\documentclass[11pt,a4paper]{article}

% File: 085SvAv2.tex
% Compile with:
%   pdflatex 085SvAv2.tex
%   pdflatex 085SvAv2.tex
%
% Figure files retain their original basenames:
%   fig1, fig2, fig3, fig4, fig5, fig6, fig7.
% Place the supplied graphics in the same directory as this file.
% The loader below accepts PDF, PNG, JPG, or JPEG, in that order.
%
% Figure numbering follows first appearance:
%   Figure 1 = fig3  (stencil and iterated reachability)
%   Figure 2 = fig4  (orthant network)
%   Figure 3 = fig1  (diamond and Lorentz cones)
%   Figure 4 = fig2  (compact affine bases)
%   Figure 5 = fig5  (sparse Lorentz sample)
%   Figure 6 = fig6  (nondifferentiable chronological automorphism)
%   Figure 7 = fig7  (conditional implication diagram)
%
% Mathematical corrections to annotations in the supplied artwork
% are stated explicitly in the corresponding captions. In particular:
% - the upper pointed caps in fig1 are not parts of the future cones;
% - the formula printed below fig5 is not the event set or Lorentz cone;
% - fig6 has F([-1,1]^3)=[-1,2]x[-1,1]^2, not [-2,2]x[-1,1]^2;
% - a closed cube is the closure of an Alexandrov interval;
% - the vertical arrows in fig7 indicate reading order, not implications
%   between the conclusions of successive boxes.
%
% Missing graphics produce visible placeholders and package warnings,
% rather than stopping compilation. Such placeholders must be resolved
% before submission.
%
% No external bibliography or shell escape is required.
% Compilation has not been executed in this interface.

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{booktabs,array,longtable,tabularx}
\usepackage{enumitem}
\usepackage{microtype}
\usepackage{graphicx}
\usepackage{flafter}
\usepackage[section]{placeins}
\usepackage{caption}
\usepackage[hidelinks]{hyperref}

\setlength{\emergencystretch}{3em}
\setlist[enumerate]{itemsep=0.3em}
\setlist[itemize]{itemsep=0.25em}
\renewcommand{\arraystretch}{1.12}
\newcolumntype{Y}{>{\raggedright\arraybackslash}X}
\captionsetup{font=small,labelfont=bf}
\setcounter{topnumber}{3}
\setcounter{bottomnumber}{2}
\setcounter{totalnumber}{5}
\renewcommand{\topfraction}{0.9}
\renewcommand{\bottomfraction}{0.8}
\renewcommand{\textfraction}{0.08}
\renewcommand{\floatpagefraction}{0.65}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{assumption}[theorem]{Assumption}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\HH}{\mathbb{H}}
\newcommand{\OO}{\mathbb{O}}
\newcommand{\cl}{\operatorname{cl}}
\newcommand{\interior}{\operatorname{int}}
\newcommand{\cone}{\operatorname{cone}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\FV}{\operatorname{FV}}
\newcommand{\Past}{\operatorname{Past}}
\newcommand{\Future}{\operatorname{Future}}
\newcommand{\dom}{\operatorname{dom}}
\newcommand{\rel}{\mathrel{\triangleleft}}
\newcommand{\hor}{\mathrel{\longrightarrow}}
\newcommand{\Aff}{\mathsf{Aff}}
\newcommand{\Sym}{\mathsf{S}}
\newcommand{\Quad}{\mathsf{Q}}
\newcommand{\Boost}{\mathsf{B}}
\newcommand{\Lor}{L}
\newcommand{\ip}[2]{\langle #1,#2\rangle}

% Preserve figure basenames while allowing common pdflatex formats.
\newcommand{\MissingFigure}[1]{%
  \PackageWarning{085SvAv2}{%
    Missing figure #1. Supply #1.pdf, #1.png, #1.jpg, or #1.jpeg%
  }%
  \fbox{%
    \parbox[c][0.14\textheight][c]{0.9\linewidth}{%
      \centering
      \textbf{Figure file not found}\\[0.5em]
      \texttt{\detokenize{#1}}\\[0.5em]
      Place the supplied PDF, PNG, JPG, or JPEG file
      in the manuscript directory.
    }%
  }%
}
\newcommand{\ManuscriptGraphic}[2][]{%
  \IfFileExists{#2.pdf}{%
    \includegraphics[#1]{#2.pdf}%
  }{%
    \IfFileExists{#2.png}{%
      \includegraphics[#1]{#2.png}%
    }{%
      \IfFileExists{#2.jpg}{%
        \includegraphics[#1]{#2.jpg}%
      }{%
        \IfFileExists{#2.jpeg}{%
          \includegraphics[#1]{#2.jpeg}%
        }{%
          \MissingFigure{#2}%
        }%
      }%
    }%
  }%
}

\title{Scheduler Consistency and Lorentz-Cone Representation\\
\large Countermodels, Explicit Geometric Hypotheses, and Causal Typing}
\author{Ngo Cheung}
\date{}

\begin{document}

\maketitle

\begin{abstract}
Can consistency of local reductions force a Lorentzian causal geometry?
We separate event order, operational influence, cone representation,
and conformal reconstruction. Acyclic dependency generates a partial
order, and compositional commuting diamonds give schedule-independent
records. Neither conclusion selects a geometric cone.

We construct bounded-degree record networks with finite support,
finitely many reduction shapes, terminating finite experiments,
exact operational influence, and schedule-independent records.
Their standard affine realizations have either Euclidean-ball
one-step lattice sections or polyhedral reachability cones.
We also exhibit a distinguishing chronology on a smooth
three-dimensional manifold whose chronological automorphisms need
not be differentiable.

A matching positive result identifies sufficient additional
hypotheses. Translation and positive-dilation invariance produce
convex cone orders. An irreducible, homogeneous, self-dual, strictly
convex proper cone is linearly isomorphic to a future Lorentz cone;
the proof applies the classical classification of symmetric cones.
A stronger published characterization allows self-duality and
irreducibility to be omitted. We state the operational interpretation
and the unproved representation content of these assumptions
separately. The theorem excludes the specified polyhedral affine
realizations, not the underlying rewrite systems under arbitrary
re-embeddings.

The cone determines a Lorentzian quadratic form up to positive scale.
A causal-origin refinement gives an exact typing criterion, while
general conformal reconstruction still requires a distinguishing
Lorentzian realization. Scheduler consistency alone supplies neither
the affine structure nor the cone symmetries used in the positive
theorem.
\end{abstract}

\tableofcontents

\section{Introduction and principal results}

A consistency condition on a rewrite system can preserve an event
order: incompatible dependencies are excluded, independent operations
commute, and records do not depend on their legal interleaving.
The further claim that this mechanism selects a Lorentzian light cone
requires additional mathematics.

A scheduler chooses an evaluation order. Interchanging independent
reductions does not, by itself, define a boost, a linear action on
events, or a transformation of clocks and rulers. Distributed
computation already provides a setting in which causal ordering is
meaningful without a spacetime metric \cite{Lamport}.

This paper gives an obstruction theorem and a matching conditional
representation theorem. The latter uses an established cone
classification, not a new characterization of Lorentz cones.
Its contribution here is the precise separation of the operational
premises from the geometric hypotheses required to apply that
classification.

\subsection{Scope and conventions}

The motivating proposal refers to an Enter calculus, an ECB confluence
result, and a companion manuscript entitled
\emph{From the Enter Calculus to Relativistic Causal Structure}.
Their complete syntax, residual theory, and proof texts are not
premises of this paper. Consequently:
\begin{enumerate}
\item the operational model below is self-contained;
\item no result is attributed to an unspecified Enter type system;
\item any application to a particular calculus must verify the stated
hypotheses;
\item geometric representation assumptions are not presented as
consequences of confluence.
\end{enumerate}

We use \(\N=\{0,1,2,\ldots\}\). All vector spaces are finite-dimensional
and real unless stated otherwise. A \emph{proper cone} is a closed,
convex, pointed cone with nonempty interior in its ambient space.
Here pointed means \(K\cap(-K)=\{0\}\).
The notation \(\cone A\) denotes all finite nonnegative linear
combinations of elements of \(A\).

\subsection{Results at a glance}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.19\linewidth}Y@{}}
\toprule
Question & Mathematical status\\
\midrule
Order formation &
Acyclic dependency gives a partial order. Statistical sensitivity
requires separate treatment.\\
Schedule independence &
Compositional independent diamonds suffice on finite histories.\\
Cone selection &
False from scheduler consistency alone:
Propositions~\ref{prop:ball} and~\ref{prop:orthant}.\\
Lorentz cone &
True for an affine cone representation satisfying
\(\Sym\): Theorem~\ref{thm:representation}.\\
Quadratic signature &
A pointed closure of a full negative component selects Lorentzian
signature: Theorem~\ref{thm:signature}.\\
Sharp propagation &
A geometric boundary and operational attainability are distinct
requirements.\\
Conformal reconstruction &
Requires Lorentzian realizability, not merely a smooth Alexandrov
topology.\\
Causal typing &
An explicit origin refinement excludes unavailable free uses,
not unused context entries.\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Obstruction and representation}

\begin{theorem}[Failure of reconstruction from scheduler consistency]
\label{thm:main}
There are local record networks with:
\begin{enumerate}
\item acyclic dependency;
\item bounded degree, finite support, and finitely many local
reduction shapes;
\item termination of every finite experiment;
\item identical joint records under every legal interleaving;
\item exact operational sensitivity equal to strict reachability;
\item no intervention-induced change outside the dependency future;
\item local point distinguishability and finite pasts for finite
collections of records.
\end{enumerate}
These properties do not imply that:
\begin{enumerate}[label=(\alph*)]
\item a one-step spatial section cannot be a Euclidean metric ball
on a discrete slice;
\item a generated reachability cone is quadratic or Lorentzian;
\item interpolation or a manifold topology exists;
\item an otherwise unspecified type system enforces causal
availability.
\end{enumerate}
\end{theorem}

\begin{proof}
The network construction is Theorem~\ref{thm:network}.
The ball and orthant realizations are
Propositions~\ref{prop:ball} and~\ref{prop:orthant}.
A two-event chain already lacks interpolation.
The operational assumptions do not specify typing rules;
Section~\ref{sec:typing} gives a refinement for which the desired
availability statement can actually be proved.
\end{proof}

\begin{theorem}[Matching conditional representation]
\label{thm:mainpositive}
Let \(N\) be a record network of Theorem~\ref{thm:network}.
Suppose its event order has a generated affine cone representation
\((\iota,V,K)\) in the sense of
Definition~\ref{def:affine}, with \(\dim V=m\geq3\).
Assume that \(K\):
\begin{enumerate}
\item is homogeneous under its linear automorphism group;
\item is self-dual for some positive-definite inner product;
\item is irreducible;
\item is strictly convex in the cone sense of
Definition~\ref{def:strict}.
\end{enumerate}
Then an invertible linear map \(T:V\to\R\times\R^{m-1}\) satisfies
\[
T(K)=\Lor_m
:=\{(t,x):t\geq\|x\|_2\}.
\]
In these coordinates,
\[
p\rightsquigarrow q
\quad\Longleftrightarrow\quad
T\bigl(\iota(q)-\iota(p)\bigr)\in\Lor_m
\qquad(p\ne q).
\]
The cone determines a Lorentzian quadratic form up to positive
multiplication. The standard generated affine realizations of the
ball and orthant networks, and their invertible affine changes of
coordinates, do not satisfy these additional hypotheses.
\end{theorem}

\begin{proof}
Apply Theorem~\ref{thm:representation},
Corollary~\ref{cor:networkrepresentation}, and
Proposition~\ref{prop:quadraticunique}.
The exclusion statement is
Corollary~\ref{cor:exclude}.
\end{proof}

The last theorem is conditional on geometric data.
It neither constructs an affine space from a raw event poset nor
proves that cone automorphisms lift to automorphisms of the record
network. These distinctions prevent a vacuous identification of
scheduler changes with changes of observer.

\section{Events, dependency, and operational influence}
\label{sec:events}

\subsection{Occurrence identity}

\begin{definition}[Event occurrence]
An event is a labelled occurrence of a completed local operation.
Its label is retained when independent operations are interchanged.
Identical local inputs, outputs, support, or residual states do not
by themselves identify two occurrences.
\end{definition}

For example, identifying \(a_0\) and \(a_1\) in
\[
a_0\rel b\rel a_1
\]
creates a cycle in the quotient although the occurrence graph is
acyclic. A concrete rewrite calculus therefore needs an occurrence
construction and a residual theorem, not merely equality of local
states.

\subsection{Three relations}

\begin{definition}[Dependency and reachability]
Let \(E\) be an event set and \(\rel\) a directed dependency relation.
Write \(p<q\) when a nonempty finite dependency path leads from \(p\)
to \(q\), and write \(p\leq q\) when \(p=q\) or \(p<q\).
\end{definition}

\begin{definition}[Operational sensitivity]
Fix admissible experiments and interventions.
For distinct events, write \(p\rightsquigarrow q\) if two admissible
choices localized at \(p\), with all other experimental choices held
fixed, give different marginal observation laws at \(q\).
\end{definition}

This definition concerns interventions, not conditioning on outcomes.
Correlation alone is not signaling.

\begin{proposition}[Nonconstant channels need not compose nonconstantly]
\label{prop:sensitivity}
Composability of channels does not imply transitivity of operational
sensitivity.
\end{proposition}

\begin{proof}
Let \(X\in\{0,1\}\), let \(N\) be an independent fair bit, and consider
\[
K:X\longmapsto(X,N),
\qquad
L:(a,b)\longmapsto a\oplus b.
\]
Both channels are nonconstant, but \(LK(X)=X\oplus N\) has the same
fair-bit law for either value of \(X\).
An intervention on the intermediate pair can change the terminal
output although changing \(X\) cannot change its law.
\end{proof}

If sensitivity is existential over protocols, transitivity requires
compatible witnesses or an additional routing theorem. The network
below has that property by direct calculation.

\subsection{Order formation}

\begin{proposition}[Acyclic dependency produces an order]
\label{prop:order}
If the dependency graph is acyclic, then \(<\) is a strict partial
order and \(\leq\) is a partial order.
\end{proposition}

\begin{proof}
Concatenation proves transitivity. A path from an event to itself
would give a cycle, so \(<\) is irreflexive. Its reflexive closure
is therefore antisymmetric and transitive.
\end{proof}

\begin{proposition}[Strongly connected quotient]
For an arbitrary directed graph, define
\[
p\sim q
\quad\Longleftrightarrow\quad
p\leq q\ \text{and}\ q\leq p.
\]
Reachability induces a partial order on \(E/{\sim}\).
\end{proposition}

\begin{proof}
Reflexivity and transitivity make \(\sim\) an equivalence relation.
If \(p\sim p'\), \(q\sim q'\), and \(p\leq q\), then
\(p'\leq p\leq q\leq q'\). Thus the quotient relation is
well-defined. Mutual quotient reachability means equality of
equivalence classes.
\end{proof}

This quotient collapses cyclic components; it does not prove that
cycles are precisely inconsistent records. For example, the Boolean
constraints \(a=b,\ b=a\) are consistent, whereas
\(a=1-b,\ b=a\) are not. Strong normalization concerns legal
reduction sequences and cannot substitute for an analysis of such
constraint systems.

\section{Legal schedules and a local record model}
\label{sec:networks}

\subsection{Finite histories}

For an acyclic graph, put
\[
\Past(e)=\{p:p\leq e\}.
\]

\begin{definition}[Admissible graph and experiment]
A graph is admissible if every principal past is finite and there
is a common finite bound on indegree and outdegree.
A finite experiment is a finite down-set \(D\subseteq E\).
A schedule is a linear extension of the induced order on \(D\).
\end{definition}

Past-finiteness is stronger than finite support of one reduction.
It is an explicit assumption of the countermodels.

A state records a completed down-set \(H\subseteq D\).
An event in \(D\setminus H\) is enabled when all its immediate
predecessors are completed.

\begin{lemma}[Adjacent interchange]
\label{lem:interchange}
Any two linear extensions of a finite partial order are connected
by swaps of adjacent incomparable elements.
\end{lemma}

\begin{proof}
Find the first element of the second extension in the first.
Every element preceding it there is incomparable with it: a
predecessor would also have to precede it in the second extension,
and a successor could not precede it in the first.
Move it to the front by adjacent swaps, remove it, and apply
induction.
\end{proof}

\begin{corollary}[Compositional diamonds imply schedule independence]
\label{cor:schedules}
Suppose swapping independent enabled operations preserves their
joint record law and leaves equivalent residual configurations for
all subsequent operations. All schedules of a fixed finite history
then have the same joint record law.
\end{corollary}

\begin{proof}
Apply Lemma~\ref{lem:interchange} to the complete computation.
The residual condition makes each interchange valid in context.
\end{proof}

Equality of isolated marginals is insufficient: the diamond must
preserve the joint information used by the remaining computation.

\subsection{Classical linear wires}

Assign a distinct channel name \(x_{wv}\) to each edge \(w\rel v\).
Only \(w\) produces it and only \(v\) consumes it.
At event \(v\), a controllable local bit \(u_v\) and the incoming bits
determine
\begin{equation}
b_v=u_v\lor\bigvee_{w\rel v}b_w.
\label{eq:or}
\end{equation}
The event writes \(b_v\) to a private record and creates a fresh
outgoing channel carrying \(b_v\) for each outgoing edge.
The empty disjunction is zero.

The intervention class consists initially of changes to the
controllable bits, with the remaining bits fixed.
Observations include the private record.
Each event also carries an identifier; a uniform equality-test
primitive with an identifier parameter separates any two events.
Identifiers are data, not different reduction shapes.

A degree bound gives finitely many possible arities.
Copying a classical bit to fresh channels does not duplicate an owned
channel name and makes no claim about cloning quantum states.

\begin{theorem}[Exact operational realization]
\label{thm:network}
On every admissible graph, the network \eqref{eq:or} satisfies:
\begin{enumerate}
\item every reduction touches finitely many linearly owned channels;
\item every finite experiment terminates, and every maximal legal
schedule completes it;
\item distinct simultaneously enabled operations have disjoint
channel ownership and commute exactly;
\item all schedules produce the same joint records;
\item for each event,
\begin{equation}
b_v=\bigvee_{p\leq v}u_p;
\label{eq:ancestoror}
\end{equation}
\item for distinct events, \(p\rightsquigarrow q\) if and only if
\(p<q\);
\item an intervention outside the dependency past of an observation
does not change it;
\item every finite collection of records is determined by a finite
down-set.
\end{enumerate}
\end{theorem}

\begin{proof}
The degree bound bounds the number of consumed and produced channels.
In a finite experiment, each reduction decreases
\(|D\setminus H|\) by one. If this set is nonempty, one of its minimal
elements is enabled. Hence every maximal legal schedule terminates
with all events completed.

Two distinct simultaneously enabled events cannot be connected by
a dependency edge. Unique producers, unique consumers, and distinct
edge names make their channel operations disjoint. They also write
distinct private records, so they commute exactly.
Corollary~\ref{cor:schedules} gives schedule independence.

Induct along a topological order of the finite past of \(v\).
Every strict ancestor of \(v\) is an ancestor of an immediate
predecessor. Associativity and idempotence of disjunction give
\eqref{eq:ancestoror}.

If \(p<q\), set all controls other than \(u_p\) to zero.
Changing \(u_p\) changes \(b_q\). If \(p\not\leq q\), the expression
for \(b_q\) contains no \(u_p\), proving the converse.
The same dependency induction proves invariance under changes outside
the past of the observed records. It also allows replacement
operations, provided they preserve the specified channel interface
and introduce no external dependencies.

Finally, the union of the principal pasts of finitely many observed
events is a finite down-set.
\end{proof}

\begin{remark}[Probability and finite programs]
The deterministic laws are point masses. Any fixed joint law on the
initial controls gives schedule-independent output laws by
pushforward. Interventions are compared with the nonintervened law
held fixed. An infinite graph describes compatible finite experiments,
not one finite term that completes infinitely many reductions.
\end{remark}

\subsection{Incomparability and ownership}

No-signaling alone does not imply disjoint ownership. Two operations
may access the same cell while writing the same constant, or may be
mutually exclusive alternatives rather than co-occurring events.
In the present model, simultaneous enabledness and the channel
discipline prove disjointness. An application to another calculus
must supply the corresponding ownership theorem.

\section{Countermodels to cone selection}
\label{sec:countermodels}

\subsection{A Euclidean-ball one-step section}

\begin{proposition}[Ball sections are compatible with confluence]
\label{prop:ball}
There is an admissible record network in two spatial dimensions
whose one-step spatial reachability section is exactly a closed
Euclidean unit ball for the metric restricted to a lattice.
\end{proposition}

\begin{proof}
Set
\[
E=\N\times\Z^2,\qquad
S=\{(0,0),(1,0),(-1,0),(0,1),(0,-1)\},
\]
and put
\[
(t,x)\rel(t+1,x+s)\qquad(s\in S).
\]
Time strictly increases along edges. The indegree is at most five
and the outdegree is five. Each principal past is finite, since
there are finitely many step sequences of each possible length.
Theorem~\ref{thm:network} applies.

On the next slice, the reachable sites are
\[
x+S=\{y\in\Z^2:\|y-x\|_2\leq1\}.
\]
The one-step section is the discrete Euclidean unit ball; Minkowski
iteration produces the discrete diamond \(\|z\|_1\leq n\), as proved
below and illustrated in Figure~\ref{fig:stencil}.
Away from the initial layer every event has the same input and
output arities.
\end{proof}

% First appearance: original file fig3 becomes manuscript Figure 1.
\begin{figure}[!htbp]
\centering
\ManuscriptGraphic[
  width=\linewidth,
  height=0.48\textheight,
  keepaspectratio
]{fig3}
\caption[One-step stencil and its Minkowski iterates]{%
The spatial stencil \(S\) and its iterated displacement set
\(S^{(n)}=\{z\in\Z^2:\|z\|_1\leq n\}\).
The loop at the origin denotes zero spatial displacement during
one positive time step, not a dependency cycle.
The panel is schematic: the five elements of \(S\) are exactly
the lattice points of the Euclidean unit ball.
The right-hand formula applies for arbitrary \(n\); the number of
drawn grid subdivisions should not be read as a literal
two-step computation.}
\label{fig:stencil}
\end{figure}

The exact \(n\)-step displacement set is
\begin{equation}
S^{(n)}
=\underbrace{S+\cdots+S}_{n\text{ summands}}
=\{z\in\Z^2:\|z\|_1\leq n\}.
\label{eq:diamond}
\end{equation}
One inclusion follows because each step has \(\ell^1\)-length at
most one. For the other, use coordinate steps and pad with zero
steps.

The closed cone generated by the spacetime increments \((1,s)\) is
\begin{equation}
K_{\diamond}
=\{(t,x)\in\R\times\R^2:t\geq\|x\|_1\}.
\label{eq:diamondcone}
\end{equation}
Indeed, the convex hull of \(S\) is the \(\ell^1\) unit ball.
For the standard embedding, reachability is exactly membership of
the event difference in \(K_{\diamond}\).

Finite support rules out a finite target set being a positive-radius
ball in continuous \(\R^2\), but not a metric ball on a discrete
slice. Moreover, Euclidean spatial balls are themselves sections of
a Lorentz cone. The obstruction here is the \emph{generated
polyhedral cone}, not the word ``ball.''

\subsection{Orthant reachability}

\begin{proposition}[A homogeneous non-Lorentzian cone]
\label{prop:orthant}
For each \(m\geq3\), there is an admissible record network whose
reachability is coordinatewise order on \(\N^m\). Its standard
generated cone is
\[
K_{\mathrm{orth}}=\R_{\geq0}^m,
\]
which is not linearly isomorphic to a Lorentz cone.
\end{proposition}

\begin{proof}
Take edges
\[
z\rel z+e_i,\qquad 1\leq i\leq m.
\]
The coordinate sum increases along each edge, every principal past
lies in a finite coordinate box
(Figure~\ref{fig:orthantnetwork}), and the degree is bounded.
Theorem~\ref{thm:network} applies.

% Second appearance: original file fig4 becomes manuscript Figure 2.
\begin{figure}[!htbp]
\centering
\ManuscriptGraphic[
  width=\linewidth,
  height=0.45\textheight,
  keepaspectratio
]{fig4}
\caption[The orthant network and a finite principal past]{%
Schematic three-dimensional orthant network with edges
\(z\rel z+e_i\).
Its principal past at \((2,2,2)\) is
\(P_{(2,2,2)}=\Past((2,2,2))=\{0,1,2\}^3\), containing \(27\)
events.
The shaded coordinate box illustrates past-finiteness; its lattice
points, rather than the solid continuous box, are the network
events. The edge rule specifies the directed order, independently
of the perspective drawing.}
\label{fig:orthantnetwork}
\end{figure}

A path only increases coordinates, and any prescribed nonnegative
integer coordinate increments can be taken successively. Hence
\(z\leq w\) exactly when \(w-z\in\N^m\).
The generated real cone is the positive orthant.

It has exactly \(m\) extreme rays. In contrast, each
\((1,u)\), \(\|u\|_2=1\), generates an extreme ray of \(\Lor_m\).
Equality in the Euclidean triangle inequality proves extremality.
For \(m\geq3\) there are infinitely many such rays, and an invertible
linear map preserves extreme rays.
\end{proof}

\begin{lemma}[Diagnostic comparison]
\label{lem:diagnostic}
The preceding networks share the operational properties of
Theorem~\ref{thm:network}, but their standard generated cones have
the properties in Table~\ref{tab:diagnostic}.
Neither cone is strictly convex or boundary-transitive.
The orthant is homogeneous but reducible.
The diamond cone is irreducible but not homogeneous; its
contrast with the Lorentz cone is illustrated in
Figure~\ref{fig:cones}.
\end{lemma}

\begin{proof}
The operational assertions follow from the construction.
Both cones have proper faces of dimension at least two.
For the diamond cone, its four extreme generators can be written
\[
r_1=(1,1,0),\quad r_2=(1,-1,0),\quad
r_3=(1,0,1),\quad r_4=(1,0,-1).
\]
Their relation space is one-dimensional, generated by
\(r_1+r_2-r_3-r_4=0\).
A cone automorphism permutes the extreme rays.
An automorphism fixing all four rays must scale them by the same
positive factor, by uniqueness of that relation.
Thus every orbit is contained in a finite union of rays after
choosing representatives of the finite permutation action.
It cannot be the three-dimensional interior.

A nontrivial decomposition of a three-dimensional proper cone has
factor dimensions one and two. Such a product has three extreme
rays, since a two-dimensional proper cone has two.
The four-ray diamond cone is therefore irreducible.

Positive diagonal matrices act transitively on the interior of the
orthant, which is a product of one-dimensional cones.
For both examples, a boundary extreme-ray point cannot be mapped
to a point in the relative interior of a higher-dimensional proper
face. This proves failure of boundary transitivity.

For the standard Euclidean pairing,
\(K_{\diamond}^*=\{(t,y):t\geq\|y\|_\infty\}\), whereas the orthant
is self-dual. The claim about the diamond pairing does not assert
failure for every possible inner product.

Finally, a nondegenerate quadratic form in dimension at least three
cannot vanish on an open subset of a hyperplane: polynomial
restriction would make it vanish on the whole hyperplane, forcing
a linear factor and rank at most two. The facets therefore preclude
a nondegenerate quadratic boundary.
\end{proof}

\begin{table}[!htbp]
\centering
\small
\begin{tabularx}{\linewidth}{@{}p{0.43\linewidth}YY@{}}
\toprule
Property & Lattice-ball model & Orthant model\\
\midrule
Acyclic, finite support, finite shapes & Yes & Yes\\
Exact diamonds and schedule independence & Yes & Yes\\
Sensitivity equals strict reachability & Yes & Yes\\
Constant discrete increment rule & Yes & Yes\\
Same arity away from initial boundaries & Yes & Yes\\
Standard generated cone & \(t\geq\|x\|_1\) & \(\R_{\geq0}^m\)\\
Number of extreme rays & \(4\) & \(m\)\\
Interior-transitive cone automorphism group & No & Yes\\
Boundary-transitive cone automorphism group & No & No\\
Self-dual for the standard Euclidean pairing & No & Yes\\
Irreducible cone & Yes & No\\
Strictly convex cone & No & No\\
Nondegenerate quadratic boundary & No & No\\
\bottomrule
\end{tabularx}
\caption{Operational uniformity does not select a Lorentz cone.
``Constant increment rule'' is not a transitive translation group on
a network with an initial boundary.}
\label{tab:diagnostic}
\end{table}

\begin{remark}[Dimension two]
The cone \(t\geq|x|\) is a Lorentz cone and is linearly isomorphic
to \(\R_{\geq0}^2\), through \((t,x)\mapsto(t+x,t-x)\).
The exclusion statements therefore require dimension at least three;
Figure~\ref{fig:cones} illustrates the three-dimensional contrast.
\end{remark}

% Third appearance: original file fig1 becomes manuscript Figure 3.
\begin{figure}[!htbp]
\centering
\ManuscriptGraphic[
  width=\linewidth,
  height=0.47\textheight,
  keepaspectratio
]{fig1}
\caption[Diamond and Lorentz cones and their spatial sections]{%
Diamond and Lorentz spatial sections:
\(K_{\diamond}\cap\{t=1\}\) is an \(\ell^1\) unit ball,
whereas \(\Lor_3\cap\{t=1\}\) is a Euclidean unit ball.
Their boundaries respectively contain line segments and are strictly
convex.
In the supplied schematic, only the portions from \(O\) to the
section \(t=1\) represent truncated future cones.
The upper pointed caps are not parts of either cone:
both actual cones continue unbounded beyond \(t=1\), with expanding
sections. Coordinates on the left are ordered as \((t,x,y)\).}
\label{fig:cones}
\end{figure}

\subsection{Finite touch is not a speed bound}

\begin{proposition}[Unconstrained auxiliary distances]
Finite support and a uniform reduction shape do not imply a uniform
speed bound in an independently assigned spatial metric.
\end{proposition}

\begin{proof}
Use the independent chains
\[
E=\N\times\Z^2,\qquad (k,j)\rel(k+1,j).
\]
Assign \(T(k,j)=k\) and \(X(k,j)=2^k j\).
Each edge has unit time increment and spatial displacement
\(2^k\|j\|_2\), which is unbounded.
The operational assumptions do not constrain these auxiliary
coordinates.
\end{proof}

A speed statement requires an axiom relating transitions to geometry.

\subsection{A degenerate through-source slice condition}

\begin{proposition}
\label{prop:sliceball}
If an antichain \(\Sigma\) contains \(e\), every directed path
starting at \(e\) meets \(\Sigma\) only at \(e\).
Thus requiring such a path to meet \(\Sigma\) within distance
\(\rho\geq0\) of \(e\) does not distinguish one-step targets from
later reachable targets.
\end{proposition}

\begin{proof}
Every subsequent path vertex is strictly above \(e\), so cannot
belong to the antichain. The initial meeting has distance zero.
\end{proof}

Without a separate path-existence requirement, unreachable targets
also satisfy a universal condition over paths vacuously.
A meaningful spatial neighborhood needs a separately specified
cut, projection, and metric.

\subsection{Why scheduler swaps are not boosts}

The graph and increments in both countermodels remain unchanged
when independent operations are interchanged. No such interchange
transforms a spatial slice into a new inertial frame.
Section~\ref{sec:operationalaxioms} identifies the stronger
representation and symmetry hypotheses that these examples fail.

\section{Reachability sums and affine cone representations}
\label{sec:cones}

\subsection{Iteration}

Let \(R_n(e)\) be the endpoints of paths of exactly \(n\) edges,
with \(R_0(e)=\{e\}\). Then
\begin{equation}
R_{n+1}(e)=\bigcup_{v\in R_n(e)}R_1(v).
\label{eq:recurrence}
\end{equation}
For at most \(n\) edges use \(R_{\leq n}(e)=\bigcup_{k=0}^nR_k(e)\).
These notions differ unless waiting has been specified.

\begin{proposition}[When Minkowski sums apply]
If edges have exactly the form
\[
(t,x)\rel(t+1,x+s),\qquad s\in S,
\]
for a fixed increment set \(S\), then
\[
R_n(t,x)=\{(t+n,x+s):s\in S^{(n)}\}.
\]
\end{proposition}

\begin{proof}
A path contributes a sequence of increments from \(S\).
Conversely, every such sequence specifies a permitted path.
\end{proof}

This argument uses a translation-invariant increment rule, not
commutation of independent reductions.

\subsection{An algebraic cone theorem}

\begin{theorem}[Translation and dilation produce a cone order]
\label{thm:conefromsymmetry}
Suppose a partial order on a real vector space \(V\) satisfies
\[
x\leq y\iff x+a\leq y+a,
\qquad
x\leq y\iff \lambda x\leq\lambda y
\quad(\lambda>0).
\]
Then \(K=\{v:0\leq v\}\) is a pointed convex cone and
\[
x\leq y\iff y-x\in K.
\]
If the order graph is closed, \(K\) is closed.
Conversely, a pointed convex cone defines such an order.
\end{theorem}

\begin{proof}
Reflexivity gives \(0\in K\). If \(u,v\in K\), translation gives
\(u\leq u+v\), and transitivity gives \(u+v\in K\).
Positive scaling and addition imply convexity.
If \(v,-v\in K\), antisymmetry gives \(v=0\).
Translation by \(-x\) gives the difference formula.
Closedness follows by taking the inverse image of the order graph
under \(v\mapsto(0,v)\).

Conversely, membership of zero, addition, and pointedness establish
reflexivity, transitivity, and antisymmetry; the difference formula
establishes both invariances.
\end{proof}

Nonempty interior is an additional full-dimensionality condition.
The theorem permits orthants and other non-Lorentzian cones.

\subsection{Discrete samples versus completed cone orders}

\begin{definition}[Generated affine cone representation]
\label{def:affine}
A generated affine cone representation of an event order is a triple
\((\iota,V,K)\), where \(\iota:E\to V\) is injective, \(K\) is proper,
and
\begin{align}
p\leq q
&\iff \iota(q)-\iota(p)\in K,
\label{eq:orderfaithful}\\
K
&=\cl\cone\{\iota(q)-\iota(p):p<q\}.
\label{eq:generated}
\end{align}
The existence of this data is the representation hypothesis
\(\Aff\). The corresponding \emph{completed order} is the order
on all of \(V\) defined by differences in \(K\).
\end{definition}

Equation~\eqref{eq:orderfaithful} prevents a representation from
introducing false comparisons among actual events.
Equation~\eqref{eq:generated} prevents enlargement to a convenient
ambient cone unsupported by the represented displacements.
Neither equation asserts that every point of \(V\) is an event,
or that every boundary displacement is operationally attained.

In a continuum model one may instead start with an order on all of
\(V\). Theorem~\ref{thm:conefromsymmetry}, closedness, and nonempty
interior then provide the proper cone. With \(\iota\) the identity,
\eqref{eq:generated} is automatic.

\begin{proposition}[No continuum identification for a past-finite model]
\label{prop:pastfinite}
A nontrivial proper cone order on all of \(V\) cannot be identified
bijectively with a past-finite event order.
\end{proposition}

\begin{proof}
Choose \(u\in\interior K\). The distinct points \(p-nu\), \(n\geq1\),
all precede \(p\). Thus its principal past is infinite.
\end{proof}

Consequently, continuous translations in the completed space must
not silently be called automorphisms of a past-finite network.

\begin{proposition}[Finite increment obstruction]
\label{prop:finitestencil}
Suppose a generated affine representation uses only finitely many
edge displacement vectors. Its generated cone is finitely
generated. In dimension at least three it is not linearly
isomorphic to a Lorentz cone.
\end{proposition}

\begin{proof}
Every reachable displacement is a sum of edge displacements, and
every edge displacement is reachable. Thus the two sets generate
the same cone. A finitely generated cone is closed and polyhedral;
equivalently, intersecting a pointed such cone with a strictly
positive affine section gives the convex hull of finitely many
normalized generators. It has finitely many extreme rays.
A Lorentz cone in dimension at least three has infinitely many,
as proved in Proposition~\ref{prop:orthant}.
\end{proof}

A finite menu of operation shapes is not a finite menu of geometric
displacements. Confusing these conditions would make a claimed
exact Lorentz representation impossible for every fixed finite
stencil.

\section{Chronology, horismos, and operational sharpness}
\label{sec:kp}

\subsection{A causal order does not specify chronology}

The distinction between causal and chronological precedence is
central to the causal-space approach \cite{KP}.

\begin{definition}[Causal-space properties used here]
Let \(J\) be a partial order and let \(I\subseteq J\) be irreflexive.
Assume the push-up implications
\[
xJy,\ yIz\Rightarrow xIz,
\qquad
xIy,\ yJz\Rightarrow xIz.
\]
Write \(x\leq y\) for \(J\), \(x\ll y\) for \(I\), and define
horismos by \(H=J\setminus I\), written \(x\hor y\).
\end{definition}

These are the order and push-up properties needed below;
no stronger causal-space axioms are used.

\begin{proposition}[Nonuniqueness of chronology]
\label{prop:chronnonunique}
Every partial order admits both
\[
I=\varnothing
\quad\text{and}\quad
I=J\setminus\{(x,x):x\in E\}
\]
with these properties.
\end{proposition}

\begin{proof}
The empty case is immediate. For strict order, if
\(x\leq y<z\), then \(x\leq z\). Equality \(x=z\) would contradict
antisymmetry, so \(x<z\). The other push-up implication is analogous.
\end{proof}

\subsection{The proper-cone construction}

\begin{theorem}[Chronology from a proper cone]
\label{thm:kpcone}
For a proper cone \(K\subset V\) of positive dimension, define
\[
p\leq_Kq\iff q-p\in K,
\qquad
p\ll_Kq\iff q-p\in\interior K.
\]
These relations have the stated causal-space properties, and
\[
p\hor_Kq\iff q-p\in\partial K.
\]
Every positive multiple of a nonzero boundary vector remains on
the boundary.
\end{theorem}

\begin{proof}
Addition and pointedness give the partial order.
The origin is not interior, since a neighborhood of zero would
contain a nonzero vector and its negative.

If \(a\in\interior K\) and \(b\in K\), choose
\(B(a,\epsilon)\subset K\). Then
\(B(a+b,\epsilon)\subset K+K\subset K\).
Thus \(\interior K+K\subseteq\interior K\), proving push-up.
Closedness gives \(K\setminus\interior K=\partial K\).
Positive dilation is a homeomorphism preserving both \(K\) and
its interior, hence its boundary.
\end{proof}

Boundary rays need not be extreme. For example,
\((1,1,0)\in\partial\R_{\geq0}^3\) is the sum of two nonproportional
boundary vectors.

\subsection{Attainability and robust propagation}

\begin{definition}[Faithful operational realization]
An operational system on \(E\subset V\) faithfully realizes the
restriction of a cone order if
\[
p\rightsquigarrow q\iff q-p\in K
\qquad(p,q\in E,\ p\ne q),
\]
with the left side witnessed by an admissible protocol.
\end{definition}

\begin{proposition}[Boundary attainability for actual event pairs]
\label{prop:relay}
In a faithful realization, every distinct event pair with
\(q-p\in\partial K\) admits an influence-witnessing protocol and is
horismotic for the chronology induced by \(\interior K\).
\end{proposition}

\begin{proof}
The boundary is contained in \(K\), so faithfulness supplies the
protocol. Theorem~\ref{thm:kpcone} supplies the horismos statement.
\end{proof}

This does not assert the existence of an event pair on every
boundary ray. Closure alone does not supply protocols.
For example, the displacement relation
\[
K'=\{0\}\cup\interior\Lor_m
\]
is a cone order but attains no nonzero displacement on
\(\partial\Lor_m\).

There is also an important distinction between null propagation and
indivisibility.

\begin{proposition}[Same-ray indivisibility is not horismos]
\label{prop:raytest}
In a faithful continuum realization of a cone order, every
nonzero reachable displacement \(v\) has reachable displacements
\(sv\) for every \(0<s<1\). This includes boundary displacements.
\end{proposition}

\begin{proof}
Cone scaling gives \(sv\in K\setminus\{0\}\), and continuum
faithfulness supplies the corresponding events and protocols.
\end{proof}

Thus ``no reachable intermediate event on the same ray'' cannot
define null propagation.

A viable geometric operational test instead requires a topology and
a continuum of controllable endpoints. Let \(\mathcal R\subset V\)
be the reachable displacement set, including zero.
Call \(v\ne0\) \emph{robustly reachable} when an open neighborhood
of \(v\) is contained in \(\mathcal R\), and \emph{sharp} when it
is reachable but not robustly reachable. If
\(\mathcal R=K\) is proper, the sharp set is exactly
\(\partial K\setminus\{0\}\). Applying this definition directly to a
sparse event sample would generally make every displacement
nonrobust; passing to a completion is an additional assumption.

\section{Signature and Lorentz-cone selection}
\label{sec:selection}

\subsection{Conditional quadratic signature}

\begin{theorem}[Quadratic component classification]
\label{thm:signature}
Let \(Q\) be a nondegenerate quadratic form on \(V\), with
\(\dim V=m\geq3\). Suppose \(\{Q>0\}\ne\varnothing\), and a
connected component \(F\) of \(\{Q<0\}\) has pointed closure.
Then \(Q\) has negative index one and positive index \(m-1\).

Conversely, for this signature, \(\{Q<0\}\) has two components,
each an open convex cone with proper closure.
\end{theorem}

\begin{proof}
By a linear change of coordinates,
\[
Q(u,w)=-\|u\|_2^2+\|w\|_2^2,
\qquad u\in\R^k,\quad w\in\R^\ell.
\]
The assumptions imply \(k,\ell\geq1\).
The map
\[
S^{k-1}\times\R^\ell\times(0,\infty)
\longrightarrow\{Q<0\},
\qquad
(\theta,w,a)\longmapsto
\bigl(\sqrt{\|w\|_2^2+a^2}\,\theta,w\bigr)
\]
is a homeomorphism, with inverse
\[
(u,w)\longmapsto
\left(\frac{u}{\|u\|_2},w,
\sqrt{\|u\|_2^2-\|w\|_2^2}\right).
\]
If \(k\geq2\), the negative region is connected and contains both
\(v\) and \(-v\) for each of its vectors. Its closure is not pointed.
Thus \(k=1\).

Writing \(u=t\), the components are
\[
F_+=\{(t,w):t>\|w\|_2\},\qquad F_-=-F_+.
\]
The triangle inequality proves convexity. Their closures
\(t\geq\|w\|_2\) and \(t\leq-\|w\|_2\) are pointed, closed, and
have nonempty interior.
\end{proof}

The full negative region is not convex in the Lorentzian case:
one first chooses a time-oriented component.
Likewise, the two closed sheets of \(\{Q\leq0\}\) meet at zero,
so a future cone must not be defined as a connected component of
that closed set.

A higher-index quadratic form can contain smaller pointed cones.
For example,
\[
Q(t_1,t_2,x)=-t_1^2-t_2^2+\|x\|_2^2
\]
is nonpositive on
\[
\left\{t_1\geq\sqrt{t_2^2+\|x\|_2^2}\right\}.
\]
The theorem concerns a \emph{full component}, not an arbitrarily
chosen subcone.

\subsection{Operational content of the extra axiom}
\label{sec:operationalaxioms}

\paragraph{Affine structure and translations.}
Hypothesis \(\Aff\) is a declared representation hypothesis,
not a consequence of equal local arity. An operational derivation
would require coherent displacement composition and a compatible
real affine completion. At the completed level, translations form
a simply transitive action of the additive group \(V\).
If that action is to describe operations rather than coordinates,
it must be lifted to a family of record-preserving experiments.
An abstract abelian group alone does not supply real scalar
multiplication or a topology. By
Proposition~\ref{prop:pastfinite}, the full continuous action cannot
be identified with a transitive action on the past-finite event
set used in the countermodels.

\paragraph{Homogeneity.}
Interior homogeneity means that any two interior displacements are
related by an invertible linear map preserving the completed cone.
Its stronger operational reading is a corresponding symmetry of
experiments, including channel interfaces, admissible interventions,
and records. This lift is additional data. Positive diagonal maps
make the real orthant homogeneous, but generally do not act
bijectively on its integer event sample. Cone homogeneity,
graph homogeneity, and uniform reduction shapes are three different
conditions.

\paragraph{Self-duality.}
The pairing in self-duality is an explicitly assumed
positive-definite inner product. A proposed derivation from record
correlations or protocol costs would have to prove bilinearity,
positive definiteness, and the stated primal--dual equality;
none is proved here. The pairing is Euclidean auxiliary data, not
the Lorentzian metric eventually obtained. The weaker published
selection theorem in Theorem~\ref{thm:weaker} avoids assuming a
pairing, but still requires homogeneous and strictly convex
completed influence geometry.

\paragraph{Irreducibility and absence of faces.}
Irreducibility excludes a decomposition of the completed order
into independent product orders. The orthant is the canonical
counterexample. Strict convexity is stronger: any two
nonproportional feasible displacements have an interior sum.
Operationally, when displacement concatenation and robust endpoint
control are available, two different limiting propagation
directions compose to a robustly feasible displacement.
This interpretation requires that completion and control structure;
it is not an assertion about two arbitrary finite protocols.
Neither exclusion follows from commuting reductions.

\subsection{The symmetric-cone representation theorem}
\label{sec:representation}

\begin{definition}[Faces and strict convexity of a cone]
\label{def:strict}
A face of \(K\) is a convex subcone \(F\subseteq K\) such that
\(u,v\in K\) and \(u+v\in F\) imply \(u,v\in F\).
It is proper when \(F\ne K\).
A proper cone is \emph{strictly convex} if
\begin{equation}
u,v\in K\setminus\{0\},\quad
u\notin\R_{>0}v
\quad\Longrightarrow\quad
u+v\in\interior K.
\label{eq:strict}
\end{equation}
Equivalently, every nonzero proper face is an extreme ray.
\end{definition}

To verify the equivalence, if a sum in \eqref{eq:strict} lies on
the boundary, a supporting functional gives a proper exposed face
containing both summands. Conversely, two nonproportional elements
of a proper face violate \eqref{eq:strict}.
The word ``proper'' is essential: \(K\) itself is a face.
A condition about all linear subspaces meeting the boundary would
not be an equivalent definition.

\begin{assumption}[Symmetric-cone selection axiom \(\Sym\)]
\label{ass:S}
For a proper cone \(K\subset V\), assume:
\begin{enumerate}
\item \(\Aut(K)=\{A\in\mathrm{GL}(V):AK=K\}\) acts transitively on
\(\interior K\);
\item there is a positive-definite inner product \(h\) such that
\[
K=K^{*h}
:=\{y:h(y,x)\geq0\text{ for all }x\in K\};
\]
\item there is no decomposition \(V=V_1\oplus V_2\) into
positive-dimensional subspaces with
\(K=K_1\oplus K_2\), where \(K_i\) is proper in \(V_i\);
\item \(K\) is strictly convex.
\end{enumerate}
\end{assumption}

In dimension at least three, strict convexity already excludes
nontrivial products: one factor would have dimension at least two
and would be a higher-dimensional proper face. Irreducibility is
retained in \(\Sym\) to make its operational meaning explicit.

\begin{theorem}[Representation from cone order and \(\Sym\)]
\label{thm:representation}
Let \(K\) be a proper cone in \(V\), with \(m=\dim V\geq3\).
If \(\Sym\) holds, there is a linear isomorphism
\(T:V\to\R\times\R^{m-1}\) such that
\[
T(K)=\Lor_m=\{(t,x):t\geq\|x\|_2\}.
\]
\end{theorem}

\begin{proof}
Homogeneity and self-duality make \(K\) a symmetric cone.
The Koecher--Vinberg correspondence identifies symmetric cones
with cones of squares in Euclidean Jordan algebras
\cite{Koecher,Vinberg,FK}. Irreducibility makes the algebra simple.
The classification lists the one-dimensional algebra, the
Hermitian matrix algebras over \(\R,\C,\HH\), spin factors, and
the exceptional algebra of \(3\times3\) Hermitian octonionic
matrices \cite[Chapter~V]{FK}; see also \cite{JNW}.

We now apply the strict-convexity hypothesis.
In each matrix algebra of size \(n\geq3\), let
\(a=\operatorname{diag}(1,0,\ldots,0)\) and
\(b=\operatorname{diag}(0,1,0,\ldots,0)\).
They are nonproportional positive elements, but \(a+b\) has a
zero eigenvalue and is on the boundary of the positive cone.
The same diagonal elements in the exceptional rank-three algebra
have a boundary sum, by the Jordan spectral criterion for
positivity \cite[Chapter~III]{FK}.
These cases violate \eqref{eq:strict}.

The one-dimensional case is excluded by \(m\geq3\).
The remaining cases are spin factors, including the
\(2\times2\) Hermitian cases. Explicitly,
\[
\begin{pmatrix}a&z\\ \bar z&b\end{pmatrix}
\longmapsto
\left(\frac{a+b}{2},\frac{a-b}{2},z\right)
\]
identifies their positive cones with
\(t\geq\sqrt{s^2+|z|^2}\), treating \(z\) in real coordinates.

For completeness, the spin factor on \(\R\oplus\R^{m-1}\) has
product
\[
(t,x)\circ(s,y)
=(ts+x\cdot y,\ ty+sx).
\]
A square is
\[
(t,x)^2=(t^2+\|x\|_2^2,\ 2tx),
\]
which belongs to \(\Lor_m\).
Conversely, if \(a\geq\|b\|_2\), put
\[
\alpha=\sqrt{a+\|b\|_2},
\qquad
\beta=\sqrt{a-\|b\|_2}.
\]
For \(b\ne0\), the element
\[
\left(\frac{\alpha+\beta}{2},
      \frac{\alpha-\beta}{2}\frac{b}{\|b\|_2}\right)
\]
squares to \((a,b)\); the case \(b=0\) is immediate.
Thus the cone of squares is exactly \(\Lor_m\).
\end{proof}

\begin{corollary}[Application to records]
\label{cor:networkrepresentation}
Under the hypotheses of Theorem~\ref{thm:mainpositive},
\[
p\rightsquigarrow q
\iff
T(\iota(q)-\iota(p))\in\Lor_m
\qquad(p\ne q).
\]
\end{corollary}

\begin{proof}
Theorem~\ref{thm:network} identifies sensitivity with strict
reachability, \eqref{eq:orderfaithful} identifies reachability
with the restricted cone order, and
Theorem~\ref{thm:representation} identifies that cone with \(\Lor_m\).
\end{proof}

\begin{corollary}[Exclusion of the polyhedral realizations]
\label{cor:exclude}
The standard generated representations of
Propositions~\ref{prop:ball} and~\ref{prop:orthant}, and every
invertible affine change of either representation, fail \(\Sym\).
More generally, an exact generated representation with a finite
edge-increment set cannot satisfy \(\Sym\) in dimension at least
three.
\end{corollary}

\begin{proof}
Both standard cones fail strict convexity.
The orthant additionally fails irreducibility, while
\(K_{\diamond}\) fails interior homogeneity
(Lemma~\ref{lem:diagnostic}).
These properties are invariant under linear isomorphism, and an
affine change transforms displacement vectors by its linear part.
The final statement follows from
Proposition~\ref{prop:finitestencil} and
Theorem~\ref{thm:representation}.
\end{proof}

This excludes the stated \emph{geometric realizations}.
It does not prove that the underlying event posets admit no
unrelated nonlinear embedding into a Lorentzian order.
Nor does it prohibit Euclidean-ball sections of a Lorentz cone.
The compact affine bases emphasize the presence of boundary
segments in the orthant base and their absence in the Lorentz-cone
base (Figure~\ref{fig:bases}).

% Fourth appearance: original file fig2 becomes manuscript Figure 4.
\begin{figure}[!htbp]
\centering
\ManuscriptGraphic[
  width=\linewidth,
  height=0.46\textheight,
  keepaspectratio
]{fig2}
\caption[Compact affine bases of the orthant and Lorentz cones]{%
Compact affine bases in dimension three.
An orthant base is a triangle; a compact affine base of a Lorentz
cone is an ellipsoid.
The former has boundary segments, whereas the latter is strictly
convex. The planar coordinates in the artwork are schematic affine
coordinates, not a common metrically calibrated coordinate system.}
\label{fig:bases}
\end{figure}

\subsection{A weaker published characterization}

\begin{theorem}[Homogeneity and strict convexity suffice]
\label{thm:weaker}
A homogeneous strictly convex proper cone in dimension \(m\geq3\)
is linearly isomorphic to \(\Lor_m\).
\end{theorem}

\begin{proof}[Application of a published characterization]
Ito and Louren\c{c}o classify homogeneous cones with strictly convex
closure: up to linear isomorphism the possibilities are
\(\{0\}\), \(\R_{\geq0}\), \(\R_{\geq0}^2\), and Lorentz cones
\cite[Theorem~7]{ItoLourenco}.
Apply their result to the homogeneous cone \(K\), which equals
its closure. Full dimension \(m\geq3\) excludes the first three
possibilities.
\end{proof}

Thus self-duality and irreducibility are not logically necessary
for cone selection once homogeneity and strict convexity are
assumed. We retain the proof under \(\Sym\) because it exposes the
pairing assumption and the exclusion of higher-rank positive cones.
The weaker theorem provides a precise alternative without invoking
an unproved ``homogeneous oval'' argument or a gauge-Hessian
calculation.

\begin{proposition}[Boundary transitivity as a sufficient replacement]
\label{prop:boundarytransitive}
For a proper cone \(K\) of dimension at least two, if
\(\Aut(K)\) acts transitively on
\(\partial K\setminus\{0\}\), then \(K\) is strictly convex.
Consequently, in Theorem~\ref{thm:representation} its
strict-convexity clause may be replaced by boundary transitivity.
\end{proposition}

\begin{proof}
Choose a compact base of \(K\) using a strictly positive linear
functional. An extreme point of that base gives an extreme ray,
which lies on the boundary in dimension at least two.
Linear cone automorphisms preserve extremality.
Transitivity therefore makes every nonzero boundary point generate
an extreme ray.

If nonproportional \(u,v\in K\setminus\{0\}\) had
\(u+v\in\partial K\), that boundary ray would not be extreme.
This contradicts the preceding conclusion.
\end{proof}

Boundary transitivity is a sufficient condition, not a weaker
condition than strict convexity for arbitrary cones.

\subsection{Pairings and uniqueness of the quadratic form}

\begin{lemma}[Change of a self-dualizing pairing]
\label{lem:pairing}
Suppose two positive-definite inner products \(h_1,h_2\) make
the same cone \(K\) self-dual. There is an \(h_1\)-self-adjoint,
positive-definite operator \(A\in\Aut(K)\) such that
\[
h_2(x,y)=h_1(Ax,y).
\]
\end{lemma}

\begin{proof}
Finite-dimensional linear algebra gives the stated positive
operator \(A\). Its self-adjointness implies
\[
K^{*h_2}=A^{-1}K^{*h_1}.
\]
Both dual cones equal \(K\), so \(AK=K\).
\end{proof}

This statement concerns auxiliary Euclidean pairings.
The following independent argument establishes uniqueness of the
Lorentzian conformal quadratic form.

\begin{proposition}[The cone determines a quadratic form up to scale]
\label{prop:quadraticunique}
Let \(T(K)=\Lor_m\), \(m\geq3\), and set
\[
Q_T(v)=-(Tv)_0^2+\sum_{i=1}^{m-1}(Tv)_i^2.
\]
Then \(Q_T\) is nondegenerate of signature \((1,m-1)\).
For every other linear isomorphism \(T'\) with
\(T'(K)=\Lor_m\), there is \(c>0\) such that
\(Q_{T'}=cQ_T\).
\end{proposition}

\begin{proof}
Only uniqueness needs proof. Work in the \(T\)-coordinates and
write the second quadratic form as
\[
q(t,x)=a t^2+2t\,b\cdot x+x^{\mathsf T}Cx,
\]
where \(C\) is symmetric.
It vanishes at every \((1,u)\), \(\|u\|_2=1\), because the two
maps send the same boundary to the null boundary.
Comparing \(u\) and \(-u\) gives \(b=0\).
Then \(u^{\mathsf T}Cu=-a\) on the unit sphere, so \(C=-aI\).
Hence \(q=(-a)(-t^2+\|x\|_2^2)\).
The second form is negative on the cone interior, so \(a<0\).
\end{proof}

The form obtained here satisfies the hypotheses of
Theorem~\ref{thm:signature}: its future component has closure \(K\)
and its exterior is nonempty. The signature theorem and the
symmetric-cone theorem therefore agree, but only the latter
selects quadraticity from the nonquadratic assumptions in \(\Sym\).

\subsection{A nonvacuous discrete example and its limitation}

\begin{example}[A generated Lorentz cone without continuum events]
\label{ex:sparseLorentz}
Fix \(m\geq3\). Choose a family \((z_n)_{n\in\Z}\) dense in the
open unit ball of \(\R^{m-1}\), and a fixed spatial unit vector \(e_1\).
Take disjoint two-event chains \(a_n\rel b_n\), represented by
\[
\iota(a_n)=(0,10n e_1),\qquad
\iota(b_n)=(1,10n e_1+z_n),
\qquad n\in\Z.
\]
This graph is admissible. Its order is exactly the restriction
of the Lorentz order. The event sample is a disjoint union of
two-event chains; its generated cone is \(\Lor_m\), but no
nonzero boundary displacement is realized
(Figure~\ref{fig:sparselorentz}).
\end{example}

% Fifth appearance: original file fig5 remains manuscript Figure 5.
\begin{figure}[!htbp]
\centering
\ManuscriptGraphic[
  width=\linewidth,
  height=0.42\textheight,
  keepaspectratio
]{fig5}
\caption[Sparse event sample generating a Lorentz cone]{%
Schematic arrangement of separated two-event chains.
The exact event set is
\(\iota(E)=
\{(0,10n e_1),(1,10n e_1+z_n):n\in\Z\}\),
with \(\|z_n\|_2<1\).
The closure of the cone generated by their differences
\((1,z_n)\) is \(\Lor_m\), although no chain realizes a nonzero
null displacement.
The artwork is not a metrically faithful spacetime plot:
all upper events have time coordinate \(1\).
Its upper point labels are schematic identifiers, and the formula
printed below the drawing is incorrect and is replaced by the
event-set formula in this caption.
In particular, the event set is not itself \(\Lor_m\), and the
condition \(\tau=1,\ s^2+\tau^2<1\) printed in the artwork has no
solutions.}
\label{fig:sparselorentz}
\end{figure}

\begin{proof}
The intended difference is \((1,z_n)\in\interior\Lor_m\).
Events within either time slice are incomparable.
For \(n\ne k\), the spatial difference between \(a_n\) and \(b_k\)
has norm at least \(10|k-n|-1>1\), so those pairs are incomparable
as well.

The closure of the cone generated by \((1,z_n)\) is \(\Lor_m\):
the closure of their convex hull in the slice \(t=1\) is the closed
unit ball. Thus \(\Aff\) holds with \(K=\Lor_m\), which satisfies
\(\Sym\).
\end{proof}

This example shows that the network representation theorem is not
vacuous. It also shows exactly what it does not prove.
No nonzero boundary displacement is realized by an event pair, and
continuous Lorentz transformations need not preserve the event
sample. Generated geometry is not a continuum operational theory.

\subsection{Quadratic and boost alternatives}
\label{sec:alternatives}

\begin{assumption}[Quadratic representation hypothesis \(\Quad\)]
There is a nondegenerate quadratic form \(Q\), with
\(\{Q>0\}\ne\varnothing\), and a connected component
\(F\subset\{Q<0\}\) such that \(K=\cl F\).
\end{assumption}

For a proper cone, \(\Quad\) implies the Lorentz-cone conclusion
by Theorem~\ref{thm:signature}. Its boundary is then contained in
\(\{Q=0\}\). If sharp operational displacements are to mean that
boundary, the robust-endpoint interpretation in
Section~\ref{sec:kp} must also be supplied.
Quadraticity remains a representation input in this route.

A boost route is also possible, but it adds explicit continuous
symmetries.

\begin{assumption}[Calibrated boost hypothesis \(\Boost\)]
There is a decomposition \(V=\R e_0\oplus W\), a positive-definite
inner product on \(W\), and \(e_0\in\interior K\).
For every unit \(n\in W\) and \(s\in\R\), the map
\[
B_n(s):
\begin{cases}
t'=t\cosh s+(x\cdot n)\sinh s,\\
x'=x+\bigl(t\sinh s+(x\cdot n)(\cosh s-1)\bigr)n
\end{cases}
\]
belongs to \(\Aut(K)\).
\end{assumption}

\begin{proposition}[Boosts select the cone]
\label{prop:boost}
A proper cone satisfying \(\Boost\), with \(\dim W\geq1\), equals
\(\{te_0+x:t\geq\|x\|_2\}\).
\end{proposition}

\begin{proof}
Positive multiples of \(B_n(s)e_0\) fill the future timelike
region. Closedness therefore gives \(\Lor_m\subseteq K\).

Suppose \(v=(t,x)\in K\setminus\Lor_m\).
If \(x=0\), then \(t<0\), contradicting pointedness because
\(e_0\in K\). Otherwise set \(a=\|x\|_2>0\), \(n=x/a\).
Then \(t<a\), and invariance and scaling give
\(e^sB_n(s)v\in K\). As \(s\to-\infty\),
\[
e^sB_n(s)v\longrightarrow
\frac{t-a}{2}(e_0-n).
\]
This is a negative nonzero multiple of a vector already in
\(\Lor_m\subseteq K\), again contradicting pointedness.
\end{proof}

Every two-dimensional proper cone is linearly isomorphic to a
two-dimensional Lorentz cone. Merely imposing that property on
plane sections would therefore not replace the calibrated,
compatible boost action in \(\Boost\).

If a graph automorphism \(\phi\) lifts one of these symmetries, the
OR network has the precise relabeling action
\[
u'_{\phi(v)}=u_v,\qquad
b'_{\phi(v)}=b_v,\qquad
x_{vw}\longmapsto x_{\phi(v)\phi(w)}.
\]
Equation~\eqref{eq:or} is equivariant under this action.
Producing such a \(\phi\), however, is a separate theorem:
a general cone automorphism does not preserve the discrete event
set or its edges. No continuous boost lift is inferred here from
a commuting diamond.

\section{Alexandrov intervals and manifold structure}
\label{sec:topology}

For an irreflexive transitive relation \(\ll\), define
\[
I^+(p)=\{q:p\ll q\},\qquad
I^-(p)=\{q:q\ll p\},\qquad
I(a,b)=I^+(a)\cap I^-(b).
\]
The smallest topology containing these intervals always exists.
Whether the intervals themselves form a basis is a separate
question. Here ``Alexandrov topology'' refers to this chronological
construction, not the unrelated arbitrary-intersection convention
in general topology.

\subsection{A sufficient basis criterion}

\begin{proposition}[Interval basis criterion]
\label{prop:basis}
Suppose:
\begin{enumerate}
\item every point belongs to an interval;
\item if \(a_1\ll x\) and \(a_2\ll x\), there exists \(a\) with
\(a_1\ll a,\ a_2\ll a,\ a\ll x\);
\item if \(x\ll b_1\) and \(x\ll b_2\), there exists \(b\) with
\(x\ll b,\ b\ll b_1,\ b\ll b_2\).
\end{enumerate}
Then the intervals form a basis.
\end{proposition}

\begin{proof}
The first condition gives coverage. If \(x\) lies in the
intersection of two intervals, the other conditions produce
\(a,b\) with \(x\in I(a,b)\) contained in that intersection,
by transitivity.
\end{proof}

\subsection{Interpolation alone is insufficient}

\begin{proposition}[A dense distinguishing order without an interval basis]
\label{prop:densebranch}
There is an interpolating order whose strict pasts and strict
futures each distinguish points, and whose points all have
predecessors and successors, but whose chronological intervals
do not form a basis.
\end{proposition}

\begin{proof}
Use
\[
E=(\Q_{<0}\times\{A,B\})\sqcup\Q_{\geq0}.
\]
Order each negative branch by its rational coordinate, put both
branches below the nonnegative chain, and order that chain
normally. The two negative branches are mutually incomparable.
Let \(\ll\) be strict order.

Rational density gives interpolation, including across the joining
point by choosing a later negative point on the same branch.
Pasts and futures distinguish points within a branch by density,
and distinguish different branches through their negative elements.
Every point has predecessors and successors.

For \(a=(-1,A)\), \(b=(-1,B)\), and \(c=1\),
\[
I(a,c)\cap I(b,c)=\Q\cap[0,1).
\]
Every interval containing \(0\) has a lower endpoint on one
negative branch, so contains negative points of that branch.
No such interval is contained in the displayed intersection.
\end{proof}

Local measurements that distinguish events do not imply that
chronological intervals separate them.

\subsection{Discreteness and continuum assumptions}

A two-event chain has no intermediate event.
More generally, an order with finite intervals cannot interpolate
unconditionally between all distinct comparable points:
repeated interpolation would produce infinitely many elements
inside one interval.

Even an interval basis and density do not imply a manifold.
The ordered rational line has both, but every neighborhood is
countable and therefore cannot be homeomorphic to a nonempty open
subset of \(\R^d\), \(d\geq1\).
The distinction between a discrete order and its possible continuum
approximation is also fundamental in causal-set theory
\cite{Bombelli}.

\section{Causal typing as an explicit refinement}
\label{sec:typing}

\subsection{An affine first-order interface}

Fix an event order \((E,\leq)\), types, and typed primitives
\[
\mathsf{op}:A_1\times\cdots\times A_k\longrightarrow B,
\]
including nullary constants. Terms are
\[
t::=x\mid\mathsf{op}(t_1,\ldots,t_k).
\]
An origin-annotated context is a finite map
\[
\Gamma=\{x_1:A_1@e_1,\ldots,x_n:A_n@e_n\}.
\]
The annotation records the capability's origin, not a claim of
physical storage at every subsequent event.
Judgments are \(\Gamma\vdash_f t:A\).

The rules are
\[
\frac{e\leq f}{x:A@e\vdash_f x:A}
\quad\textsc{Var},
\]
\[
\frac{
\Gamma_i\vdash_f t_i:A_i\ (1\leq i\leq k)
\qquad
\dom\Gamma_i\cap\dom\Gamma_j=\varnothing\ (i\ne j)
}{
\biguplus_{i=1}^k\Gamma_i
\vdash_f\mathsf{op}(t_1,\ldots,t_k):B
}
\quad\textsc{Op},
\]
and
\[
\frac{\Gamma\vdash_f t:A\qquad x\notin\dom\Gamma}
{\Gamma,x:B@e\vdash_f t:A}
\quad\textsc{Weak}.
\]
A nullary operation has empty context; there is no contraction rule.

Let \(|\Gamma|\) erase origins. The ordinary affine system
\(|\Gamma|\vdash_{\mathrm{aff}}t:A\) has the same rules without
event annotations or the variable side condition.

\begin{theorem}[Exact availability criterion]
\label{thm:typing}
One has \(\Gamma\vdash_f t:A\) if and only if
\[
|\Gamma|\vdash_{\mathrm{aff}}t:A
\quad\text{and}\quad
\forall x\in\FV(t),\
\operatorname{origin}_{\Gamma}(x)\leq f.
\]
Consequently, an unavailable context entry may occur only if its
name is unused.
\end{theorem}

\begin{proof}
Erasing an annotated derivation gives an affine derivation.
Induction proves the origin condition: it is the variable side
condition, is inherited through operations, and is unaffected by
weakening.

Conversely, restore each context entry's origin in an affine
derivation. Every variable leaf is a free occurrence in the
enclosing first-order term, so the assumed origin condition
validates its annotated variable rule. Disjointness of contexts
is unchanged, and weakening places no condition on an unused
entry. Induction reconstructs the annotated derivation.
\end{proof}

\subsection{Limits of the interpretation}

The theorem excludes unavailable \emph{uses}, not every context
containing an unavailable name. It is also important to retain the
direction:
\[
e\leq f
\quad\Longleftrightarrow\quad
\text{the origin precedes the use}.
\]
Availability alone does not establish adequate transport resources
for several competing uses. An implementation may additionally
require path certificates and exclusive channel ownership.

\begin{corollary}[Geometric reading]
In a cone representation, a used name originating at \(e\) and
used at \(f\) satisfies
\[
\iota(f)-\iota(e)\in K.
\]
Under Theorem~\ref{thm:representation}, this difference is
future causal for the resulting Lorentz form.
\end{corollary}

\begin{proof}
Apply Theorem~\ref{thm:typing} and the order-faithfulness condition
\eqref{eq:orderfaithful}.
\end{proof}

This is type safety relative to a specified order, not a derivation
of its geometry.

\section{Conditional conformal reconstruction}
\label{sec:reconstruction}

\subsection{A smooth chronological counterexample}

\begin{theorem}
\label{thm:smoothcounter}
There is an interpolating, past- and future-distinguishing
causal-space structure on the smooth manifold \(\R^3\) whose
Alexandrov intervals form a basis for its usual topology, but
which admits a chronological automorphism that is not
differentiable.
\end{theorem}

\begin{proof}
Put
\[
x\leq y\iff x_i\leq y_i\ \text{for all }i,
\qquad
x\ll y\iff x_i<y_i\ \text{for all }i.
\]
This is the proper-orthant construction.
Midpoints give interpolation, and intervals are the open boxes
\(\prod_i(a_i,b_i)\).
Strict futures \(\prod_i(x_i,\infty)\) and strict pasts determine
all coordinates, so both distinguishability conditions hold.

The strictly increasing bijection
\[
h(s)=
\begin{cases}
s,&s\leq0,\\
2s,&s>0
\end{cases}
\]
induces
\[
F(x_1,x_2,x_3)=(h(x_1),x_2,x_3).
\]
The piecewise-linear stretch \(h\) induces a chronological
automorphism of \((\R^3,\ll)\) that fails to be differentiable on
the plane \(x_1=0\) (Figure~\ref{fig:stretch}).
Indeed, strict monotonicity preserves each coordinate inequality
in both directions, while the two one-sided derivatives in the
first coordinate are \(1\) and \(2\).

For the open Alexandrov interval \(U=(-1,1)^3\), one has
\[
F(U)=(-1,2)\times(-1,1)^2.
\]
Writing \(S=\cl U=[-1,1]^3\) gives
\[
F(S)=[-1,2]\times[-1,1]^2.
\]
Thus the map preserves the interval basis, but not smoothness.
\end{proof}

% Sixth appearance: original file fig6 remains manuscript Figure 6.
\begin{figure}[!htbp]
\centering
\ManuscriptGraphic[
  width=\linewidth,
  height=0.49\textheight,
  keepaspectratio
]{fig6}
\caption[A chronological automorphism that is not differentiable]{%
The coordinate stretch \(F(x_1,x_2,x_3)=(h(x_1),x_2,x_3)\)
preserves strict coordinatewise chronology but is not differentiable
on \(x_1=0\).
Corrections to the supplied artwork:
\(F([-1,1]^3)=[-1,2]\times[-1,1]^2\), not
\([-2,2]\times[-1,1]^2\).
The closed cube \(S=[-1,1]^3\) is the closure of the Alexandrov
interval \(U=(-1,1)^3\), not itself an open Alexandrov interval.
The graph of \(h\) has its kink at \((0,0)\); its drawn vertical
offset and the box proportions are schematic rather than exact.}
\label{fig:stretch}
\end{figure}

Thus a smooth atlas compatible with Alexandrov topology is not
a substitute for Lorentzian realizability.

\subsection{The actual reconstruction input}

\begin{assumption}[Lorentzian realization]
\label{ass:lorentz}
A chronology \((E,\ll)\) has a Lorentzian realization if a
bijection \(\iota:E\to M\) to a smooth time-oriented Lorentzian
manifold satisfies
\[
p\ll q
\iff
\iota(q)\in I_g^+(\iota(p)).
\]
For the next theorem, the manifold has dimension at least three
and is both past and future distinguishing.
\end{assumption}

\begin{theorem}[Classical chronological reconstruction]
\label{thm:malament}
Let \((M,g)\) and \((M',g')\) be smooth time-oriented Lorentzian
manifolds of the same dimension at least three, both past and
future distinguishing. A bijection preserving chronology in both
directions is a smooth conformal diffeomorphism:
\[
F^*g'=\Omega^2g
\]
for a smooth positive function \(\Omega\).
\end{theorem}

\begin{proof}[Attribution]
This is the chronological reconstruction result of Malament,
using the causal, differential, and conformal rigidity results
associated with Hawking, King, and McCarthy
\cite{Malament,HKM}. The distinguishing formulation and its
relation to causal bijections are discussed in
\cite{ParrikarSurya}.
\end{proof}

\begin{corollary}[Uniqueness after realization]
Two realizations of the same chronology satisfying
Assumption~\ref{ass:lorentz}, in the same dimension, are related by
a conformal diffeomorphism.
\end{corollary}

\begin{proof}
Compose one realization map with the inverse of the other and
apply Theorem~\ref{thm:malament}.
\end{proof}

If the completed event space is all of \(V\), cone selection
already gives a flat Lorentzian realization of its interior-cone
chronology. A sparse sample such as
Example~\ref{ex:sparseLorentz} is not thereby a manifold.
A curved realization requires additional compatible local data.

Strong causality, not merely past and future distinguishability,
is the condition relevant to equality of manifold and Alexandrov
topologies \cite{ParrikarSurya}. The path topology in
\cite{HKM} is not simply another name for the Alexandrov topology.

The Alexandrov and Zeeman rigidity results likewise concern
transformations of an already specified causal geometry
\cite{AlexandrovOvchinnikova,Zeeman}. They are not used to select
a cone from scheduler consistency.

\subsection{Null directions and horismos}

The selected conformal class determines null directions.
For \(g'=\Omega^2g\), the connections satisfy
\[
\nabla'_k k
=
\nabla_k k
+2k(\log\Omega)k
-g(k,k)\operatorname{grad}_g(\log\Omega).
\]
If \(k\) is tangent to a \(g\)-null geodesic, the final term
vanishes and the difference is tangent to the curve.
Reparametrization therefore preserves the unparametrized null
geodesic.

To identify an abstract \(J\setminus I\) with geometric horismos,
one must additionally identify \(J\) with Lorentzian causal
reachability; a realization of \(I\) alone does not verify an
arbitrarily supplied \(J\).
A null geodesic also need not remain an achronal boundary generator
throughout its entire extension.
No global identification of all null geodesics with all
horismotic pairs is asserted.

\section{An audited reconstruction sequence}
\label{sec:audit}

The logical dependencies are:
\[
\begin{array}{c}
\text{specified event occurrences and dependency}\\
+\ \text{acyclicity}
\end{array}
\quad\Longrightarrow\quad
\text{partial order},
\]
\[
\begin{array}{c}
\text{finite histories}\\
+\ \text{compositional independent diamonds}
\end{array}
\quad\Longrightarrow\quad
\text{schedule-independent records},
\]
\[
\begin{array}{c}
\text{vector-space order}\\
+\ \text{translation and positive-dilation invariance}
\end{array}
\quad\Longrightarrow\quad
\text{convex cone order},
\]
\[
\text{proper cone order}+\Sym
\quad\Longrightarrow\quad
\text{Lorentz cone},
\]
\[
\begin{array}{c}
\text{full quadratic component}\\
+\ \text{pointed closure and nonempty exterior}
\end{array}
\quad\Longrightarrow\quad
\text{Lorentzian signature},
\]
and
\[
\begin{array}{c}
\text{distinguishing Lorentzian realizations}\\
+\ \text{chronological isomorphism}
\end{array}
\quad\Longrightarrow\quad
\text{conformal diffeomorphism}.
\]

These are different conditional implications, not a chain in which
each conclusion alone supplies the next set of hypotheses.
Figure~\ref{fig:ladder} summarizes their order of presentation.
No arrow is obtained by renaming an interleaving as a change of
observer. We do not assert logical independence of all six results.

% Seventh appearance: original file fig7 remains manuscript Figure 7.
\begin{figure}[!htbp]
\centering
\ManuscriptGraphic[
  width=0.94\linewidth,
  height=0.70\textheight,
  keepaspectratio
]{fig7}
\caption[Six conditional implications and their separate hypotheses]{%
Six conditional results used in the manuscript.
Each horizontal arrow inside a box represents an implication with
the hypotheses written in that box.
The vertical connectors indicate reading order only: they do not
assert that one box's conclusion entails the next box's hypotheses.
For box~(5), the full assumptions also include nondegeneracy of the
quadratic form, nonempty exterior \(\{Q>0\}\), and dimension at
least three, as in Theorem~\ref{thm:signature}.
Box~(6) has the smoothness, dimension, time-orientation, and
distinguishability hypotheses of Theorem~\ref{thm:malament}.}
\label{fig:ladder}
\end{figure}

\subsection{Audit of the lemma ladder}

The following checklist records the proof obligations underlying
the conditional results summarized in Figure~\ref{fig:ladder}.

\begin{longtable}{@{}p{0.09\linewidth}p{0.85\linewidth}@{}}
\toprule
Step & Required statement\\
\midrule
\endfirsthead
\toprule
Step & Required statement\\
\midrule
\endhead
L1 &
Retain occurrence identity. Equal support and residual state do
not identify distinct events.\\
L2 &
Use dependency reachability as an order candidate. Prove
sensitivity--reachability equivalence separately.\\
L3 &
Acyclicity yields an order. Strongly connected quotienting also
does so, but neither proves that cycles are exactly inconsistent
records.\\
L4 &
Derive ownership disjointness from channel discipline and
simultaneous enabledness, not from no-signaling alone.\\
L5 &
Use \eqref{eq:recurrence}. Minkowski addition needs a fixed
increment representation.\\
L6 &
The ball and orthant constructions refute selection from the
operational assumptions alone.\\
L7 &
Prove a relay for actual event pairs. Closure of a displacement
set does not supply protocols.\\
L8 &
A causal order does not uniquely determine chronology.
Proper cones give a conditional construction.\\
L9 &
Interpolation alone gives neither an interval basis nor a
manifold.\\
L10 &
Quadratic signature selection requires a full negative component
with pointed closure and a nonempty positive region.\\
L10a &
Uniform local shapes are not \(\Sym\).
Apply Theorem~\ref{thm:representation}, or the precisely cited
weaker Theorem~\ref{thm:weaker}.\\
L10b &
Separate cone automorphisms from their possible lifts to records,
and generated-cone boundaries from attained displacements.\\
L11 &
Use explicit causal-origin rules. The criterion concerns free
uses, not unused context entries.\\
L12 &
Apply conformal reconstruction only after Lorentzian
realizability.\\
\bottomrule
\end{longtable}

\subsection{Relation to finite-range models}

A specified finite-range schedule may prove a domain-of-dependence
bound and admit a saturating relay. Such a theorem concerns the
geometry already encoded by its allowed increments.

The present countermodels do not refute a stronger, separately
specified calculus. They show that the operational axiom list
alone cannot select its geometry.
A relay from another model transfers only after its operations,
interfaces, and placement have been shown legal.
Furthermore, an exact fixed finite increment set has a polyhedral
generated cone, by Proposition~\ref{prop:finitestencil}; a Lorentzian
limit would require an additional limiting construction rather
than an exact finite-stencil identification.

\subsection{Requirements for an Enter-specific theorem}

A formal application would require:
\begin{enumerate}
\item full configuration and event-occurrence definitions;
\item a residual theorem validating swaps without collapsing
occurrences;
\item a precise intervention and observation semantics;
\item a sensitivity--reachability or routing theorem;
\item a compositional probabilistic diamond;
\item an ownership theorem for simultaneously enabled operations;
\item the representation data \(\Aff\), including its completion
and faithfulness claims;
\item a sound implementation of causal-origin typing and
transport;
\item record-preserving symmetry constructions supporting
\(\Sym\), its homogeneous strictly convex weakening, or
\(\Boost\), together with a proof that the polyhedral embeddings
are not equivariant realizations of those symmetries.
\end{enumerate}

For a discrete calculus, the final item may require a completion
or limit of families of experiments. It must not impose a
transitive continuous event action incompatible with past-finiteness.

\section{Conclusion and open problems}

Scheduler consistency has an exact consequence: legal evaluations
of a fixed finite history produce the same records.
Acyclic dependency has another: reachability is a partial order.
The countermodels show that neither conclusion chooses a light cone.

The matching positive theorem is also exact. An affine realization
with a homogeneous, self-dual, irreducible, strictly convex proper
influence cone has Lorentz-cone geometry.
The standard polyhedral embeddings fail explicit clauses of this
hypothesis. The sharper homogeneous strictly convex characterization
shows that the pairing need not be assumed if one uses the stronger
published selection theorem.

The remaining issue is derivation, not terminology:
\begin{quote}
Scheduler consistency plus an affine homogeneous strictly convex
self-dual influence cone yields a Lorentz cone.
Scheduler consistency alone does not produce the affine structure,
the automorphism action, or the absence of faces.
\end{quote}

The resulting research program has identifiable tasks:
\begin{enumerate}
\item construct affine displacement data, or a justified affine
completion, from a specified residual theory;
\item prove that operational sensitivity has the required routing
and compositional properties;
\item derive homogeneous strictly convex influence geometry, or
the explicit axiom \(\Sym\), from the rewrite system rather than
assume it;
\item distinguish symmetries of a completed cone from implementable
symmetries of records;
\item prove boundary attainability and, where appropriate,
Lorentzian manifold realizability;
\item establish transport-aware causal typing for actual programs.
\end{enumerate}

Constructing \(V\) from a raw poset is a separate representation
problem. Possible approaches through coherent displacement data or
simply transitive translation actions remain open here.
The present theorem excludes the countermodels as the specified
geometric realizations, not as abstract rewrite systems under all
conceivable embeddings.

No numerical value of \(c\), Einstein equation, preferred spacetime
dimension, continuum limit, or quantum measurement rule follows.
Even after selecting a cone written as
\[
t\geq\frac{\|x\|_2}{c},
\]
its numerical slope changes under independent choices of spatial
and temporal units. Selecting the Lorentz cone and calibrating
physical units are different tasks.

\paragraph{Source and dependency note.}
The cone-selection step uses classical symmetric-cone theory and,
for its weakening, the precise result of Ito and Louren\c{c}o.
The causal reconstruction results are invoked only with their
Lorentzian hypotheses. No unavailable Enter lemma, residual
construction, or companion relay is used as a proof dependency.

\begin{thebibliography}{99}

\bibitem{Lamport}
L.~Lamport,
\emph{Time, clocks, and the ordering of events in a distributed system},
Communications of the ACM \textbf{21} (1978), no.~7, 558--565.
doi:10.1145/359545.359563.

\bibitem{KP}
E.~H.~Kronheimer and R.~Penrose,
\emph{On the structure of causal spaces},
Proceedings of the Cambridge Philosophical Society
\textbf{63} (1967), no.~2, 481--501.
doi:10.1017/S030500410004144X.

\bibitem{Koecher}
M.~Koecher,
\emph{Positivit\"atsbereiche im \(\R^n\)},
American Journal of Mathematics \textbf{79} (1957), no.~3,
575--596.
doi:10.2307/2372563.

\bibitem{Vinberg}
E.~B.~Vinberg,
\emph{The theory of homogeneous convex cones},
Trudy Moskovskogo Matematicheskogo Obshchestva
\textbf{12} (1963), 303--358 (Russian).
English translation:
Transactions of the Moscow Mathematical Society,
1963 volume, 340--403, published 1965.

\bibitem{FK}
J.~Faraut and A.~Kor\'anyi,
\emph{Analysis on Symmetric Cones},
Oxford Mathematical Monographs,
Clarendon Press, Oxford, 1994.
Chapters III and V.
doi:10.1093/oso/9780198534778.001.0001.

\bibitem{JNW}
P.~Jordan, J.~von Neumann, and E.~Wigner,
\emph{On an algebraic generalization of the quantum mechanical
formalism},
Annals of Mathematics, Second Series,
\textbf{35} (1934), no.~1, 29--64.
doi:10.2307/1968117.

\bibitem{ItoLourenco}
M.~Ito and B.~F.~Louren\c{c}o,
\emph{The \(p\)-cones in dimension \(n\geq3\) are not homogeneous
when \(p\ne2\)},
Linear Algebra and its Applications \textbf{533} (2017), 326--335.
doi:10.1016/j.laa.2017.07.029.
arXiv:1701.05801.

\bibitem{AlexandrovOvchinnikova}
A.~D.~Alexandrov and V.~V.~Ovchinnikova,
\emph{Remarks on the foundations of the theory of relativity},
Vestnik Leningradskogo Universiteta (1953), no.~11, 95--110
(Russian).

\bibitem{Zeeman}
E.~C.~Zeeman,
\emph{Causality implies the Lorentz group},
Journal of Mathematical Physics \textbf{5} (1964), no.~4,
490--493.
doi:10.1063/1.1704140.

\bibitem{HKM}
S.~W.~Hawking, A.~R.~King, and P.~J.~McCarthy,
\emph{A new topology for curved space--time which incorporates the
causal, differential, and conformal structures},
Journal of Mathematical Physics \textbf{17} (1976), no.~2,
174--181.
doi:10.1063/1.522874.

\bibitem{Malament}
D.~B.~Malament,
\emph{The class of continuous timelike curves determines the
topology of spacetime},
Journal of Mathematical Physics \textbf{18} (1977), no.~7,
1399--1404.
doi:10.1063/1.523436.

\bibitem{Bombelli}
L.~Bombelli, J.~Lee, D.~Meyer, and R.~D.~Sorkin,
\emph{Space-time as a causal set},
Physical Review Letters \textbf{59} (1987), no.~5, 521--524.
doi:10.1103/PhysRevLett.59.521.

\bibitem{ParrikarSurya}
O.~Parrikar and S.~Surya,
\emph{Causal topology in future and past distinguishing spacetimes},
Classical and Quantum Gravity \textbf{28} (2011), 155020.
doi:10.1088/0264-9381/28/15/155020.
arXiv:1102.0936.

\end{thebibliography}

\end{document}