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rangle","display":"\\rangle","kind":"use","occurrences":[281,289]},{"id":"symbol-cb89ae5751eaa9b5","tex":"\\rangle_\\mu","aliases":"⟩_μ rangle mu","display":"\\rangle_\\mu","kind":"use","occurrences":[281,289]}],"contexts":["\\section{The claim worth formalizing}\nThe motivating idea is not that different models of set theory are secretly identical. It is that a family of models may possess a meaningful transformation structure, and that this structure deserves its own calculus. The primitive question changes from ``Which world is the correct one?'' to the more local and testable question\n\\[\n\\text{Which observables are preserved, and what exactly changes along a specified transformation?}\n\\]\nThis does not replace the ordinary semantics of a model. It organizes comparisons between models.","\\section{Worlds, presentations, and the ambient theory}\\label{sec:worlds}\nFix a first-order language $L$ and an $L$-theory $T$. The intended case is the membership language with $T=\\ZFC$, but the algebraic results do not depend on that choice. Work in an external metatheory sufficient to discuss sets, categories, and satisfaction for \\emph{set-sized} structures.","A world is an $L$-structure $M\\models T$. A chosen system of worlds is a small category $\\mathcal C$. Its arrows are specified transformations, not automatically elementary embeddings, isomorphisms, or inclusions. Their meaning must be supplied by an instance of the framework.","\\begin{proposition}[Coordinate changes preserve truth]\\label{prop:coordinates}\nIf $h:M\\cong N$ is an isomorphism of $L$-structures, then for every formula $\\varphi(\\vec x)$ and tuple $\\vec a$ in $M$,\n\\[\nM\\models\\varphi(\\vec a)\\quad\\Longleftrightarrow\\quad\nN\\models\\varphi(h\\vec a).\n\\]\nConsequently, two worlds that disagree on an $L$-sentence are not related by a mere isomorphic change of coordinates in that language.\n\\end{proposition}\n\\begin{proof}\nInduct on formulas. Atomic relations are preserved by the definition of an isomorphism. Boolean connectives commute with the induction. For existential quantification, transport a witness through $h$ in one direction and through $h^{-1}$ in the other.\n\\end{proof}","w\\quad\nN\\models\\varphi(h\\vec a).\n\\]\nConsequently, two worlds that disagree on an $L$-sentence are not related by a mere isomorphic change of coordinates in that language.\n\\end{proposition}\n\\begin{proof}\nInduct on formulas. Atomic relations are preserved by the definition of an isomorphism. Boolean connectives commute with the induction. For existential quantification, transport a witness through $h$ in one direction and through $h^{-1}$ in the other.\n\\end{proof}","\\]\nConsequently, two worlds that disagree on an $L$-sentence are not related by a mere isomorphic change of coordinates in that language.\n\\end{proposition}\n\\begin{proof}\nInduct on formulas. Atomic relations are preserved by the definition of an isomorphism. Boolean connectives commute with the induction. For existential quantification, transport a witness through $h$ in one direction and through $h^{-1}$ in the other.\n\\end{proof}","There is an equally important distinction between transformation and information refinement. In the information order on $\\{\\bot,0,1\\}$, where $\\bot\\leq0$ and $\\bot\\leq1$ but $0$ and $1$ are incomparable, a monotone refinement cannot change a settled $0$ into a settled $1$. A truth-changing world transformation is therefore not a monotone refinement of a fixed partial truth assignment. It may instead change the quantifier domain, the admissible objects, or the semantic realization.","\\section{A common presentation exists, but need not explain anything}\n\\begin{theorem}[Boolean product presentation]\\label{thm:product}\nAssume the axiom of choice in the metatheory. Let $I$ be a nonempty set and let $(M_i)_{i\\in I}$ be a set-indexed family of nonempty $L$-structures. There is a full $\\Pow(I)$-valued structure $\\mathcal M$ such that for each $i\\in I$, quotienting at the principal ultrafilter\n\\[\nU_i=\\{S\\subseteq I:i\\in S\\}\n\\]\nrecovers $M_i$ up to isomorphism. If every $M_i\\models T$, every axiom of $T$ has Boolean value $I$ in $\\mathcal M$.\n\\end{theorem}\n\\begin{proof}\nTake the domain $X=\\prod_{i\\in I}|M_i|$. Interpret function symbols coordinatewise. Set\n\\[\n\\bnorm{x=y}=\\{i:x_i=y_i\\},\\qquad\n\\bnorm{R(\\vec x)}=\\{i:M_i\\models R(\\vec x_i)\\}.\n\\]\nThe equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in","\\section{A common presentation exists, but need not explain anything}\n\\begin{theorem}[Boolean product presentation]\\label{thm:product}\nAssume the axiom of choice in the metatheory. Let $I$ be a nonempty set and let $(M_i)_{i\\in I}$ be a set-indexed family of nonempty $L$-structures. There is a full $\\Pow(I)$-valued structure $\\mathcal M$ such that for each $i\\in I$, quotienting at the principal ultrafilter\n\\[\nU_i=\\{S\\subseteq I:i\\in S\\}\n\\]\nrecovers $M_i$ up to isomorphism. If every $M_i\\models T$, every axiom of $T$ has Boolean value $I$ in $\\mathcal M$.\n\\end{theorem}\n\\begin{proof}\nTake the domain $X=\\prod_{i\\in I}|M_i|$. Interpret function symbols coordinatewise. Set\n\\[\n\\bnorm{x=y}=\\{i:x_i=y_i\\},\\qquad\n\\bnorm{R(\\vec x)}=\\{i:M_i\\models R(\\vec x_i)\\}.\n\\]\nThe equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in $\\Pow(I)$. An induction on formulas gives\n\\","\\section{A common presentation exists, but need not explain anything}\n\\begin{theorem}[Boolean product presentation]\\label{thm:product}\nAssume the axiom of choice in the metatheory. Let $I$ be a nonempty set and let $(M_i)_{i\\in I}$ be a set-indexed family of nonempty $L$-structures. There is a full $\\Pow(I)$-valued structure $\\mathcal M$ such that for each $i\\in I$, quotienting at the principal ultrafilter\n\\[\nU_i=\\{S\\subseteq I:i\\in S\\}\n\\]\nrecovers $M_i$ up to isomorphism. If every $M_i\\models T$, every axiom of $T$ has Boolean value $I$ in $\\mathcal M$.\n\\end{theorem}\n\\begin{proof}\nTake the domain $X=\\prod_{i\\in I}|M_i|$. Interpret function symbols coordinatewise. Set\n\\[\n\\bnorm{x=y}=\\{i:x_i=y_i\\},\\qquad\n\\bnorm{R(\\vec x)}=\\{i:M_i\\models R(\\vec x_i)\\}.\n\\]\nThe equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in $\\Pow(I)$. An induction on formulas gives\n\\begin{equation}\\label{eq:productsemantic","\\section{A common presentation exists, but need not explain anything}\n\\begin{theorem}[Boolean product presentation]\\label{thm:product}\nAssume the axiom of choice in the metatheory. Let $I$ be a nonempty set and let $(M_i)_{i\\in I}$ be a set-indexed family of nonempty $L$-structures. There is a full $\\Pow(I)$-valued structure $\\mathcal M$ such that for each $i\\in I$, quotienting at the principal ultrafilter\n\\[\nU_i=\\{S\\subseteq I:i\\in S\\}\n\\]\nrecovers $M_i$ up to isomorphism. If every $M_i\\models T$, every axiom of $T$ has Boolean value $I$ in $\\mathcal M$.\n\\end{theorem}\n\\begin{proof}\nTake the domain $X=\\prod_{i\\in I}|M_i|$. Interpret function symbols coordinatewise. Set\n\\[\n\\bnorm{x=y}=\\{i:x_i=y_i\\},\\qquad\n\\bnorm{R(\\vec x)}=\\{i:M_i\\models R(\\vec x_i)\\}.\n\\]\nThe equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in $\\Pow(I)$. An induction on formulas gives\n\\begin{equation}\\label{eq:productsemantics}\n\\bnorm{\\varphi(\\vec x)}=\\{i:M_i\\mod","\\section{A common presentation exists, but need not explain anything}\n\\begin{theorem}[Boolean product presentation]\\label{thm:product}\nAssume the axiom of choice in the metatheory. Let $I$ be a nonempty set and let $(M_i)_{i\\in I}$ be a set-indexed family of nonempty $L$-structures. There is a full $\\Pow(I)$-valued structure $\\mathcal M$ such that for each $i\\in I$, quotienting at the principal ultrafilter\n\\[\nU_i=\\{S\\subseteq I:i\\in S\\}\n\\]\nrecovers $M_i$ up to isomorphism. If every $M_i\\models T$, every axiom of $T$ has Boolean value $I$ in $\\mathcal M$.\n\\end{theorem}\n\\begin{proof}\nTake the domain $X=\\prod_{i\\in I}|M_i|$. Interpret function symbols coordinatewise. Set\n\\[\n\\bnorm{x=y}=\\{i:x_i=y_i\\},\\qquad\n\\bnorm{R(\\vec x)}=\\{i:M_i\\models R(\\vec x_i)\\}.\n\\]\nThe equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in $\\Pow(I)$. An induction on formulas gives\n\\begin{equation}\\label{eq:productsemantics}\n\\bnorm{\\varphi(\\vec x)}=\\{i:M_i\\models\\varphi(\\vec x_i)\\}.\n\\end{e","\\section{A common presentation exists, but need not explain anything}\n\\begin{theorem}[Boolean product presentation]\\label{thm:product}\nAssume the axiom of choice in the metatheory. Let $I$ be a nonempty set and let $(M_i)_{i\\in I}$ be a set-indexed family of nonempty $L$-structures. There is a full $\\Pow(I)$-valued structure $\\mathcal M$ such that for each $i\\in I$, quotienting at the principal ultrafilter\n\\[\nU_i=\\{S\\subseteq I:i\\in S\\}\n\\]\nrecovers $M_i$ up to isomorphism. If every $M_i\\models T$, every axiom of $T$ has Boolean value $I$ in $\\mathcal M$.\n\\end{theorem}\n\\begin{proof}\nTake the domain $X=\\prod_{i\\in I}|M_i|$. Interpret function symbols coordinatewise. Set\n\\[\n\\bnorm{x=y}=\\{i:x_i=y_i\\},\\qquad\n\\bnorm{R(\\vec x)}=\\{i:M_i\\models R(\\vec x_i)\\}.\n\\]\nThe equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in $\\Pow(I)$. An induction on formulas gives\n\\begin{equation}\\label{eq:productsemantics}\n\\bnorm{\\varphi(\\vec x)}=\\{i:M_i\\models\\varphi(\\vec x_i)\\}.\n\\end{equation}\nOnly the quantifier","ommon presentation exists, but need not explain anything}\n\\begin{theorem}[Boolean product presentation]\\label{thm:product}\nAssume the axiom of choice in the metatheory. Let $I$ be a nonempty set and let $(M_i)_{i\\in I}$ be a set-indexed family of nonempty $L$-structures. There is a full $\\Pow(I)$-valued structure $\\mathcal M$ such that for each $i\\in I$, quotienting at the principal ultrafilter\n\\[\nU_i=\\{S\\subseteq I:i\\in S\\}\n\\]\nrecovers $M_i$ up to isomorphism. If every $M_i\\models T$, every axiom of $T$ has Boolean value $I$ in $\\mathcal M$.\n\\end{theorem}\n\\begin{proof}\nTake the domain $X=\\prod_{i\\in I}|M_i|$. Interpret function symbols coordinatewise. Set\n\\[\n\\bnorm{x=y}=\\{i:x_i=y_i\\},\\qquad\n\\bnorm{R(\\vec x)}=\\{i:M_i\\models R(\\vec x_i)\\}.\n\\]\nThe equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in $\\Pow(I)$. An induction on formulas gives\n\\begin{equation}\\label{eq:productsemantics}\n\\bnorm{\\varphi(\\vec x)}=\\{i:M_i\\models\\varphi(\\vec x_i)\\}.\n\\end{equation}\nOnly the quantifier step requires comment. A coordinate $i$ belon…","plain anything}\n\\begin{theorem}[Boolean product presentation]\\label{thm:product}\nAssume the axiom of choice in the metatheory. Let $I$ be a nonempty set and let $(M_i)_{i\\in I}$ be a set-indexed family of nonempty $L$-structures. There is a full $\\Pow(I)$-valued structure $\\mathcal M$ such that for each $i\\in I$, quotienting at the principal ultrafilter\n\\[\nU_i=\\{S\\subseteq I:i\\in S\\}\n\\]\nrecovers $M_i$ up to isomorphism. If every $M_i\\models T$, every axiom of $T$ has Boolean value $I$ in $\\mathcal M$.\n\\end{theorem}\n\\begin{proof}\nTake the domain $X=\\prod_{i\\in I}|M_i|$. Interpret function symbols coordinatewise. Set\n\\[\n\\bnorm{x=y}=\\{i:x_i=y_i\\},\\qquad\n\\bnorm{R(\\vec x)}=\\{i:M_i\\models R(\\vec x_i)\\}.\n\\]\nThe equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in $\\Pow(I)$. An induction on formulas gives\n\\begin{equation}\\label{eq:productsemantics}\n\\bnorm{\\varphi(\\vec x)}=\\{i:M_i\\models\\varphi(\\vec x_i)\\}.\n\\end{equation}\nOnly the quantifier step requires comment. A coordinate $i$ belongs to $\\bigcup_{y\\in X}\\bnorm{\\varphi(y,\\v…","olean product presentation]\\label{thm:product}\nAssume the axiom of choice in the metatheory. Let $I$ be a nonempty set and let $(M_i)_{i\\in I}$ be a set-indexed family of nonempty $L$-structures. There is a full $\\Pow(I)$-valued structure $\\mathcal M$ such that for each $i\\in I$, quotienting at the principal ultrafilter\n\\[\nU_i=\\{S\\subseteq I:i\\in S\\}\n\\]\nrecovers $M_i$ up to isomorphism. If every $M_i\\models T$, every axiom of $T$ has Boolean value $I$ in $\\mathcal M$.\n\\end{theorem}\n\\begin{proof}\nTake the domain $X=\\prod_{i\\in I}|M_i|$. Interpret function symbols coordinatewise. Set\n\\[\n\\bnorm{x=y}=\\{i:x_i=y_i\\},\\qquad\n\\bnorm{R(\\vec x)}=\\{i:M_i\\models R(\\vec x_i)\\}.\n\\]\nThe equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in $\\Pow(I)$. An induction on formulas gives\n\\begin{equation}\\label{eq:productsemantics}\n\\bnorm{\\varphi(\\vec x)}=\\{i:M_i\\models\\varphi(\\vec x_i)\\}.\n\\end{equation}\nOnly the quantifier step requires comment. A coordinate $i$ belongs to $\\bigcup_{y\\in X}\\bnorm{\\varphi(y,\\vec x)}$ precisely when a witness e…","el{thm:product}\nAssume the axiom of choice in the metatheory. Let $I$ be a nonempty set and let $(M_i)_{i\\in I}$ be a set-indexed family of nonempty $L$-structures. There is a full $\\Pow(I)$-valued structure $\\mathcal M$ such that for each $i\\in I$, quotienting at the principal ultrafilter\n\\[\nU_i=\\{S\\subseteq I:i\\in S\\}\n\\]\nrecovers $M_i$ up to isomorphism. If every $M_i\\models T$, every axiom of $T$ has Boolean value $I$ in $\\mathcal M$.\n\\end{theorem}\n\\begin{proof}\nTake the domain $X=\\prod_{i\\in I}|M_i|$. Interpret function symbols coordinatewise. Set\n\\[\n\\bnorm{x=y}=\\{i:x_i=y_i\\},\\qquad\n\\bnorm{R(\\vec x)}=\\{i:M_i\\models R(\\vec x_i)\\}.\n\\]\nThe equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in $\\Pow(I)$. An induction on formulas gives\n\\begin{equation}\\label{eq:productsemantics}\n\\bnorm{\\varphi(\\vec x)}=\\{i:M_i\\models\\varphi(\\vec x_i)\\}.\n\\end{equation}\nOnly the quantifier step requires comment. A coordinate $i$ belongs to $\\bigcup_{y\\in X}\\bnorm{\\varphi(y,\\vec x)}$ precisely when a witness exists in $M_i$. To see this fro…","the axiom of choice in the metatheory. Let $I$ be a nonempty set and let $(M_i)_{i\\in I}$ be a set-indexed family of nonempty $L$-structures. There is a full $\\Pow(I)$-valued structure $\\mathcal M$ such that for each $i\\in I$, quotienting at the principal ultrafilter\n\\[\nU_i=\\{S\\subseteq I:i\\in S\\}\n\\]\nrecovers $M_i$ up to isomorphism. If every $M_i\\models T$, every axiom of $T$ has Boolean value $I$ in $\\mathcal M$.\n\\end{theorem}\n\\begin{proof}\nTake the domain $X=\\prod_{i\\in I}|M_i|$. Interpret function symbols coordinatewise. Set\n\\[\n\\bnorm{x=y}=\\{i:x_i=y_i\\},\\qquad\n\\bnorm{R(\\vec x)}=\\{i:M_i\\models R(\\vec x_i)\\}.\n\\]\nThe equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in $\\Pow(I)$. An induction on formulas gives\n\\begin{equation}\\label{eq:productsemantics}\n\\bnorm{\\varphi(\\vec x)}=\\{i:M_i\\models\\varphi(\\vec x_i)\\}.\n\\end{equation}\nOnly the quantifier step requires comment. A coordinate $i$ belongs to $\\bigcup_{y\\in X}\\bnorm{\\varphi(y,\\vec x)}$ precisely when a witness exists in $M_i$. To see this from right to left, fill t","iom of choice in the metatheory. Let $I$ be a nonempty set and let $(M_i)_{i\\in I}$ be a set-indexed family of nonempty $L$-structures. There is a full $\\Pow(I)$-valued structure $\\mathcal M$ such that for each $i\\in I$, quotienting at the principal ultrafilter\n\\[\nU_i=\\{S\\subseteq I:i\\in S\\}\n\\]\nrecovers $M_i$ up to isomorphism. If every $M_i\\models T$, every axiom of $T$ has Boolean value $I$ in $\\mathcal M$.\n\\end{theorem}\n\\begin{proof}\nTake the domain $X=\\prod_{i\\in I}|M_i|$. Interpret function symbols coordinatewise. Set\n\\[\n\\bnorm{x=y}=\\{i:x_i=y_i\\},\\qquad\n\\bnorm{R(\\vec x)}=\\{i:M_i\\models R(\\vec x_i)\\}.\n\\]\nThe equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in $\\Pow(I)$. An induction on formulas gives\n\\begin{equation}\\label{eq:productsemantics}\n\\bnorm{\\varphi(\\vec x)}=\\{i:M_i\\models\\varphi(\\vec x_i)\\}.\n\\end{equation}\nOnly the quantifier step requires comment. A coordinate $i$ belongs to $\\bigcup_{y\\in X}\\bnorm{\\varphi(y,\\vec x)}$ precisely when a witness exists in $M_i$. To see this from right to left, fill the oth…","and let $(M_i)_{i\\in I}$ be a set-indexed family of nonempty $L$-structures. There is a full $\\Pow(I)$-valued structure $\\mathcal M$ such that for each $i\\in I$, quotienting at the principal ultrafilter\n\\[\nU_i=\\{S\\subseteq I:i\\in S\\}\n\\]\nrecovers $M_i$ up to isomorphism. If every $M_i\\models T$, every axiom of $T$ has Boolean value $I$ in $\\mathcal M$.\n\\end{theorem}\n\\begin{proof}\nTake the domain $X=\\prod_{i\\in I}|M_i|$. Interpret function symbols coordinatewise. Set\n\\[\n\\bnorm{x=y}=\\{i:x_i=y_i\\},\\qquad\n\\bnorm{R(\\vec x)}=\\{i:M_i\\models R(\\vec x_i)\\}.\n\\]\nThe equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in $\\Pow(I)$. An induction on formulas gives\n\\begin{equation}\\label{eq:productsemantics}\n\\bnorm{\\varphi(\\vec x)}=\\{i:M_i\\models\\varphi(\\vec x_i)\\}.\n\\end{equation}\nOnly the quantifier step requires comment. A coordinate $i$ belongs to $\\bigcup_{y\\in X}\\bnorm{\\varphi(y,\\vec x)}$ precisely when a witness exists in $M_i$. To see this from right to left, fill the other coordinates using a chosen family of default elements. T…","res. There is a full $\\Pow(I)$-valued structure $\\mathcal M$ such that for each $i\\in I$, quotienting at the principal ultrafilter\n\\[\nU_i=\\{S\\subseteq I:i\\in S\\}\n\\]\nrecovers $M_i$ up to isomorphism. If every $M_i\\models T$, every axiom of $T$ has Boolean value $I$ in $\\mathcal M$.\n\\end{theorem}\n\\begin{proof}\nTake the domain $X=\\prod_{i\\in I}|M_i|$. Interpret function symbols coordinatewise. Set\n\\[\n\\bnorm{x=y}=\\{i:x_i=y_i\\},\\qquad\n\\bnorm{R(\\vec x)}=\\{i:M_i\\models R(\\vec x_i)\\}.\n\\]\nThe equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in $\\Pow(I)$. An induction on formulas gives\n\\begin{equation}\\label{eq:productsemantics}\n\\bnorm{\\varphi(\\vec x)}=\\{i:M_i\\models\\varphi(\\vec x_i)\\}.\n\\end{equation}\nOnly the quantifier step requires comment. A coordinate $i$ belongs to $\\bigcup_{y\\in X}\\bnorm{\\varphi(y,\\vec x)}$ precisely when a witness exists in $M_i$. To see this from right to left, fill the other coordinates using a chosen family of default elements. To obtain fullness, choose witnesses simultaneously at every coordinate w…","M_i\\models T$, every axiom of $T$ has Boolean value $I$ in $\\mathcal M$.\n\\end{theorem}\n\\begin{proof}\nTake the domain $X=\\prod_{i\\in I}|M_i|$. Interpret function symbols coordinatewise. Set\n\\[\n\\bnorm{x=y}=\\{i:x_i=y_i\\},\\qquad\n\\bnorm{R(\\vec x)}=\\{i:M_i\\models R(\\vec x_i)\\}.\n\\]\nThe equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in $\\Pow(I)$. An induction on formulas gives\n\\begin{equation}\\label{eq:productsemantics}\n\\bnorm{\\varphi(\\vec x)}=\\{i:M_i\\models\\varphi(\\vec x_i)\\}.\n\\end{equation}\nOnly the quantifier step requires comment. A coordinate $i$ belongs to $\\bigcup_{y\\in X}\\bnorm{\\varphi(y,\\vec x)}$ precisely when a witness exists in $M_i$. To see this from right to left, fill the other coordinates using a chosen family of default elements. To obtain fullness, choose witnesses simultaneously at every coordinate where one exists, and default elements elsewhere. The resulting $y\\in X$ attains the Boolean value of the existential formula.","n $\\mathcal M$.\n\\end{theorem}\n\\begin{proof}\nTake the domain $X=\\prod_{i\\in I}|M_i|$. Interpret function symbols coordinatewise. Set\n\\[\n\\bnorm{x=y}=\\{i:x_i=y_i\\},\\qquad\n\\bnorm{R(\\vec x)}=\\{i:M_i\\models R(\\vec x_i)\\}.\n\\]\nThe equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in $\\Pow(I)$. An induction on formulas gives\n\\begin{equation}\\label{eq:productsemantics}\n\\bnorm{\\varphi(\\vec x)}=\\{i:M_i\\models\\varphi(\\vec x_i)\\}.\n\\end{equation}\nOnly the quantifier step requires comment. A coordinate $i$ belongs to $\\bigcup_{y\\in X}\\bnorm{\\varphi(y,\\vec x)}$ precisely when a witness exists in $M_i$. To see this from right to left, fill the other coordinates using a chosen family of default elements. To obtain fullness, choose witnesses simultaneously at every coordinate where one exists, and default elements elsewhere. The resulting $y\\in X$ attains the Boolean value of the existential formula.","},\\qquad\n\\bnorm{R(\\vec x)}=\\{i:M_i\\models R(\\vec x_i)\\}.\n\\]\nThe equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in $\\Pow(I)$. An induction on formulas gives\n\\begin{equation}\\label{eq:productsemantics}\n\\bnorm{\\varphi(\\vec x)}=\\{i:M_i\\models\\varphi(\\vec x_i)\\}.\n\\end{equation}\nOnly the quantifier step requires comment. A coordinate $i$ belongs to $\\bigcup_{y\\in X}\\bnorm{\\varphi(y,\\vec x)}$ precisely when a witness exists in $M_i$. To see this from right to left, fill the other coordinates using a chosen family of default elements. To obtain fullness, choose witnesses simultaneously at every coordinate where one exists, and default elements elsewhere. The resulting $y\\in X$ attains the Boolean value of the existential formula.","{R(\\vec x)}=\\{i:M_i\\models R(\\vec x_i)\\}.\n\\]\nThe equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in $\\Pow(I)$. An induction on formulas gives\n\\begin{equation}\\label{eq:productsemantics}\n\\bnorm{\\varphi(\\vec x)}=\\{i:M_i\\models\\varphi(\\vec x_i)\\}.\n\\end{equation}\nOnly the quantifier step requires comment. A coordinate $i$ belongs to $\\bigcup_{y\\in X}\\bnorm{\\varphi(y,\\vec x)}$ precisely when a witness exists in $M_i$. To see this from right to left, fill the other coordinates using a chosen family of default elements. To obtain fullness, choose witnesses simultaneously at every coordinate where one exists, and default elements elsewhere. The resulting $y\\in X$ attains the Boolean value of the existential formula.","ms hold coordinatewise. Extend the semantics by complements, intersections, and joins in $\\Pow(I)$. An induction on formulas gives\n\\begin{equation}\\label{eq:productsemantics}\n\\bnorm{\\varphi(\\vec x)}=\\{i:M_i\\models\\varphi(\\vec x_i)\\}.\n\\end{equation}\nOnly the quantifier step requires comment. A coordinate $i$ belongs to $\\bigcup_{y\\in X}\\bnorm{\\varphi(y,\\vec x)}$ precisely when a witness exists in $M_i$. To see this from right to left, fill the other coordinates using a chosen family of default elements. To obtain fullness, choose witnesses simultaneously at every coordinate where one exists, and default elements elsewhere. The resulting $y\\in X$ attains the Boolean value of the existential formula.","on}\nOnly the quantifier step requires comment. A coordinate $i$ belongs to $\\bigcup_{y\\in X}\\bnorm{\\varphi(y,\\vec x)}$ precisely when a witness exists in $M_i$. To see this from right to left, fill the other coordinates using a chosen family of default elements. To obtain fullness, choose witnesses simultaneously at every coordinate where one exists, and default elements elsewhere. The resulting $y\\in X$ attains the Boolean value of the existential formula.","Modulo $U_i$, equality is exactly equality of the $i$th coordinate. The map $[x]_{U_i}\\mapsto x_i$ is well-defined, injective, and surjective. It preserves functions and relations, so it is an isomorphism to $M_i$. Equation~\\eqref{eq:productsemantics} also gives Boolean value $I$ to each axiom satisfied by every coordinate structure.\n\\end{proof}","\\begin{corollary}\\label{cor:resolution-not-explanation}\nEvery supplied set-indexed family of models of $T$ can be regarded as a family of two-valued resolutions of a richer Boolean-valued structure. This conclusion alone imposes no transformation relation on distinct models in the family.\n\\end{corollary}\n\\begin{proof}\nThe construction requires only the family of structures. It does not use any maps between them. Consequently, its existence cannot by itself establish additional relationships among the coordinates.\n\\end{proof}","\\begin{definition}[Resolution datum]\\label{def:datum}\nA resolution datum consists of a small category $\\mathcal C$, a unital Boolean algebra $A$, a functor\n\\[\nB:\\mathcal C\\longrightarrow\\Bool,\n\\]\nand a unital Boolean homomorphism $v_M:A\\to B_M$ for each object $M$. Write $B_f:B_M\\to B_N$ for the map attached to $f:M\\to N$.\n\\end{definition}\nThe category $\\Bool$ here uses ordinary Boolean homomorphisms. It does \\emph{not} require preservation of arbitrary joins. In Boolean-ring notation, $+$ is symmetric difference, multiplication is conjunction, and $\\neg a=1+a$. In particular, $a+a=0$.","category $\\mathcal C$, a unital Boolean algebra $A$, a functor\n\\[\nB:\\mathcal C\\longrightarrow\\Bool,\n\\]\nand a unital Boolean homomorphism $v_M:A\\to B_M$ for each object $M$. Write $B_f:B_M\\to B_N$ for the map attached to $f:M\\to N$.\n\\end{definition}\nThe category $\\Bool$ here uses ordinary Boolean homomorphisms. It does \\emph{not} require preservation of arbitrary joins. In Boolean-ring notation, $+$ is symmetric difference, multiplication is conjunction, and $\\neg a=1+a$. In particular, $a+a=0$.","\\[\nB:\\mathcal C\\longrightarrow\\Bool,\n\\]\nand a unital Boolean homomorphism $v_M:A\\to B_M$ for each object $M$. Write $B_f:B_M\\to B_N$ for the map attached to $f:M\\to N$.\n\\end{definition}\nThe category $\\Bool$ here uses ordinary Boolean homomorphisms. It does \\emph{not} require preservation of arbitrary joins. In Boolean-ring notation, $+$ is symmetric difference, multiplication is conjunction, and $\\neg a=1+a$. In particular, $a+a=0$.","w\\Bool,\n\\]\nand a unital Boolean homomorphism $v_M:A\\to B_M$ for each object $M$. Write $B_f:B_M\\to B_N$ for the map attached to $f:M\\to N$.\n\\end{definition}\nThe category $\\Bool$ here uses ordinary Boolean homomorphisms. It does \\emph{not} require preservation of arbitrary joins. In Boolean-ring notation, $+$ is symmetric difference, multiplication is conjunction, and $\\neg a=1+a$. In particular, $a+a=0$.","For sentences, $A$ can be the Lindenbaum Boolean algebra of $L$-sentences modulo provable equivalence over $T$. An ordinary model gives $B_M=\\F$ and $v_M([\\varphi])=1$ precisely when $M\\models\\varphi$. The datum can also be studied without claiming that its valuations arise from models of ZFC.","\\begin{definition}[Transport defect]\nFor $f:M\\to N$ define\n\\begin{equation}\\label{eq:defect}\nD_f(a)=B_f(v_M(a))+v_N(a)\\in B_N.\n\\end{equation}\n\\end{definition}\nThe terms on the right live in the same algebra. A proposed residual without this typing condition is not a mathematical definition.","\\begin{theorem}[Defect calculus]\\label{thm:calculus}\nPut $u=B_f\\circ v_M$ and $w=v_N$. Then\n\\begin{align}\nD_f(0)&=D_f(1)=0,\\qquad D_f(\\neg a)=D_f(a),","\\begin{theorem}[Defect calculus]\\label{thm:calculus}\nPut $u=B_f\\circ v_M$ and $w=v_N$. Then\n\\begin{align}\nD_f(0)&=D_f(1)=0,\\qquad D_f(\\neg a)=D_f(a),\\\\\nD_f(a+b)&=D_f(a)+D_f(b),\\\\\nD_f(ab)&=D_f(a)u(b)+w(a)D_f(b).\\label{eq:leibniz}\n\\end{align}\nFor composable arrows $M\\xrightarrow{f}N\\xrightarrow{g}P$,\n\\begin{equation}\\label{eq:cocycle}\nD_{gf}(a)=B_g(D_f(a))+D_g(a),\\qquad D_{\\id_M}(a)=0.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nBoth $u$ and $w$ preserve $0$, $1$, addition, and multiplication. The first identities follow immediately, including\n\\[\nD_f(1+a)=(1+u(a))+(1+w(a))=D_f(a).\n\\]\nFor the product rule, expand\n\\[\n(u(a)+w(a))u(b)+w(a)(u(b)+w(b))\n=u(a)u(b)+w(a)w(b).\n\\]\nThe mixed terms cancel in characteristic two. For composition, functoriality and additivity give\n\\begin{align*}\nB_g(D_f(a))+D_g(a)\n&=B_gB_fv_M(a)+B_gv_N(a)+B_gv_N(a)+v_P(a)","D_f(ab)&=D_f(a)u(b)+w(a)D_f(b).\\label{eq:leibniz}\n\\end{align}\nFor composable arrows $M\\xrightarrow{f}N\\xrightarrow{g}P$,\n\\begin{equation}\\label{eq:cocycle}\nD_{gf}(a)=B_g(D_f(a))+D_g(a),\\qquad D_{\\id_M}(a)=0.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nBoth $u$ and $w$ preserve $0$, $1$, addition, and multiplication. The first identities follow immediately, including\n\\[\nD_f(1+a)=(1+u(a))+(1+w(a))=D_f(a).\n\\]\nFor the product rule, expand\n\\[\n(u(a)+w(a))u(b)+w(a)(u(b)+w(b))\n=u(a)u(b)+w(a)w(b).\n\\]\nThe mixed terms cancel in characteristic two. For composition, functoriality and additivity give\n\\begin{align*}\nB_g(D_f(a))+D_g(a)\n&=B_gB_fv_M(a)+B_gv_N(a)+B_gv_N(a)+v_P(a)",":leibniz}\n\\end{align}\nFor composable arrows $M\\xrightarrow{f}N\\xrightarrow{g}P$,\n\\begin{equation}\\label{eq:cocycle}\nD_{gf}(a)=B_g(D_f(a))+D_g(a),\\qquad D_{\\id_M}(a)=0.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nBoth $u$ and $w$ preserve $0$, $1$, addition, and multiplication. The first identities follow immediately, including\n\\[\nD_f(1+a)=(1+u(a))+(1+w(a))=D_f(a).\n\\]\nFor the product rule, expand\n\\[\n(u(a)+w(a))u(b)+w(a)(u(b)+w(b))\n=u(a)u(b)+w(a)w(b).\n\\]\nThe mixed terms cancel in characteristic two. For composition, functoriality and additivity give\n\\begin{align*}\nB_g(D_f(a))+D_g(a)\n&=B_gB_fv_M(a)+B_gv_N(a)+B_gv_N(a)+v_P(a)","\\end{equation}\n\\end{theorem}\n\\begin{proof}\nBoth $u$ and $w$ preserve $0$, $1$, addition, and multiplication. The first identities follow immediately, including\n\\[\nD_f(1+a)=(1+u(a))+(1+w(a))=D_f(a).\n\\]\nFor the product rule, expand\n\\[\n(u(a)+w(a))u(b)+w(a)(u(b)+w(b))\n=u(a)u(b)+w(a)w(b).\n\\]\nThe mixed terms cancel in characteristic two. For composition, functoriality and additivity give\n\\begin{align*}\nB_g(D_f(a))+D_g(a)\n&=B_gB_fv_M(a)+B_gv_N(a)+B_gv_N(a)+v_P(a)\\\\\n&=B_{gf}v_M(a)+v_P(a).\n\\end{align*}\nThe identity case follows from $B_{\\id_M}=\\id$.\n\\end{proof}","&=B_{gf}v_M(a)+v_P(a).\n\\end{align*}\nThe identity case follows from $B_{\\id_M}=\\id$.\n\\end{proof}","The product rule is a twisted derivation identity between two homomorphisms. It does not make $D_f$ a ring homomorphism or an ordinary derivation relative to one fixed algebra action.","\\begin{proposition}[Product telescope]\\label{prop:producttelescope}\nFor $a_1,\\ldots,a_n\\in A$,\n\\[\nD_f\\left(\\prod_{i=1}^n a_i\\right)\n=\\sum_{k=1}^n\\left(\\prod_{i<k}w(a_i)\\right)\nD_f(a_k)\\left(\\prod_{i>k}u(a_i)\\right).\n\\]\n\\end{proposition}\n\\begin{proof}\nExpand $D_f(a_k)=u(a_k)+w(a_k)$. Every intermediate product occurs twice, once with transition index $k$ and once with transition index $k+1$. They cancel, leaving the all-$u$ product plus the all-$w$ product. This also gives an induction on $n$, with the empty product case equal to $D_f(1)=0$.\n\\end{proof}","uct telescope]\\label{prop:producttelescope}\nFor $a_1,\\ldots,a_n\\in A$,\n\\[\nD_f\\left(\\prod_{i=1}^n a_i\\right)\n=\\sum_{k=1}^n\\left(\\prod_{i<k}w(a_i)\\right)\nD_f(a_k)\\left(\\prod_{i>k}u(a_i)\\right).\n\\]\n\\end{proposition}\n\\begin{proof}\nExpand $D_f(a_k)=u(a_k)+w(a_k)$. Every intermediate product occurs twice, once with transition index $k$ and once with transition index $k+1$. They cancel, leaving the all-$u$ product plus the all-$w$ product. This also gives an induction on $n$, with the empty product case equal to $D_f(1)=0$.\n\\end{proof}",":producttelescope}\nFor $a_1,\\ldots,a_n\\in A$,\n\\[\nD_f\\left(\\prod_{i=1}^n a_i\\right)\n=\\sum_{k=1}^n\\left(\\prod_{i<k}w(a_i)\\right)\nD_f(a_k)\\left(\\prod_{i>k}u(a_i)\\right).\n\\]\n\\end{proposition}\n\\begin{proof}\nExpand $D_f(a_k)=u(a_k)+w(a_k)$. Every intermediate product occurs twice, once with transition index $k$ and once with transition index $k+1$. They cancel, leaving the all-$u$ product plus the all-$w$ product. This also gives an induction on $n$, with the empty product case equal to $D_f(1)=0$.\n\\end{proof}","\\[\nD_f\\left(\\prod_{i=1}^n a_i\\right)\n=\\sum_{k=1}^n\\left(\\prod_{i<k}w(a_i)\\right)\nD_f(a_k)\\left(\\prod_{i>k}u(a_i)\\right).\n\\]\n\\end{proposition}\n\\begin{proof}\nExpand $D_f(a_k)=u(a_k)+w(a_k)$. Every intermediate product occurs twice, once with transition index $k$ and once with transition index $k+1$. They cancel, leaving the all-$u$ product plus the all-$w$ product. This also gives an induction on $n$, with the empty product case equal to $D_f(1)=0$.\n\\end{proof}","m_{k=1}^n\\left(\\prod_{i<k}w(a_i)\\right)\nD_f(a_k)\\left(\\prod_{i>k}u(a_i)\\right).\n\\]\n\\end{proposition}\n\\begin{proof}\nExpand $D_f(a_k)=u(a_k)+w(a_k)$. Every intermediate product occurs twice, once with transition index $k$ and once with transition index $k+1$. They cancel, leaving the all-$u$ product plus the all-$w$ product. This also gives an induction on $n$, with the empty product case equal to $D_f(1)=0$.\n\\end{proof}","\\subsection{Transported syntax and parameters}\nFor parameters or changing languages, replace $A$ by a functor $A_M$ with translation maps $\\alpha_f:A_M\\to A_N$. The correct defect is\n\\begin{equation}\\label{eq:typeddefect}\nD_f(a)=B_fv_M(a)+v_N\\alpha_f(a),\\qquad a\\in A_M.\n\\end{equation}\nAssuming both transport systems are functorial, the composition law becomes\n\\[\nD_{gf}(a)=B_gD_f(a)+D_g(\\alpha_f(a)).\n\\]\nThe same cancellation proves it. The fixed-$A$ theory is the special case $\\alpha_f=\\id$. A single aggregate quotient of $A$ will only be used in that special case. This prevents a silent comparison of formulas whose parameters have no specified transport.","parameters or changing languages, replace $A$ by a functor $A_M$ with translation maps $\\alpha_f:A_M\\to A_N$. The correct defect is\n\\begin{equation}\\label{eq:typeddefect}\nD_f(a)=B_fv_M(a)+v_N\\alpha_f(a),\\qquad a\\in A_M.\n\\end{equation}\nAssuming both transport systems are functorial, the composition law becomes\n\\[\nD_{gf}(a)=B_gD_f(a)+D_g(\\alpha_f(a)).\n\\]\nThe same cancellation proves it. The fixed-$A$ theory is the special case $\\alpha_f=\\id$. A single aggregate quotient of $A$ will only be used in that special case. This prevents a silent comparison of formulas whose parameters have no specified transport.","es, replace $A$ by a functor $A_M$ with translation maps $\\alpha_f:A_M\\to A_N$. The correct defect is\n\\begin{equation}\\label{eq:typeddefect}\nD_f(a)=B_fv_M(a)+v_N\\alpha_f(a),\\qquad a\\in A_M.\n\\end{equation}\nAssuming both transport systems are functorial, the composition law becomes\n\\[\nD_{gf}(a)=B_gD_f(a)+D_g(\\alpha_f(a)).\n\\]\nThe same cancellation proves it. The fixed-$A$ theory is the special case $\\alpha_f=\\id$. A single aggregate quotient of $A$ will only be used in that special case. This prevents a silent comparison of formulas whose parameters have no specified transport.","tion maps $\\alpha_f:A_M\\to A_N$. The correct defect is\n\\begin{equation}\\label{eq:typeddefect}\nD_f(a)=B_fv_M(a)+v_N\\alpha_f(a),\\qquad a\\in A_M.\n\\end{equation}\nAssuming both transport systems are functorial, the composition law becomes\n\\[\nD_{gf}(a)=B_gD_f(a)+D_g(\\alpha_f(a)).\n\\]\nThe same cancellation proves it. The fixed-$A$ theory is the special case $\\alpha_f=\\id$. A single aggregate quotient of $A$ will only be used in that special case. This prevents a silent comparison of formulas whose parameters have no specified transport.","\\section{What the invariant core actually is}\\label{sec:core}\nFor the fixed-$A$ datum set\n\\[\nC^0=\\prod_{M\\in\\Ob(\\mathcal C)}B_M,\\qquad\nC^1=\\prod_{f\\in\\Mor(\\mathcal C)}B_{\\operatorname{target}(f)}.\n\\]\nAs additive groups these are vector spaces over $\\F$. Define\n\\[\n(d^0c)_f=B_f(c_M)+c_N,\\qquad\n\\Ev(a)=(v_M(a))_M,\\qquad D=d^0\\Ev.\n\\]\nHere $\\Ev$ is a Boolean homomorphism, while $d^0$ is generally only additive.","\\Needspace{12\\baselineskip}\n\\begin{theorem}[Invariant algebra and residual quotient]\\label{thm:exactsequence}\nThe set\n\\[\n\\Inv=\\ker D=\\bigcap_f\\{a:B_fv_M(a)=v_N(a)\\}\n\\]\nis a unital Boolean subalgebra of $A$. There is a natural exact sequence of $\\F$-vector spaces\n\\begin{equation}\\label{eq:exactsequence}\n0\\longrightarrow\\Inv\\longrightarrow A\\xrightarrow{D}\\im D\\longrightarrow0,\n\\end{equation}\nand therefore a natural additive isomorphism\n\\[\nA/\\Inv\\ \\cong\\ \\im D.\n\\]\nTwo observables have the same residual profile exactly when their symmetric difference belongs to $\\Inv$.\n\\end{theorem}\n\\begin{proof}\nEach defining condition for $\\Inv$ is an equalizer of two Boolean homomorphisms. Equalizers are closed under $0$, $1$, addition, multiplication, and complement. Additivity of $D$ gives the exact sequence and the first isomorphism theorem for vector spaces. Finally","\\Needspace{12\\baselineskip}\n\\begin{theorem}[Invariant algebra and residual quotient]\\label{thm:exactsequence}\nThe set\n\\[\n\\Inv=\\ker D=\\bigcap_f\\{a:B_fv_M(a)=v_N(a)\\}\n\\]\nis a unital Boolean subalgebra of $A$. There is a natural exact sequence of $\\F$-vector spaces\n\\begin{equation}\\label{eq:exactsequence}\n0\\longrightarrow\\Inv\\longrightarrow A\\xrightarrow{D}\\im D\\longrightarrow0,\n\\end{equation}\nand therefore a natural additive isomorphism\n\\[\nA/\\Inv\\ \\cong\\ \\im D.\n\\]\nTwo observables have the same residual profile exactly when their symmetric difference belongs to $\\Inv$.\n\\end{theorem}\n\\begin{proof}\nEach defining condition for $\\Inv$ is an equalizer of two Boolean homomorphisms. Equalizers are closed under $0$, $1$, addition, multiplication, and complement. Additivity of $D$ gives the exact sequence and the first isomorphism theorem for vector spaces. Finally $D(a)=D(b)$ holds exactly when $D(a+b)","\\Needspace{12\\baselineskip}\n\\begin{theorem}[Invariant algebra and residual quotient]\\label{thm:exactsequence}\nThe set\n\\[\n\\Inv=\\ker D=\\bigcap_f\\{a:B_fv_M(a)=v_N(a)\\}\n\\]\nis a unital Boolean subalgebra of $A$. There is a natural exact sequence of $\\F$-vector spaces\n\\begin{equation}\\label{eq:exactsequence}\n0\\longrightarrow\\Inv\\longrightarrow A\\xrightarrow{D}\\im D\\longrightarrow0,\n\\end{equation}\nand therefore a natural additive isomorphism\n\\[\nA/\\Inv\\ \\cong\\ \\im D.\n\\]\nTwo observables have the same residual profile exactly when their symmetric difference belongs to $\\Inv$.\n\\end{theorem}\n\\begin{proof}\nEach defining condition for $\\Inv$ is an equalizer of two Boolean homomorphisms. Equalizers are closed under $0$, $1$, addition, multiplication, and complement. Additivity of $D$ gives the exact sequence and the first isomorphism theorem for vector spaces. Finally $D(a)=D(b)$ holds exactly when $D(a+b)=0$.\n\\end{proof}","m}[Invariant algebra and residual quotient]\\label{thm:exactsequence}\nThe set\n\\[\n\\Inv=\\ker D=\\bigcap_f\\{a:B_fv_M(a)=v_N(a)\\}\n\\]\nis a unital Boolean subalgebra of $A$. There is a natural exact sequence of $\\F$-vector spaces\n\\begin{equation}\\label{eq:exactsequence}\n0\\longrightarrow\\Inv\\longrightarrow A\\xrightarrow{D}\\im D\\longrightarrow0,\n\\end{equation}\nand therefore a natural additive isomorphism\n\\[\nA/\\Inv\\ \\cong\\ \\im D.\n\\]\nTwo observables have the same residual profile exactly when their symmetric difference belongs to $\\Inv$.\n\\end{theorem}\n\\begin{proof}\nEach defining condition for $\\Inv$ is an equalizer of two Boolean homomorphisms. Equalizers are closed under $0$, $1$, addition, multiplication, and complement. Additivity of $D$ gives the exact sequence and the first isomorphism theorem for vector spaces. Finally $D(a)=D(b)$ holds exactly when $D(a+b)=0$.\n\\end{proof}","]\nis a unital Boolean subalgebra of $A$. There is a natural exact sequence of $\\F$-vector spaces\n\\begin{equation}\\label{eq:exactsequence}\n0\\longrightarrow\\Inv\\longrightarrow A\\xrightarrow{D}\\im D\\longrightarrow0,\n\\end{equation}\nand therefore a natural additive isomorphism\n\\[\nA/\\Inv\\ \\cong\\ \\im D.\n\\]\nTwo observables have the same residual profile exactly when their symmetric difference belongs to $\\Inv$.\n\\end{theorem}\n\\begin{proof}\nEach defining condition for $\\Inv$ is an equalizer of two Boolean homomorphisms. Equalizers are closed under $0$, $1$, addition, multiplication, and complement. Additivity of $D$ gives the exact sequence and the first isomorphism theorem for vector spaces. Finally $D(a)=D(b)$ holds exactly when $D(a+b)=0$.\n\\end{proof}","t sequence of $\\F$-vector spaces\n\\begin{equation}\\label{eq:exactsequence}\n0\\longrightarrow\\Inv\\longrightarrow A\\xrightarrow{D}\\im D\\longrightarrow0,\n\\end{equation}\nand therefore a natural additive isomorphism\n\\[\nA/\\Inv\\ \\cong\\ \\im D.\n\\]\nTwo observables have the same residual profile exactly when their symmetric difference belongs to $\\Inv$.\n\\end{theorem}\n\\begin{proof}\nEach defining condition for $\\Inv$ is an equalizer of two Boolean homomorphisms. Equalizers are closed under $0$, $1$, addition, multiplication, and complement. Additivity of $D$ gives the exact sequence and the first isomorphism theorem for vector spaces. Finally $D(a)=D(b)$ holds exactly when $D(a+b)=0$.\n\\end{proof}","ightarrow\\Inv\\longrightarrow A\\xrightarrow{D}\\im D\\longrightarrow0,\n\\end{equation}\nand therefore a natural additive isomorphism\n\\[\nA/\\Inv\\ \\cong\\ \\im D.\n\\]\nTwo observables have the same residual profile exactly when their symmetric difference belongs to $\\Inv$.\n\\end{theorem}\n\\begin{proof}\nEach defining condition for $\\Inv$ is an equalizer of two Boolean homomorphisms. Equalizers are closed under $0$, $1$, addition, multiplication, and complement. Additivity of $D$ gives the exact sequence and the first isomorphism theorem for vector spaces. Finally $D(a)=D(b)$ holds exactly when $D(a+b)=0$.\n\\end{proof}","rrow\\Inv\\longrightarrow A\\xrightarrow{D}\\im D\\longrightarrow0,\n\\end{equation}\nand therefore a natural additive isomorphism\n\\[\nA/\\Inv\\ \\cong\\ \\im D.\n\\]\nTwo observables have the same residual profile exactly when their symmetric difference belongs to $\\Inv$.\n\\end{theorem}\n\\begin{proof}\nEach defining condition for $\\Inv$ is an equalizer of two Boolean homomorphisms. Equalizers are closed under $0$, $1$, addition, multiplication, and complement. Additivity of $D$ gives the exact sequence and the first isomorphism theorem for vector spaces. Finally $D(a)=D(b)$ holds exactly when $D(a+b)=0$.\n\\end{proof}",",\n\\end{equation}\nand therefore a natural additive isomorphism\n\\[\nA/\\Inv\\ \\cong\\ \\im D.\n\\]\nTwo observables have the same residual profile exactly when their symmetric difference belongs to $\\Inv$.\n\\end{theorem}\n\\begin{proof}\nEach defining condition for $\\Inv$ is an equalizer of two Boolean homomorphisms. Equalizers are closed under $0$, $1$, addition, multiplication, and complement. Additivity of $D$ gives the exact sequence and the first isomorphism theorem for vector spaces. Finally $D(a)=D(b)$ holds exactly when $D(a+b)=0$.\n\\end{proof}","Two observables have the same residual profile exactly when their symmetric difference belongs to $\\Inv$.\n\\end{theorem}\n\\begin{proof}\nEach defining condition for $\\Inv$ is an equalizer of two Boolean homomorphisms. Equalizers are closed under $0$, $1$, addition, multiplication, and complement. Additivity of $D$ gives the exact sequence and the first isomorphism theorem for vector spaces. Finally $D(a)=D(b)$ holds exactly when $D(a+b)=0$.\n\\end{proof}","esidual profile exactly when their symmetric difference belongs to $\\Inv$.\n\\end{theorem}\n\\begin{proof}\nEach defining condition for $\\Inv$ is an equalizer of two Boolean homomorphisms. Equalizers are closed under $0$, $1$, addition, multiplication, and complement. Additivity of $D$ gives the exact sequence and the first isomorphism theorem for vector spaces. Finally $D(a)=D(b)$ holds exactly when $D(a+b)=0$.\n\\end{proof}","\\textbf{The quotient in \\eqref{eq:exactsequence} is not a Boolean-algebra quotient.} The subalgebra $\\Inv$ contains $1$. Unless $\\Inv=A$, it cannot be an ideal, since an ideal containing $1$ is the whole ring. In particular, a nonzero residual profile does not inherit a well-defined conjunction from this quotient.","An immediate information loss is $D(a)=D(\\neg a)$. A residual detects change, not its absolute orientation. It cannot replace the original observable without boundary data or invariant information.","\\begin{proposition}[Generating transformations suffice]\\label{prop:generators}\nIf a family $\\Gamma$ generates the arrows of $\\mathcal C$ under composition, then\n\\[\n\\Inv=\\bigcap_{f\\in\\Gamma}\\ker D_f.\n\\]\nAdding admissible transformations can only shrink the invariant core.\n\\end{proposition}\n\\begin{proof}\nThe composition law propagates vanishing along finite composites. Identity defects vanish. The monotonicity statement is an immediate property of intersections.\n\\end{proof}","\\subsection{Coherent sections and definability}\nThe limit $\\varprojlim_{\\mathcal C}B$ consists of families $(b_M)_M$ satisfying $B_f(b_M)=b_N$. Thus $a\\in\\Inv$ precisely when $\\Ev(a)$ is a coherent family. But not every coherent family need be represented by a sentence, and distinct sentences can have the same evaluations.","\\begin{proposition}[The precise section statement]\\label{prop:sections}\nLet $K=\\bigcap_M\\ker v_M$. Then $K$ is a Boolean ideal contained in $\\Inv$, and\n\\[\n\\Inv/K\\ \\cong\\ \\im\\bigl(\\Ev|_{\\Inv}\\bigr)\n\\ \\subseteq\\ \\varprojlim_{\\mathcal C}B\n\\]\nas Boolean algebras.\n\\end{proposition}\n\\begin{proof}\nThe kernel of a Boolean-ring homomorphism is an ideal. An observable whose evaluations are all zero has zero defect, so $K\\subseteq\\Inv$. Restrict $\\Ev$ to $\\Inv$ and apply the Boolean-ring first isomorphism theorem.\n\\end{proof}","ition}[The precise section statement]\\label{prop:sections}\nLet $K=\\bigcap_M\\ker v_M$. Then $K$ is a Boolean ideal contained in $\\Inv$, and\n\\[\n\\Inv/K\\ \\cong\\ \\im\\bigl(\\Ev|_{\\Inv}\\bigr)\n\\ \\subseteq\\ \\varprojlim_{\\mathcal C}B\n\\]\nas Boolean algebras.\n\\end{proposition}\n\\begin{proof}\nThe kernel of a Boolean-ring homomorphism is an ideal. An observable whose evaluations are all zero has zero defect, so $K\\subseteq\\Inv$. Restrict $\\Ev$ to $\\Inv$ and apply the Boolean-ring first isomorphism theorem.\n\\end{proof}","statement]\\label{prop:sections}\nLet $K=\\bigcap_M\\ker v_M$. Then $K$ is a Boolean ideal contained in $\\Inv$, and\n\\[\n\\Inv/K\\ \\cong\\ \\im\\bigl(\\Ev|_{\\Inv}\\bigr)\n\\ \\subseteq\\ \\varprojlim_{\\mathcal C}B\n\\]\nas Boolean algebras.\n\\end{proposition}\n\\begin{proof}\nThe kernel of a Boolean-ring homomorphism is an ideal. An observable whose evaluations are all zero has zero defect, so $K\\subseteq\\Inv$. Restrict $\\Ev$ to $\\Inv$ and apply the Boolean-ring first isomorphism theorem.\n\\end{proof}","]\\label{prop:sections}\nLet $K=\\bigcap_M\\ker v_M$. Then $K$ is a Boolean ideal contained in $\\Inv$, and\n\\[\n\\Inv/K\\ \\cong\\ \\im\\bigl(\\Ev|_{\\Inv}\\bigr)\n\\ \\subseteq\\ \\varprojlim_{\\mathcal C}B\n\\]\nas Boolean algebras.\n\\end{proposition}\n\\begin{proof}\nThe kernel of a Boolean-ring homomorphism is an ideal. An observable whose evaluations are all zero has zero defect, so $K\\subseteq\\Inv$. Restrict $\\Ev$ to $\\Inv$ and apply the Boolean-ring first isomorphism theorem.\n\\end{proof}","For example, a countable sentence algebra cannot cover all coherent families over a countably infinite \\emph{discrete} diagram with $B_M=\\F$, since that limit is $\\F^{\\mathbb N}$ and is uncountable. The correct phrase is therefore \\emph{definable coherent observables}, not ``all global sections.'' No sheaf has yet been defined. A sheaf additionally requires a site and a gluing condition, neither of which follows from a limit alone.","\\section{Finite integrability and reconstruction}\\label{sec:graph}\nLet $G=(W,E)$ be a finite simple undirected graph. Edges record which worlds are directly compared. Choose an orientation only to index them. For $a\\in\\F^W$ set\n\\[\n(da)_{\\{u,v\\}}=a_u+a_v.\n\\]\nUnlike the semantic algebra $A$, the space $\\F^W$ contains every binary labeling. An actual family of models and a selected language may realize only a proper subalgebra of these labelings.","\\begin{theorem}[Finite residual integrability]\\label{thm:integrability}\nSuppose $G$ has $n$ vertices, $m$ edges, and $c$ connected components.\n\\begin{enumerate}\n\\item $\\ker d$ consists exactly of the componentwise constant labelings and has dimension $c$.\n\\item An edge field $r\\in\\F^E$ equals $da$ for some $a$ if and only if its sum around every cycle is zero.\n\\item Every integrable $r$ has exactly $2^c$ reconstructions. Fixing one vertex value per component makes the reconstruction unique.\n\\item $\\dim\\im d=n-c$ and $\\dim(\\F^E/\\im d)=m-n+c$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf $da=0$, endpoints of every edge have equal values, so values agree along each path. Conversely componentwise constants have zero derivative. This proves the first statement.","gin{theorem}[Finite residual integrability]\\label{thm:integrability}\nSuppose $G$ has $n$ vertices, $m$ edges, and $c$ connected components.\n\\begin{enumerate}\n\\item $\\ker d$ consists exactly of the componentwise constant labelings and has dimension $c$.\n\\item An edge field $r\\in\\F^E$ equals $da$ for some $a$ if and only if its sum around every cycle is zero.\n\\item Every integrable $r$ has exactly $2^c$ reconstructions. Fixing one vertex value per component makes the reconstruction unique.\n\\item $\\dim\\im d=n-c$ and $\\dim(\\F^E/\\im d)=m-n+c$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf $da=0$, endpoints of every edge have equal values, so values agree along each path. Conversely componentwise constants have zero derivative. This proves the first statement.","m$ edges, and $c$ connected components.\n\\begin{enumerate}\n\\item $\\ker d$ consists exactly of the componentwise constant labelings and has dimension $c$.\n\\item An edge field $r\\in\\F^E$ equals $da$ for some $a$ if and only if its sum around every cycle is zero.\n\\item Every integrable $r$ has exactly $2^c$ reconstructions. Fixing one vertex value per component makes the reconstruction unique.\n\\item $\\dim\\im d=n-c$ and $\\dim(\\F^E/\\im d)=m-n+c$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf $da=0$, endpoints of every edge have equal values, so values agree along each path. Conversely componentwise constants have zero derivative. This proves the first statement.","nnected components.\n\\begin{enumerate}\n\\item $\\ker d$ consists exactly of the componentwise constant labelings and has dimension $c$.\n\\item An edge field $r\\in\\F^E$ equals $da$ for some $a$ if and only if its sum around every cycle is zero.\n\\item Every integrable $r$ has exactly $2^c$ reconstructions. Fixing one vertex value per component makes the reconstruction unique.\n\\item $\\dim\\im d=n-c$ and $\\dim(\\F^E/\\im d)=m-n+c$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf $da=0$, endpoints of every edge have equal values, so values agree along each path. Conversely componentwise constants have zero derivative. This proves the first statement.","the componentwise constant labelings and has dimension $c$.\n\\item An edge field $r\\in\\F^E$ equals $da$ for some $a$ if and only if its sum around every cycle is zero.\n\\item Every integrable $r$ has exactly $2^c$ reconstructions. Fixing one vertex value per component makes the reconstruction unique.\n\\item $\\dim\\im d=n-c$ and $\\dim(\\F^E/\\im d)=m-n+c$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nIf $da=0$, endpoints of every edge have equal values, so values agree along each path. Conversely componentwise constants have zero derivative. This proves the first statement.","For any cycle, summing $da$ counts every vertex on the cycle twice, hence gives zero. Conversely choose a spanning forest and a root value in each tree. Define $a_v$ by adding the edge values along the unique tree path from its root. This gives $da=r$ on tree edges. A non-tree edge and the tree path between its endpoints form a cycle. Its zero-sum condition gives the desired equation on that edge as well.","The difference of any two reconstructions lies in $\\ker d$. There are $2^c$ such differences. The rank and quotient-dimension statements now follow from rank-nullity.\n\\end{proof}","The quotient $\\F^E/\\im d$ is the first graph cohomology group when the graph is regarded as a one-dimensional complex. A nonzero class obstructs integrating \\emph{proposed} edge data. This does not make an observed truth differential curved: an observed differential already lies in $\\im d$ and has zero class.","\\subsection{A worked diamond}\nTake edges $01,02,13,23$ and a labeling\n\\[\na=(0,1,0,1).\n\\]\nThen\n\\[\nda=(1,0,0,1).\n\\]\nThe path $0\\to1\\to3$ has accumulated defect $1+0=1$. The path $0\\to2\\to3$ has defect $0+1=1$. The unique cycle has sum $1+0+1+0=0$.","With root value $a_0=0$ the labeling above is recovered. With root value $a_0=1$ the reconstruction is $(1,0,1,0)$. The derivative cannot distinguish these two complementary labelings. That ambiguity is exactly the one-dimensional invariant sector of this connected graph.","By contrast, assigning $r=(1,0,0)$ around a triangle is impossible. The cycle sum is $1$. No collection of vertex truth values has that derivative.","\\subsection{Reconstruction with nontrivial transport}\nFor a path $M_0\\xrightarrow{f_1}\\cdots\\xrightarrow{f_n}M_n$, repeated use of \\eqref{eq:defect} gives\n\\begin{equation}\\label{eq:pathreconstruction}\nv_{M_n}(a)=B_{f_n\\cdots f_1}v_{M_0}(a)\n+\\sum_{k=1}^n B_{f_n\\cdots f_{k+1}}D_{f_k}(a),\n\\end{equation}\nwhere the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the transported residuals on a directed path. It does not claim that every target has such a path or that a general arrow can be reversed.","\\section{Projectors: existence, obstruction, and a canonical substitute}\\label{sec:projectors}\nThe formula $a=P(a)+R(a)$ is meaningful only after the type and existence of $P$ have been established.","\\begin{proposition}[Additive splitting]\\label{prop:additivesplit}\nOver a metatheory with choice, the inclusion $\\Inv\\subseteq A$ always admits an $\\F$-linear retraction $r:A\\to\\Inv$. For $P=\\iota r$ and $R=\\id+P$,\n\\[\nP^2=P,\\quad R^2=R,\\quad PR=RP=0,\\quad a=P(a)+R(a).\n\\]\nThis is a vector-space decomposition. The retraction need not preserve multiplication or be canonical.\n\\end{proposition}\n\\begin{proof}\nChoose a vector-space basis of $\\Inv$ and extend it to a basis of $A$. Let $r$ fix the first basis and kill the added vectors. Then $P^2=P$. In characteristic two,\n$(\\id+P)^2=\\id+P$, and $P(\\id+P)=(\\id+P)P=0$.\n\\end{proof}","]\\label{prop:additivesplit}\nOver a metatheory with choice, the inclusion $\\Inv\\subseteq A$ always admits an $\\F$-linear retraction $r:A\\to\\Inv$. For $P=\\iota r$ and $R=\\id+P$,\n\\[\nP^2=P,\\quad R^2=R,\\quad PR=RP=0,\\quad a=P(a)+R(a).\n\\]\nThis is a vector-space decomposition. The retraction need not preserve multiplication or be canonical.\n\\end{proposition}\n\\begin{proof}\nChoose a vector-space basis of $\\Inv$ and extend it to a basis of $A$. Let $r$ fix the first basis and kill the added vectors. Then $P^2=P$. In characteristic two,\n$(\\id+P)^2=\\id+P$, and $P(\\id+P)=(\\id+P)P=0$.\n\\end{proof}","metatheory with choice, the inclusion $\\Inv\\subseteq A$ always admits an $\\F$-linear retraction $r:A\\to\\Inv$. For $P=\\iota r$ and $R=\\id+P$,\n\\[\nP^2=P,\\quad R^2=R,\\quad PR=RP=0,\\quad a=P(a)+R(a).\n\\]\nThis is a vector-space decomposition. The retraction need not preserve multiplication or be canonical.\n\\end{proposition}\n\\begin{proof}\nChoose a vector-space basis of $\\Inv$ and extend it to a basis of $A$. Let $r$ fix the first basis and kill the added vectors. Then $P^2=P$. In characteristic two,\n$(\\id+P)^2=\\id+P$, and $P(\\id+P)=(\\id+P)P=0$.\n\\end{proof}","y with choice, the inclusion $\\Inv\\subseteq A$ always admits an $\\F$-linear retraction $r:A\\to\\Inv$. For $P=\\iota r$ and $R=\\id+P$,\n\\[\nP^2=P,\\quad R^2=R,\\quad PR=RP=0,\\quad a=P(a)+R(a).\n\\]\nThis is a vector-space decomposition. The retraction need not preserve multiplication or be canonical.\n\\end{proposition}\n\\begin{proof}\nChoose a vector-space basis of $\\Inv$ and extend it to a basis of $A$. Let $r$ fix the first basis and kill the added vectors. Then $P^2=P$. In characteristic two,\n$(\\id+P)^2=\\id+P$, and $P(\\id+P)=(\\id+P)P=0$.\n\\end{proof}","its an $\\F$-linear retraction $r:A\\to\\Inv$. For $P=\\iota r$ and $R=\\id+P$,\n\\[\nP^2=P,\\quad R^2=R,\\quad PR=RP=0,\\quad a=P(a)+R(a).\n\\]\nThis is a vector-space decomposition. The retraction need not preserve multiplication or be canonical.\n\\end{proposition}\n\\begin{proof}\nChoose a vector-space basis of $\\Inv$ and extend it to a basis of $A$. Let $r$ fix the first basis and kill the added vectors. Then $P^2=P$. In characteristic two,\n$(\\id+P)^2=\\id+P$, and $P(\\id+P)=(\\id+P)P=0$.\n\\end{proof}",":A\\to\\Inv$. For $P=\\iota r$ and $R=\\id+P$,\n\\[\nP^2=P,\\quad R^2=R,\\quad PR=RP=0,\\quad a=P(a)+R(a).\n\\]\nThis is a vector-space decomposition. The retraction need not preserve multiplication or be canonical.\n\\end{proposition}\n\\begin{proof}\nChoose a vector-space basis of $\\Inv$ and extend it to a basis of $A$. Let $r$ fix the first basis and kill the added vectors. Then $P^2=P$. In characteristic two,\n$(\\id+P)^2=\\id+P$, and $P(\\id+P)=(\\id+P)P=0$.\n\\end{proof}","a r$ and $R=\\id+P$,\n\\[\nP^2=P,\\quad R^2=R,\\quad PR=RP=0,\\quad a=P(a)+R(a).\n\\]\nThis is a vector-space decomposition. The retraction need not preserve multiplication or be canonical.\n\\end{proposition}\n\\begin{proof}\nChoose a vector-space basis of $\\Inv$ and extend it to a basis of $A$. Let $r$ fix the first basis and kill the added vectors. Then $P^2=P$. In characteristic two,\n$(\\id+P)^2=\\id+P$, and $P(\\id+P)=(\\id+P)P=0$.\n\\end{proof}","If $r$ is a Boolean retraction, its kernel is an ideal $J$ and the additive decomposition is $A=\\Inv\\oplus J$. In these coordinates multiplication is\n\\[\n(i,x)(j,y)=(ij,\\,iy+jx+xy).\n\\]\nIt is not generally a direct-product multiplication. Writing a direct sum symbol must not silently erase these cross terms.","\\begin{example}[Completeness of the ambient algebra is not enough]\\label{ex:noretraction}\nLet $A=\\Pow(\\mathbb N)$ and let $\\Inv$ be the finite-cofinite subalgebra. There is no Boolean retraction $r:A\\to\\Inv$ fixing $\\Inv$.\nIndeed, for every $S\\subseteq\\mathbb N$ and $n\\in\\mathbb N$,\n\\[\nr(S)\\cap\\{n\\}=r(S\\cap\\{n\\})=S\\cap\\{n\\}.\n\\]\nThus $r(S)$ has exactly the same members as $S$, so $r(S)=S$. An infinite coinfinite $S$ gives a contradiction.\n\\end{example}","Completeness of the ambient algebra is not enough]\\label{ex:noretraction}\nLet $A=\\Pow(\\mathbb N)$ and let $\\Inv$ be the finite-cofinite subalgebra. There is no Boolean retraction $r:A\\to\\Inv$ fixing $\\Inv$.\nIndeed, for every $S\\subseteq\\mathbb N$ and $n\\in\\mathbb N$,\n\\[\nr(S)\\cap\\{n\\}=r(S\\cap\\{n\\})=S\\cap\\{n\\}.\n\\]\nThus $r(S)$ has exactly the same members as $S$, so $r(S)=S$. An infinite coinfinite $S$ gives a contradiction.\n\\end{example}","This example can occur as an invariant core of a resolution datum, rather than merely as an arbitrary subalgebra. Take worlds to be the nonprincipal ultrafilters on $\\mathbb N$, join every ordered pair by an arrow, and use $v_U(S)=1$ iff $S\\in U$. All these worlds agree on finite and cofinite sets. Every infinite coinfinite set belongs to some nonprincipal ultrafilter and has its complement in another, by the ultrafilter extension argument. Hence the invariant core is exactly the finite-cofinite algebra. These worlds are propositional valuations, not asserted models of ZFC.","\\subsection{A two-world obstruction}\n\\begin{theorem}[No symmetric Boolean projector]\\label{thm:nocanonical}\nLet $A=\\F^2$ and $\\Inv=\\{(0,0),(1,1)\\}$. There is no Boolean retraction $P:A\\to\\Inv$ that commutes with swapping the two coordinates. There is not even a swap-equivariant additive retraction onto $\\Inv$.\n\\end{theorem}\n\\begin{proof}\nWrite $e_1=(1,0)$ and $e_2=(0,1)$. Swapping coordinates fixes every element of $\\Inv$, so equivariance forces $P(e_1)=P(e_2)$. Additivity then gives\n\\[\nP(1,1)=P(e_1+e_2)=P(e_1)+P(e_2)=0,\n\\]\ncontrary to the retraction condition $P(1,1)=(1,1)$. Every Boolean retraction is additive, so both claims follow.\n\\end{proof}","rld obstruction}\n\\begin{theorem}[No symmetric Boolean projector]\\label{thm:nocanonical}\nLet $A=\\F^2$ and $\\Inv=\\{(0,0),(1,1)\\}$. There is no Boolean retraction $P:A\\to\\Inv$ that commutes with swapping the two coordinates. There is not even a swap-equivariant additive retraction onto $\\Inv$.\n\\end{theorem}\n\\begin{proof}\nWrite $e_1=(1,0)$ and $e_2=(0,1)$. Swapping coordinates fixes every element of $\\Inv$, so equivariance forces $P(e_1)=P(e_2)$. Additivity then gives\n\\[\nP(1,1)=P(e_1+e_2)=P(e_1)+P(e_2)=0,\n\\]\ncontrary to the retraction condition $P(1,1)=(1,1)$. Every Boolean retraction is additive, so both claims follow.\n\\end{proof}","}[No symmetric Boolean projector]\\label{thm:nocanonical}\nLet $A=\\F^2$ and $\\Inv=\\{(0,0),(1,1)\\}$. There is no Boolean retraction $P:A\\to\\Inv$ that commutes with swapping the two coordinates. There is not even a swap-equivariant additive retraction onto $\\Inv$.\n\\end{theorem}\n\\begin{proof}\nWrite $e_1=(1,0)$ and $e_2=(0,1)$. Swapping coordinates fixes every element of $\\Inv$, so equivariance forces $P(e_1)=P(e_2)$. Additivity then gives\n\\[\nP(1,1)=P(e_1+e_2)=P(e_1)+P(e_2)=0,\n\\]\ncontrary to the retraction condition $P(1,1)=(1,1)$. Every Boolean retraction is additive, so both claims follow.\n\\end{proof}","thm:nocanonical}\nLet $A=\\F^2$ and $\\Inv=\\{(0,0),(1,1)\\}$. There is no Boolean retraction $P:A\\to\\Inv$ that commutes with swapping the two coordinates. There is not even a swap-equivariant additive retraction onto $\\Inv$.\n\\end{theorem}\n\\begin{proof}\nWrite $e_1=(1,0)$ and $e_2=(0,1)$. Swapping coordinates fixes every element of $\\Inv$, so equivariance forces $P(e_1)=P(e_2)$. Additivity then gives\n\\[\nP(1,1)=P(e_1+e_2)=P(e_1)+P(e_2)=0,\n\\]\ncontrary to the retraction condition $P(1,1)=(1,1)$. Every Boolean retraction is additive, so both claims follow.\n\\end{proof}","retraction $P:A\\to\\Inv$ that commutes with swapping the two coordinates. There is not even a swap-equivariant additive retraction onto $\\Inv$.\n\\end{theorem}\n\\begin{proof}\nWrite $e_1=(1,0)$ and $e_2=(0,1)$. Swapping coordinates fixes every element of $\\Inv$, so equivariance forces $P(e_1)=P(e_2)$. Additivity then gives\n\\[\nP(1,1)=P(e_1+e_2)=P(e_1)+P(e_2)=0,\n\\]\ncontrary to the retraction condition $P(1,1)=(1,1)$. Every Boolean retraction is additive, so both claims follow.\n\\end{proof}","\\subsection{The finite resolution envelope theorem}\nLet $\\Pi$ be a partition of a finite nonempty set $W$. Let $\\Inv_\\Pi\\subseteq\\F^W$ be the Boolean algebra of functions constant on each block. For $w\\in C\\in\\Pi$ define\n\\[\nL_\\Pi(a)_w=\\bigwedge_{u\\in C}a_u,\\qquad\nU_\\Pi(a)_w=\\bigvee_{u\\in C}a_u.\n\\]\nThese lower and upper approximations are the classical equivalence-relation rough-set operators \\cite{pawlak}. Here they give a precise replacement for a missing canonical projector.","\\begin{theorem}[Finite resolution envelope]\\label{thm:envelope}\nEvery Boolean retraction $P:\\F^W\\to\\Inv_\\Pi$ is obtained by choosing one representative $s(C)\\in C$ per block and setting\n\\begin{equation}\\label{eq:selector}\nP_s(a)_w=a_{s(C)}\\quad(w\\in C).\n\\end{equation}\nFor each fixed observable $a$, its possible projected values are exactly\n\\begin{equation}\\label{eq:envelope}\n\\{P_s(a):s\\text{ chooses one point in each block}\\}\n=\\{b\\in\\Inv_\\Pi:L_\\Pi(a)\\leq b\\leq U_\\Pi(a)\\}.\n\\end{equation}\nConsequently,\n\\[\nL_\\Pi(a)=\\bigwedge_sP_s(a),\\qquad\nU_\\Pi(a)=\\bigvee_sP_s(a).\n\\]\nIf $a$ is mixed on exactly $k$ blocks, there are exactly $2^k$ possible invariant outputs for that $a$.\n\\end{theorem}\n\\begin{proof}\nA unital Boolean homomorphism $h:\\F^W\\to\\F$ must evaluate at exactly one point. To see this, the single","\\begin{theorem}[Finite resolution envelope]\\label{thm:envelope}\nEvery Boolean retraction $P:\\F^W\\to\\Inv_\\Pi$ is obtained by choosing one representative $s(C)\\in C$ per block and setting\n\\begin{equation}\\label{eq:selector}\nP_s(a)_w=a_{s(C)}\\quad(w\\in C).\n\\end{equation}\nFor each fixed observable $a$, its possible projected values are exactly\n\\begin{equation}\\label{eq:envelope}\n\\{P_s(a):s\\text{ chooses one point in each block}\\}\n=\\{b\\in\\Inv_\\Pi:L_\\Pi(a)\\leq b\\leq U_\\Pi(a)\\}.\n\\end{equation}\nConsequently,\n\\[\nL_\\Pi(a)=\\bigwedge_sP_s(a),\\qquad\nU_\\Pi(a)=\\bigvee_sP_s(a).\n\\]\nIf $a$ is mixed on exactly $k$ blocks, there are exactly $2^k$ possible invariant outputs for that $a$.\n\\end{theorem}\n\\begin{proof}\nA unital Boolean homomorphism $h:\\F^W\\to\\F$ must evaluate at exactly one point. To see this, the singleton indicators are pairwise disjoint and their join is","\\begin{theorem}[Finite resolution envelope]\\label{thm:envelope}\nEvery Boolean retraction $P:\\F^W\\to\\Inv_\\Pi$ is obtained by choosing one representative $s(C)\\in C$ per block and setting\n\\begin{equation}\\label{eq:selector}\nP_s(a)_w=a_{s(C)}\\quad(w\\in C).\n\\end{equation}\nFor each fixed observable $a$, its possible projected values are exactly\n\\begin{equation}\\label{eq:envelope}\n\\{P_s(a):s\\text{ chooses one point in each block}\\}\n=\\{b\\in\\Inv_\\Pi:L_\\Pi(a)\\leq b\\leq U_\\Pi(a)\\}.\n\\end{equation}\nConsequently,\n\\[\nL_\\Pi(a)=\\bigwedge_sP_s(a),\\qquad\nU_\\Pi(a)=\\bigvee_sP_s(a).\n\\]\nIf $a$ is mixed on exactly $k$ blocks, there are exactly $2^k$ possible invariant outputs for that $a$.\n\\end{theorem}\n\\begin{proof}\nA unital Boolean homomorphism $h:\\F^W\\to\\F$ must evaluate at exactly one point. To see this, the singleton indicators are pairwise disjoint and their join is $1$. Their images are disjoint elements of $\\F$ whose join is $1$, so exactly one is $1$. Th","\\begin{theorem}[Finite resolution envelope]\\label{thm:envelope}\nEvery Boolean retraction $P:\\F^W\\to\\Inv_\\Pi$ is obtained by choosing one representative $s(C)\\in C$ per block and setting\n\\begin{equation}\\label{eq:selector}\nP_s(a)_w=a_{s(C)}\\quad(w\\in C).\n\\end{equation}\nFor each fixed observable $a$, its possible projected values are exactly\n\\begin{equation}\\label{eq:envelope}\n\\{P_s(a):s\\text{ chooses one point in each block}\\}\n=\\{b\\in\\Inv_\\Pi:L_\\Pi(a)\\leq b\\leq U_\\Pi(a)\\}.\n\\end{equation}\nConsequently,\n\\[\nL_\\Pi(a)=\\bigwedge_sP_s(a),\\qquad\nU_\\Pi(a)=\\bigvee_sP_s(a).\n\\]\nIf $a$ is mixed on exactly $k$ blocks, there are exactly $2^k$ possible invariant outputs for that $a$.\n\\end{theorem}\n\\begin{proof}\nA unital Boolean homomorphism $h:\\F^W\\to\\F$ must evaluate at exactly one point. To see this, the singleton indicators are pairwise disjoint and their join is $1$. Their images are disjoint elements of $\\F$ whose join is $1$, so exactly one is $1$. The image of every subset is then determined","\\begin{theorem}[Finite resolution envelope]\\label{thm:envelope}\nEvery Boolean retraction $P:\\F^W\\to\\Inv_\\Pi$ is obtained by choosing one representative $s(C)\\in C$ per block and setting\n\\begin{equation}\\label{eq:selector}\nP_s(a)_w=a_{s(C)}\\quad(w\\in C).\n\\end{equation}\nFor each fixed observable $a$, its possible projected values are exactly\n\\begin{equation}\\label{eq:envelope}\n\\{P_s(a):s\\text{ chooses one point in each block}\\}\n=\\{b\\in\\Inv_\\Pi:L_\\Pi(a)\\leq b\\leq U_\\Pi(a)\\}.\n\\end{equation}\nConsequently,\n\\[\nL_\\Pi(a)=\\bigwedge_sP_s(a),\\qquad\nU_\\Pi(a)=\\bigvee_sP_s(a).\n\\]\nIf $a$ is mixed on exactly $k$ blocks, there are exactly $2^k$ possible invariant outputs for that $a$.\n\\end{theorem}\n\\begin{proof}\nA unital Boolean homomorphism $h:\\F^W\\to\\F$ must evaluate at exactly one point. To see this, the singleton indicators are pairwise disjoint and their join is $1$. Their images are disjoint elements of $\\F$ whose join is $1$, so exactly one is $1$. The image of every subset is then determined by whether it contains that point.","is obtained by choosing one representative $s(C)\\in C$ per block and setting\n\\begin{equation}\\label{eq:selector}\nP_s(a)_w=a_{s(C)}\\quad(w\\in C).\n\\end{equation}\nFor each fixed observable $a$, its possible projected values are exactly\n\\begin{equation}\\label{eq:envelope}\n\\{P_s(a):s\\text{ chooses one point in each block}\\}\n=\\{b\\in\\Inv_\\Pi:L_\\Pi(a)\\leq b\\leq U_\\Pi(a)\\}.\n\\end{equation}\nConsequently,\n\\[\nL_\\Pi(a)=\\bigwedge_sP_s(a),\\qquad\nU_\\Pi(a)=\\bigvee_sP_s(a).\n\\]\nIf $a$ is mixed on exactly $k$ blocks, there are exactly $2^k$ possible invariant outputs for that $a$.\n\\end{theorem}\n\\begin{proof}\nA unital Boolean homomorphism $h:\\F^W\\to\\F$ must evaluate at exactly one point. To see this, the singleton indicators are pairwise disjoint and their join is $1$. Their images are disjoint elements of $\\F$ whose join is $1$, so exactly one is $1$. The image of every subset is then determined by whether it contains that point.","d setting\n\\begin{equation}\\label{eq:selector}\nP_s(a)_w=a_{s(C)}\\quad(w\\in C).\n\\end{equation}\nFor each fixed observable $a$, its possible projected values are exactly\n\\begin{equation}\\label{eq:envelope}\n\\{P_s(a):s\\text{ chooses one point in each block}\\}\n=\\{b\\in\\Inv_\\Pi:L_\\Pi(a)\\leq b\\leq U_\\Pi(a)\\}.\n\\end{equation}\nConsequently,\n\\[\nL_\\Pi(a)=\\bigwedge_sP_s(a),\\qquad\nU_\\Pi(a)=\\bigvee_sP_s(a).\n\\]\nIf $a$ is mixed on exactly $k$ blocks, there are exactly $2^k$ possible invariant outputs for that $a$.\n\\end{theorem}\n\\begin{proof}\nA unital Boolean homomorphism $h:\\F^W\\to\\F$ must evaluate at exactly one point. To see this, the singleton indicators are pairwise disjoint and their join is $1$. Their images are disjoint elements of $\\F$ whose join is $1$, so exactly one is $1$. The image of every subset is then determined by whether it contains that point.","n}\\label{eq:selector}\nP_s(a)_w=a_{s(C)}\\quad(w\\in C).\n\\end{equation}\nFor each fixed observable $a$, its possible projected values are exactly\n\\begin{equation}\\label{eq:envelope}\n\\{P_s(a):s\\text{ chooses one point in each block}\\}\n=\\{b\\in\\Inv_\\Pi:L_\\Pi(a)\\leq b\\leq U_\\Pi(a)\\}.\n\\end{equation}\nConsequently,\n\\[\nL_\\Pi(a)=\\bigwedge_sP_s(a),\\qquad\nU_\\Pi(a)=\\bigvee_sP_s(a).\n\\]\nIf $a$ is mixed on exactly $k$ blocks, there are exactly $2^k$ possible invariant outputs for that $a$.\n\\end{theorem}\n\\begin{proof}\nA unital Boolean homomorphism $h:\\F^W\\to\\F$ must evaluate at exactly one point. To see this, the singleton indicators are pairwise disjoint and their join is $1$. Their images are disjoint elements of $\\F$ whose join is $1$, so exactly one is $1$. The image of every subset is then determined by whether it contains that point.","=a_{s(C)}\\quad(w\\in C).\n\\end{equation}\nFor each fixed observable $a$, its possible projected values are exactly\n\\begin{equation}\\label{eq:envelope}\n\\{P_s(a):s\\text{ chooses one point in each block}\\}\n=\\{b\\in\\Inv_\\Pi:L_\\Pi(a)\\leq b\\leq U_\\Pi(a)\\}.\n\\end{equation}\nConsequently,\n\\[\nL_\\Pi(a)=\\bigwedge_sP_s(a),\\qquad\nU_\\Pi(a)=\\bigvee_sP_s(a).\n\\]\nIf $a$ is mixed on exactly $k$ blocks, there are exactly $2^k$ possible invariant outputs for that $a$.\n\\end{theorem}\n\\begin{proof}\nA unital Boolean homomorphism $h:\\F^W\\to\\F$ must evaluate at exactly one point. To see this, the singleton indicators are pairwise disjoint and their join is $1$. Their images are disjoint elements of $\\F$ whose join is $1$, so exactly one is $1$. The image of every subset is then determined by whether it contains that point.","each fixed observable $a$, its possible projected values are exactly\n\\begin{equation}\\label{eq:envelope}\n\\{P_s(a):s\\text{ chooses one point in each block}\\}\n=\\{b\\in\\Inv_\\Pi:L_\\Pi(a)\\leq b\\leq U_\\Pi(a)\\}.\n\\end{equation}\nConsequently,\n\\[\nL_\\Pi(a)=\\bigwedge_sP_s(a),\\qquad\nU_\\Pi(a)=\\bigvee_sP_s(a).\n\\]\nIf $a$ is mixed on exactly $k$ blocks, there are exactly $2^k$ possible invariant outputs for that $a$.\n\\end{theorem}\n\\begin{proof}\nA unital Boolean homomorphism $h:\\F^W\\to\\F$ must evaluate at exactly one point. To see this, the singleton indicators are pairwise disjoint and their join is $1$. Their images are disjoint elements of $\\F$ whose join is $1$, so exactly one is $1$. The image of every subset is then determined by whether it contains that point.","xactly\n\\begin{equation}\\label{eq:envelope}\n\\{P_s(a):s\\text{ chooses one point in each block}\\}\n=\\{b\\in\\Inv_\\Pi:L_\\Pi(a)\\leq b\\leq U_\\Pi(a)\\}.\n\\end{equation}\nConsequently,\n\\[\nL_\\Pi(a)=\\bigwedge_sP_s(a),\\qquad\nU_\\Pi(a)=\\bigvee_sP_s(a).\n\\]\nIf $a$ is mixed on exactly $k$ blocks, there are exactly $2^k$ possible invariant outputs for that $a$.\n\\end{theorem}\n\\begin{proof}\nA unital Boolean homomorphism $h:\\F^W\\to\\F$ must evaluate at exactly one point. To see this, the singleton indicators are pairwise disjoint and their join is $1$. Their images are disjoint elements of $\\F$ whose join is $1$, so exactly one is $1$. The image of every subset is then determined by whether it contains that point.","q U_\\Pi(a)\\}.\n\\end{equation}\nConsequently,\n\\[\nL_\\Pi(a)=\\bigwedge_sP_s(a),\\qquad\nU_\\Pi(a)=\\bigvee_sP_s(a).\n\\]\nIf $a$ is mixed on exactly $k$ blocks, there are exactly $2^k$ possible invariant outputs for that $a$.\n\\end{theorem}\n\\begin{proof}\nA unital Boolean homomorphism $h:\\F^W\\to\\F$ must evaluate at exactly one point. To see this, the singleton indicators are pairwise disjoint and their join is $1$. Their images are disjoint elements of $\\F$ whose join is $1$, so exactly one is $1$. The image of every subset is then determined by whether it contains that point.","\\[\nL_\\Pi(a)=\\bigwedge_sP_s(a),\\qquad\nU_\\Pi(a)=\\bigvee_sP_s(a).\n\\]\nIf $a$ is mixed on exactly $k$ blocks, there are exactly $2^k$ possible invariant outputs for that $a$.\n\\end{theorem}\n\\begin{proof}\nA unital Boolean homomorphism $h:\\F^W\\to\\F$ must evaluate at exactly one point. To see this, the singleton indicators are pairwise disjoint and their join is $1$. Their images are disjoint elements of $\\F$ whose join is $1$, so exactly one is $1$. The image of every subset is then determined by whether it contains that point.","ge_sP_s(a),\\qquad\nU_\\Pi(a)=\\bigvee_sP_s(a).\n\\]\nIf $a$ is mixed on exactly $k$ blocks, there are exactly $2^k$ possible invariant outputs for that $a$.\n\\end{theorem}\n\\begin{proof}\nA unital Boolean homomorphism $h:\\F^W\\to\\F$ must evaluate at exactly one point. To see this, the singleton indicators are pairwise disjoint and their join is $1$. Their images are disjoint elements of $\\F$ whose join is $1$, so exactly one is $1$. The image of every subset is then determined by whether it contains that point.","(a)=\\bigvee_sP_s(a).\n\\]\nIf $a$ is mixed on exactly $k$ blocks, there are exactly $2^k$ possible invariant outputs for that $a$.\n\\end{theorem}\n\\begin{proof}\nA unital Boolean homomorphism $h:\\F^W\\to\\F$ must evaluate at exactly one point. To see this, the singleton indicators are pairwise disjoint and their join is $1$. Their images are disjoint elements of $\\F$ whose join is $1$, so exactly one is $1$. The image of every subset is then determined by whether it contains that point.","For a retraction $P$, evaluation of its output on a block $C$ is such a homomorphism, hence selects a point $s(C)\\in W$. Since $P$ fixes the indicator of $C$, this point must belong to $C$. This proves the classification. Conversely every choice in \\eqref{eq:selector} gives a Boolean homomorphism that fixes blockwise constants.","For fixed $a$, a selected value on $C$ lies between its minimum and maximum there. Conversely, take $b\\in\\Inv_\\Pi$ in that interval. When $a$ is constant on $C$, $b$ has that same value. When $a$ is mixed, choose a point of $C$ at which $a$ has the value prescribed by $b$. These independent choices produce $P_s(a)=b$. Each mixed block allows precisely two outputs and every other block allows only one.\n\\end{proof}","The extrema are canonical even though an individual retraction need not be. They preserve order and satisfy\n\\[\nL_\\Pi(a)\\leq a\\leq U_\\Pi(a),\\qquad\nL_\\Pi(\\neg a)=\\neg U_\\Pi(a).\n\\]\nThe lower map preserves meets and the upper map preserves joins. Neither generally preserves both Boolean operations. Their role is a certified interval of invariant summaries, not a new binary truth valuation.","\\begin{proposition}[Adjoint characterization]\\label{prop:adjoints}\nFor $b\\in\\Inv_\\Pi$,\n\\[\nU_\\Pi(a)\\leq b\\ \\Longleftrightarrow\\ a\\leq b,\\qquad\nb\\leq L_\\Pi(a)\\ \\Longleftrightarrow\\ b\\leq a.\n\\]\nThus $U_\\Pi$ is left adjoint to the inclusion and $L_\\Pi$ is right adjoint to it, in the order-theoretic sense.\n\\end{proposition}\n\\begin{proof}\n$U_\\Pi(a)$ is exactly the smallest union of blocks containing the support of $a$. Dually $L_\\Pi(a)$ is the largest union of blocks contained in that support.\n\\end{proof}","ition}[Adjoint characterization]\\label{prop:adjoints}\nFor $b\\in\\Inv_\\Pi$,\n\\[\nU_\\Pi(a)\\leq b\\ \\Longleftrightarrow\\ a\\leq b,\\qquad\nb\\leq L_\\Pi(a)\\ \\Longleftrightarrow\\ b\\leq a.\n\\]\nThus $U_\\Pi$ is left adjoint to the inclusion and $L_\\Pi$ is right adjoint to it, in the order-theoretic sense.\n\\end{proposition}\n\\begin{proof}\n$U_\\Pi(a)$ is exactly the smallest union of blocks containing the support of $a$. Dually $L_\\Pi(a)$ is the largest union of blocks contained in that support.\n\\end{proof}","nt characterization]\\label{prop:adjoints}\nFor $b\\in\\Inv_\\Pi$,\n\\[\nU_\\Pi(a)\\leq b\\ \\Longleftrightarrow\\ a\\leq b,\\qquad\nb\\leq L_\\Pi(a)\\ \\Longleftrightarrow\\ b\\leq a.\n\\]\nThus $U_\\Pi$ is left adjoint to the inclusion and $L_\\Pi$ is right adjoint to it, in the order-theoretic sense.\n\\end{proposition}\n\\begin{proof}\n$U_\\Pi(a)$ is exactly the smallest union of blocks containing the support of $a$. Dually $L_\\Pi(a)$ is the largest union of blocks contained in that support.\n\\end{proof}","\\begin{theorem}[Residual support determines ambiguity]\\label{thm:ambiguity}\nTake $\\Pi$ to be the connected components of a finite graph $G$. Define\n\\[\n\\mathcal A_\\Pi(a)=U_\\Pi(a)+L_\\Pi(a).\n\\]\nThen $\\mathcal A_\\Pi(a)$ is the indicator of the union of components containing a nonzero edge of $da$:\n\\[\n\\supp\\mathcal A_\\Pi(a)=\\Sat_\\Pi(\\supp da).\n\\]\nHere edge support is saturated by taking all vertices in components containing a supported edge.\n\\end{theorem}\n\\begin{proof}\nA component is ambiguous exactly when it contains both truth values. If it contains an edge with unequal endpoint values, it is mixed. Conversely a path from a $0$ vertex to a $1$ vertex must contain an edge where the value changes. This proves both implications.\n\\end{proof}","or of the union of components containing a nonzero edge of $da$:\n\\[\n\\supp\\mathcal A_\\Pi(a)=\\Sat_\\Pi(\\supp da).\n\\]\nHere edge support is saturated by taking all vertices in components containing a supported edge.\n\\end{theorem}\n\\begin{proof}\nA component is ambiguous exactly when it contains both truth values. If it contains an edge with unequal endpoint values, it is mixed. Conversely a path from a $0$ vertex to a $1$ vertex must contain an edge where the value changes. This proves both implications.\n\\end{proof}","of components containing a nonzero edge of $da$:\n\\[\n\\supp\\mathcal A_\\Pi(a)=\\Sat_\\Pi(\\supp da).\n\\]\nHere edge support is saturated by taking all vertices in components containing a supported edge.\n\\end{theorem}\n\\begin{proof}\nA component is ambiguous exactly when it contains both truth values. If it contains an edge with unequal endpoint values, it is mixed. Conversely a path from a $0$ vertex to a $1$ vertex must contain an edge where the value changes. This proves both implications.\n\\end{proof}","The three objects now have distinct roles: $da$ locates transitions, $(L_\\Pi(a),U_\\Pi(a))$ describes all invariant Boolean summaries, and a chosen base value or selector supplies reconstruction data. None of the three should be silently substituted for the others.","\\section{A real-valued realization of Komorebi multiresolution}\\label{sec:real}\nThe Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let $W$ be finite and assign positive weights $\\mu_w$. On $\\mathbb R^W$ use\n\\[\n\\langle f,g\\rangle_\\mu=\\sum_{w\\in W}\\mu_w f_wg_w.\n\\]\nFor a partition $\\Pi$, define\n\\[\n(E_\\Pi f)_w=\n\\frac{\\sum_{u\\in C}\\mu_u f_u}{\\sum_{u\\in C}\\mu_u}\n\\quad(w\\in C\\in\\Pi).\n\\]","\\begin{theorem}[Weighted invariant projection]\\label{thm:weighted}\n$E_\\Pi$ is the unique orthogonal projector onto the real vector space of blockwise constant functions. It is positive, fixes $1$, and is equivariant under weight-preserving permutations that preserve the partition. For binary $a$,\n\\[\nL_\\Pi(a)\\leq E_\\Pi(a)\\leq U_\\Pi(a).\n\\]\n\\end{theorem}\n\\begin{proof}\nThe formula is blockwise constant and fixes such functions. For every blockwise constant $g$,\n\\[\n\\langle f-E_\\Pi f,g\\rangle_\\mu\n=\\sum_{C\\in\\Pi}g_C\\sum_{w\\in C}\\mu_w(f_w-(E_\\Pi f)_w)=0.\n\\]\nThis characterizes the orthogonal projector uniquely. Positivity, equivariance, and the interval bounds follow directly from a positive weighted average.\n\\end{proof}","eighted}\n$E_\\Pi$ is the unique orthogonal projector onto the real vector space of blockwise constant functions. It is positive, fixes $1$, and is equivariant under weight-preserving permutations that preserve the partition. For binary $a$,\n\\[\nL_\\Pi(a)\\leq E_\\Pi(a)\\leq U_\\Pi(a).\n\\]\n\\end{theorem}\n\\begin{proof}\nThe formula is blockwise constant and fixes such functions. For every blockwise constant $g$,\n\\[\n\\langle f-E_\\Pi f,g\\rangle_\\mu\n=\\sum_{C\\in\\Pi}g_C\\sum_{w\\in C}\\mu_w(f_w-(E_\\Pi f)_w)=0.\n\\]\nThis characterizes the orthogonal projector uniquely. Positivity, equivariance, and the interval bounds follow directly from a positive weighted average.\n\\end{proof}","d}\n$E_\\Pi$ is the unique orthogonal projector onto the real vector space of blockwise constant functions. It is positive, fixes $1$, and is equivariant under weight-preserving permutations that preserve the partition. For binary $a$,\n\\[\nL_\\Pi(a)\\leq E_\\Pi(a)\\leq U_\\Pi(a).\n\\]\n\\end{theorem}\n\\begin{proof}\nThe formula is blockwise constant and fixes such functions. For every blockwise constant $g$,\n\\[\n\\langle f-E_\\Pi f,g\\rangle_\\mu\n=\\sum_{C\\in\\Pi}g_C\\sum_{w\\in C}\\mu_w(f_w-(E_\\Pi f)_w)=0.\n\\]\nThis characterizes the orthogonal projector uniquely. Positivity, equivariance, and the interval bounds follow directly from a positive weighted average.\n\\end{proof}","This average is not a Boolean truth assignment. With equal weights on two points, $(0,1)$ projects to $(1/2,1/2)$. Also, for the disjoint singleton indicators $e_1,e_2$, one has $E(e_1e_2)=0$ while $(Ee_1)(Ee_2)$ is the constant $1/4$. Conjunction is not preserved.","\\begin{theorem}[Nested resolution decomposition]\\label{thm:multiresolution}\nLet $\\Pi_0,\\ldots,\\Pi_m$ be successively finer partitions, using the same weights. Write $E_k=E_{\\Pi_k}$ and $Q_k=E_k-E_{k-1}$ for $1\\leq k\\leq m$. Then\n\\[\nE_iE_j=E_{\\min(i,j)},\\qquad Q_k^2=Q_k,\\qquad Q_kQ_l=0\\ (k\\ne l),\n\\]\nand every $f\\in\\mathbb R^W$ has the orthogonal decomposition\n\\begin{equation}\\label{eq:multiresolution}\nf=E_0f+\\sum_{k=1}^mQ_kf+(\\id-E_m)f.\n\\end{equation}\nIn particular the squared norm is the sum of the squared norms of these terms.\n\\end{theorem}\n\\begin{proof}\nThe blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives $E_iE_j=E_i$ for $i\\leq j$, and the reverse product agrees by self-adjointness. Expanding products proves the projector and orthogonality iden","\\begin{theorem}[Nested resolution decomposition]\\label{thm:multiresolution}\nLet $\\Pi_0,\\ldots,\\Pi_m$ be successively finer partitions, using the same weights. Write $E_k=E_{\\Pi_k}$ and $Q_k=E_k-E_{k-1}$ for $1\\leq k\\leq m$. Then\n\\[\nE_iE_j=E_{\\min(i,j)},\\qquad Q_k^2=Q_k,\\qquad Q_kQ_l=0\\ (k\\ne l),\n\\]\nand every $f\\in\\mathbb R^W$ has the orthogonal decomposition\n\\begin{equation}\\label{eq:multiresolution}\nf=E_0f+\\sum_{k=1}^mQ_kf+(\\id-E_m)f.\n\\end{equation}\nIn particular the squared norm is the sum of the squared norms of these terms.\n\\end{theorem}\n\\begin{proof}\nThe blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives $E_iE_j=E_i$ for $i\\leq j$, and the reverse product agrees by self-adjointness. Expanding products proves the projector and orthogonality identities for the $Q_k$. Equation~\\eqref{eq:multiresolution} telescopes. The norm i","\\begin{theorem}[Nested resolution decomposition]\\label{thm:multiresolution}\nLet $\\Pi_0,\\ldots,\\Pi_m$ be successively finer partitions, using the same weights. Write $E_k=E_{\\Pi_k}$ and $Q_k=E_k-E_{k-1}$ for $1\\leq k\\leq m$. Then\n\\[\nE_iE_j=E_{\\min(i,j)},\\qquad Q_k^2=Q_k,\\qquad Q_kQ_l=0\\ (k\\ne l),\n\\]\nand every $f\\in\\mathbb R^W$ has the orthogonal decomposition\n\\begin{equation}\\label{eq:multiresolution}\nf=E_0f+\\sum_{k=1}^mQ_kf+(\\id-E_m)f.\n\\end{equation}\nIn particular the squared norm is the sum of the squared norms of these terms.\n\\end{theorem}\n\\begin{proof}\nThe blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives $E_iE_j=E_i$ for $i\\leq j$, and the reverse product agrees by self-adjointness. Expanding products proves the projector and orthogonality identities for the $Q_k$. Equation~\\eqref{eq:multiresolution} telescopes. The norm identity is the Pythago","\\begin{theorem}[Nested resolution decomposition]\\label{thm:multiresolution}\nLet $\\Pi_0,\\ldots,\\Pi_m$ be successively finer partitions, using the same weights. Write $E_k=E_{\\Pi_k}$ and $Q_k=E_k-E_{k-1}$ for $1\\leq k\\leq m$. Then\n\\[\nE_iE_j=E_{\\min(i,j)},\\qquad Q_k^2=Q_k,\\qquad Q_kQ_l=0\\ (k\\ne l),\n\\]\nand every $f\\in\\mathbb R^W$ has the orthogonal decomposition\n\\begin{equation}\\label{eq:multiresolution}\nf=E_0f+\\sum_{k=1}^mQ_kf+(\\id-E_m)f.\n\\end{equation}\nIn particular the squared norm is the sum of the squared norms of these terms.\n\\end{theorem}\n\\begin{proof}\nThe blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives $E_iE_j=E_i$ for $i\\leq j$, and the reverse product agrees by self-adjointness. Expanding products proves the projector and orthogonality identities for the $Q_k$. Equation~\\eqref{eq:multiresolution} telescopes. The norm identity is the Pythagorean theorem for the","\\begin{theorem}[Nested resolution decomposition]\\label{thm:multiresolution}\nLet $\\Pi_0,\\ldots,\\Pi_m$ be successively finer partitions, using the same weights. Write $E_k=E_{\\Pi_k}$ and $Q_k=E_k-E_{k-1}$ for $1\\leq k\\leq m$. Then\n\\[\nE_iE_j=E_{\\min(i,j)},\\qquad Q_k^2=Q_k,\\qquad Q_kQ_l=0\\ (k\\ne l),\n\\]\nand every $f\\in\\mathbb R^W$ has the orthogonal decomposition\n\\begin{equation}\\label{eq:multiresolution}\nf=E_0f+\\sum_{k=1}^mQ_kf+(\\id-E_m)f.\n\\end{equation}\nIn particular the squared norm is the sum of the squared norms of these terms.\n\\end{theorem}\n\\begin{proof}\nThe blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives $E_iE_j=E_i$ for $i\\leq j$, and the reverse product agrees by self-adjointness. Expanding products proves the projector and orthogonality identities for the $Q_k$. Equation~\\eqref{eq:multiresolution} telescopes. The norm identity is the Pythagorean theorem for these orthogonal summands.\n\\end{proof}","d Q_k^2=Q_k,\\qquad Q_kQ_l=0\\ (k\\ne l),\n\\]\nand every $f\\in\\mathbb R^W$ has the orthogonal decomposition\n\\begin{equation}\\label{eq:multiresolution}\nf=E_0f+\\sum_{k=1}^mQ_kf+(\\id-E_m)f.\n\\end{equation}\nIn particular the squared norm is the sum of the squared norms of these terms.\n\\end{theorem}\n\\begin{proof}\nThe blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives $E_iE_j=E_i$ for $i\\leq j$, and the reverse product agrees by self-adjointness. Expanding products proves the projector and orthogonality identities for the $Q_k$. Equation~\\eqref{eq:multiresolution} telescopes. The norm identity is the Pythagorean theorem for these orthogonal summands.\n\\end{proof}","d Q_kQ_l=0\\ (k\\ne l),\n\\]\nand every $f\\in\\mathbb R^W$ has the orthogonal decomposition\n\\begin{equation}\\label{eq:multiresolution}\nf=E_0f+\\sum_{k=1}^mQ_kf+(\\id-E_m)f.\n\\end{equation}\nIn particular the squared norm is the sum of the squared norms of these terms.\n\\end{theorem}\n\\begin{proof}\nThe blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives $E_iE_j=E_i$ for $i\\leq j$, and the reverse product agrees by self-adjointness. Expanding products proves the projector and orthogonality identities for the $Q_k$. Equation~\\eqref{eq:multiresolution} telescopes. The norm identity is the Pythagorean theorem for these orthogonal summands.\n\\end{proof}","_{k=1}^mQ_kf+(\\id-E_m)f.\n\\end{equation}\nIn particular the squared norm is the sum of the squared norms of these terms.\n\\end{theorem}\n\\begin{proof}\nThe blockwise constant subspaces are nested. Orthogonal projection onto nested subspaces gives $E_iE_j=E_i$ for $i\\leq j$, and the reverse product agrees by self-adjointness. Expanding products proves the projector and orthogonality identities for the $Q_k$. Equation~\\eqref{eq:multiresolution} telescopes. The norm identity is the Pythagorean theorem for these orthogonal summands.\n\\end{proof}","For the original Boolean bounds, finer partitions instead give monotone information:\n\\[\nL_{\\Pi_0}(a)\\leq L_{\\Pi_1}(a)\\leq\\cdots\\leq a\n\\leq\\cdots\\leq U_{\\Pi_1}(a)\\leq U_{\\Pi_0}(a).\n\\]\nThe ambiguity support shrinks as blocks are refined. This is an order-theoretic resolution process, not an additive Boolean detail expansion.","\\section{Stabilization requires a path quantifier}\\label{sec:stability}\nFor a path $M_0\\to M_1\\to\\cdots$ with two-valued observations, write $a_n=v_{M_n}(a)$. Along that path, eventual stability means\n\\[\n\\exists N\\ \\forall n\\geq N\\quad a_n=a_N,\n\\]\nequivalently $a_n+a_{n+1}=0$ eventually. This differs from requiring stability on every continuation from some reachable world.","On a reflexive accessibility relation, $\\Diamond\\Box\\varphi$ says that a world can be reached from which every later accessible world satisfies $\\varphi$. A path that happens to stay at a $\\varphi$-world does not establish this. A two-state switch system with arrows in both directions and identity loops already separates the notions.","\\begin{theorem}[Finite all-path stability criterion]\\label{thm:stability}\nLet $G$ be a finite directed graph with a binary vertex labeling. Every infinite directed path is eventually constant in label if and only if every strongly connected component is monochromatic. When this holds, every path has at most $k-1$ label changes, where $k$ is the number of strongly connected components.\n\\end{theorem}\n\\begin{proof}\nIf one strongly connected component contains both labels, travel from a $0$ vertex to a $1$ vertex and back. Repeating the resulting closed walk gives an infinite path with infinitely many changes.","nite directed graph with a binary vertex labeling. Every infinite directed path is eventually constant in label if and only if every strongly connected component is monochromatic. When this holds, every path has at most $k-1$ label changes, where $k$ is the number of strongly connected components.\n\\end{theorem}\n\\begin{proof}\nIf one strongly connected component contains both labels, travel from a $0$ vertex to a $1$ vertex and back. Repeating the resulting closed walk gives an infinite path with infinitely many changes.","aph with a binary vertex labeling. Every infinite directed path is eventually constant in label if and only if every strongly connected component is monochromatic. When this holds, every path has at most $k-1$ label changes, where $k$ is the number of strongly connected components.\n\\end{theorem}\n\\begin{proof}\nIf one strongly connected component contains both labels, travel from a $0$ vertex to a $1$ vertex and back. Repeating the resulting closed walk gives an infinite path with infinitely many changes.","Conversely, a path can leave a strongly connected component only by moving forward in the finite acyclic condensation graph. It can never return to a component it has left. There are at most $k-1$ such transitions. Within each component the label is constant. Hence an infinite path eventually has constant label and has at most the asserted number of changes.\n\\end{proof}","There need not be a uniform \\emph{time index} for the last change. A path may remain at a self-looped $0$ vertex for arbitrarily many steps before moving once to an absorbing $1$ vertex. Thus a bound on the number of residual events is not a bound on when the last residual occurs.","\\section{Forcing as a conditional set-theoretic instance}\\label{sec:forcing}\nFix externally a countable transitive $M\\models\\ZFC$. Let $\\mathbb B\\in M$ be a Boolean algebra that $M$ regards as complete, and let $G$ be an $M$-generic ultrafilter. For a sentence $\\varphi$ with names $\\vec\\tau$, the standard forcing truth lemma gives\n\\begin{equation}\\label{eq:truthlemma}\nM[G]\\models\\varphi(\\vec\\tau_G)\n\\quad\\Longleftrightarrow\\quad\n\\bnorm{\\varphi(\\vec\\tau)}_{\\mathbb B}^{M}\\in G.\n\\end{equation}\nBoolean-valued forcing and the role of generic quotient models are developed in \\cite{hamkinsseabold}. The completeness assumption here is internal to $M$. The map $b\\mapsto\\mathbf1_{b\\in G}$ is a Boolean homomorphism but need not preserve every externally given infinite join.","or a sentence $\\varphi$ with names $\\vec\\tau$, the standard forcing truth lemma gives\n\\begin{equation}\\label{eq:truthlemma}\nM[G]\\models\\varphi(\\vec\\tau_G)\n\\quad\\Longleftrightarrow\\quad\n\\bnorm{\\varphi(\\vec\\tau)}_{\\mathbb B}^{M}\\in G.\n\\end{equation}\nBoolean-valued forcing and the role of generic quotient models are developed in \\cite{hamkinsseabold}. The completeness assumption here is internal to $M$. The map $b\\mapsto\\mathbf1_{b\\in G}$ is a Boolean homomorphism but need not preserve every externally given infinite join.","$\\varphi$ with names $\\vec\\tau$, the standard forcing truth lemma gives\n\\begin{equation}\\label{eq:truthlemma}\nM[G]\\models\\varphi(\\vec\\tau_G)\n\\quad\\Longleftrightarrow\\quad\n\\bnorm{\\varphi(\\vec\\tau)}_{\\mathbb B}^{M}\\in G.\n\\end{equation}\nBoolean-valued forcing and the role of generic quotient models are developed in \\cite{hamkinsseabold}. The completeness assumption here is internal to $M$. The map $b\\mapsto\\mathbf1_{b\\in G}$ is a Boolean homomorphism but need not preserve every externally given infinite join.","Equation~\\eqref{eq:truthlemma} is a precise local instance of ``richer data resolved into ordinary truth.'' It is not permission to replace $G$ by an arbitrary ultrafilter and still claim an ordinary generic extension of the same ground model. Nor is the atomic product presentation of Section~3 the same construction as a forcing extension.","\\subsection{A changing observable and an invariant fragment}\nTaking two-valued fibers and an arrow $M\\to M[G]$ gives\n\\[\nD_f(\\varphi)=\\mathbf1_{M\\models\\varphi}\n+\\mathbf1_{M[G]\\models\\varphi}.\n\\]\nFor suitable forcing, CH can change, so it is not invariant under unrestricted forcing. Both CH and its negation are forceable in the relevant forcing setting \\cite{asperoviale}. This is not a contradiction: the quantifiers of CH are interpreted in worlds with different available sets.","There is also a direct invariant fragment that requires no new theorem.\n\\begin{proposition}[Bounded-formula invariance]\\label{prop:delta0}\nIf $M\\subseteq N$ are transitive membership structures and all parameters belong to $M$, they agree on every bounded formula, namely a formula whose quantifiers are all of the form $\\exists x\\in y$ or $\\forall x\\in y$.\n\\end{proposition}\n\\begin{proof}\nInduct on the formula. Atomic membership and equality are the same actual relations in both structures. Boolean connectives preserve agreement. A bounded quantifier over $y\\in M$ ranges over the same elements in both structures, because transitivity puts every member of $y$ in $M$.\n\\end{proof}\nThis gives zero defect for the specified formulas and parameter transports. Stronger generic-absoluteness results require their own hypotheses and semantic fragments. The","There is also a direct invariant fragment that requires no new theorem.\n\\begin{proposition}[Bounded-formula invariance]\\label{prop:delta0}\nIf $M\\subseteq N$ are transitive membership structures and all parameters belong to $M$, they agree on every bounded formula, namely a formula whose quantifiers are all of the form $\\exists x\\in y$ or $\\forall x\\in y$.\n\\end{proposition}\n\\begin{proof}\nInduct on the formula. Atomic membership and equality are the same actual relations in both structures. Boolean connectives preserve agreement. A bounded quantifier over $y\\in M$ ranges over the same elements in both structures, because transitivity puts every member of $y$ in $M$.\n\\end{proof}\nThis gives zero defect for the specified formulas and parameter transports. Stronger generic-absoluteness results require their own hypotheses and semantic fragments. They are not consequences of merely naming an invariant core.","transitive membership structures and all parameters belong to $M$, they agree on every bounded formula, namely a formula whose quantifiers are all of the form $\\exists x\\in y$ or $\\forall x\\in y$.\n\\end{proposition}\n\\begin{proof}\nInduct on the formula. Atomic membership and equality are the same actual relations in both structures. Boolean connectives preserve agreement. A bounded quantifier over $y\\in M$ ranges over the same elements in both structures, because transitivity puts every member of $y$ in $M$.\n\\end{proof}\nThis gives zero defect for the specified formulas and parameter transports. Stronger generic-absoluteness results require their own hypotheses and semantic fragments. They are not consequences of merely naming an invariant core.","a, namely a formula whose quantifiers are all of the form $\\exists x\\in y$ or $\\forall x\\in y$.\n\\end{proposition}\n\\begin{proof}\nInduct on the formula. Atomic membership and equality are the same actual relations in both structures. Boolean connectives preserve agreement. A bounded quantifier over $y\\in M$ ranges over the same elements in both structures, because transitivity puts every member of $y$ in $M$.\n\\end{proof}\nThis gives zero defect for the specified formulas and parameter transports. Stronger generic-absoluteness results require their own hypotheses and semantic fragments. They are not consequences of merely naming an invariant core.","ly a formula whose quantifiers are all of the form $\\exists x\\in y$ or $\\forall x\\in y$.\n\\end{proposition}\n\\begin{proof}\nInduct on the formula. Atomic membership and equality are the same actual relations in both structures. Boolean connectives preserve agreement. A bounded quantifier over $y\\in M$ ranges over the same elements in both structures, because transitivity puts every member of $y$ in $M$.\n\\end{proof}\nThis gives zero defect for the specified formulas and parameter transports. Stronger generic-absoluteness results require their own hypotheses and semantic fragments. They are not consequences of merely naming an invariant core.","\\subsection{Modal structure is established prior mathematics}\nThe modal interpretation of forcing uses $\\Diamond\\varphi$ for forceability and $\\Box\\varphi$ for truth in every forcing extension. Assuming consistency, the ZFC-provably valid modal principles for unrestricted forcing are exactly S4.2 \\cite{hamkinslowe}. Restricted forcing classes have different structural behavior \\cite{hamkinsleibmanlowe}. KRS does not reprove these results. It records individual transport defects and their reconstruction alongside the existing modal language.","\\section{Where nontrivial transport geometry can enter}\\label{sec:geometry}\nFor constant $\\F$ fibers, a truth differential is a coboundary. On every closed comparison cycle its sum is zero. This is an exact identity, not evidence that every richer world system is geometrically trivial.","To retain information about structures rather than just truth values, give each world a fiber $X_M$ and each arrow an actual transport map $T_f:X_M\\to X_N$. Parallel paths $p,q:M\\to N$ can then be compared only because their source and target types agree. For linear fibers define the path discrepancy\n\\[\n\\Delta_{p,q}=T_p-T_q.\n\\]\nIf the paths are declared to be the \\emph{same morphism} in the source category and $T$ is a functor, this discrepancy must vanish. A nonzero value means that the paths are distinct morphisms, or that a claimed relation is not respected. It cannot be described as curvature of an already commuting strict functor square.","mation about structures rather than just truth values, give each world a fiber $X_M$ and each arrow an actual transport map $T_f:X_M\\to X_N$. Parallel paths $p,q:M\\to N$ can then be compared only because their source and target types agree. For linear fibers define the path discrepancy\n\\[\n\\Delta_{p,q}=T_p-T_q.\n\\]\nIf the paths are declared to be the \\emph{same morphism} in the source category and $T$ is a functor, this discrepancy must vanish. A nonzero value means that the paths are distinct morphisms, or that a claimed relation is not respected. It cannot be described as curvature of an already commuting strict functor square.","\\begin{example}[Nontrivial holonomy with an exact defect]\\label{ex:holonomy}\nLet the one-object category be the group with generator $s$ and relation $s^2=1$. Put $X=\\F^2$ and let $T_s$ swap coordinates. This is a valid nontrivial action. Its invariant vectors are $(0,0)$ and $(1,1)$.\nFor $x=(1,0)$,\n\\[\nT_sx+x=(1,1)\\ne0,\n\\]\nbut\n\\[\nT_s(T_sx+x)+(T_sx+x)=0.\n\\]\nThe second equation is the transported composition law for $s^2=1$. Thus nontrivial fiber transport is compatible with an exact defect. No contradiction or unexplained curvature is involved.\n\\end{example}","trivial holonomy with an exact defect]\\label{ex:holonomy}\nLet the one-object category be the group with generator $s$ and relation $s^2=1$. Put $X=\\F^2$ and let $T_s$ swap coordinates. This is a valid nontrivial action. Its invariant vectors are $(0,0)$ and $(1,1)$.\nFor $x=(1,0)$,\n\\[\nT_sx+x=(1,1)\\ne0,\n\\]\nbut\n\\[\nT_s(T_sx+x)+(T_sx+x)=0.\n\\]\nThe second equation is the transported composition law for $s^2=1$. Thus nontrivial fiber transport is compatible with an exact defect. No contradiction or unexplained curvature is involved.\n\\end{example}","The graph tests compare direct enumeration of all gradients with an independent propagation-based integrator and with Gaussian elimination over $\\F$. They verify reconstruction counts, kernel dimension, and ambiguity support. The partition tests compare every selector output with the interval in Theorem~\\ref{thm:envelope}, and independently enumerate coordinate homomorphisms through four points. The two-world test enumerates all functions $\\F^2\\to\\F$ and finds exactly two Boolean retractions, neither swap-equivariant.","f all gradients with an independent propagation-based integrator and with Gaussian elimination over $\\F$. They verify reconstruction counts, kernel dimension, and ambiguity support. The partition tests compare every selector output with the interval in Theorem~\\ref{thm:envelope}, and independently enumerate coordinate homomorphisms through four points. The two-world test enumerates all functions $\\F^2\\to\\F$ and finds exactly two Boolean retractions, neither swap-equivariant.","\\subsection{Why the symbolic checks support universal identities}\nA Boolean polynomial is represented as a finite sum of squarefree monomials over $\\F$. Addition cancels duplicate monomials. Multiplication unions their variable supports, using $x_i^2=x_i$. These rules are valid in every Boolean ring. A polynomial identity whose difference reduces to the empty normal form therefore has an algebraic certificate applicable to every Boolean-ring substitution.","The most useful central equation is therefore conditional rather than rhetorical:\n\\[\n\\boxed{\\text{transported source data}+\\text{certified residual}=\\text{target data}.}\n\\]\nWhen an invariant projector exists, it gives a stronger decomposition. When it does not, the theory retains canonical bounds and records the obstruction. This is the foundation of a workable resolution semantics, and a precise starting point for testing a broader Komorebi calculus of mathematical worlds.","midrule\\endhead\n$M\\models T$ & A supplied model, not an assertion that the ambient theory proves such a model exists.","$\\mathcal C$ & A small category of typed world transformations.","$A$ & A Boolean algebra of observables, often sentences modulo provable equivalence.","$B_M$ & The Boolean algebra of values at one world.","$v_M$ & Semantic evaluation of observables.","$D_f$ & Additive defect between transported source evaluation and target evaluation.","$\\Inv$ & Zero-defect observables. A subalgebra, generally not an ideal.","$K$ & Observables vanishing at every world. An ideal.","$A/\\Inv$ & Additive residual quotient. Not a Boolean quotient in general.","$\\Inv/K$ & Definable coherent evaluations, a Boolean algebra.","$L_\\Pi,U_\\Pi$ & Canonical lower and upper invariant approximations.","$P_s$ & Boolean retraction determined by a representative in each finite block.","$E_\\Pi$ & Weighted real orthogonal projection. Not a Boolean homomorphism.","$Q_k$ & Real multiresolution detail projector for nested partitions.","$\\Delta_{p,q}$ & Typed discrepancy between parallel transport paths."]}