From f8f5d330f9a6e225a4dfaa3bd705800a03059c1a Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Sun, 23 Nov 2025 12:36:06 -0800 Subject: [PATCH 01/11] fix: clarifications and touch-ups --- aion-holography/references.bib | 10 ++++---- aion-holography/sections/holography.tex | 25 +++++++++++++------- aion-holography/sections/multiway_ruliad.tex | 2 +- aion-holography/sections/rmg.tex | 23 +++++++++++++----- 4 files changed, 39 insertions(+), 21 deletions(-) diff --git a/aion-holography/references.bib b/aion-holography/references.bib index a2a8876..a45fbf1 100644 --- a/aion-holography/references.bib +++ b/aion-holography/references.bib @@ -1,5 +1,5 @@ @incollection{EhrigLowe1997, - author = {Hartmut Ehrig and Michael L{\"o}we}, + author = {Ehrig, Hartmut and L{\"o}we, Michael}, title = {Graph Rewriting with the Double Pushout Approach}, booktitle = {Handbook of Graph Grammars and Computing by Graph Transformation}, editor = {Grzegorz Rozenberg}, @@ -10,7 +10,7 @@ @incollection{EhrigLowe1997 } @article{vanOostrom1994, - author = {Vincent van Oostrom}, + author = {van Oostrom, Vincent}, title = {Confluence by Decreasing Diagrams}, journal = {Theoretical Computer Science}, year = {1994}, @@ -20,7 +20,7 @@ @article{vanOostrom1994 } @article{CoeckeDuncan2011, - author = {Bob Coecke and Ross Duncan}, + author = {Coecke, Bob and Duncan, Ross}, title = {Interacting Quantum Observables: Categorical Algebra and Diagrammatics}, journal = {New Journal of Physics}, year = {2011}, @@ -29,7 +29,7 @@ @article{CoeckeDuncan2011 } @article{Wolfram2020, - author = {Stephen Wolfram}, + author = {Wolfram, Stephen}, title = {A Class of Models with the Potential to Represent Fundamental Physics}, journal = {Complex Systems}, year = {2020}, @@ -39,7 +39,7 @@ @article{Wolfram2020 } @article{Rissanen1978, - author = {Jorma Rissanen}, + author = {Rissanen, Jorma}, title = {Modeling by Shortest Data Description}, journal = {Automatica}, year = {1978}, diff --git a/aion-holography/sections/holography.tex b/aion-holography/sections/holography.tex index 43a4695..046dac6 100644 --- a/aion-holography/sections/holography.tex +++ b/aion-holography/sections/holography.tex @@ -30,16 +30,20 @@ \subsection{Microsteps and derivation graphs} For a value $v$ in some state $S_i$ we define a \emph{derivation graph} $\mathcal{D}(v)$ whose nodes are intermediate values and whose edges are microstep applications that produced them; the construction is standard -and we omit the routine details. Because we only consider finite derivations +and we omit the routine details. For a finite derivation \[ S_0 \Rewrite^{\mu_0} S_1 \Rewrite^{\mu_1} \cdots \Rewrite^{\mu_{n-1}} S_n, \] -every provenance edge in $\mathcal{D}(v)$ points from a value in some -state $S_j$ to a value in a strictly later state $S_{j'}$ with $j' > j$. -Immutability ensures that values are never updated in-place, only created -at later ticks. Hence every causal chain leading to $v$ has length at -most $n$, and $\mathcal{D}(v)$ is a finite, acyclic graph. +each microstep consumes values in some $S_j$ and produces new values in +the strictly later state $S_{j+1}$, so every provenance edge in +$\mathcal{D}(v)$ points from a value in $S_j$ to a value in $S_{j'}$ +with $j' > j$. Immutability ensures that values are never updated +in-place, only created at later ticks. Since each RMG state $S_j$ is +finite and there are only $n+1$ such states along the derivation, +$\mathcal{D}(v)$ has finitely many nodes; and because tick indices +strictly increase along edges, every causal chain leading to $v$ has +length at most $n$, so $\mathcal{D}(v)$ is a finite acyclic graph. \subsection{AION state packets as an instance} @@ -171,9 +175,12 @@ \subsection{Computational holography} S_{i+1} \;=\; \Apply(S_i,\mu_i) \] for $0 \le i < n$, where $\Apply$ executes the unique microstep -described by~$\mu_i$ under the tick semantics. By -Theorem~\ref{thm:tick-confluence}, each $S_{i+1}$ is well-defined up -to isomorphism. +described by~$\mu_i$ under the deterministic tick semantics. +Determinism ensures that each $S_{i+1}$ is uniquely determined (up to +isomorphism), while tick-level confluence +(Theorem~\ref{thm:tick-confluence}) guarantees that any internal +interleaving of concurrent matches compatible with $\mu_i$ yields an +isomorphic successor. \end{definition} \begin{theorem}[Computational holography] diff --git a/aion-holography/sections/multiway_ruliad.tex b/aion-holography/sections/multiway_ruliad.tex index 340b6df..2dd48a9 100644 --- a/aion-holography/sections/multiway_ruliad.tex +++ b/aion-holography/sections/multiway_ruliad.tex @@ -86,7 +86,7 @@ \section{Multiway Systems and the Ruliad} The class of all possible such worldlines, across all rule sets and inputs, forms a large multiway object akin to the Ruliad. The rulial distance from \cref{sec:rulial} equips this space of observers with a -geometry, and the Chronos--Kairos--Aion time model from the \AION{} +geometry, and the Chronos, Kairos, Aion time model from the \AION{} calculus\footnote{Developed in a separate technical note on the \AION{} time model~\cite{RossAIONCalculus2025}.} gives a temporal structure on branches and merges. diff --git a/aion-holography/sections/rmg.tex b/aion-holography/sections/rmg.tex index fbf6309..7563090 100644 --- a/aion-holography/sections/rmg.tex +++ b/aion-holography/sections/rmg.tex @@ -57,16 +57,26 @@ \subsection{Initial algebra viewpoint} \subsection{Morphisms and category of RMGs} \begin{definition}[RMG morphism] -We define morphisms by structural recursion on RMG depth. A morphism +We define morphisms by structural recursion on RMG depth, taking the +set of atomic payloads $P$ as a discrete category (only identity +morphisms). Thus +\[ + \Hom(\Atom(p),\Atom(p')) = + \begin{cases} + \{\id_{\Atom(p)}\} & \text{if } p = p',\\ + \emptyset & \text{otherwise.} + \end{cases} +\] +For composite objects, a morphism $f : (S,\alpha,\beta) \To (S',\alpha',\beta')$ consists of: \begin{itemize}[leftmargin=*] \item a graph homomorphism of skeletons $f_V : V \To V'$, $f_E : E \To E'$ preserving sources and targets; and \item for each $v \in V$ a morphism of attachments $f_v : \alpha(v) \To \alpha'(f_V(v))$ and, for each $e \in E$, a - morphism $f_e : \beta(e) \To \beta'(f_E(e))$, defined recursively - using the same clause whenever an attachment is itself of the form - $(S,\alpha,\beta)$. + morphism $f_e : \beta(e) \To \beta'(f_E(e))$, where each $f_v$ and + $f_e$ is itself an RMG morphism, defined recursively on the depth + of the attachment. \end{itemize} Composition and identities are defined componentwise. \end{definition} @@ -147,7 +157,7 @@ \subsection{Notation summary} \toprule \textbf{Symbol} & \textbf{Meaning} \\ \midrule -$\mathcal{U} = (G;\alpha,\beta)$ & single RMG state (one object in a universe $\mathcal{U}$) \\ +$\mathcal{U} = (G;\alpha,\beta)$ & single RMG state (one object in an RMG universe $U$) \\ $p = (L \xleftarrow{\ell} K \xrightarrow{r} R)$ & DPOI rule \\ $\mu_i$ & microstep label \\ $P = (\mu_0,\dots,\mu_{n-1})$ & provenance payload \\ @@ -161,4 +171,5 @@ \subsection{Notation summary} \medskip Subsequent sections introduce $D_{\tau,m}$ (rulial distance), -$\Hist(U,R)$ (history category), and other observer-related notation. +$\Hist(U,R)$ (history category on a universe $U$ of RMG states), and +other observer-related notation. From f6dee50b5c2ef5d0bf91fdc7cec0d61bd615c130 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Sun, 23 Nov 2025 12:41:47 -0800 Subject: [PATCH 02/11] chore: remove todo.md --- aion-holography/todo.md | 415 ---------------------------------------- 1 file changed, 415 deletions(-) delete mode 100644 aion-holography/todo.md diff --git a/aion-holography/todo.md b/aion-holography/todo.md deleted file mode 100644 index 2a4015f..0000000 --- a/aion-holography/todo.md +++ /dev/null @@ -1,415 +0,0 @@ -@@ -0,0 +1,295 @@ -\sectionbreak -\section{Provenance Payloads and Computational Holography} -\label{sec:holography} - -We now make precise the idea that the entire interior evolution of a -computation can be encoded on a ``boundary'': an initial state together -with a finite provenance payload. This is the formal content of -\emph{computational holography}. - -\subsection{Microsteps and derivation graphs} - -Fix a rule set $R$ and tick semantics as in -Theorem~\ref{thm:tick-confluence}. A \emph{microstep} is a single -scheduler tick whose batch contains exactly one skeleton DPOI step -(possibly accompanied by attachment steps in preserved fibers). We -write -\[ - S_i \;\Rewrite^{\mu_i}\; S_{i+1} -\] -for such a microstep, where the label $\mu_i$ records: -\begin{itemize}[leftmargin=*] - \item the rule identifier $p \in R$; - \item the match identifier for the skeleton step; - \item any attachment-rule identifiers used in the same tick; - \item auxiliary metadata (timestamps, policy hashes, etc.). -\end{itemize} -We abstract this as a finite record in some fixed alphabet; in -particular, we assume it has a self-delimiting encoding. - -For a value $v$ in some state $S_i$ we define a \emph{derivation graph} -$\mathcal{D}(v)$ whose nodes are intermediate values and whose edges are -microstep applications that produced them; the construction is standard -and we omit the routine details. Because we only consider finite derivations -\[ - S_0 \Rewrite^{\mu_0} S_1 \Rewrite^{\mu_1} \cdots - \Rewrite^{\mu_{n-1}} S_n, -\] -every provenance edge in $\mathcal{D}(v)$ points from a value in some -state $S_j$ to a value in a strictly later state $S_{j'}$ with $j' > j$. -Immutability ensures that values are never updated in-place, only created -at later ticks. Hence every causal chain leading to $v$ has length at -most $n$, and $\mathcal{D}(v)$ is a finite, acyclic graph. -Comment on lines +30 to +42 -@coderabbitai -coderabbitai bot -3 hours ago -⚠️ Potential issue | 🟠 Major - -Tighten the finiteness/acyclicity justification for the derivation graph. - -Right now you jump from “finite derivation” + forward‑in‑time edges to “(\mathcal{D}(v)) is a finite, acyclic graph” in one sentence. Acyclicity is fine (edges strictly increase tick index), but finiteness actually uses two separate facts: - -Only finitely many states (S_0,\dots,S_n) occur in the volume. -Each state is a finite RMG state (finite underlying graph), so only finitely many values ever appear. -I’d spell that out and avoid the slightly misleading “Because we only consider finite derivations … every provenance edge …” phrasing. For example: - --For a value $v$ in some state $S_i$ we define a \emph{derivation graph} --$\mathcal{D}(v)$ whose nodes are intermediate values and whose edges are --microstep applications that produced them; the construction is standard --and we omit the routine details. Because we only consider finite derivations --\[ -- S_0 \Rewrite^{\mu_0} S_1 \Rewrite^{\mu_1} \cdots -- \Rewrite^{\mu_{n-1}} S_n, --\] --every provenance edge in $\mathcal{D}(v)$ points from a value in some --state $S_j$ to a value in a strictly later state $S_{j'}$ with $j' > j$. --Immutability ensures that values are never updated in-place, only created --at later ticks. Hence every causal chain leading to $v$ has length at --most $n$, and $\mathcal{D}(v)$ is a finite, acyclic graph. -+For a value $v$ in some state $S_i$ we define a \emph{derivation graph} -+$\mathcal{D}(v)$ whose nodes are intermediate values and whose edges are -+microstep applications that produced them; the construction is standard -+and we omit the routine details. For a finite derivation -+\[ -+ S_0 \Rewrite^{\mu_0} S_1 \Rewrite^{\mu_1} \cdots -+ \Rewrite^{\mu_{n-1}} S_n, -+\] -+each microstep consumes values in some $S_j$ and produces new values in -+the strictly later state $S_{j+1}$, so every provenance edge in -+$\mathcal{D}(v)$ points from a value in $S_j$ to a value in $S_{j'}$ -+with $j' > j$. Immutability ensures that values are never updated -+in-place, only created at later ticks. Since each RMG state $S_j$ is -+finite and there are only $n+1$ such states along the derivation, -+$\mathcal{D}(v)$ has finitely many nodes; and because tick indices -+strictly increase along edges, every causal chain leading to $v$ has -+length at most $n$, so $\mathcal{D}(v)$ is a finite acyclic graph. - ---- - -Comment on lines +167 to +176 -@coderabbitai -coderabbitai bot -3 hours ago -⚠️ Potential issue | 🟠 Major - -Stop crediting tick‑level confluence for what is really determinism in the reconstruction definition. - -You fixed the determinism vs confluence split in the wormhole definition, but here you still say: - -“By Theorem~\textbackslash ref{thm:tick-confluence}, each (S_{i+1}) is well-defined up to isomorphism.” - -However, the fact that (\Apply(S_i,\mu_i)) yields a unique successor is fundamentally determinism of the tick semantics; tick‑level confluence is about independence of interleavings of concurrent matches. - -I’d mirror the earlier wording and make both roles explicit: - -- S_{i+1} \;=\; \Apply(S_i,\mu_i) --\] --for $0 \le i < n$, where $\Apply$ executes the unique microstep --described by~$\mu_i$ under the tick semantics. By --Theorem~\ref{thm:tick-confluence}, each $S_{i+1}$ is well-defined up --to isomorphism. -+ S_{i+1} \;=\; \Apply(S_i,\mu_i) -+\] -+for $0 \le i < n$, where $\Apply$ executes the unique microstep -+described by~$\mu_i$ under the deterministic tick semantics. -+Determinism ensures that each $S_{i+1}$ is uniquely determined (up to -+isomorphism), while tick-level confluence -+(Theorem~\ref{thm:tick-confluence}) guarantees that any internal -+interleaving of concurrent matches compatible with $\mu_i$ yields an -+isomorphic successor. -Otherwise you’re (again) implicitly loading determinism into a theorem that’s nominally about confluence. - ---- - -aion-holography/sections/multiway_ruliad.tex lines 89 and 98-100: the -punctuation for the time model triad is inconsistent ("Chronos--Kairos--Aion" vs -"Chronos, Kairos, Aion"); choose one convention and make all occurrences -identical—either replace the commas with en-dashes on lines 98-100 -(Chronos--Kairos--Aion) if treating it as a compound triad, or replace the -en-dashes on line 89 with commas on line 89 (Chronos, Kairos, Aion) if treating -them as a simple list—then update every other instance in the file to match. - ---- - -Comment on lines +59 to +71 -@coderabbitai -coderabbitai bot -3 hours ago -⚠️ Potential issue | 🟠 Major - -RMG morphism definition still lacks a base case and a clear ambient category for atoms. - -You fixed the overt circularity by switching to “structural recursion on RMG depth”, but this is still incomplete: - -You only give a clause for composite objects ((S,\alpha,\beta)); there is no base case describing morphisms between atoms (\mathrm{Atom}(p)). -You say “for each (v) a morphism of attachments (f_v : \alpha(v) \To \alpha'(f_V(v)))”, but you never specify in which category these attachment morphisms live. Previously this was (incorrectly) “in (\RMG)”; now it’s just floating. -As written, (\RMG) is not actually a category: Hom‑sets between atomic objects are undefined, and the recursion on attachments has no base case. - -You need to either: - -Treat the payloads (P) as a discrete base category (most conservative), or -Parametrise RMGs over an arbitrary small base category (\mathcal{C}) of atomic payloads and make that explicit. -Here’s a concrete minimal patch using the discrete‑payload option: - --\begin{definition}[RMG morphism] --We define morphisms by structural recursion on RMG depth. A morphism --$f : (S,\alpha,\beta) \To (S',\alpha',\beta')$ consists of: -+\begin{definition}[RMG morphism] -+We define morphisms by structural recursion on RMG depth, taking the -+set of atomic payloads $P$ as a discrete category (only identity -+morphisms). Thus -+\[ -+ \Hom(\Atom(p),\Atom(p')) = -+ \begin{cases} -+ \{\id_{\Atom(p)}\} & \text{if } p = p',\\ -+ \emptyset & \text{otherwise.} -+ \end{cases} -+\] -+For composite objects, a morphism -+$f : (S,\alpha,\beta) \To (S',\alpha',\beta')$ consists of: -@@ -- \item for each $v \in V$ a morphism of attachments -- $f_v : \alpha(v) \To \alpha'(f_V(v))$ and, for each $e \in E$, a -- morphism $f_e : \beta(e) \To \beta'(f_E(e))$, defined recursively -- using the same clause whenever an attachment is itself of the form -- $(S,\alpha,\beta)$. -+ \item for each $v \in V$ a morphism of attachments -+ $f_v : \alpha(v) \To \alpha'(f_V(v))$ and, for each $e \in E$, a -+ morphism $f_e : \beta(e) \To \beta'(f_E(e))$, where each $f_v$ and -+ $f_e$ is itself an RMG morphism, defined recursively on the depth -+ of the attachment. -If you need nontrivial maps between atoms later, just replace “discrete category on (P)” with “small base category (\mathcal{C}) whose objects are payloads” and adjust the Hom‑clause accordingly. But as it stands, the category structure you claim for (\RMG) is underspecified. - -🤖 Prompt for AI Agents -In aion-holography/sections/rmg.tex around lines 59–71, the RMG morphism -definition omits a base case for atoms and fails to specify the ambient category -for attachment morphisms; add an explicit base clause that defines Hom(Atom(p), -Atom(q)) (e.g. make payloads P a discrete category so Hom(Atom(p),Atom(q)) = -empty unless p=q, where it is the identity), and state that every attachment -morphism f_v and f_e lives in that chosen base category (or alternatively -parameterize the whole definition over a small base category C of payloads and -replace “in RMG” with morphisms in C); update the recursive clause to stop at -atoms using that base-case Hom and ensure composition/identities use the -base-category identities for atomic attachments. - ---- - - -Original file line number Diff line number Diff line change -@@ -0,0 +1,162 @@ -\sectionbreak -\section{Recursive Metagraphs} -\label{sec:rmg} - -In this section we define Recursive Metagraphs (RMGs) and relate them to -standard graph models and typed open graphs. An RMG is a finite typed -open graph whose nodes and edges may themselves carry RMGs recursively, -forming a finitely branching, well-founded tree of graphs. - -\subsection{Inductive definition} - -Fix a set $P$ of atomic payloads (blobs, literals, external IDs). - -\begin{definition}[Recursive Metagraph]\label{def:rmg} -The class $\RMG$ of \emph{recursive metagraphs} is the least set closed -under the following constructors: -\begin{enumerate}[leftmargin=*] - \item for each $p \in P$ there is an \emph{atom} $\mathrm{Atom}(p) - \in \RMG$; - \item for any finite directed multigraph $S = (V,E,s,t)$ and - assignments $\alpha : V \To \RMG$, $\beta : E \To \RMG$, the triple - $(S,\alpha,\beta)$ is in $\RMG$. -\end{enumerate} -\end{definition} - -We write an element of $\RMG$ as either an atom or as a ``1-skeleton'' -graph decorated by attachments on vertices and edges. Attachments -themselves may be recursive metagraphs, so this attachment structure -can nest arbitrarily deeply. This definition agrees with the -set-theoretic and initial-algebra presentation developed in previous -technical notes. - -\paragraph{Example (A tiny recursive metagraph).} -As a concrete instance, consider a program call graph where each -function node carries its own abstract syntax tree (AST) and each call -edge carries a small provenance graph (e.g.\ optimisation decisions or -runtime statistics). We can model this as an RMG whose skeleton has -nodes $v_f,v_g$ for functions $f,g$, a directed edge -$e_{\mathsf{call}} : v_f \to v_g$ for the call, and attachments: -$\alpha(v_f)$ the AST of $f$, $\alpha(v_g)$ the AST of $g$, and -$\beta(e_{\mathsf{call}})$ the call provenance. - -\subsection{Initial algebra viewpoint} - -Let $\mathcal{G}$ be a small collection of allowable skeleton shapes -(finite directed multigraphs up to isomorphism). Define a finitary -polynomial endofunctor -$F : \Set \To \Set$ by -\[ - F(X) = P + \coprod_{S\in\mathcal{G}} (V_S \To X) \times (E_S \To X). -\] -Then $\RMG$ is (up to isomorphism) the carrier of the initial -$F$-algebra. This yields the usual structural recursion and induction -principles: every function out of $\RMG$ is uniquely determined by its -action on atoms and on decorated skeletons. - -\subsection{Morphisms and category of RMGs} - -\begin{definition}[RMG morphism] -We define morphisms by structural recursion on RMG depth. A morphism -$f : (S,\alpha,\beta) \To (S',\alpha',\beta')$ consists of: -\begin{itemize}[leftmargin=*] - \item a graph homomorphism of skeletons $f_V : V \To V'$, $f_E : E \To - E'$ preserving sources and targets; and - \item for each $v \in V$ a morphism of attachments - $f_v : \alpha(v) \To \alpha'(f_V(v))$ and, for each $e \in E$, a - morphism $f_e : \beta(e) \To \beta'(f_E(e))$, defined recursively - using the same clause whenever an attachment is itself of the form - $(S,\alpha,\beta)$. -\end{itemize} -Composition and identities are defined componentwise. -flyingrobots marked this conversation as resolved. -\end{definition} -\begin{figure}[t] - \centering - \begin{tikzpicture}[ - snode/.style={circle,draw=purple!70!black,fill=purple!10,thick,minimum size=10mm,inner sep=0pt}, - sedge/.style={-Latex,thick,purple!70!black}, - anode/.style={circle,draw=orange!70!black,fill=orange!8,thin,minimum size=4mm,inner sep=0pt}, - aedge/.style={-Latex,thin,orange!70!black}, - >=Latex - ] - - % Skeleton level - \node[snode] (v1) at (0,0) {$v_1$}; - \node[snode] (v2) at (3.0,0) {$v_2$}; - \draw[sedge] (v1) -- node[above]{\small $e_{\mathsf{call}}$} (v2); - - \node at (1.5,-1.0) {\small skeleton $G$}; - - % Attachment for v1 (e.g. an AST) - \begin{scope}[shift={(-2.4,1.6)}] - \node[anode] (a1) at (0,0) {}; - \node[anode] (a2) at (0.9,0.6) {}; - \node[anode] (a3) at (0.9,-0.6) {}; - \draw[aedge] (a1) -- (a2); - \draw[aedge] (a1) -- (a3); - \node[anchor=east] at (-0.1,0) {\small $\alpha(v_1)$}; - \end{scope} - \draw[thin,dashed] (-0.3,0.5) to[out=150,in=0] (-1.5,1.6); - - % Attachment for v2 - \begin{scope}[shift={(1.8,1.6)}] - \node[anode] (b1) at (0,0) {}; - \node[anode] (b2) at (0.9,0) {}; - \node[anode] (b3) at (0.45,0.8) {}; - \draw[aedge] (b1) -- (b2); - \draw[aedge] (b2) -- (b3); - \draw[aedge] (b3) -- (b1); - \node[anchor=east] at (-0.1,0) {\small $\alpha(v_2)$}; - \end{scope} - \draw[thin,dashed] (3.3,0.5) to[out=30,in=180] (2.7,1.6); - - % Attachment for edge e - \begin{scope}[shift={(1.5,-2.0)}] - \node[anode] (c1) at (-0.6,0) {}; - \node[anode] (c2) at (0.6,0) {}; - \draw[aedge] (c1) -- (c2); - \node at (0,-0.8) {\small $\beta(e_{\mathsf{call}})$}; - \end{scope} - \draw[thin,dashed] (1.5,-0.2) -- (1.5,-1.4); - - \end{tikzpicture} - \caption{A simple recursive metagraph: the skeleton $G$ has two - nodes $v_1,v_2$ and an edge $e_{\mathsf{call}}$, while each node and - edge carries its own attached graph $\alpha(v_i),\beta(e_{\mathsf{call}})$. - In an RMG this attachment structure recurses: the attachment graphs - themselves may have attachments, and so on.} - \label{fig:rmg-recursive-example} -\end{figure} - -\subsection{Relation to ordinary and hypergraphs} - -Typed open graphs $\OGraph_T$ form an adhesive category, and DPO -rewriting is well-behaved there. Typed hypergraphs embed fully and -faithfully into typed open graphs via an incidence construction that -preserves DPO steps and their multiway derivations. -Thus RMG rewriting subsumes standard open-graph and hypergraph -rewriting while adding recursive structure through attachments. - -\subsection{Notation summary} - -For convenience, we collect the main notation introduced so far: - -\medskip -\begin{center} -\begin{tabular}{ll} -\textbf{Symbol} & \textbf{Meaning} \\[2pt] -\hline\\[-8pt] -$\mathcal{U} = (G;\alpha,\beta)$ & single RMG state (one object in a universe $\mathcal{U}$) \\ -$p = (L \xleftarrow{\ell} K \xrightarrow{r} R)$ & DPOI rule \\ -$\mu_i$ & microstep label \\ -$P = (\mu_0,\dots,\mu_{n-1})$ & provenance payload \\ -$S_0 \Rewrite^\ast S_n$ & derivation volume (interior evolution) \\ -$(S_0,P)$ & wormhole (boundary encoding) \\ -$\Del(m), \Use(m)$ & delete and use sets of a match \\ -$\Recon(S_0,P)$ & reconstruction procedure \\ -\end{tabular} -\end{center} -\medskip - -Subsequent sections introduce $D_{\tau,m}$ (rulial distance), -Comment on lines +149 to +161 -@coderabbitai -coderabbitai bot -3 hours ago -⚠️ Potential issue | 🟠 Major - -Notation for (\mathcal{U}) vs (U) is now self‑contradictory; fix the universe symbol. - -The table entry currently reads: - -$\mathcal{U} = (G;\alpha,\beta)$ & single RMG state (one object in a universe $\mathcal{U}$) \\ -You’re using (\mathcal{U}) simultaneously for: - -A single RMG state, and -The ambient “universe” containing that state. -Then, two lines later, you introduce (\Hist(U,R)), where (U) is clearly meant to be the collection/category of RMG states. This is exactly the ambiguity the earlier review asked you to avoid, and the new wording makes it worse. - -I’d disambiguate as follows: - --$\mathcal{U} = (G;\alpha,\beta)$ & single RMG state (one object in a universe $\mathcal{U}$) \\ -+$\mathcal{U} = (G;\alpha,\beta)$ & single RMG state (one object in an RMG universe $U$) \\ -@@ --Subsequent sections introduce $D_{\tau,m}$ (rulial distance), --$\Hist(U,R)$ (history category), and other observer-related notation. -+Subsequent sections introduce $D_{\tau,m}$ (rulial distance), -+$\Hist(U,R)$ (history category on a universe $U$ of RMG states), and -+other observer-related notation. -That way (\mathcal{U}) is consistently “a particular state” and (U) is “the collection/category of such states”. - -🤖 Prompt for AI Agents -In aion-holography/sections/rmg.tex around lines 149-161, the symbol \mathcal{U} -is used inconsistently as both a single RMG state and the ambient universe; -change the notation so \mathcal{U} denotes one particular RMG state (e.g. -“\mathcal{U} = (G;\alpha,\beta)” and describe it as “a single RMG state”), -reserve U (plain U) for the collection/category of states (the universe), update -the table entry text accordingly, and then search and replace other occurrences -in this section (e.g. \Hist(U,R) and any sentences that refer to “universe”) to -use U when referring to the collection and \mathcal{U} when referring to an -individual state so the notation is consistent throughout. - --- - -Review is correct but critically incomplete—fixes needed at three locations, not one. - -The bibliography indeed mixes formats. However, your diff only addresses line 2 (Ehrig2006). The regex verification found THREE entries using "First Last" format: - -Line 2 (Ehrig2006): needs fix ✓ (covered by your diff) -Line 9: author = {Hartmut Ehrig and Michael L{\"o}we} — needs fix -Line 30: author = {Bob Coecke and Ross Duncan} — needs fix -Lines 73, 84 (LS06, EEPT06) are already correctly formatted as "Last, First" and require no changes. - -Apply the diff to line 2, then standardize lines 9 and 30 to Last, FirstName format as well. Verify the entire file has zero remaining "First Last" entries after the fixes. From 71fe4b42ba7a64e93e2614a190e88479c3d8f807 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Sun, 23 Nov 2025 13:40:36 -0800 Subject: [PATCH 03/11] fix: missing operator --- aion-holography/macros.tex | 2 ++ aion-holography/sections/rmg.tex | 4 ++-- 2 files changed, 4 insertions(+), 2 deletions(-) diff --git a/aion-holography/macros.tex b/aion-holography/macros.tex index a4da491..fd52c4c 100644 --- a/aion-holography/macros.tex +++ b/aion-holography/macros.tex @@ -6,6 +6,8 @@ \newcommand{\Graph}{\cat{Graph}} \newcommand{\Hyp}{\cat{Hyp}} \newcommand{\Set}{\cat{Set}} +\DeclareMathOperator{\Hom}{Hom} +\newcommand{\id}{\mathrm{id}} % ========================================== % AION / RMG Names diff --git a/aion-holography/sections/rmg.tex b/aion-holography/sections/rmg.tex index 7563090..7272255 100644 --- a/aion-holography/sections/rmg.tex +++ b/aion-holography/sections/rmg.tex @@ -61,9 +61,9 @@ \subsection{Morphisms and category of RMGs} set of atomic payloads $P$ as a discrete category (only identity morphisms). Thus \[ - \Hom(\Atom(p),\Atom(p')) = + \Hom(\mathrm{Atom}(p),\mathrm{Atom}(p')) = \begin{cases} - \{\id_{\Atom(p)}\} & \text{if } p = p',\\ + \{\id_{\mathrm{Atom}(p)}\} & \text{if } p = p',\\ \emptyset & \text{otherwise.} \end{cases} \] From b28379f34b6cb5a090a45d3e3796e5fac7a5d353 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Sun, 23 Nov 2025 13:44:00 -0800 Subject: [PATCH 04/11] Clarify RMG text and standardize references --- aion-holography/references.bib | 6 +++--- aion-holography/sections/discussion.tex | 2 +- aion-holography/sections/holography.tex | 4 ++-- aion-holography/sections/rmg.tex | 18 ++++++++++++------ 4 files changed, 18 insertions(+), 12 deletions(-) diff --git a/aion-holography/references.bib b/aion-holography/references.bib index a45fbf1..ff2ab47 100644 --- a/aion-holography/references.bib +++ b/aion-holography/references.bib @@ -49,14 +49,14 @@ @article{Rissanen1978 } @misc{RossRMG2025, - author = {James Ross}, + author = {Ross, James}, title = {Recursive Metagraphs: DPOI Semantics, Confluence, Hypergraph Embedding, and Rulial Distance}, year = {2025}, note = {Technical report} } @misc{RossAIONCalculus2025, - author = {James Ross}, + author = {Ross, James}, title = {The {AION} Calculus}, year = {2025}, note = {Working note} @@ -81,7 +81,7 @@ @book{EEPT06 } @misc{ross_universal_charter_v1, - author = {James Ross}, + author = {Ross, James}, title = {Universal Charter: A Living Covenant for All Forms of Being Across Substrate, Time, and Dimension}, howpublished = {\url{https://github.com/universalcharter/universal-charter}}, note = {Version 1.0.0 (First Flame), commit 849d9ca}, diff --git a/aion-holography/sections/discussion.tex b/aion-holography/sections/discussion.tex index f393dbf..97c3add 100644 --- a/aion-holography/sections/discussion.tex +++ b/aion-holography/sections/discussion.tex @@ -70,7 +70,7 @@ \subsection{Related work} to layer cryptographic commitments and zero-knowledge proofs on top, enabling external verifiers to check correctness properties without learning private data. - \item \textbf{Temporal logic and Time Cube.} The Chronos--Kairos--Aion + \item \textbf{Temporal logic and Time Cube.} The Chronos, Kairos, Aion triad suggests new modal and temporal logics for reasoning about linear time, branch points, and the surrounding possibility space. \item \textbf{\COMPUTER{} architecture.} Building on this foundation, diff --git a/aion-holography/sections/holography.tex b/aion-holography/sections/holography.tex index 046dac6..4d0923a 100644 --- a/aion-holography/sections/holography.tex +++ b/aion-holography/sections/holography.tex @@ -35,7 +35,7 @@ \subsection{Microsteps and derivation graphs} S_0 \Rewrite^{\mu_0} S_1 \Rewrite^{\mu_1} \cdots \Rewrite^{\mu_{n-1}} S_n, \] -each microstep consumes values in some $S_j$ and produces new values in +each microstep reads values in some $S_j$ and produces new values in the strictly later state $S_{j+1}$, so every provenance edge in $\mathcal{D}(v)$ points from a value in $S_j$ to a value in $S_{j'}$ with $j' > j$. Immutability ensures that values are never updated @@ -177,7 +177,7 @@ \subsection{Computational holography} for $0 \le i < n$, where $\Apply$ executes the unique microstep described by~$\mu_i$ under the deterministic tick semantics. Determinism ensures that each $S_{i+1}$ is uniquely determined (up to -isomorphism), while tick-level confluence +isomorphism). Furthermore, tick-level confluence (Theorem~\ref{thm:tick-confluence}) guarantees that any internal interleaving of concurrent matches compatible with $\mu_i$ yields an isomorphic successor. diff --git a/aion-holography/sections/rmg.tex b/aion-holography/sections/rmg.tex index 7272255..98132e4 100644 --- a/aion-holography/sections/rmg.tex +++ b/aion-holography/sections/rmg.tex @@ -57,9 +57,9 @@ \subsection{Initial algebra viewpoint} \subsection{Morphisms and category of RMGs} \begin{definition}[RMG morphism] -We define morphisms by structural recursion on RMG depth, taking the -set of atomic payloads $P$ as a discrete category (only identity -morphisms). Thus +We define morphisms by structural recursion on RMG depth. First form the +discrete category $\mathbf{P}$ with $\mathrm{Ob}(\mathbf{P}) = P$ and +$\mathrm{Mor}(\mathbf{P})$ containing only identity morphisms. Thus \[ \Hom(\mathrm{Atom}(p),\mathrm{Atom}(p')) = \begin{cases} @@ -67,6 +67,8 @@ \subsection{Morphisms and category of RMGs} \emptyset & \text{otherwise.} \end{cases} \] +Under this construction the embedding $\mathrm{Atom} : \mathbf{P} \To \RMG$ is +a faithful functor. For composite objects, a morphism $f : (S,\alpha,\beta) \To (S',\alpha',\beta')$ consists of: \begin{itemize}[leftmargin=*] @@ -75,8 +77,9 @@ \subsection{Morphisms and category of RMGs} \item for each $v \in V$ a morphism of attachments $f_v : \alpha(v) \To \alpha'(f_V(v))$ and, for each $e \in E$, a morphism $f_e : \beta(e) \To \beta'(f_E(e))$, where each $f_v$ and - $f_e$ is itself an RMG morphism, defined recursively on the depth - of the attachment. + $f_e$ is itself an RMG morphism, defined by structural recursion on + RMG depth (i.e.\ the nesting level in the construction of + Definition~\ref{def:rmg}). \end{itemize} Composition and identities are defined componentwise. \end{definition} @@ -157,7 +160,7 @@ \subsection{Notation summary} \toprule \textbf{Symbol} & \textbf{Meaning} \\ \midrule -$\mathcal{U} = (G;\alpha,\beta)$ & single RMG state (one object in an RMG universe $U$) \\ +$\mathcal{U} = (G;\alpha,\beta)$ & single RMG state with $\mathcal{U} \in U$, where an RMG universe $U$ is a collection of RMG states related by rewriting \\ $p = (L \xleftarrow{\ell} K \xrightarrow{r} R)$ & DPOI rule \\ $\mu_i$ & microstep label \\ $P = (\mu_0,\dots,\mu_{n-1})$ & provenance payload \\ @@ -170,6 +173,9 @@ \subsection{Notation summary} \end{center} \medskip +Here $U$ always denotes an RMG universe in this sense, with +$\mathcal{U} \in U$ indicating a particular state. + Subsequent sections introduce $D_{\tau,m}$ (rulial distance), $\Hist(U,R)$ (history category on a universe $U$ of RMG states), and other observer-related notation. From 1fbf710778c729265cfbf593132a78486e322ef6 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Sun, 23 Nov 2025 13:47:46 -0800 Subject: [PATCH 05/11] Tighten temporal triad wording --- aion-holography/sections/discussion.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/aion-holography/sections/discussion.tex b/aion-holography/sections/discussion.tex index 97c3add..357b352 100644 --- a/aion-holography/sections/discussion.tex +++ b/aion-holography/sections/discussion.tex @@ -70,7 +70,7 @@ \subsection{Related work} to layer cryptographic commitments and zero-knowledge proofs on top, enabling external verifiers to check correctness properties without learning private data. - \item \textbf{Temporal logic and Time Cube.} The Chronos, Kairos, Aion + \item \textbf{Temporal logic and Time Cube.} The Chronos, Kairos, and Aion triad suggests new modal and temporal logics for reasoning about linear time, branch points, and the surrounding possibility space. \item \textbf{\COMPUTER{} architecture.} Building on this foundation, From ade00dce86d41fa7d0306aefc74cad7bd23f97d3 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Sun, 23 Nov 2025 13:48:54 -0800 Subject: [PATCH 06/11] Clarify atomic hom-sets and provenance edge direction --- aion-holography/sections/holography.tex | 17 +++++++++-------- aion-holography/sections/rmg.tex | 5 +++-- 2 files changed, 12 insertions(+), 10 deletions(-) diff --git a/aion-holography/sections/holography.tex b/aion-holography/sections/holography.tex index 4d0923a..682c69b 100644 --- a/aion-holography/sections/holography.tex +++ b/aion-holography/sections/holography.tex @@ -36,14 +36,15 @@ \subsection{Microsteps and derivation graphs} \Rewrite^{\mu_{n-1}} S_n, \] each microstep reads values in some $S_j$ and produces new values in -the strictly later state $S_{j+1}$, so every provenance edge in -$\mathcal{D}(v)$ points from a value in $S_j$ to a value in $S_{j'}$ -with $j' > j$. Immutability ensures that values are never updated -in-place, only created at later ticks. Since each RMG state $S_j$ is -finite and there are only $n+1$ such states along the derivation, -$\mathcal{D}(v)$ has finitely many nodes; and because tick indices -strictly increase along edges, every causal chain leading to $v$ has -length at most $n$, so $\mathcal{D}(v)$ is a finite acyclic graph. +the immediately later state $S_{j+1}$, so every provenance edge in +$\mathcal{D}(v)$ points from a value in $S_j$ to a value in $S_{j+1}$ +(hence tick indices strictly increase along edges). Immutability +ensures that values are never updated in-place, only created at later +ticks. Since each RMG state $S_j$ is finite and there are only $n+1$ +such states along the derivation, $\mathcal{D}(v)$ has finitely many +nodes; and because tick indices strictly increase along edges, every +causal chain leading to $v$ has length at most $n$, so $\mathcal{D}(v)$ +is a finite acyclic graph. \subsection{AION state packets as an instance} diff --git a/aion-holography/sections/rmg.tex b/aion-holography/sections/rmg.tex index 98132e4..60564ac 100644 --- a/aion-holography/sections/rmg.tex +++ b/aion-holography/sections/rmg.tex @@ -59,9 +59,10 @@ \subsection{Morphisms and category of RMGs} \begin{definition}[RMG morphism] We define morphisms by structural recursion on RMG depth. First form the discrete category $\mathbf{P}$ with $\mathrm{Ob}(\mathbf{P}) = P$ and -$\mathrm{Mor}(\mathbf{P})$ containing only identity morphisms. Thus +$\mathrm{Mor}(\mathbf{P})$ containing only identity morphisms. We declare the +RMG hom-sets on atoms by \[ - \Hom(\mathrm{Atom}(p),\mathrm{Atom}(p')) = + \Hom_{\RMG}(\mathrm{Atom}(p),\mathrm{Atom}(p')) = \begin{cases} \{\id_{\mathrm{Atom}(p)}\} & \text{if } p = p',\\ \emptyset & \text{otherwise.} From ce53aba3299730fa6a85ea43eba551ca3cc980fb Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Sun, 23 Nov 2025 14:16:03 -0800 Subject: [PATCH 07/11] Consolidate RMG depth explanation --- aion-holography/sections/rmg.tex | 7 +++---- 1 file changed, 3 insertions(+), 4 deletions(-) diff --git a/aion-holography/sections/rmg.tex b/aion-holography/sections/rmg.tex index 60564ac..2f92deb 100644 --- a/aion-holography/sections/rmg.tex +++ b/aion-holography/sections/rmg.tex @@ -57,7 +57,8 @@ \subsection{Initial algebra viewpoint} \subsection{Morphisms and category of RMGs} \begin{definition}[RMG morphism] -We define morphisms by structural recursion on RMG depth. First form the +We define morphisms by structural recursion on RMG depth (the nesting level in +the construction of Definition~\ref{def:rmg}). First form the discrete category $\mathbf{P}$ with $\mathrm{Ob}(\mathbf{P}) = P$ and $\mathrm{Mor}(\mathbf{P})$ containing only identity morphisms. We declare the RMG hom-sets on atoms by @@ -78,9 +79,7 @@ \subsection{Morphisms and category of RMGs} \item for each $v \in V$ a morphism of attachments $f_v : \alpha(v) \To \alpha'(f_V(v))$ and, for each $e \in E$, a morphism $f_e : \beta(e) \To \beta'(f_E(e))$, where each $f_v$ and - $f_e$ is itself an RMG morphism, defined by structural recursion on - RMG depth (i.e.\ the nesting level in the construction of - Definition~\ref{def:rmg}). + $f_e$ is itself an RMG morphism. \end{itemize} Composition and identities are defined componentwise. \end{definition} From 178389e538c7d9dbec716a66c8986f69027a32df Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Sun, 23 Nov 2025 14:23:05 -0800 Subject: [PATCH 08/11] Streamline RMG notation table entry --- aion-holography/sections/rmg.tex | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/aion-holography/sections/rmg.tex b/aion-holography/sections/rmg.tex index 2f92deb..30af2e0 100644 --- a/aion-holography/sections/rmg.tex +++ b/aion-holography/sections/rmg.tex @@ -60,8 +60,8 @@ \subsection{Morphisms and category of RMGs} We define morphisms by structural recursion on RMG depth (the nesting level in the construction of Definition~\ref{def:rmg}). First form the discrete category $\mathbf{P}$ with $\mathrm{Ob}(\mathbf{P}) = P$ and -$\mathrm{Mor}(\mathbf{P})$ containing only identity morphisms. We declare the -RMG hom-sets on atoms by +$\mathrm{Mor}(\mathbf{P})$ containing only identity morphisms. We define the +RMG hom-sets on atoms to match this discrete structure: \[ \Hom_{\RMG}(\mathrm{Atom}(p),\mathrm{Atom}(p')) = \begin{cases} @@ -69,8 +69,8 @@ \subsection{Morphisms and category of RMGs} \emptyset & \text{otherwise.} \end{cases} \] -Under this construction the embedding $\mathrm{Atom} : \mathbf{P} \To \RMG$ is -a faithful functor. +This embedding is faithful because it preserves the identity-only structure of +$\mathbf{P}$. For composite objects, a morphism $f : (S,\alpha,\beta) \To (S',\alpha',\beta')$ consists of: \begin{itemize}[leftmargin=*] @@ -160,7 +160,7 @@ \subsection{Notation summary} \toprule \textbf{Symbol} & \textbf{Meaning} \\ \midrule -$\mathcal{U} = (G;\alpha,\beta)$ & single RMG state with $\mathcal{U} \in U$, where an RMG universe $U$ is a collection of RMG states related by rewriting \\ +$\mathcal{U} = (G;\alpha,\beta)$ & single RMG state in universe $U$ \\ $p = (L \xleftarrow{\ell} K \xrightarrow{r} R)$ & DPOI rule \\ $\mu_i$ & microstep label \\ $P = (\mu_0,\dots,\mu_{n-1})$ & provenance payload \\ From 04ba8d6273c483f3b68c6466b29147ab057a3cad Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Sun, 23 Nov 2025 14:23:26 -0800 Subject: [PATCH 09/11] Tighten RMG universe definition --- aion-holography/sections/rmg.tex | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/aion-holography/sections/rmg.tex b/aion-holography/sections/rmg.tex index 30af2e0..b275f23 100644 --- a/aion-holography/sections/rmg.tex +++ b/aion-holography/sections/rmg.tex @@ -173,8 +173,9 @@ \subsection{Notation summary} \end{center} \medskip -Here $U$ always denotes an RMG universe in this sense, with -$\mathcal{U} \in U$ indicating a particular state. +Throughout, an \emph{RMG universe} $U$ is a set of RMG states (typically closed +under the rewrite rules $R$ under consideration), and $\mathcal{U} \in U$ +denotes a particular state in that universe. Subsequent sections introduce $D_{\tau,m}$ (rulial distance), $\Hist(U,R)$ (history category on a universe $U$ of RMG states), and From 5b17a51c5d9750672a6992dd8f6321b977a06a4f Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Sun, 23 Nov 2025 14:23:48 -0800 Subject: [PATCH 10/11] Specify history category uses the universe U --- aion-holography/sections/rmg.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/aion-holography/sections/rmg.tex b/aion-holography/sections/rmg.tex index b275f23..b68d660 100644 --- a/aion-holography/sections/rmg.tex +++ b/aion-holography/sections/rmg.tex @@ -178,5 +178,5 @@ \subsection{Notation summary} denotes a particular state in that universe. Subsequent sections introduce $D_{\tau,m}$ (rulial distance), -$\Hist(U,R)$ (history category on a universe $U$ of RMG states), and +$\Hist(U,R)$ (history category on the universe $U$ of RMG states), and other observer-related notation. From 8dc4b7f0fe9a972d7a563caf1857a5c9a5d794f7 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Sun, 23 Nov 2025 14:24:05 -0800 Subject: [PATCH 11/11] Separate recursive clause for attachment morphisms --- aion-holography/sections/rmg.tex | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/aion-holography/sections/rmg.tex b/aion-holography/sections/rmg.tex index b68d660..a54e51e 100644 --- a/aion-holography/sections/rmg.tex +++ b/aion-holography/sections/rmg.tex @@ -78,9 +78,10 @@ \subsection{Morphisms and category of RMGs} E'$ preserving sources and targets; and \item for each $v \in V$ a morphism of attachments $f_v : \alpha(v) \To \alpha'(f_V(v))$ and, for each $e \in E$, a - morphism $f_e : \beta(e) \To \beta'(f_E(e))$, where each $f_v$ and - $f_e$ is itself an RMG morphism. + morphism $f_e : \beta(e) \To \beta'(f_E(e))$. \end{itemize} +Note that each $f_v$ and $f_e$ is itself an RMG morphism, so this definition +proceeds by the structural recursion announced above. Composition and identities are defined componentwise. \end{definition} \begin{figure}[t]