Scheduler Consistency and Lorentz-Cone Representation Countermodels, Explicit Geometric Hypotheses, and Causal Typing

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.

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 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.

The motivating proposal refers to an Enter calculus, an ECB confluence result, and a companion manuscript entitled From the Enter Calculus to Relativistic Causal Structure. Their complete syntax, residual theory, and proof texts are not premises of this paper. Consequently:

1. the operational model below is self-contained;

2. no result is attributed to an unspecified Enter type system;

3. any application to a particular calculus must verify the stated hypotheses;

4. geometric representation assumptions are not presented as consequences of confluence.

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

There are local record networks with:

1. acyclic dependency;

2. bounded degree, finite support, and finitely many local reduction shapes;

3. termination of every finite experiment;

4. identical joint records under every legal interleaving;

5. exact operational sensitivity equal to strict reachability;

6. no intervention-induced change outside the dependency future;

7. local point distinguishability and finite pasts for finite collections of records.

These properties do not imply that:

1. a one-step spatial section cannot be a Euclidean metric ball on a discrete slice;

2. a generated reachability cone is quadratic or Lorentzian;

3. interpolation or a manifold topology exists;

4. an otherwise unspecified type system enforces causal availability.

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

Let $N$ be a record network of Theorem thm:network. Suppose its event order has a generated affine cone representation $(\iota ,V,K)$ in the sense of Definition def:affine, with $\dim V=m\ge 3$. Assume that $K$:

1. is homogeneous under its linear automorphism group;

2. is self-dual for some positive-definite inner product;

3. is irreducible;

4. is strictly convex in the cone sense of Definition def:strict.

Then an invertible linear map $T:V\to \mathbb{R}\times {\mathbb{R}}^{m-1}$ satisfies

In these coordinates,

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.

Proof. Apply Theorem thm:representation, Corollary cor:networkrepresentation, and Proposition prop:quadraticunique. The exclusion statement is Corollary cor:exclude.

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.

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.

For example, identifying $a_0$ and $a_1$ in

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.

Let $E$ be an event set and $◃$ a directed dependency relation. Write $p<q$ when a nonempty finite dependency path leads from $p$ to $q$, and write $p\le q$ when $p=q$ or $p<q$.

Fix admissible experiments and interventions. For distinct events, write $p⇝q$ if two admissible choices localized at $p$, with all other experimental choices held fixed, give different marginal observation laws at $q$.

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

Composability of channels does not imply transitivity of operational sensitivity.

Proof. Let $X\in \{0,1\}$, let $N$ be an independent fair bit, and consider

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.

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

If the dependency graph is acyclic, then $<$ is a strict partial order and $\le$ is a partial order.

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.

For an arbitrary directed graph, define

Reachability induces a partial order on $E/\sim$.

Proof. Reflexivity and transitivity make $\sim$ an equivalence relation. If $p\sim p^{\prime }$, $q\sim q^{\prime }$, and $p\le q$, then $p^{\prime }\le p\le q\le q^{\prime }$. Thus the quotient relation is well-defined. Mutual quotient reachability means equality of equivalence classes.

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.

For an acyclic graph, put

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 $D$.

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.

Any two linear extensions of a finite partial order are connected by swaps of adjacent incomparable elements.

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.

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.

Proof. Apply Lemma lem:interchange to the complete computation. The residual condition makes each interchange valid in context.

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

Assign a distinct channel name $x_{wv}$ to each edge $w◃v$. Only $w$ produces it and only $v$ consumes it. At event $v$, a controllable local bit $u_v$ and the incoming bits determine

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.

On every admissible graph, the network (eq:or) satisfies:

1. every reduction touches finitely many linearly owned channels;

2. every finite experiment terminates, and every maximal legal schedule completes it;

3. distinct simultaneously enabled operations have disjoint channel ownership and commute exactly;

4. all schedules produce the same joint records;

5. for each event,

   $$b_v=\underset{p\le v}{\bigvee }{u}_p;$$

6. for distinct events, $p⇝q$ if and only if $p<q$;

7. an intervention outside the dependency past of an observation does not change it;

8. every finite collection of records is determined by a finite down-set.

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 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 (eq:ancestoror).

If $p<q$, set all controls other than $u_p$ to zero. Changing $u_p$ changes $b_q$. If $p\nleqq q$, 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.

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.

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.

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.

Proof. Set

and put

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 thm:network applies.

On the next slice, the reachable sites are

The one-step section is the discrete Euclidean unit ball; Minkowski iteration produces the discrete diamond $\Vert z{\Vert }_1\le n$, as proved below and illustrated in Figure fig:stencil. Away from the initial layer every event has the same input and output arities.

The exact $n$-step displacement set is

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

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 ${\mathbb{R}}^2$, 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 generated polyhedral cone, not the word “ball.”

For each $m\ge 3$, there is an admissible record network whose reachability is coordinatewise order on ${\mathbb{N}}^m$. Its standard generated cone is

which is not linearly isomorphic to a Lorentz cone.

Proof. Take edges

The coordinate sum increases along each edge, every principal past lies in a finite coordinate box (Figure fig:orthantnetwork), and the degree is bounded. Theorem thm:network applies.

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

It has exactly $m$ extreme rays. In contrast, each $(1,u)$, $\Vert u{\Vert }_2=1$, generates an extreme ray of $L_m$. Equality in the Euclidean triangle inequality proves extremality. For $m\ge 3$ there are infinitely many such rays, and an invertible linear map preserves extreme rays.

The preceding networks share the operational properties of Theorem thm:network, but their standard generated cones have the properties in Table 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 fig:cones.

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

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, $K_{\diamond }^{\ast }=\{(t,y):t\ge \Vert y{\Vert }_{\infty }\}$, 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.

uniformity does not select a Lorentz cone. “Constant increment rule” is not a transitive translation group on a network with an initial boundary.

The cone $t\ge |x|$ is a Lorentz cone and is linearly isomorphic to ${\mathbb{R}}_{\ge 0}^2$, through $(t,x)\mapsto (t+x,t-x)$. The exclusion statements therefore require dimension at least three; Figure fig:cones illustrates the three-dimensional contrast.

Finite support and a uniform reduction shape do not imply a uniform speed bound in an independently assigned spatial metric.

Proof. Use the independent chains

Assign $T(k,j)=k$ and $X(k,j)=2^{k}j$. Each edge has unit time increment and spatial displacement $2^k\Vert j{\Vert }_2$, which is unbounded. The operational assumptions do not constrain these auxiliary coordinates.

A speed statement requires an axiom relating transitions to geometry.

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 $\rho \ge 0$ of $e$ does not distinguish one-step targets from later reachable targets.

Proof. Every subsequent path vertex is strictly above $e$, so cannot belong to the antichain. The initial meeting has distance zero.

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.

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 sec:operationalaxioms identifies the stronger representation and symmetry hypotheses that these examples fail.

Let $R_n(e)$ be the endpoints of paths of exactly $n$ edges, with $R_0(e)=\{e\}$. Then

For at most $n$ edges use $R_{\le n}(e)={{\bigcup }_{k=0}}^n{R}_k(e)$. These notions differ unless waiting has been specified.

If edges have exactly the form

for a fixed increment set $S$, then

Proof. A path contributes a sequence of increments from $S$. Conversely, every such sequence specifies a permitted path.

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

Suppose a partial order on a real vector space $V$ satisfies

Then $K=\{v:0\le v\}$ is a pointed convex cone and

If the order graph is closed, $K$ is closed. Conversely, a pointed convex cone defines such an order.

Proof. Reflexivity gives $0\in K$. If $u,v\in K$, translation gives $u\le u+v$, 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.

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

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

The existence of this data is the representation hypothesis $\mathsf{Aff}$. The corresponding completed order is the order on all of $V$ defined by differences in $K$.

Equation (eq:orderfaithful) prevents a representation from introducing false comparisons among actual events. Equation (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 thm:conefromsymmetry, closedness, and nonempty interior then provide the proper cone. With $\iota$ the identity, (eq:generated) is automatic.

A nontrivial proper cone order on all of $V$ cannot be identified bijectively with a past-finite event order.

Proof. Choose $u\in \mathrm{int}K$. The distinct points $p-nu$, $n\ge 1$, all precede $p$. Thus its principal past is infinite.

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

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.

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 prop:orthant.

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.

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

Let $J$ be a partial order and let $I\subseteq J$ be irreflexive. Assume the push-up implications

Write $x\le y$ for $J$, $x\ll y$ for $I$, and define horismos by $H=J\setminus I$, written $x⟶y$.

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

Every partial order admits both

with these properties.

Proof. The empty case is immediate. For strict order, if $x\le y<z$, then $x\le z$. Equality $x=z$ would contradict antisymmetry, so $x<z$. The other push-up implication is analogous.

For a proper cone $K\subset V$ of positive dimension, define

These relations have the stated causal-space properties, and

Every positive multiple of a nonzero boundary vector remains on the boundary.

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

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

An operational system on $E\subset V$ faithfully realizes the restriction of a cone order if

with the left side witnessed by an admissible protocol.

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 $\mathrm{int}K$.

Proof. The boundary is contained in $K$, so faithfulness supplies the protocol. Theorem thm:kpcone supplies the horismos statement.

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

is a cone order but attains no nonzero displacement on $\partial L_m$.

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

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.

Proof. Cone scaling gives $sv\in K\setminus \{0\}$, and continuum faithfulness supplies the corresponding events and protocols.

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 $v\ne 0$ robustly reachable when an open neighborhood of $v$ is contained in $\mathcal{R}$, and 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.

Let $Q$ be a nondegenerate quadratic form on $V$, with $\dim V=m\ge 3$. 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.

Proof. By a linear change of coordinates,

The assumptions imply $k,\ell \ge 1$. The map

is a homeomorphism, with inverse

If $k\ge 2$, 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

The triangle inequality proves convexity. Their closures $t\ge \Vert w{\Vert }_2$ and $t\le -\Vert w{\Vert }_2$ are pointed, closed, and have nonempty interior.

The full negative region is not convex in the Lorentzian case: one first chooses a time-oriented component. Likewise, the two closed sheets of $\{Q\le 0\}$ 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,

is nonpositive on

The theorem concerns a full component, not an arbitrarily chosen subcone.

structure and translations. Hypothesis $\mathsf{Aff}$ 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 prop:pastfinite, the full continuous action cannot be identified with a transitive action on the past-finite event set used in the countermodels.

. 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.

-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 thm:weaker avoids assuming a pairing, but still requires homogeneous and strictly convex completed influence geometry.

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.

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 strictly convex if

Equivalently, every nonzero proper face is an extreme ray.

To verify the equivalence, if a sum in (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 (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.

For a proper cone $K\subset V$, assume:

1. $\mathrm{Aut}(K)=\{A\in \mathrm{GL}(V):AK=K\}$ acts transitively on $\mathrm{int}K$;

2. there is a positive-definite inner product $h$ such that

   $$K=K^{\ast h}:=\{y:h(y,x)\ge 0\text{ for all }x\in K\};$$

3. 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 $V_i$;

4. $K$ is strictly convex.

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 $\mathsf{S}$ to make its operational meaning explicit.

Let $K$ be a proper cone in $V$, with $m=\dim V\ge 3$. If $\mathsf{S}$ holds, there is a linear isomorphism $T:V\to \mathbb{R}\times {\mathbb{R}}^{m-1}$ such that

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 Koecher,Vinberg,FK. Irreducibility makes the algebra simple. The classification lists the one-dimensional algebra, the Hermitian matrix algebras over $\mathbb{R},\mathbb{C},\mathbb{H}$, spin factors, and the exceptional algebra of $3\times 3$ Hermitian octonionic matrices FK (Chapter V); see also JNW.

We now apply the strict-convexity hypothesis. In each matrix algebra of size $n\ge 3$, let $a=\mathrm{diag}(1,0,\dots ,0)$ and $b=\mathrm{diag}(0,1,0,\dots ,0)$. 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 FK (Chapter III). These cases violate (eq:strict).

The one-dimensional case is excluded by $m\ge 3$. The remaining cases are spin factors, including the $2\times 2$ Hermitian cases. Explicitly,

identifies their positive cones with $t\ge \sqrt{s^2+|z{|}^2}$, treating $z$ in real coordinates.

For completeness, the spin factor on $\mathbb{R}\oplus {\mathbb{R}}^{m-1}$ has product

A square is

which belongs to $L_m$. Conversely, if $a\ge \Vert b{\Vert }_2$, put

For $b\ne 0$, the element

squares to $(a,b)$; the case $b=0$ is immediate. Thus the cone of squares is exactly $L_m$.

Under the hypotheses of Theorem thm:mainpositive,

Proof. Theorem thm:network identifies sensitivity with strict reachability, (eq:orderfaithful) identifies reachability with the restricted cone order, and Theorem thm:representation identifies that cone with $L_m$.

The standard generated representations of Propositions prop:ball and prop:orthant, and every invertible affine change of either representation, fail $\mathsf{S}$. More generally, an exact generated representation with a finite edge-increment set cannot satisfy $\mathsf{S}$ in dimension at least three.

Proof. Both standard cones fail strict convexity. The orthant additionally fails irreducibility, while $K_{\diamond }$ fails interior homogeneity (Lemma 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 prop:finitestencil and Theorem thm:representation.

This excludes the stated 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 fig:bases).

A homogeneous strictly convex proper cone in dimension $m\ge 3$ is linearly isomorphic to $L_m$.

Proof. Ito and Lourenço classify homogeneous cones with strictly convex closure: up to linear isomorphism the possibilities are $\{0\}$, ${\mathbb{R}}_{\ge 0}$, ${\mathbb{R}}_{\ge 0}^2$, and Lorentz cones ItoLourenco (Theorem 7). Apply their result to the homogeneous cone $K$, which equals its closure. Full dimension $m\ge 3$ excludes the first three possibilities.

Thus self-duality and irreducibility are not logically necessary for cone selection once homogeneity and strict convexity are assumed. We retain the proof under $\mathsf{S}$ 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.

For a proper cone $K$ of dimension at least two, if $\mathrm{Aut}(K)$ acts transitively on $\partial K\setminus \{0\}$, then $K$ is strictly convex. Consequently, in Theorem thm:representation its strict-convexity clause may be replaced by boundary transitivity.

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.

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

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 $A\in \mathrm{Aut}(K)$ such that

Proof. Finite-dimensional linear algebra gives the stated positive operator $A$. Its self-adjointness implies

Both dual cones equal $K$, so $AK=K$.

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

Let $T(K)=L_m$, $m\ge 3$, and set

Then $Q_T$ is nondegenerate of signature $(1,m-1)$. For every other linear isomorphism $T^{\prime }$ with $T^{\prime }(K)=L_m$, there is $c>0$ such that $Q_{T^{\prime }}=c{Q}_T$.

Proof. Only uniqueness needs proof. Work in the $T$-coordinates and write the second quadratic form as

where $C$ is symmetric. It vanishes at every $(1,u)$, $\Vert u{\Vert }_2=1$, 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 $q=(-a)(-t^2+\Vert x{\Vert }_2^2)$. The second form is negative on the cone interior, so $a<0$.

The form obtained here satisfies the hypotheses of Theorem 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 $\mathsf{S}$.

Fix $m\ge 3$. Choose a family $(z_n{)}_{n\in \mathbb{Z}}$ dense in the open unit ball of ${\mathbb{R}}^{m-1}$, and a fixed spatial unit vector $e_1$. Take disjoint two-event chains $a_n◃b_n$, represented by

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 $L_m$, but no nonzero boundary displacement is realized (Figure fig:sparselorentz).

Proof. The intended difference is $(1,z_n)\in \mathrm{int}{L}_m$. 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 $L_m$: the closure of their convex hull in the slice $t=1$ is the closed unit ball. Thus $\mathsf{Aff}$ holds with $K=L_m$, which satisfies $\mathsf{S}$.

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.

There is a nondegenerate quadratic form $Q$, with $\{Q>0\}\ne \varnothing$, and a connected component $F\subset \{Q<0\}$ such that $K=\mathrm{cl}F$.

For a proper cone, $\mathsf{Q}$ implies the Lorentz-cone conclusion by Theorem 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 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.

There is a decomposition $V=\mathbb{R}{e}_0\oplus W$, a positive-definite inner product on $W$, and $e_0\in \mathrm{int}K$. For every unit $n\in W$ and $s\in \mathbb{R}$, the map

belongs to $\mathrm{Aut}(K)$.

A proper cone satisfying $\mathsf{B}$, with $\dim W\ge 1$, equals $\{t{e}_0+x:t\ge \Vert x{\Vert }_2\}$.

Proof. Positive multiples of $B_n(s)e_0$ fill the future timelike region. Closedness therefore gives $L_m\subseteq K$.

Suppose $v=(t,x)\in K\setminus L_m$. If $x=0$, then $t<0$, contradicting pointedness because $e_0\in K$. Otherwise set $a=\Vert x{\Vert }_2>0$, $n=x/a$. Then $t<a$, and invariance and scaling give $e^s{B}_n(s)v\in K$. As $s\to -\infty$,

This is a negative nonzero multiple of a vector already in $L_m\subseteq K$, again contradicting pointedness.

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 $\mathsf{B}$.

If a graph automorphism $\varphi$ lifts one of these symmetries, the OR network has the precise relabeling action

Equation (eq:or) is equivariant under this action. Producing such a $\varphi$, 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.

For an irreflexive transitive relation $\ll$, define

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.

Suppose:

1. every point belongs to an interval;

2. if $a_1\ll x$ and $a_2\ll x$, there exists $a$ with $a_1\ll a,a_2\ll a,a\ll x$;

3. if $x\ll b_1$ and $x\ll b_2$, there exists $b$ with $x\ll b,b\ll b_1,b\ll b_2$.

Then the intervals form a basis.

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.

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.

Proof. Use

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 $c=1$,

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.

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

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 ${\mathbb{R}}^d$, $d\ge 1$. The distinction between a discrete order and its possible continuum approximation is also fundamental in causal-set theory Bombelli.

Fix an event order $(E,\le )$, types, and typed primitives

including nullary constants. Terms are

An origin-annotated context is a finite map

The annotation records the capability's origin, not a claim of physical storage at every subsequent event. Judgments are $\Gamma {⊢}_{f}t:A$.

The rules are

and

A nullary operation has empty context; there is no contraction rule.

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

One has $\Gamma {⊢}_{f}t:A$ if and only if

Consequently, an unavailable context entry may occur only if its name is unused.

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.

The theorem excludes unavailable uses, not every context containing an unavailable name. It is also important to retain the direction:

Availability alone does not establish adequate transport resources for several competing uses. An implementation may additionally require path certificates and exclusive channel ownership.

In a cone representation, a used name originating at $e$ and used at $f$ satisfies

Under Theorem thm:representation, this difference is future causal for the resulting Lorentz form.

Proof. Apply Theorem thm:typing and the order-faithfulness condition (eq:orderfaithful).

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

There is an interpolating, past- and future-distinguishing causal-space structure on the smooth manifold ${\mathbb{R}}^3$ whose Alexandrov intervals form a basis for its usual topology, but which admits a chronological automorphism that is not differentiable.

Proof. Put

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

induces

The piecewise-linear stretch $h$ induces a chronological automorphism of $({\mathbb{R}}^3,\ll )$ that fails to be differentiable on the plane $x_1=0$ (Figure 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

Writing $S=\mathrm{cl}U=[-1,1{]}^3$ gives

Thus the map preserves the interval basis, but not smoothness.

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

A chronology $(E,\ll )$ has a Lorentzian realization if a bijection $\iota :E\to M$ to a smooth time-oriented Lorentzian manifold satisfies

For the next theorem, the manifold has dimension at least three and is both past and future distinguishing.

Let $(M,g)$ and $(M^{\prime },g^{\prime })$ 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:

for a smooth positive function $\Omega$.

Proof. This is the chronological reconstruction result of Malament, using the causal, differential, and conformal rigidity results associated with Hawking, King, and McCarthy Malament,HKM. The distinguishing formulation and its relation to causal bijections are discussed in ParrikarSurya.

Two realizations of the same chronology satisfying Assumption ass:lorentz, in the same dimension, are related by a conformal diffeomorphism.

Proof. Compose one realization map with the inverse of the other and apply Theorem thm:malament.

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 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 ParrikarSurya. The path topology in 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 AlexandrovOvchinnikova,Zeeman. They are not used to select a cone from scheduler consistency.

The selected conformal class determines null directions. For $g^{\prime }={\Omega }^{2}g$, the connections satisfy

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.

The logical dependencies are:

and

These are different conditional implications, not a chain in which each conclusion alone supplies the next set of hypotheses. Figure 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.

The following checklist records the proof obligations underlying the conditional results summarized in Figure fig:ladder.

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 prop:finitestencil; a Lorentzian limit would require an additional limiting construction rather than an exact finite-stencil identification.

A formal application would require:

1. full configuration and event-occurrence definitions;

2. a residual theorem validating swaps without collapsing occurrences;

3. a precise intervention and observation semantics;

4. a sensitivity–reachability or routing theorem;

5. a compositional probabilistic diamond;

6. an ownership theorem for simultaneously enabled operations;

7. the representation data $\mathsf{Aff}$, including its completion and faithfulness claims;

8. a sound implementation of causal-origin typing and transport;

9. record-preserving symmetry constructions supporting $\mathsf{S}$, its homogeneous strictly convex weakening, or $\mathsf{B}$, together with a proof that the polyhedral embeddings are not equivariant realizations of those symmetries.

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.

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:

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.

The resulting research program has identifiable tasks:

1. construct affine displacement data, or a justified affine completion, from a specified residual theory;

2. prove that operational sensitivity has the required routing and compositional properties;

3. derive homogeneous strictly convex influence geometry, or the explicit axiom $\mathsf{S}$, from the rewrite system rather than assume it;

4. distinguish symmetries of a completed cone from implementable symmetries of records;

5. prove boundary attainability and, where appropriate, Lorentzian manifold realizability;

6. establish transport-aware causal typing for actual programs.

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

its numerical slope changes under independent choices of spatial and temporal units. Selecting the Lorentz cone and calibrating physical units are different tasks.

and dependency note. The cone-selection step uses classical symmetric-cone theory and, for its weakening, the precise result of Ito and Lourenço. 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.

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