fix: clarifications and touch-ups

Author: flyingrobots

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},

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]

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

@@ -57,16 +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.