From dbc403367467221f1f6288bfb08a516c60a878de Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 17:18:30 -0800 Subject: [PATCH 01/24] Flatten LaTeX sources for arXiv upload --- aion-holography/Makefile | 5 ++- .../{sections => }/assumptions.tex | 0 .../{sections => }/determinism_confluence.tex | 0 aion-holography/{sections => }/discussion.tex | 0 aion-holography/{sections => }/dpo_rmg.tex | 0 aion-holography/{sections => }/ethics.tex | 0 aion-holography/{sections => }/holography.tex | 0 aion-holography/{sections => }/intro.tex | 0 aion-holography/macros.tex | 2 +- aion-holography/main.tex | 32 +++++++++++-------- .../{sections => }/multiway_ruliad.tex | 0 aion-holography/{sections => }/rmg.tex | 0 .../{sections => }/rulial_distance.tex | 0 aion-holography/{sections => }/wormholes.tex | 0 14 files changed, 24 insertions(+), 15 deletions(-) rename aion-holography/{sections => }/assumptions.tex (100%) rename aion-holography/{sections => }/determinism_confluence.tex (100%) rename aion-holography/{sections => }/discussion.tex (100%) rename aion-holography/{sections => }/dpo_rmg.tex (100%) rename aion-holography/{sections => }/ethics.tex (100%) rename aion-holography/{sections => }/holography.tex (100%) rename aion-holography/{sections => }/intro.tex (100%) rename aion-holography/{sections => }/multiway_ruliad.tex (100%) rename aion-holography/{sections => }/rmg.tex (100%) rename aion-holography/{sections => }/rulial_distance.tex (100%) rename aion-holography/{sections => }/wormholes.tex (100%) diff --git a/aion-holography/Makefile b/aion-holography/Makefile index a1ba595..5223ffe 100644 --- a/aion-holography/Makefile +++ b/aion-holography/Makefile @@ -8,7 +8,10 @@ MAIN = main all: $(MAIN).pdf -$(MAIN).pdf: $(MAIN).tex sections/*.tex macros.tex references.bib + +TEX_SRCS = $(filter-out $(MAIN).tex,$(wildcard *.tex)) + +$(MAIN).pdf: $(MAIN).tex $(TEX_SRCS) references.bib $(TEX) $(MAIN) $(BIB) $(MAIN) $(TEX) $(MAIN) diff --git a/aion-holography/sections/assumptions.tex b/aion-holography/assumptions.tex similarity index 100% rename from aion-holography/sections/assumptions.tex rename to aion-holography/assumptions.tex diff --git a/aion-holography/sections/determinism_confluence.tex b/aion-holography/determinism_confluence.tex similarity index 100% rename from aion-holography/sections/determinism_confluence.tex rename to aion-holography/determinism_confluence.tex diff --git a/aion-holography/sections/discussion.tex b/aion-holography/discussion.tex similarity index 100% rename from aion-holography/sections/discussion.tex rename to aion-holography/discussion.tex diff --git a/aion-holography/sections/dpo_rmg.tex b/aion-holography/dpo_rmg.tex similarity index 100% rename from aion-holography/sections/dpo_rmg.tex rename to aion-holography/dpo_rmg.tex diff --git a/aion-holography/sections/ethics.tex b/aion-holography/ethics.tex similarity index 100% rename from aion-holography/sections/ethics.tex rename to aion-holography/ethics.tex diff --git a/aion-holography/sections/holography.tex b/aion-holography/holography.tex similarity index 100% rename from aion-holography/sections/holography.tex rename to aion-holography/holography.tex diff --git a/aion-holography/sections/intro.tex b/aion-holography/intro.tex similarity index 100% rename from aion-holography/sections/intro.tex rename to aion-holography/intro.tex diff --git a/aion-holography/macros.tex b/aion-holography/macros.tex index fd52c4c..ee03914 100644 --- a/aion-holography/macros.tex +++ b/aion-holography/macros.tex @@ -22,7 +22,7 @@ \providecommand{\AIONInline}{\textrm{AION}} \providecommand{\AIONSignature}[1][1]{\AIONWordmarkSerif} \newcommand{\PaperDate}{\today} -\newcommand{\AIONProjectURL}{\url{https://flyingrobots.dev}} +\newcommand{\AIONProjectURL}{\url{https://github.com/flyingrobots/aion}} % ========================================== % Rewrite / Footprint helpers diff --git a/aion-holography/main.tex b/aion-holography/main.tex index a35906a..0096319 100644 --- a/aion-holography/main.tex +++ b/aion-holography/main.tex @@ -27,6 +27,7 @@ <8> stmary8 <9> stmary9 <10> stmary10 + <10.5> stmary10 <10.95> stmary10 <12> stmary10 <14.4> stmary10 @@ -62,10 +63,10 @@ \begin{document} +\hypersetup{pageanchor=false} +\pagenumbering{roman} \input{titlepage} -\maketitle - \begin{abstract} We develop a formal model of \emph{computational holography}: a way of representing a computation so that its entire interior evolution is @@ -84,19 +85,24 @@ observers. \end{abstract} +\clearpage \tableofcontents -\input{sections/assumptions} -\input{sections/intro} -\input{sections/rmg} -\input{sections/dpo_rmg} -\input{sections/determinism_confluence} -\input{sections/holography} -\input{sections/wormholes} -\input{sections/rulial_distance} -\input{sections/multiway_ruliad} -\input{sections/ethics} -\input{sections/discussion} +\clearpage +\hypersetup{pageanchor=true} +\pagenumbering{arabic} + +\input{assumptions} +\input{intro} +\input{rmg} +\input{dpo_rmg} +\input{determinism_confluence} +\input{holography} +\input{wormholes} +\input{rulial_distance} +\input{multiway_ruliad} +\input{ethics} +\input{discussion} \bibliographystyle{alpha} \bibliography{references} diff --git a/aion-holography/sections/multiway_ruliad.tex b/aion-holography/multiway_ruliad.tex similarity index 100% rename from aion-holography/sections/multiway_ruliad.tex rename to aion-holography/multiway_ruliad.tex diff --git a/aion-holography/sections/rmg.tex b/aion-holography/rmg.tex similarity index 100% rename from aion-holography/sections/rmg.tex rename to aion-holography/rmg.tex diff --git a/aion-holography/sections/rulial_distance.tex b/aion-holography/rulial_distance.tex similarity index 100% rename from aion-holography/sections/rulial_distance.tex rename to aion-holography/rulial_distance.tex diff --git a/aion-holography/sections/wormholes.tex b/aion-holography/wormholes.tex similarity index 100% rename from aion-holography/sections/wormholes.tex rename to aion-holography/wormholes.tex From f3ba2e14b52cdab33700b5626a0a356af9c08843 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 17:36:06 -0800 Subject: [PATCH 02/24] Add citation to space-time reversible graph rewriting --- aion-holography/determinism_confluence.tex | 4 +++- aion-holography/discussion.tex | 9 +++++++++ aion-holography/intro.tex | 2 +- aion-holography/references.bib | 10 ++++++++++ aion-holography/rmg.tex | 19 ++++++++++++++++--- 5 files changed, 39 insertions(+), 5 deletions(-) diff --git a/aion-holography/determinism_confluence.tex b/aion-holography/determinism_confluence.tex index 5c09631..f93a8b3 100644 --- a/aion-holography/determinism_confluence.tex +++ b/aion-holography/determinism_confluence.tex @@ -94,7 +94,9 @@ \subsection{Tick semantics and scheduler confluence} $\{m_{i,S} : L_{i,S} \hookrightarrow G_S\}$ are pairwise independent in the sense of \cref{def:independence}. By the Concurrency / Parallel Independence Theorem for DPO rewriting in adhesive categories -(see, e.g.,~\cite{EEPT06}), parallel independent steps commute: for +(see, e.g.,~\cite{EEPT06}; similar categorical techniques appear in the +space--time reversible graph-rewriting setting of~\cite{Arrighi2025Reversible}), +parallel independent steps commute: for any $i \neq j$ we have a diagram \[ G_S \;\Rightarrow_{(p_i,m_{i,S})}\; G_i diff --git a/aion-holography/discussion.tex b/aion-holography/discussion.tex index 357b352..9173e35 100644 --- a/aion-holography/discussion.tex +++ b/aion-holography/discussion.tex @@ -22,6 +22,15 @@ \subsection{Related work} theorem (\cref{thm:tick-confluence}) is a specialization of the standard concurrency theorem for adhesive systems. +Category-theoretic graph rewriting has also been applied to +discretised space--time models in +Arrighi--Costes--Maignan~\cite{Arrighi2025Reversible}, which uses DPO +rewriting in an adhesive setting to obtain space--time deterministic +and reversible graph dynamics. Our use of the same machinery is +conceptually similar, but we focus on deterministic multiway semantics +and holographic provenance in a computational setting rather than on +physical geometry. + \paragraph{Confluence and termination.} The critical-pair lemma and Newman's lemma are classical tools in term rewriting; for decreasing-diagram techniques, see van diff --git a/aion-holography/intro.tex b/aion-holography/intro.tex index 64ef72c..6c7c532 100644 --- a/aion-holography/intro.tex +++ b/aion-holography/intro.tex @@ -27,7 +27,7 @@ \section{Introduction} The technical starting point is algebraic graph transformation using the double--pushout (DPO) approach in adhesive categories, together with the -Recursive Metagraph (RMG) object model developed in earlier work. +Recursive Metagraph (RMG) object model we define in Section~\ref{sec:rmg}. We extend this setting with: \begin{enumerate}[leftmargin=*] diff --git a/aion-holography/references.bib b/aion-holography/references.bib index ff2ab47..a4652a5 100644 --- a/aion-holography/references.bib +++ b/aion-holography/references.bib @@ -48,6 +48,16 @@ @article{Rissanen1978 pages = {465--471} } +@misc{Arrighi2025Reversible, + author = {Arrighi, Pablo and Costes, Marin and Maignan, Luidnel}, + title = {Space-time Reversible Graph Rewriting}, + year = {2025}, + eprint = {2510.03296}, + archivePrefix = {arXiv}, + primaryClass = {cs.DM}, + url = {https://arxiv.org/abs/2510.03296} +} + @misc{RossRMG2025, author = {Ross, James}, title = {Recursive Metagraphs: DPOI Semantics, Confluence, Hypergraph Embedding, and Rulial Distance}, diff --git a/aion-holography/rmg.tex b/aion-holography/rmg.tex index a54e51e..4c6ae29 100644 --- a/aion-holography/rmg.tex +++ b/aion-holography/rmg.tex @@ -26,9 +26,9 @@ \subsection{Inductive 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. +can nest arbitrarily deeply. This definition matches the +set-theoretic and initial-algebra presentation used in our internal +design notes. \paragraph{Example (A tiny recursive metagraph).} As a concrete instance, consider a program call graph where each @@ -54,6 +54,19 @@ \subsection{Initial algebra viewpoint} principles: every function out of $\RMG$ is uniquely determined by its action on atoms and on decorated skeletons. +\subsection{Unfoldings and recursion schemes} + +The depth of an RMG $X$ is the length of the longest attachment chain +in $X$; finite depth follows from the inductive definition. For +$k\in\mathbb{N}$, the \emph{$k$-unfolding} of $X$, written $\mathrm{unf}_k(X)$, +replaces each attachment below depth $k$ by an opaque atom while leaving +shallower structure intact. The \emph{infinite unfolding} +$\mathrm{unf}_\infty(X)$ is the direct limit of the $k$-unfoldings; it is a +possibly infinite tree that still admits the same structural recursion +principles because $\RMG$ is initial for $F$. These unfoldings let us +reason about recursion schemes (catamorphisms, anamorphisms) over RMGs +without leaving the typed open-graph setting. + \subsection{Morphisms and category of RMGs} \begin{definition}[RMG morphism] From 381c06bd0083f7c6800a5f6a9b3f261d8c5db0da Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 17:44:19 -0800 Subject: [PATCH 03/24] Cite reversible spacetime rewriting and tidy assumptions --- aion-holography/assumptions.tex | 10 +++++----- aion-holography/determinism_confluence.tex | 13 ++++++++----- aion-holography/discussion.tex | 6 ++++++ aion-holography/dpo_rmg.tex | 2 +- aion-holography/intro.tex | 9 +++++++-- 5 files changed, 27 insertions(+), 13 deletions(-) diff --git a/aion-holography/assumptions.tex b/aion-holography/assumptions.tex index 9b2f6af..86af208 100644 --- a/aion-holography/assumptions.tex +++ b/aion-holography/assumptions.tex @@ -11,16 +11,16 @@ \section*{Standing Assumptions} \textbf{Assumption} & \textbf{Where used} \\ \midrule Skeleton independence via footprints & -\cref{def:footprint,def:independence,def:batch}; -\cref{thm:tick-confluence} \\ -No-delete/no-clone under descent (ND/NC) & -\cref{def:no-delete}; \cref{thm:two-plane,thm:global} \\ +Definitions~\ref{def:footprint}--\ref{def:batch}; +Theorem~\ref{thm:tick-confluence} \\ +No-delete/no-clone-under-descent (ND/NC) & +Definition~\ref{def:no-delete}; Theorems~\ref{thm:two-plane} and~\ref{thm:global} \\ Termination / decreasing diagrams on the skeleton & \cref{thm:global} (conditional global confluence) \\ No re-derivation (single producer) & \cref{sec:holography}, \cref{thm:backward} (backward provenance) \\ Budgeted translators and 1-Lipschitz distortion & -\cref{sec:rulial}, \cref{lem:rulial-basic,thm:rulial-triangle} (rulial distance) \\ +\cref{sec:rulial}, Lemma~\ref{lem:rulial-basic} and Theorem~\ref{thm:rulial-triangle} (rulial distance) \\ \bottomrule \end{tabular} \end{center} diff --git a/aion-holography/determinism_confluence.tex b/aion-holography/determinism_confluence.tex index f93a8b3..aa17382 100644 --- a/aion-holography/determinism_confluence.tex +++ b/aion-holography/determinism_confluence.tex @@ -2,6 +2,9 @@ \section{Determinism and Confluence} \label{sec:determinism} +Throughout this section we work under the standing assumptions +summarised on page~\texttt{ii}. + We sketch the concurrency discipline, define independence, and state the main confluence theorems for tick-level execution, working with RMG states and tick semantics as introduced in \cref{def:rmg-state,def:rmg-tick}. @@ -179,7 +182,7 @@ \subsection{Two-plane commutation via a fibration} $\pi^{-1}(G)$; a \emph{skeleton step} is a DPOI step in the base $\OGraph_T$. Both are built from pushouts along monos. -\begin{definition}[No-delete/no-clone under descent]\label{def:no-delete} +\begin{definition}[No-delete/no-clone-under-descent]\label{def:no-delete} Consider a tick on an RMG state $\mathcal{U} = (G;\alpha,\beta)$ consisting of \begin{enumerate}[leftmargin=*] @@ -190,8 +193,8 @@ \subsection{Two-plane commutation via a fibration} $G_S \Rewrite_S G_S'$ induced by a rule and match $m_S : L_S \to G_S$. \end{enumerate} - We say that this tick satisfies \emph{no-delete/no-clone under - descent} if: + We say that this tick satisfies \emph{no-delete/no-clone-under-descent} + if: \begin{enumerate}[leftmargin=*] \item[(ND)] (\emph{No delete under descent.}) If a skeleton vertex or edge $x\in G_S$ is deleted by the @@ -206,7 +209,7 @@ \subsection{Two-plane commutation via a fibration} steps in the same tick. In particular, no attachment object is copied to multiple descendants of $x$. \end{enumerate} - A rule pack $R$ satisfies no-delete/no-clone under descent if every + A rule pack $R$ satisfies no-delete/no-clone-under-descent if every tick generated from $R$ has this property. \end{definition} @@ -333,7 +336,7 @@ \subsection{Global confluence} This theorem applies directly to the skeleton plane; together with the no-delete/no-clone-under-descent invariant and two-plane commutation, it yields uniqueness of \emph{worldlines} at the level of RMG states -when the rule pack satisfies these conditions. +for rule packs satisfying these termination / critical-pair conditions. Under the hypotheses of the global confluence theorem, any two complete derivations from a fixed initial state are joinable and yield diff --git a/aion-holography/discussion.tex b/aion-holography/discussion.tex index 9173e35..d1f1c4d 100644 --- a/aion-holography/discussion.tex +++ b/aion-holography/discussion.tex @@ -93,3 +93,9 @@ \subsection{Related work} standard as content-addressing and version control are today: every nontrivial system records not just \emph{what} happened, but a compact, verifiable encoding of \emph{how} it happened. + +In the concrete \AION{} runtime (Echo), these semantic assumptions are +enforced as determinism invariants---world-hash equivalence, merge +determinism, temporal stability, and replay integrity---which ensure +that the abstract confluence properties proven here manifest as +bit-level reproducibility in practice. diff --git a/aion-holography/dpo_rmg.tex b/aion-holography/dpo_rmg.tex index f5dd739..c96a397 100644 --- a/aion-holography/dpo_rmg.tex +++ b/aion-holography/dpo_rmg.tex @@ -76,7 +76,7 @@ \subsection{RMG states as two-plane objects} of a finite family of attachment steps in the fibres over $G$ followed by a finite family of skeleton steps on $G$, chosen by the scheduler. In \cref{sec:determinism} we impose additional conditions (independence, -scheduler--admissible batches, and the no-delete/no-clone under descent +scheduler--admissible batches, and the no-delete/no-clone-under-descent invariant) on the ticks generated by the runtime. \end{definition} diff --git a/aion-holography/intro.tex b/aion-holography/intro.tex index 6c7c532..6f2dd14 100644 --- a/aion-holography/intro.tex +++ b/aion-holography/intro.tex @@ -16,7 +16,7 @@ \section{Introduction} \item all evolution of that state is given by well-typed graph rewrites; \item under an explicit independence discipline and - no-delete/no-clone under descent invariants, the operational + no-delete/no-clone-under-descent invariants, the operational semantics is deterministic (up to typed open-graph isomorphism) and confluent at the level of ``ticks'' of computation (\cref{thm:tick-confluence,thm:two-plane,thm:global}); and @@ -35,7 +35,9 @@ \section{Introduction} \item a two-plane concurrent operational semantics with attachment--then--skeleton publication, together with confluence results: tick-level determinism and two-plane - commutation, plus conditions for global confluence; + commutation (Theorems~\ref{thm:tick-confluence} and + \ref{thm:two-plane}), and, under standard rewrite-theory + hypotheses, global confluence (Theorem~\ref{thm:global}); \item a provenance payload calculus giving \emph{computational holography}; \item an MDL-based quasi-pseudometric on observers, the \emph{rulial @@ -62,6 +64,9 @@ \section{Introduction} \item \emph{Two-plane commutation} (Theorem~\ref{thm:two-plane}): attachment and skeleton updates can be applied in either order up to isomorphism, via a fibration structure; + \item \emph{Worldline uniqueness} (Corollary~\ref{cor:worldline-uniqueness}): + every schedule of a given tick produces the same successor up to + isomorphism; \item \emph{Computational holography} (Theorem~\ref{thm:holography}): the boundary data $(S_0,P)$ is information-complete with respect to the interior evolution, enabling reconstruction of the full From eff4a6fa85b4db41ab4042b291e580fc8c84717e Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 17:50:04 -0800 Subject: [PATCH 04/24] Rename rulial basics as theorem and clarify universe notation --- aion-holography/assumptions.tex | 2 +- aion-holography/determinism_confluence.tex | 12 +++++------ aion-holography/dpo_rmg.tex | 4 ++-- aion-holography/rmg.tex | 11 ++++++----- aion-holography/rulial_distance.tex | 23 +++++++++++----------- 5 files changed, 27 insertions(+), 25 deletions(-) diff --git a/aion-holography/assumptions.tex b/aion-holography/assumptions.tex index 86af208..b590f62 100644 --- a/aion-holography/assumptions.tex +++ b/aion-holography/assumptions.tex @@ -20,7 +20,7 @@ \section*{Standing Assumptions} No re-derivation (single producer) & \cref{sec:holography}, \cref{thm:backward} (backward provenance) \\ Budgeted translators and 1-Lipschitz distortion & -\cref{sec:rulial}, Lemma~\ref{lem:rulial-basic} and Theorem~\ref{thm:rulial-triangle} (rulial distance) \\ +\cref{sec:rulial}, Theorems~\ref{thm:rulial-basic} and~\ref{thm:rulial-triangle} (rulial distance) \\ \bottomrule \end{tabular} \end{center} diff --git a/aion-holography/determinism_confluence.tex b/aion-holography/determinism_confluence.tex index aa17382..727cccf 100644 --- a/aion-holography/determinism_confluence.tex +++ b/aion-holography/determinism_confluence.tex @@ -14,7 +14,7 @@ \subsection{Footprints and Independence on the Skeleton Plane} We work at the level of the skeleton plane. Write $G_S$ for the skeleton component of $G$, and likewise $L_S$, $K_S$, $R_S$ for the underlying skeleton graphs of a rule. Let -$\mathcal{U} = (G;\alpha,\beta)$ be an RMG state and let +$U = (G;\alpha,\beta)$ be an RMG state and let $p = (L \xleftarrow{\ell} K \xrightarrow{r} R)$ be a DPOI rule. A \emph{skeleton match} is a mono $m_S : L_S \hookrightarrow G_S$ in $\OGraph_T$ satisfying the usual gluing conditions. @@ -52,7 +52,7 @@ \subsection{Footprints and Independence on the Skeleton Plane} \subsubsection{Scheduler--admissible batches} \begin{definition}[Scheduler--admissible batch]\label{def:batch} - Let $\mathcal{U} = (G;\alpha,\beta)$ be an RMG state. A finite + Let $U = (G;\alpha,\beta)$ be an RMG state. A finite family of skeleton matches $B = \{m_{i,S} : L_{i,S} \to G_S\}_{i\in I}$ is \emph{scheduler--admissible} if the matches are pairwise @@ -85,7 +85,7 @@ \subsection{Tick semantics and scheduler confluence} $\Use\cup\Del$; we do not repeat the implementation details here. \begin{theorem}[Skeleton-plane tick confluence]\label{thm:tick-confluence} - Let $\mathcal{U} = (G;\alpha,\beta)$ be an RMG state and let + Let $U = (G;\alpha,\beta)$ be an RMG state and let $B = \{m_{i,S} : L_{i,S} \hookrightarrow G_S\}_{i\in I}$ be a scheduler--admissible batch for a family of DPOI rules. Then any two sequentialisations of the corresponding DPOI steps yield @@ -183,7 +183,7 @@ \subsection{Two-plane commutation via a fibration} $\OGraph_T$. Both are built from pushouts along monos. \begin{definition}[No-delete/no-clone-under-descent]\label{def:no-delete} - Consider a tick on an RMG state $\mathcal{U} = (G;\alpha,\beta)$ +Consider a tick on an RMG state $U = (G;\alpha,\beta)$ consisting of \begin{enumerate}[leftmargin=*] \item a family of attachment--plane DPOI steps, each acting inside @@ -216,8 +216,8 @@ \subsection{Two-plane commutation via a fibration} \begin{theorem}[Two-plane commutation]\label{thm:two-plane} Let $R$ be a rule pack satisfying the no-delete/no-clone-under-descent invariant of \cref{def:no-delete}. Let -$\mathcal{U} = (G;\alpha,\beta)$ be an RMG state generated from $R$. -Let $A : \mathcal{U} \Rewrite \mathcal{U}_A$ be a finite composite of +$U = (G;\alpha,\beta)$ be an RMG state generated from $R$. +Let $A : U \Rewrite U_A$ be a finite composite of attachment steps in the fiber over $G$, and let $S : G \Rewrite G'$ be the skeleton DPOI step induced by the same tick such that the tick consisting of $A$ followed by $S$ satisfies the diff --git a/aion-holography/dpo_rmg.tex b/aion-holography/dpo_rmg.tex index c96a397..518a75f 100644 --- a/aion-holography/dpo_rmg.tex +++ b/aion-holography/dpo_rmg.tex @@ -50,7 +50,7 @@ \subsection{RMG states as two-plane objects} \begin{definition}[RMG state]\label{def:rmg-state} An RMG \emph{state} is a triple \[ - \mathcal{U} = (G;\alpha,\beta) + U = (G;\alpha,\beta) \] where $G \in \OGraph_T$ is the skeleton and $\alpha,\beta$ assign attachment objects in the appropriate fibres (of the forgetful functor @@ -72,7 +72,7 @@ \subsection{RMG states as two-plane objects} \end{itemize} \begin{definition}[Tick]\label{def:rmg-tick} -A \emph{tick} on an RMG state $\mathcal{U} = (G;\alpha,\beta)$ consists +A \emph{tick} on an RMG state $U = (G;\alpha,\beta)$ consists of a finite family of attachment steps in the fibres over $G$ followed by a finite family of skeleton steps on $G$, chosen by the scheduler. In \cref{sec:determinism} we impose additional conditions (independence, diff --git a/aion-holography/rmg.tex b/aion-holography/rmg.tex index 4c6ae29..d642be5 100644 --- a/aion-holography/rmg.tex +++ b/aion-holography/rmg.tex @@ -174,7 +174,7 @@ \subsection{Notation summary} \toprule \textbf{Symbol} & \textbf{Meaning} \\ \midrule -$\mathcal{U} = (G;\alpha,\beta)$ & single RMG state in universe $U$ \\ +$U = (G;\alpha,\beta)$ & single RMG state in 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 \\ @@ -187,10 +187,11 @@ \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. +Throughout, an \emph{RMG universe} $\mathcal{U}$ is a set of RMG states +(typically closed under the rewrite rules $R$ under consideration), and +$U \in \mathcal{U}$ denotes a particular state in that universe. Subsequent sections introduce $D_{\tau,m}$ (rulial distance), -$\Hist(U,R)$ (history category on the universe $U$ of RMG states), and +$\Hist(\mathcal{U},R)$ (history category on the universe $\mathcal{U}$ of +RMG states), and other observer-related notation. diff --git a/aion-holography/rulial_distance.tex b/aion-holography/rulial_distance.tex index 1316dbd..7f62ccb 100644 --- a/aion-holography/rulial_distance.tex +++ b/aion-holography/rulial_distance.tex @@ -8,11 +8,12 @@ \section{Rulial Distance: A Computable Quasi-Pseudometric on Observer Space} \subsection{Observers as functors} -Fix an RMG universe $(U,R)$ and its history category -$\Hist(U,R)$ whose objects are states and whose morphisms are derivation -paths between them. An \emph{observer} is a functor +Fix an RMG universe $\mathcal{U}$ together with a rule pack $R$ and its +history category $\Hist(\mathcal{U},R)$ whose objects are states +$U,V \in \mathcal{U}$ and whose morphisms are derivation paths between +them. An \emph{observer} is a functor \[ - O : \Hist(U,R) \To \mathcal{Y}, + O : \Hist(\mathcal{U},R) \To \mathcal{Y}, \] where $\mathcal{Y}$ is a suitable category of observations (symbol streams, trace graphs, etc.). We assume that $O$ is realised by some @@ -36,7 +37,7 @@ \subsection{Translators, MDL Complexity, and Distortion} T_{12} : O_1 \Rightarrow O_2 \] realised as a small DPOI transducer: for each history -$h \in \Hist(U,R)$ it maps the trace $O_1(h)$ to a trace +$h \in \Hist(\mathcal{U},R)$ it maps the trace $O_1(h)$ to a trace $T_{12}(O_1(h))$ in the observation category~$\mathcal{Y}$. Likewise we consider translators $T_{21} : O_2 \Rightarrow O_1$. @@ -63,10 +64,10 @@ \subsection{Translators, MDL Complexity, and Distortion} be a metric on individual traces (for example, an $L_1$ distance on symbol streams or an edit distance on labelled paths). We lift this pointwise to observers -by defining, for observers $O,O' : \Hist(U,R) \to \mathcal{Y}$, +by defining, for observers $O,O' : \Hist(\mathcal{U},R) \to \mathcal{Y}$, \[ \mathrm{Dist}(O,O') - := \sup_{h \in \Hist(U,R)} + := \sup_{h \in \Hist(\mathcal{U},R)} \mathrm{dist}_{\mathrm{tr}}\bigl(O(h),O'(h)\bigr). \] We assume all observers considered produce traces in a common metric @@ -106,11 +107,11 @@ \subsection{Translators, MDL Complexity, and Distortion} \Bigr). \] -\begin{lemma}[Basic properties of $D_{\tau,m}$]\label{lem:rulial-basic} +\begin{theorem}[Basic properties of $D_{\tau,m}$]\label{thm:rulial-basic} For all observers $O_1,O_2$ and budgets $(\tau,m)$, the distance $D_{\tau,m}(O_1,O_2)$ is nonnegative and symmetric, and $D_{\tau,m}(O,O)=0$ for every observer $O$. -\end{lemma} +\end{theorem} \begin{proof} Nonnegativity and symmetry are immediate from the definition of @@ -123,7 +124,7 @@ \subsection{Translators, MDL Complexity, and Distortion} nonnegativity implies $D_{\tau,m}(O,O)=0$. \end{proof} -\begin{theorem}[Triangle inequality for rulial distance]\label{thm:rulial-triangle} +\begin{theorem}[Main theorem on rulial distance (triangle inequality)]\label{thm:rulial-triangle} Assume: \begin{enumerate}[leftmargin=*] \item the description length $\mathrm{DL}$ is based on a prefix code @@ -146,7 +147,7 @@ \subsection{Translators, MDL Complexity, and Distortion} D_{\tau,m}(O_1,O_3) \le D_{\tau,m}(O_1,O_2) + D_{\tau,m}(O_2,O_3) + 2c. \] -In particular, together with \cref{lem:rulial-basic} this makes +In particular, together with \cref{thm:rulial-basic} this makes $D_{\tau,m}$ a quasi-pseudometric (a pseudometric up to additive slack $2c$) on observers. \end{theorem} From 10c5d82ee77079a9b01c76ff576ab703230f0e3d Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 17:53:24 -0800 Subject: [PATCH 05/24] Add footprint/independence sketch figure --- aion-holography/determinism_confluence.tex | 42 ++++++++++++++++++++++ 1 file changed, 42 insertions(+) diff --git a/aion-holography/determinism_confluence.tex b/aion-holography/determinism_confluence.tex index 727cccf..ef7ef62 100644 --- a/aion-holography/determinism_confluence.tex +++ b/aion-holography/determinism_confluence.tex @@ -49,6 +49,48 @@ \subsection{Footprints and Independence on the Skeleton Plane} written by the other. \end{definition} +\begin{figure}[t] + \centering + \begin{tikzpicture}[ + node/.style={circle,draw=gray!60,fill=gray!10,thick,minimum size=8mm}, + del/.style={fill=red!25}, + use/.style={fill=blue!20}, + edge/.style={-Latex,thick,gray!60}, + >=Latex + ] + % baseline graph + \node[node] (a) at (0,0) {$a$}; + \node[node,del] (b) at (1.6,0.7) {$b$}; + \node[node,use] (c) at (3.2,0) {$c$}; + \node[node] (d) at (1.6,-0.8) {$d$}; + \draw[edge] (a) -- (b); + \draw[edge] (b) -- (c); + \draw[edge] (a) -- (d); + \draw[edge] (d) -- (c); + + % match 1 + \draw[rounded corners,thick,teal!70] + ( -0.4, -0.4) rectangle (2.3,1.1); + \node[anchor=west,teal!70!black] at (2.35,1.05) {$m_1$}; + + % match 2 + \draw[rounded corners,thick,orange!80] + (1.0,-1.1) rectangle (3.6,0.9); + \node[anchor=west,orange!80!black] at (3.65,0.85) {$m_2$}; + + % legend + \node[node,del] (ld) at (0.2,-1.6) {}; + \node[anchor=west] at (0.45,-1.6) {delete set $\Del(m_i)$}; + \node[node,use] (lu) at (2.2,-1.6) {}; + \node[anchor=west] at (2.45,-1.6) {use set $\Use(m_i)$}; + \end{tikzpicture} + \caption{Two overlapping matches on the skeleton plane. Shaded nodes + illustrate $\Del$ (red) and $\Use$ (blue). Independence requires + $\Del(m_1)\cap\Use(m_2)=\Del(m_2)\cap\Use(m_1)=\emptyset$, ruling out + destructive interference between the batches.} + \label{fig:footprint} +\end{figure} + \subsubsection{Scheduler--admissible batches} \begin{definition}[Scheduler--admissible batch]\label{def:batch} From 8b12ead9ad8c2b721e609cd63e7f3fb75c9588d8 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 17:56:35 -0800 Subject: [PATCH 06/24] Tidy footprint figure layout --- aion-holography/determinism_confluence.tex | 22 +++++++++++----------- 1 file changed, 11 insertions(+), 11 deletions(-) diff --git a/aion-holography/determinism_confluence.tex b/aion-holography/determinism_confluence.tex index ef7ef62..a9cabb7 100644 --- a/aion-holography/determinism_confluence.tex +++ b/aion-holography/determinism_confluence.tex @@ -60,9 +60,9 @@ \subsection{Footprints and Independence on the Skeleton Plane} ] % baseline graph \node[node] (a) at (0,0) {$a$}; - \node[node,del] (b) at (1.6,0.7) {$b$}; - \node[node,use] (c) at (3.2,0) {$c$}; - \node[node] (d) at (1.6,-0.8) {$d$}; + \node[node,del] (b) at (1.8,0.9) {$b$}; + \node[node,use] (c) at (3.6,0) {$c$}; + \node[node] (d) at (1.8,-1.0) {$d$}; \draw[edge] (a) -- (b); \draw[edge] (b) -- (c); \draw[edge] (a) -- (d); @@ -70,19 +70,19 @@ \subsection{Footprints and Independence on the Skeleton Plane} % match 1 \draw[rounded corners,thick,teal!70] - ( -0.4, -0.4) rectangle (2.3,1.1); - \node[anchor=west,teal!70!black] at (2.35,1.05) {$m_1$}; + ( -0.5, -0.6) rectangle (2.6,1.3); + \node[anchor=west,teal!70!black] at (2.65,1.25) {$m_1$}; % match 2 \draw[rounded corners,thick,orange!80] - (1.0,-1.1) rectangle (3.6,0.9); - \node[anchor=west,orange!80!black] at (3.65,0.85) {$m_2$}; + (1.0,-1.4) rectangle (4.2,1.1); + \node[anchor=west,orange!80!black] at (4.25,1.05) {$m_2$}; % legend - \node[node,del] (ld) at (0.2,-1.6) {}; - \node[anchor=west] at (0.45,-1.6) {delete set $\Del(m_i)$}; - \node[node,use] (lu) at (2.2,-1.6) {}; - \node[anchor=west] at (2.45,-1.6) {use set $\Use(m_i)$}; + \node[node,del] (ld) at (0.4,-2.0) {}; + \node[anchor=west] at (0.7,-2.0) {delete set $\Del(m_i)$}; + \node[node,use] (lu) at (2.6,-2.0) {}; + \node[anchor=west] at (2.9,-2.0) {use set $\Use(m_i)$}; \end{tikzpicture} \caption{Two overlapping matches on the skeleton plane. Shaded nodes illustrate $\Del$ (red) and $\Use$ (blue). Independence requires From 04440cd303957518e9f2ef4b7fb07d8f0f831769 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 17:58:51 -0800 Subject: [PATCH 07/24] Increase spacing in footprint figure to avoid overlap --- aion-holography/determinism_confluence.tex | 20 +++++++++----------- 1 file changed, 9 insertions(+), 11 deletions(-) diff --git a/aion-holography/determinism_confluence.tex b/aion-holography/determinism_confluence.tex index a9cabb7..6a8b004 100644 --- a/aion-holography/determinism_confluence.tex +++ b/aion-holography/determinism_confluence.tex @@ -60,9 +60,9 @@ \subsection{Footprints and Independence on the Skeleton Plane} ] % baseline graph \node[node] (a) at (0,0) {$a$}; - \node[node,del] (b) at (1.8,0.9) {$b$}; - \node[node,use] (c) at (3.6,0) {$c$}; - \node[node] (d) at (1.8,-1.0) {$d$}; + \node[node,del] (b) at (2.2,1.2) {$b$}; + \node[node,use] (c) at (4.1,0.1) {$c$}; + \node[node] (d) at (2.1,-1.3) {$d$}; \draw[edge] (a) -- (b); \draw[edge] (b) -- (c); \draw[edge] (a) -- (d); @@ -70,19 +70,17 @@ \subsection{Footprints and Independence on the Skeleton Plane} % match 1 \draw[rounded corners,thick,teal!70] - ( -0.5, -0.6) rectangle (2.6,1.3); - \node[anchor=west,teal!70!black] at (2.65,1.25) {$m_1$}; + ( -0.5, -0.9) rectangle (2.9,1.6); + \node[anchor=west,teal!70!black] at (3.05,1.55) {$m_1$}; % match 2 \draw[rounded corners,thick,orange!80] - (1.0,-1.4) rectangle (4.2,1.1); - \node[anchor=west,orange!80!black] at (4.25,1.05) {$m_2$}; + (0.9,-1.7) rectangle (4.7,1.4); + \node[anchor=west,orange!80!black] at (4.85,1.35) {$m_2$}; % legend - \node[node,del] (ld) at (0.4,-2.0) {}; - \node[anchor=west] at (0.7,-2.0) {delete set $\Del(m_i)$}; - \node[node,use] (lu) at (2.6,-2.0) {}; - \node[anchor=west] at (2.9,-2.0) {use set $\Use(m_i)$}; + \node[node,del,label=right:{\scriptsize delete $\Del(m_i)$}] at (0.5,-2.4) {}; + \node[node,use,label=right:{\scriptsize use $\Use(m_i)$}] at (2.5,-2.4) {}; \end{tikzpicture} \caption{Two overlapping matches on the skeleton plane. Shaded nodes illustrate $\Del$ (red) and $\Use$ (blue). Independence requires From 7f1287e8f0537f168b829133a8da8c62eec2229d Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 18:00:14 -0800 Subject: [PATCH 08/24] Move footprint legend to the side for clarity --- aion-holography/determinism_confluence.tex | 7 +++++-- 1 file changed, 5 insertions(+), 2 deletions(-) diff --git a/aion-holography/determinism_confluence.tex b/aion-holography/determinism_confluence.tex index 6a8b004..d3cdc43 100644 --- a/aion-holography/determinism_confluence.tex +++ b/aion-holography/determinism_confluence.tex @@ -79,8 +79,11 @@ \subsection{Footprints and Independence on the Skeleton Plane} \node[anchor=west,orange!80!black] at (4.85,1.35) {$m_2$}; % legend - \node[node,del,label=right:{\scriptsize delete $\Del(m_i)$}] at (0.5,-2.4) {}; - \node[node,use,label=right:{\scriptsize use $\Use(m_i)$}] at (2.5,-2.4) {}; + % legend, off to the right + \node[node,del] (ld) at (5.1,0.6) {}; + \node[anchor=west] at (5.5,0.6) {\scriptsize delete $\Del(m_i)$}; + \node[node,use] (lu) at (5.1,-0.4) {}; + \node[anchor=west] at (5.5,-0.4) {\scriptsize use $\Use(m_i)$}; \end{tikzpicture} \caption{Two overlapping matches on the skeleton plane. Shaded nodes illustrate $\Del$ (red) and $\Use$ (blue). Independence requires From 28e3ec13b95238fb096e468132b808cfd98f08f3 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 18:02:16 -0800 Subject: [PATCH 09/24] Box legend below footprint figure --- aion-holography/determinism_confluence.tex | 13 ++++++++----- 1 file changed, 8 insertions(+), 5 deletions(-) diff --git a/aion-holography/determinism_confluence.tex b/aion-holography/determinism_confluence.tex index d3cdc43..3f76386 100644 --- a/aion-holography/determinism_confluence.tex +++ b/aion-holography/determinism_confluence.tex @@ -79,11 +79,14 @@ \subsection{Footprints and Independence on the Skeleton Plane} \node[anchor=west,orange!80!black] at (4.85,1.35) {$m_2$}; % legend - % legend, off to the right - \node[node,del] (ld) at (5.1,0.6) {}; - \node[anchor=west] at (5.5,0.6) {\scriptsize delete $\Del(m_i)$}; - \node[node,use] (lu) at (5.1,-0.4) {}; - \node[anchor=west] at (5.5,-0.4) {\scriptsize use $\Use(m_i)$}; + % legend in separate inset box below + \begin{scope}[shift={(0,-2.8)}] + \draw[rounded corners,thick,gray!40] (-0.3,-0.6) rectangle (5.0,0.6); + \node[node,del] (ld) at (0,0) {}; + \node[anchor=west] at (0.4,0) {\scriptsize delete $\Del(m_i)$}; + \node[node,use] (lu) at (2.6,0) {}; + \node[anchor=west] at (3.0,0) {\scriptsize use $\Use(m_i)$}; + \end{scope} \end{tikzpicture} \caption{Two overlapping matches on the skeleton plane. Shaded nodes illustrate $\Del$ (red) and $\Use$ (blue). Independence requires From a278a66edef7ed49195dfd3f54fa10b388fdbcd5 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 18:04:00 -0800 Subject: [PATCH 10/24] Spread legend labels and remove box --- aion-holography/determinism_confluence.tex | 7 +++---- 1 file changed, 3 insertions(+), 4 deletions(-) diff --git a/aion-holography/determinism_confluence.tex b/aion-holography/determinism_confluence.tex index 3f76386..b436c09 100644 --- a/aion-holography/determinism_confluence.tex +++ b/aion-holography/determinism_confluence.tex @@ -80,12 +80,11 @@ \subsection{Footprints and Independence on the Skeleton Plane} % legend % legend in separate inset box below - \begin{scope}[shift={(0,-2.8)}] - \draw[rounded corners,thick,gray!40] (-0.3,-0.6) rectangle (5.0,0.6); + \begin{scope}[shift={(0,-2.6)}] \node[node,del] (ld) at (0,0) {}; \node[anchor=west] at (0.4,0) {\scriptsize delete $\Del(m_i)$}; - \node[node,use] (lu) at (2.6,0) {}; - \node[anchor=west] at (3.0,0) {\scriptsize use $\Use(m_i)$}; + \node[node,use] (lu) at (2.0,0) {}; + \node[anchor=west] at (2.4,0) {\scriptsize use $\Use(m_i)$}; \end{scope} \end{tikzpicture} \caption{Two overlapping matches on the skeleton plane. Shaded nodes From b919e39c6ec0c3e4322e63a2d9e06eccf857caf2 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 18:06:58 -0800 Subject: [PATCH 11/24] Remove legend under footprint figure --- aion-holography/determinism_confluence.tex | 6 ------ 1 file changed, 6 deletions(-) diff --git a/aion-holography/determinism_confluence.tex b/aion-holography/determinism_confluence.tex index b436c09..b8239a3 100644 --- a/aion-holography/determinism_confluence.tex +++ b/aion-holography/determinism_confluence.tex @@ -80,12 +80,6 @@ \subsection{Footprints and Independence on the Skeleton Plane} % legend % legend in separate inset box below - \begin{scope}[shift={(0,-2.6)}] - \node[node,del] (ld) at (0,0) {}; - \node[anchor=west] at (0.4,0) {\scriptsize delete $\Del(m_i)$}; - \node[node,use] (lu) at (2.0,0) {}; - \node[anchor=west] at (2.4,0) {\scriptsize use $\Use(m_i)$}; - \end{scope} \end{tikzpicture} \caption{Two overlapping matches on the skeleton plane. Shaded nodes illustrate $\Del$ (red) and $\Use$ (blue). Independence requires From 13bf39c03da4eb59acee0cdc22433a231ab7d328 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 18:16:09 -0800 Subject: [PATCH 12/24] Inline Echo determinism invariants --- aion-holography/discussion.tex | 26 +++++++++++++++++++++++--- 1 file changed, 23 insertions(+), 3 deletions(-) diff --git a/aion-holography/discussion.tex b/aion-holography/discussion.tex index d1f1c4d..7a3d7e4 100644 --- a/aion-holography/discussion.tex +++ b/aion-holography/discussion.tex @@ -94,8 +94,28 @@ \subsection{Related work} nontrivial system records not just \emph{what} happened, but a compact, verifiable encoding of \emph{how} it happened. -In the concrete \AION{} runtime (Echo), these semantic assumptions are -enforced as determinism invariants---world-hash equivalence, merge -determinism, temporal stability, and replay integrity---which ensure +\paragraph{Determinism invariants (Echo).} +In the concrete \AION{} runtime (Echo), the semantic assumptions above +are enforced by a set of operational invariants; any violation aborts +the tick deterministically and emits an error node for replay analysis: +\begin{itemize}[leftmargin=*] + \item \textbf{World Equivalence:} identical diff sequences and merge + decisions yield identical world hashes. + \item \textbf{Merge Determinism:} given the same base snapshot, diffs, + and merge strategies, the resulting snapshot and diff hashes are + identical. + \item \textbf{Temporal Stability:} GC, compression, and inspector + activity do not alter logical state. + \item \textbf{Schema Consistency:} component layout hashes must match + before merges; mismatches block the merge. + \item \textbf{Causal Integrity:} writes cannot modify values they + transitively read earlier in Chronos; paradoxes are detected and + isolated. + \item \textbf{Entropy Reproducibility:} branch entropy is a + deterministic function of recorded events. + \item \textbf{Replay Integrity:} replaying from node $A$ to $B$ + produces identical world hash, event order, and PRNG draw counts. +\end{itemize} +These invariants guide both implementation and test suites and ensure that the abstract confluence properties proven here manifest as bit-level reproducibility in practice. From e587f3cf4fc28755144d60475754c5def1ce6b89 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 18:19:29 -0800 Subject: [PATCH 13/24] Freeze title date to November 26, 2025 --- aion-holography/macros.tex | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/aion-holography/macros.tex b/aion-holography/macros.tex index ee03914..9ffdeab 100644 --- a/aion-holography/macros.tex +++ b/aion-holography/macros.tex @@ -21,7 +21,8 @@ \providecommand{\AIONWordmarkSerif}{\textrm{AION}} \providecommand{\AIONInline}{\textrm{AION}} \providecommand{\AIONSignature}[1][1]{\AIONWordmarkSerif} -\newcommand{\PaperDate}{\today} +% Freeze publication date to avoid changing on re-render +\newcommand{\PaperDate}{November 26, 2025} \newcommand{\AIONProjectURL}{\url{https://github.com/flyingrobots/aion}} % ========================================== From f4abdd35b622cd8b43636427065444e988908fe5 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 18:38:21 -0800 Subject: [PATCH 14/24] Clean no-delete wording; rebuild --- aion-holography/determinism_confluence.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/aion-holography/determinism_confluence.tex b/aion-holography/determinism_confluence.tex index b8239a3..046df1e 100644 --- a/aion-holography/determinism_confluence.tex +++ b/aion-holography/determinism_confluence.tex @@ -181,7 +181,7 @@ \subsection{Tick semantics and scheduler confluence} Combining \cref{thm:tick-confluence} with the two-plane commutation result (\cref{thm:two-plane}) shows that a tick that satisfies the -no-delete/no-clone discipline has a unique outcome up to +no-delete/no-clone-under-descent invariant has a unique outcome up to isomorphism in the full RMG semantics. \begin{corollary}[Worldline uniqueness]\label{cor:worldline-uniqueness} From 19f8cce4f62f0b7ce44afa9e1e4b38364fd65667 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 18:41:03 -0800 Subject: [PATCH 15/24] Clarify def/theorem refs and ND/NC wording --- aion-holography/determinism_confluence.tex | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/aion-holography/determinism_confluence.tex b/aion-holography/determinism_confluence.tex index 046df1e..9e2d1d7 100644 --- a/aion-holography/determinism_confluence.tex +++ b/aion-holography/determinism_confluence.tex @@ -95,7 +95,7 @@ \subsubsection{Scheduler--admissible batches} family of skeleton matches $B = \{m_{i,S} : L_{i,S} \to G_S\}_{i\in I}$ is \emph{scheduler--admissible} if the matches are pairwise - independent in the sense of \cref{def:independence}, i.e. + independent in the sense of Definition~\ref{def:independence}, i.e. \[ \Del(m_{i,S}) \cap \Use(m_{j,S}) = \emptyset \quad\text{for all distinct } i,j\in I. @@ -112,7 +112,7 @@ \subsection{Tick semantics and scheduler confluence} given by the standard double square (pushout complement + pushout). We work with RMG states and tick semantics as defined in -\cref{def:rmg-state,def:rmg-tick}. In this section we analyse when +Definitions~\ref{def:rmg-state} and~\ref{def:rmg-tick}. In this section we analyse when batches of matches can be scheduled concurrently without affecting the resulting state. The deterministic properties proved here apply uniformly to all derivations; applying them to cognitive systems @@ -120,7 +120,7 @@ \subsection{Tick semantics and scheduler confluence} \cref{sec:ethics}. The scheduler computes a maximal independent set of matches in the -sense of \cref{def:batch}, using a safe over-approximation of +sense of Definition~\ref{def:batch}, using a safe over-approximation of $\Use\cup\Del$; we do not repeat the implementation details here. \begin{theorem}[Skeleton-plane tick confluence]\label{thm:tick-confluence} @@ -254,7 +254,7 @@ \subsection{Two-plane commutation via a fibration} \begin{theorem}[Two-plane commutation]\label{thm:two-plane} Let $R$ be a rule pack satisfying the no-delete/no-clone-under-descent -invariant of \cref{def:no-delete}. Let +invariant of Definition~\ref{def:no-delete}. Let $U = (G;\alpha,\beta)$ be an RMG state generated from $R$. Let $A : U \Rewrite U_A$ be a finite composite of attachment steps in the fiber over $G$, and let From 8c8507e50512bfa01f0c8e7d4cab9223b536f306 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 19:22:32 -0800 Subject: [PATCH 16/24] Rename Time Cone, inline time model, clean refs --- aion-holography/discussion.tex | 7 ++++--- aion-holography/multiway_ruliad.tex | 20 ++++++++++++++------ aion-holography/references.bib | 7 ------- 3 files changed, 18 insertions(+), 16 deletions(-) diff --git a/aion-holography/discussion.tex b/aion-holography/discussion.tex index 7a3d7e4..6167f9b 100644 --- a/aion-holography/discussion.tex +++ b/aion-holography/discussion.tex @@ -79,9 +79,10 @@ \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, and Aion - triad suggests new modal and temporal logics for reasoning about - linear time, branch points, and the surrounding possibility space. + \item \textbf{Temporal logic and the Time Cone.} The Chronos, Kairos, and + Aion triad naturally 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, the companion paper will define the \AION{} \COMPUTER{}: a machine model whose basic step is a provenance-carrying RMG rewrite, supporting diff --git a/aion-holography/multiway_ruliad.tex b/aion-holography/multiway_ruliad.tex index 2dd48a9..4211035 100644 --- a/aion-holography/multiway_ruliad.tex +++ b/aion-holography/multiway_ruliad.tex @@ -86,15 +86,23 @@ \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{} -calculus\footnote{Developed in a separate technical note on the - \AION{} time model~\cite{RossAIONCalculus2025}.} gives a temporal -structure on branches and merges. +geometry, and the Chronos, Kairos, Aion time model described below +gives a temporal structure on branches and merges. The RMG worldlines defined here thus embed naturally into the broader Ruliad framework. A more detailed study of this correspondence is left for future work, and will be taken up in the companion \COMPUTER{} paper, where we -explicitly model fork/merge operators and the Time Cube (Chronos, -Kairos, Aion) at the level of operational semantics. +explicitly model fork/merge operators and the Time Cone (Chronos, +Kairos, Aion) at the level of operational semantics. Intuitively, the +Time Cone is the forward “light-cone” of computation: Chronos gives the +linear worldline, Kairos marks branch points, and Aion is the surrounding +space of possible RMG histories. + +\subsection{Chronos, Kairos, Aion: a three-layer time model} +Chronos is the linear ordering of events inside a tick (ledger kernel). +Kairos marks branch points such as schema changes, DPO conflicts, or +rule-pack switches. Aion is the total multiway possibility space +$\Hist(\mathcal{U},R)$ ranging over all schedules, rule packs, and inputs, +supporting multiverse debugging and counterfactual replay. diff --git a/aion-holography/references.bib b/aion-holography/references.bib index a4652a5..52f8acf 100644 --- a/aion-holography/references.bib +++ b/aion-holography/references.bib @@ -65,13 +65,6 @@ @misc{RossRMG2025 note = {Technical report} } -@misc{RossAIONCalculus2025, - author = {Ross, James}, - title = {The {AION} Calculus}, - year = {2025}, - note = {Working note} -} - @article{LS06, author = {Lack, Steven and Soboci{\'n}ski, Pawel}, title = {Adhesive Categories}, From 70a4d466559ab7267cd5163cb82af732aa97099d Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 19:26:53 -0800 Subject: [PATCH 17/24] Add ORCID to title block --- aion-holography/titlepage.tex | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/aion-holography/titlepage.tex b/aion-holography/titlepage.tex index c2cf5be..8463ab7 100644 --- a/aion-holography/titlepage.tex +++ b/aion-holography/titlepage.tex @@ -26,7 +26,8 @@ % AUTHOR BLOCK % --------------------------------------------------------- {\Large James Ross}\\[0.3em] -{\normalsize Independent Researcher}\\[2em] +{\normalsize Independent Researcher}\\ +{\normalsize\texttt{https://orcid.org/0009-0006-0025-7801}}\\[2em] % --------------------------------------------------------- % DATE From 8d7f63a838d58adb5615ef569a978acc7611ecb6 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 19:28:14 -0800 Subject: [PATCH 18/24] Freeze paper date to Nov 25, 2025 --- aion-holography/macros.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/aion-holography/macros.tex b/aion-holography/macros.tex index 9ffdeab..596cda2 100644 --- a/aion-holography/macros.tex +++ b/aion-holography/macros.tex @@ -22,7 +22,7 @@ \providecommand{\AIONInline}{\textrm{AION}} \providecommand{\AIONSignature}[1][1]{\AIONWordmarkSerif} % Freeze publication date to avoid changing on re-render -\newcommand{\PaperDate}{November 26, 2025} +\newcommand{\PaperDate}{November 25, 2025} \newcommand{\AIONProjectURL}{\url{https://github.com/flyingrobots/aion}} % ========================================== From 0c76813f745452d610366421fb45f0457122b3cd Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 19:41:44 -0800 Subject: [PATCH 19/24] Add CC BY 4.0 notice to title page --- aion-holography/titlepage.tex | 2 ++ 1 file changed, 2 insertions(+) diff --git a/aion-holography/titlepage.tex b/aion-holography/titlepage.tex index 8463ab7..9bc5276 100644 --- a/aion-holography/titlepage.tex +++ b/aion-holography/titlepage.tex @@ -42,6 +42,8 @@ {\small This document is part of the \AIONInline{} Foundations Series.\\ See \AIONProjectURL{} for updates. +\\[0.6em] +\textcopyright{} 2025 James Ross. Released under the CC BY 4.0 license. } \end{center} From 2f2c99c69372017c48b3dccb63af06ee8c77e09d Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 19:50:51 -0800 Subject: [PATCH 20/24] Add arXiv rerun typeout hint --- aion-holography/main.tex | 1 + 1 file changed, 1 insertion(+) diff --git a/aion-holography/main.tex b/aion-holography/main.tex index 0096319..7e06bf3 100644 --- a/aion-holography/main.tex +++ b/aion-holography/main.tex @@ -116,3 +116,4 @@ \section*{Author's Note} on the mathematics. \end{document} +\typeout{get arXiv to do 4 passes: Label(s) may have changed. Rerun} From c1f866a3cbca8fe32b97b5fcdf2cf69e32b77fb4 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 21:06:56 -0800 Subject: [PATCH 21/24] chore: github action to publish release --- .github/workflows/release-paper.yml | 41 +++++++++++++++++++++++++++++ 1 file changed, 41 insertions(+) create mode 100644 .github/workflows/release-paper.yml diff --git a/.github/workflows/release-paper.yml b/.github/workflows/release-paper.yml new file mode 100644 index 0000000..b703578 --- /dev/null +++ b/.github/workflows/release-paper.yml @@ -0,0 +1,41 @@ +name: Release Paper PDF + +on: + push: + tags: + # Only trigger for tags ending in "-paper" + - "v*-paper" + +jobs: + release: + name: Release PDF + runs-on: ubuntu-latest + + steps: + - name: Checkout repo + uses: actions/checkout@v4 + + - name: Find PDF + id: pdf + run: | + # Adjust the filename here if needed + FILE_PATH=$(find . -maxdepth 3 -type f -iname "main.pdf") + if [ -z "$FILE_PATH" ]; then + echo "ERROR: Could not find main.pdf" + exit 1 + fi + echo "path=$FILE_PATH" >> $GITHUB_OUTPUT + + - name: Create GitHub Release + id: create_release + uses: softprops/action-gh-release@v2 + with: + tag_name: ${{ github.ref_name }} + name: Release ${{ github.ref_name }} + draft: false + prerelease: false + generate_release_notes: true + files: | + ${{ steps.pdf.outputs.path }}#AIon-Computational-Holography-${{ github.ref_name }}.pdf + env: + GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }} \ No newline at end of file From 0e5ceb400312d00132b26dc5a6e3310d59a6719d Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 21:14:48 -0800 Subject: [PATCH 22/24] Build paper PDF in release workflow --- .github/workflows/release-paper.yml | 21 ++++++++++++++------- 1 file changed, 14 insertions(+), 7 deletions(-) diff --git a/.github/workflows/release-paper.yml b/.github/workflows/release-paper.yml index b703578..6c684a5 100644 --- a/.github/workflows/release-paper.yml +++ b/.github/workflows/release-paper.yml @@ -15,16 +15,23 @@ jobs: - name: Checkout repo uses: actions/checkout@v4 - - name: Find PDF + - name: Build PDF + uses: xu-cheng/latex-action@v3 + with: + working_directory: aion-holography + root_file: main.tex + latexmk_use_xelatex: false + args: -pdf -interaction=nonstopmode -halt-on-error + + - name: Set PDF path id: pdf run: | - # Adjust the filename here if needed - FILE_PATH=$(find . -maxdepth 3 -type f -iname "main.pdf") - if [ -z "$FILE_PATH" ]; then - echo "ERROR: Could not find main.pdf" + FILE_PATH="aion-holography/main.pdf" + if [ ! -f "$FILE_PATH" ]; then + echo "ERROR: Could not find $FILE_PATH" exit 1 fi - echo "path=$FILE_PATH" >> $GITHUB_OUTPUT + echo "path=$FILE_PATH" >> "$GITHUB_OUTPUT" - name: Create GitHub Release id: create_release @@ -38,4 +45,4 @@ jobs: files: | ${{ steps.pdf.outputs.path }}#AIon-Computational-Holography-${{ github.ref_name }}.pdf env: - GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }} \ No newline at end of file + GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }} From 760f4c39f854a2de91e7d8c02852ca8fbb8a7b33 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 21:18:49 -0800 Subject: [PATCH 23/24] Clarify worldline uniqueness and open-graph wording in intro --- aion-holography/intro.tex | 12 +++++++----- 1 file changed, 7 insertions(+), 5 deletions(-) diff --git a/aion-holography/intro.tex b/aion-holography/intro.tex index 6f2dd14..1123f56 100644 --- a/aion-holography/intro.tex +++ b/aion-holography/intro.tex @@ -17,7 +17,7 @@ \section{Introduction} rewrites; \item under an explicit independence discipline and no-delete/no-clone-under-descent invariants, the operational - semantics is deterministic (up to typed open-graph isomorphism) + semantics is deterministic (up to typed open graph isomorphism) and confluent at the level of ``ticks'' of computation (\cref{thm:tick-confluence,thm:two-plane,thm:global}); and \item the \emph{entire} interior evolution of a computation is stored @@ -59,14 +59,16 @@ \section{Introduction} The main results of this paper are: \begin{enumerate}[leftmargin=5mm] \item \emph{Tick-level confluence} (Theorem~\ref{thm:tick-confluence}): - parallel independent RMG rewrites commute, yielding deterministic - semantics independent of scheduler serialization order; + parallel independent RMG rewrites commute, yielding per-tick + deterministic semantics independent of scheduler serialization + order; \item \emph{Two-plane commutation} (Theorem~\ref{thm:two-plane}): attachment and skeleton updates can be applied in either order up to isomorphism, via a fibration structure; \item \emph{Worldline uniqueness} (Corollary~\ref{cor:worldline-uniqueness}): - every schedule of a given tick produces the same successor up to - isomorphism; + any complete execution (all tick steps) yields a single worldline + up to typed open graph isomorphism, regardless of scheduler order, + extending tick-level commutation to whole runs; \item \emph{Computational holography} (Theorem~\ref{thm:holography}): the boundary data $(S_0,P)$ is information-complete with respect to the interior evolution, enabling reconstruction of the full From d7dca4aa6820f033ef1f102250771473651ec151 Mon Sep 17 00:00:00 2001 From: "J. Kirby Ross" Date: Tue, 25 Nov 2025 21:25:41 -0800 Subject: [PATCH 24/24] Polish references, metadata, and implementation notes --- aion-holography/assumptions.tex | 7 ++- aion-holography/determinism_confluence.tex | 2 +- aion-holography/discussion.tex | 58 +++++++++++----------- aion-holography/macros.tex | 7 ++- aion-holography/main.tex | 1 + aion-holography/multiway_ruliad.tex | 12 +++-- aion-holography/rmg.tex | 14 +++--- aion-holography/titlepage.tex | 6 ++- 8 files changed, 58 insertions(+), 49 deletions(-) diff --git a/aion-holography/assumptions.tex b/aion-holography/assumptions.tex index b590f62..607ab47 100644 --- a/aion-holography/assumptions.tex +++ b/aion-holography/assumptions.tex @@ -11,16 +11,15 @@ \section*{Standing Assumptions} \textbf{Assumption} & \textbf{Where used} \\ \midrule Skeleton independence via footprints & -Definitions~\ref{def:footprint}--\ref{def:batch}; -Theorem~\ref{thm:tick-confluence} \\ + \cref{def:footprint,def:batch,thm:tick-confluence} \\ No-delete/no-clone-under-descent (ND/NC) & -Definition~\ref{def:no-delete}; Theorems~\ref{thm:two-plane} and~\ref{thm:global} \\ +\cref{def:no-delete,thm:two-plane,thm:global} \\ Termination / decreasing diagrams on the skeleton & \cref{thm:global} (conditional global confluence) \\ No re-derivation (single producer) & \cref{sec:holography}, \cref{thm:backward} (backward provenance) \\ Budgeted translators and 1-Lipschitz distortion & -\cref{sec:rulial}, Theorems~\ref{thm:rulial-basic} and~\ref{thm:rulial-triangle} (rulial distance) \\ +\cref{sec:rulial,thm:rulial-basic,thm:rulial-triangle} (rulial distance) \\ \bottomrule \end{tabular} \end{center} diff --git a/aion-holography/determinism_confluence.tex b/aion-holography/determinism_confluence.tex index 9e2d1d7..1650d8d 100644 --- a/aion-holography/determinism_confluence.tex +++ b/aion-holography/determinism_confluence.tex @@ -3,7 +3,7 @@ \section{Determinism and Confluence} \label{sec:determinism} Throughout this section we work under the standing assumptions -summarised on page~\texttt{ii}. +summarised on page~\pageref{sec:assumptions}. We sketch the concurrency discipline, define independence, and state the main confluence theorems for tick-level execution, working with RMG diff --git a/aion-holography/discussion.tex b/aion-holography/discussion.tex index 6167f9b..48ab55b 100644 --- a/aion-holography/discussion.tex +++ b/aion-holography/discussion.tex @@ -9,6 +9,32 @@ \section{Discussion and Future Work} MDL-based rulial geometry on observers and connected RMG rewriting to multiway systems. +\subsection{Implementation guarantees (Echo)} + +In the concrete \AION{} runtime (Echo), the semantic assumptions above +are enforced by operational determinism invariants; any violation aborts +the tick deterministically and emits an error node for replay analysis. +These invariants are also exercised by the test suite to guarantee +bit-level reproducibility: +\begin{itemize}[leftmargin=*] + \item \textbf{World Equivalence:} identical diff sequences and merge + decisions yield identical world hashes. + \item \textbf{Merge Determinism:} given the same base snapshot, diffs, + and merge strategies, the resulting snapshot and diff hashes are + identical. + \item \textbf{Temporal Stability:} GC, compression, and inspector + activity do not alter logical state. + \item \textbf{Schema Consistency:} component layout hashes must match + before merges; mismatches block the merge. + \item \textbf{Causal Integrity:} writes cannot modify values they + transitively read earlier in Chronos; paradoxes are detected and + isolated. + \item \textbf{Entropy Reproducibility:} branch entropy is a + deterministic function of recorded events. + \item \textbf{Replay Integrity:} replaying from node $A$ to $B$ + produces identical world hash, event order, and PRNG draw counts. +\end{itemize} + \subsection{Related work} Our work builds on several research traditions: @@ -25,9 +51,9 @@ \subsection{Related work} Category-theoretic graph rewriting has also been applied to discretised space--time models in Arrighi--Costes--Maignan~\cite{Arrighi2025Reversible}, which uses DPO -rewriting in an adhesive setting to obtain space--time deterministic -and reversible graph dynamics. Our use of the same machinery is -conceptually similar, but we focus on deterministic multiway semantics +rewriting in an adhesive setting to study space--time reversible graph +rewriting. Our use of the same machinery is conceptually similar, but +we focus on deterministic multiway semantics and holographic provenance in a computational setting rather than on physical geometry. @@ -94,29 +120,3 @@ \subsection{Related work} standard as content-addressing and version control are today: every nontrivial system records not just \emph{what} happened, but a compact, verifiable encoding of \emph{how} it happened. - -\paragraph{Determinism invariants (Echo).} -In the concrete \AION{} runtime (Echo), the semantic assumptions above -are enforced by a set of operational invariants; any violation aborts -the tick deterministically and emits an error node for replay analysis: -\begin{itemize}[leftmargin=*] - \item \textbf{World Equivalence:} identical diff sequences and merge - decisions yield identical world hashes. - \item \textbf{Merge Determinism:} given the same base snapshot, diffs, - and merge strategies, the resulting snapshot and diff hashes are - identical. - \item \textbf{Temporal Stability:} GC, compression, and inspector - activity do not alter logical state. - \item \textbf{Schema Consistency:} component layout hashes must match - before merges; mismatches block the merge. - \item \textbf{Causal Integrity:} writes cannot modify values they - transitively read earlier in Chronos; paradoxes are detected and - isolated. - \item \textbf{Entropy Reproducibility:} branch entropy is a - deterministic function of recorded events. - \item \textbf{Replay Integrity:} replaying from node $A$ to $B$ - produces identical world hash, event order, and PRNG draw counts. -\end{itemize} -These invariants guide both implementation and test suites and ensure -that the abstract confluence properties proven here manifest as -bit-level reproducibility in practice. diff --git a/aion-holography/macros.tex b/aion-holography/macros.tex index 596cda2..d3311d5 100644 --- a/aion-holography/macros.tex +++ b/aion-holography/macros.tex @@ -21,8 +21,11 @@ \providecommand{\AIONWordmarkSerif}{\textrm{AION}} \providecommand{\AIONInline}{\textrm{AION}} \providecommand{\AIONSignature}[1][1]{\AIONWordmarkSerif} -% Freeze publication date to avoid changing on re-render -\newcommand{\PaperDate}{November 25, 2025} +\providecommand{\PaperVersion}{draft} % set to e.g. v1.0 for archival builds +% Publication date: defaults to dynamic \today for ongoing edits. +% For archival/release builds, override with a fixed date, e.g. +% \renewcommand{\PaperDate}{November 25, 2025} +\providecommand{\PaperDate}{\today} \newcommand{\AIONProjectURL}{\url{https://github.com/flyingrobots/aion}} % ========================================== diff --git a/aion-holography/main.tex b/aion-holography/main.tex index 7e06bf3..32cfded 100644 --- a/aion-holography/main.tex +++ b/aion-holography/main.tex @@ -116,4 +116,5 @@ \section*{Author's Note} on the mathematics. \end{document} +% Hack: force arXiv to run 4 passes by mimicking LaTeX rerun warning \typeout{get arXiv to do 4 passes: Label(s) may have changed. Rerun} diff --git a/aion-holography/multiway_ruliad.tex b/aion-holography/multiway_ruliad.tex index 4211035..1f364ff 100644 --- a/aion-holography/multiway_ruliad.tex +++ b/aion-holography/multiway_ruliad.tex @@ -87,7 +87,8 @@ \section{Multiway Systems and the Ruliad} 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 described below -gives a temporal structure on branches and merges. +(see \cref{subsec:chronos-kairos-aion}) gives a temporal structure on +branches and merges. The RMG worldlines defined here thus embed naturally into the broader Ruliad framework. @@ -96,13 +97,14 @@ \section{Multiway Systems and the Ruliad} and will be taken up in the companion \COMPUTER{} paper, where we explicitly model fork/merge operators and the Time Cone (Chronos, Kairos, Aion) at the level of operational semantics. Intuitively, the -Time Cone is the forward “light-cone” of computation: Chronos gives the -linear worldline, Kairos marks branch points, and Aion is the surrounding -space of possible RMG histories. +Time Cone is the forward ``light-cone'' of computation: Chronos gives +the linear worldline, Kairos marks branch points, and Aion is the +surrounding space of possible RMG histories. \subsection{Chronos, Kairos, Aion: a three-layer time model} +\label{subsec:chronos-kairos-aion} Chronos is the linear ordering of events inside a tick (ledger kernel). Kairos marks branch points such as schema changes, DPO conflicts, or rule-pack switches. Aion is the total multiway possibility space -$\Hist(\mathcal{U},R)$ ranging over all schedules, rule packs, and inputs, +$\Hist(\mathcal{U}, R)$ ranging over all schedules, rule packs, and inputs, supporting multiverse debugging and counterfactual replay. diff --git a/aion-holography/rmg.tex b/aion-holography/rmg.tex index d642be5..20fafc7 100644 --- a/aion-holography/rmg.tex +++ b/aion-holography/rmg.tex @@ -58,12 +58,14 @@ \subsection{Unfoldings and recursion schemes} The depth of an RMG $X$ is the length of the longest attachment chain in $X$; finite depth follows from the inductive definition. For -$k\in\mathbb{N}$, the \emph{$k$-unfolding} of $X$, written $\mathrm{unf}_k(X)$, -replaces each attachment below depth $k$ by an opaque atom while leaving -shallower structure intact. The \emph{infinite unfolding} -$\mathrm{unf}_\infty(X)$ is the direct limit of the $k$-unfoldings; it is a -possibly infinite tree that still admits the same structural recursion -principles because $\RMG$ is initial for $F$. These unfoldings let us +$k\in\mathbb{N}$, the \emph{$k$-unfolding} of $X$, written +$\mathrm{unf}_k(X)$, replaces each attachment at depth $k$ or greater +by an opaque atom while preserving all structure at depths +$0,\ldots,k{-}1$. The \emph{infinite unfolding} +$\mathrm{unf}_\infty(X)$ is the directed colimit of the +$k$-unfoldings. Because $F$ is finitary and $\RMG$ is the initial +$F$-algebra, these colimits remain finitely branching and support the +same structural recursion principles. The unfoldings thus let us reason about recursion schemes (catamorphisms, anamorphisms) over RMGs without leaving the typed open-graph setting. diff --git a/aion-holography/titlepage.tex b/aion-holography/titlepage.tex index 9bc5276..b4da0c0 100644 --- a/aion-holography/titlepage.tex +++ b/aion-holography/titlepage.tex @@ -27,7 +27,7 @@ % --------------------------------------------------------- {\Large James Ross}\\[0.3em] {\normalsize Independent Researcher}\\ -{\normalsize\texttt{https://orcid.org/0009-0006-0025-7801}}\\[2em] +{\normalsize\href{https://orcid.org/0009-0006-0025-7801}{ORCID: 0009-0006-0025-7801}}\\[2em] % --------------------------------------------------------- % DATE @@ -43,7 +43,9 @@ This document is part of the \AIONInline{} Foundations Series.\\ See \AIONProjectURL{} for updates. \\[0.6em] -\textcopyright{} 2025 James Ross. Released under the CC BY 4.0 license. +\textcopyright{} 2025 James Ross. +Released under the Creative Commons Attribution 4.0 International +(CC BY 4.0) license: \url{https://creativecommons.org/licenses/by/4.0/}. } \end{center}