500) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.00000,2.46000) 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width=.3","\\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\\sffamily\\small,line cap=round,line join=round]\n\\node[] at (0.67000,4.35000) {$A$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (0.00000,3.13000) rectangle ++(0.67000,0.67000);\n\\node[] at (0.33500,3.46500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (0.67000,3.13000) rectangle ++(0.67000,0.67000);\n\\node[] at (1.00500,3.46500) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.00000,2.46000) rectangle ++(0.67000,0.67000);\n\\node[] at (0.33500,2.79500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (0.67000,2.46000) rectangle ++(0.67000,0.67000);\n\\node[] at (1.00500,2.79500) {$1$};\n\\node[] at (3.67000,4.35000) {$B$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (3.00000,3.13000) rectangle ++(0.67000,0.67000);\n\\node[] at (3.33500,3.46500) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35","ll=black!12,draw=black!40,line width=.35pt,] (0.00000,3.13000) rectangle ++(0.67000,0.67000);\n\\node[] at (0.33500,3.46500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (0.67000,3.13000) rectangle ++(0.67000,0.67000);\n\\node[] at (1.00500,3.46500) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (0.00000,2.46000) rectangle ++(0.67000,0.67000);\n\\node[] at (0.33500,2.79500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (0.67000,2.46000) rectangle ++(0.67000,0.67000);\n\\node[] at (1.00500,2.79500) {$1$};\n\\node[] at (3.67000,4.35000) {$B$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (3.00000,3.13000) rectangle ++(0.67000,0.67000);\n\\node[] at (3.33500,3.46500) {$1$};\n\\draw[fill=black!0,draw=black!40,line width=.35pt,] (3.67000,3.13000) rectangle ++(0.67000,0.67000);\n\\node[] at (4.00500,3.46500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.3","ll=black!12,draw=black!40,line width=.35pt,] (0.67000,3.13000) rectangle ++(0.67000,0.67000);\n\\node[] at 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(11.00000,0.05000);\n\\node[] at (5.10","l=black!0,draw=black!40,line width=.35pt,] (9.00000,3.13000) rectangle ++(0.67000,0.67000);\n\\node[] at (9.33500,3.46500) {$0$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.67000,3.13000) rectangle ++(0.67000,0.67000);\n\\node[] at (10.00500,3.46500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.00000,2.46000) rectangle ++(0.67000,0.67000);\n\\node[] at (9.33500,2.79500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.67000,2.46000) rectangle ++(0.67000,0.67000);\n\\node[] at (10.00500,2.79500) {$1$};\n\\node[] at (5.10000,1.60000) {$e_0=(1,0)\\ \\mapsto\\ (1,0)\\ \\mapsto\\ (1,1)\\ \\mapsto\\ (1,1)$};\n\\node[] at (5.10000,0.75000) {$P_1(H-I)e_0=e_1\\ne0$};\n\\draw[black!30] (0.00000,0.05000) -- (11.00000,0.05000);\n\\node[] at (5.10000,-0.60000) {replace direct edge $C=I$ by $C=BA$: then $C^{-1}BA=I$};\n\\end{tikzpicture}","black!12,draw=black!40,line width=.35pt,] (9.67000,3.13000) rectangle ++(0.67000,0.67000);\n\\node[] at (10.00500,3.46500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.00000,2.46000) rectangle ++(0.67000,0.67000);\n\\node[] at (9.33500,2.79500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.67000,2.46000) rectangle ++(0.67000,0.67000);\n\\node[] at (10.00500,2.79500) {$1$};\n\\node[] at (5.10000,1.60000) {$e_0=(1,0)\\ \\mapsto\\ (1,0)\\ \\mapsto\\ (1,1)\\ \\mapsto\\ (1,1)$};\n\\node[] at (5.10000,0.75000) {$P_1(H-I)e_0=e_1\\ne0$};\n\\draw[black!30] (0.00000,0.05000) -- (11.00000,0.05000);\n\\node[] at (5.10000,-0.60000) {replace direct edge $C=I$ by $C=BA$: then $C^{-1}BA=I$};\n\\end{tikzpicture}","5pt,] (9.67000,3.13000) rectangle ++(0.67000,0.67000);\n\\node[] at (10.00500,3.46500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.00000,2.46000) rectangle ++(0.67000,0.67000);\n\\node[] at (9.33500,2.79500) {$1$};\n\\draw[fill=black!12,draw=black!40,line width=.35pt,] (9.67000,2.46000) rectangle 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