fix: clarifications and touch-ups

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

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},
@@ -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},

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,

aion-holography/sections/holography.tex

@@ -30,16 +30,21 @@ \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 reads values in some $S_j$ and produces new values in
+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}
 
@@ -171,9 +176,12 @@ \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).  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.
 \end{definition}
 
 \begin{theorem}[Computational holography]

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.

aion-holography/sections/rmg.tex

@@ -57,17 +57,31 @@ \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 (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 define the
+RMG hom-sets on atoms to match this discrete structure:
+\[
+  \Hom_{\RMG}(\mathrm{Atom}(p),\mathrm{Atom}(p')) =
+  \begin{cases}
+    \{\id_{\mathrm{Atom}(p)}\} & \text{if } p = p',\\
+    \emptyset          & \text{otherwise.}
+  \end{cases}
+\]
+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=*]
   \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 $f_e : \beta(e) \To \beta'(f_E(e))$.
 \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]
@@ -147,7 +161,7 @@ \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 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 \\
@@ -160,5 +174,10 @@ \subsection{Notation summary}
 \end{center}
 \medskip
 
+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), and other observer-related notation.
+$\Hist(U,R)$ (history category on the universe $U$ of RMG states), and
+other observer-related notation.