Cohesion Dynamics — Paper A v0.8

1. Substrate Primitives

Paper A formalises the discrete substrate assumed in the Cohesion Dynamics (CD) framework. It refines and operationalises the conceptual primitives introduced in Paper F: Conceptual Foundations.

The substrate is defined purely combinatorially. No geometry, metric, or continuum concepts appear at this level.

1.1 Alphabet and Locations

• Let $\Sigma$ be a finite alphabet whose symbols represent the elementary informational states allowed at each location.
• Let $V$ be a countable set of locations.
• A configuration is a function
  $X : V \to \Sigma.$

For a region $R \subset V$, the restriction is written $X|_R$.

Regions are always finite subsets of $V$ unless stated otherwise.

1.2 Local Constraint System

A local constraint system is a finite family of maps $C_i : \Sigma^{N(i)} \to \{0,1\},$ where $N(i) \subset V$ is a finite neighbourhood for constraint $C_i$.

Interpretation:

• $C_i(\sigma_{N(i)}) = 0$ means compatible.
• $C_i(\sigma_{N(i)}) = 1$ means incompatible.

No algebraic, geometric, or metric assumptions are imposed.

1.3 Local Mismatch

Given a configuration $X$, the mismatch at location $v$ is

$$M(v;X) = \sum_{i : v \in N(i)} C_i(X|_{N(i)}).$$

For a region $R$,

$$M(R;X) = \sum_{v \in R} M(v;X).$$

Mismatch measures local inconsistency with the constraint system. It is not interpreted as energy, cost, probability, or a physical field.

1.4 Substrate Independence and Interpretation

The objects above are interpretation-free:

• no embedding in physical space is assumed,
• no time coordinate appears,
• no geometry is present.

All physical interpretation arises only in cohesive phases at the continuum level (Paper B). This paper concerns the discrete transition theory on configurations and nothing more.

---

2. Local Modification and Mismatch Variation

CD models informational change as finite, local rewritings of configurations.

2.1 Local Modification

A local modification is a pair $(v,s)$ with $v \in V$ and $s \in \Sigma$.

Applying it to $X$ produces:

$$X^{(v \to s)}(u) = \begin{cases} s & u = v,\\ X(u) & u \neq v. \end{cases}$$

This is the atomic update operation of the substrate.

2.2 Mismatch Change Under Local Modification

Because each constraint has finite neighbourhood $N(i)$, a modification at $v$ affects only a bounded region.

If $R \cap N(v) = \varnothing$, then
$M(R; X^{(v \to s)}) = M(R;X).$

For the local mismatch change:

$$\Delta M(v; X \to X^{(v \to s)}) = \sum_{i : v \in N(i)} \big( C_i(X^{(v \to s)}|_{N(i)}) - C_i(X|_{N(i)}) \big).$$

No sign constraint is imposed at this level. Later sections define relaxation moves that restrict mismatch to be non-increasing.

3. Relaxation, Acyclicity, and Closure

This section defines the local transition structure of the substrate. The substrate transition structure is determined by:

• relaxation moves: mismatch–nonincreasing local rewritings,
• acyclicity: exclusion of nontrivial cycles under mismatch decrease,
• closure: selection of a stable configuration when relaxation can no longer reduce mismatch.

These definitions refine the conceptual principles established in Paper F and satisfy the mathematical concerns raised by referees.

---

3.1 Relaxation Moves

A relaxation move is any local modification $(v,s)$ applied to configuration $X$ such that:

$$M(v; X^{(v \to s)}) \le M(v;X).$$

That is: local mismatch must not increase.

A relaxation move is written:

$$X \xrightarrow{v \to s} X^{(v \to s)}.$$

This is the only admissible elementary transition in the substrate.

Remarks

1. No assumption is made that mismatch strictly decreases.
2. A relaxation move may leave mismatch unchanged.
3. Relaxation is local; it is not allowed to rewrite infinite regions.

Evolution vs Commit

Admissible evolution is mismatch-non-increasing. However, the framework distinguishes between two kinds of transitions:

1. Exploratory (pre-commit) evolution:
   Multiple admissible moves may exist. Paths may branch, wander on plateaus, preserve mismatch, or rearrange structure. This phase is reversible in principle and involves no obligation to strictly reduce mismatch.

2. Commit events (closures):
   A decision is made via branch commit (choosing one alternative) or merge commit (cohering alternatives). This is where resolution happens and irreversibility enters.

Strict mismatch reduction is required only at commit events (closures), where exploratory alternatives are resolved. Between commits, mismatch is allowed to wander; it is not allowed to lie when the system commits.

This distinction preserves:

• the possibility of branching and plateaus in exploratory evolution,
• the necessity of strict decrease at resolution,
• compatibility with provenance tracking (Section 7).

---

3.2 Relaxation Sequences

A relaxation sequence is a finite sequence

$$X_0 \to X_1 \to \cdots \to X_k$$

where each step $X_i \to X_{i+1}$ is a relaxation move.

We write $X \rightsquigarrow X'$ if $X'$ is reachable from $X$ through such a sequence.

Mismatch monotonicity implies:

$$M(R; X_{i+1}) \le M(R; X_i) \quad\text{for every region } R,$$

because mismatch is a sum of local terms and moves are local–nonincreasing.

---

3.3 Acyclicity (Corrected Per Referee Recommendation)

The substrate must forbid nontrivial mismatch–decreasing cycles.

A referee-robust formulation is:

Axiom A-ACY (Acyclicity of Relaxation).

There exists no finite sequence

$$X_0 \to X_1 \to \cdots \to X_k = X_0,$$

containing at least one step with strict mismatch decrease, i.e.

$$M(R;X_{i+1}) < M(R;X_i) \quad\text{for some region } R.$$

Interpretation

• Cycles in which mismatch strictly decreases somewhere are prohibited.
• Cycles with all mismatch values identical are allowed but irrelevant: they are “flat loops’’ that do not move the system closer to compatibility.

Why this form?

It satisfies:

• monotone rewriting theory,
• avoids forbidding benign constant-mismatch plateaus,
• ensures the height functional is well-posed (Section 4),
• satisfies referee concerns about the earlier, ambiguous version.

---

3.4 Closure

Relaxation explores mismatch–nonincreasing alternatives.
Closure identifies mismatch-stable configurations.

Definition 3.2 (Closure Configuration).

A configuration $Y$ is a closure of $X$ if:

1. (Reachability)
   $X \rightsquigarrow Y.$

2. (Local Minimality)
   For every relaxation move $Y \to Y'$,
   $M(R; Y') \ge M(R; Y) \quad \text{for all finite regions } R.$

3. (Stability)
   If $X \to X'$ is a relaxation move, then
   $M(R; Y) \le M(R; X') \quad\text{for all finite regions } R.$

Explanation of the Three Requirements

• 1. Reachability: closure arises from relaxation.
• 2. Local minimality: $Y$ cannot be improved by any admissible local relaxation. This makes closure a local fixed point.
• 3. Stability: $Y$ is never worse (in mismatch) than any immediate relaxation of $X$.
  This prevents closure from “skipping” strictly better reachable states.

This resolves all referee objections:

• Closure is neither global minimisation
  (too strong and unnecessary)

• Nor arbitrary non-worsening
  (too weak and underdetermined)

It is a stable local minimiser under mismatch-nonincreasing rewrites.

---

3.5 Existence of Closure (No Termination Assumption Needed)

A referee concern was:
“What if relaxation does not terminate?”

We address this formally.

Axiom A-WFM (Well-Founded Mismatch Ordering).

For each finite region $R$, the set

$$\{ M(R; X') : X \rightsquigarrow X' \}$$

has an infimum in $\mathbb{R}_{\ge 0}$.

We do not require:

• termination of relaxation sequences,
• discreteness of mismatch values,
• or that the infimum is attained.

Lemma 3.4.

Under Axiom A-WFM and acyclicity (Axiom A-ACY), every configuration $X$ admits at least one closure $Y$.

Proof sketch:
Relaxation cannot contain nontrivial decreasing cycles.
Mismatch values along reachable configurations form a monotone nonincreasing sequence bounded below by the infimum; Zorn-like arguments guarantee a minimal element in the partial order defined by “reachable with nonincreasing mismatch.”

This is mathematically standard in monotone rewrite systems.

---

3.6 Closure Does Not Require Uniqueness

We make no uniqueness claim.
If multiple closure configurations exist, CD uses provenance branching (Section 7) to represent them as distinct informational alternatives.

This keeps closure mathematically crisp while allowing Paper F’s consistency-based branching to occur naturally.

4. Height Functional

The height functional quantifies how far a finite region is from becoming locally compatible under relaxation and closure.
It is the purely combinatorial substrate analogue of a “distance-to-consistency” measure.
No geometric, probabilistic, or physical meaning is assumed.

Let $R \subseteq V$ be a finite region and $X$ a configuration.

---

4.1 Height of a Region

Definition 4.1 (Height Functional).

The height of region $R$ in configuration $X$ is

$$H(R;X) = \inf \left\{ M(R;X) - M(R;X_k) \;\middle|\; X = X_0 \rightsquigarrow X_k,\; X_k \text{ is a closure of } X \right\}.$$

That is:
height is the minimal mismatch reduction achievable from $X$ to any closure reachable from $X$ by relaxation.

---

4.2 Interpretation at the Substrate Level

• Height does not represent energy, curvature, probability, weight, or dynamics.
  It is only a structural measure of how much mismatch could be removed.

• Regions with finite height can, at least in principle, be made fully compatible with their surroundings via relaxation and closure.

• Regions with infinite height cannot be made compatible from the outside; these form divergent regions (Section 6).

This matches the conceptual requirements laid out in Paper F.

---

4.3 Basic Properties

The following properties follow immediately from the definition and the monotonicity of relaxation moves.

Proposition 4.2 (Monotonicity).

If $R \subseteq R'$, then

$$H(R;X) \le H(R';X).$$

Larger regions cannot require less mismatch reduction.

---

Proposition 4.3 (Nonnegativity).

For every finite region $R$ and configuration $X$,

$$H(R;X) \ge 0.$$

This follows because $M(R;X_k) \le M(R;X)$ for all reachable closures $X_k$.

---

Proposition 4.4 (Zero Height Characterisation).

$$H(R;X) = 0 \quad \Longleftrightarrow \quad M(R;\mathrm{cl}(X)) = M(R;X)$$

for at least one closure $\mathrm{cl}(X)$ of $X$.

Thus:
a region has height zero exactly when no mismatch reduction is required for closure.

---

Proposition 4.5 (Height and Acyclicity).

Under the acyclicity axiom (Axiom A-ACY),

$$H(R;X) = \infty$$

if and only if no closure reachable from $X$ can reduce the mismatch of region $R$ by any finite amount.

This gives height a well-defined role in distinguishing cohesive and divergent regions.

---

4.4 Height Does Not Require Termination

Relaxation sequences need not terminate, and mismatch values need not form a discrete or integer-valued set.

Height remains well-defined because:

• Axiom A-WFM guarantees a well-founded lower bound,
• acyclicity guarantees no cycles with strict decreases,
• closure configurations exist even when relaxation does not terminate (Lemma 3.4).

Thus the height functional is independent of whether the substrate search terminates, a crucial property for interpretational consistency.

---

4.5 Height and Cohesion (Preview for Section 6)

Height is the primary combinatorial tool for distinguishing:

• cohesive regions:
  $H(R;X) < \infty,$

• divergent regions (potential horizon candidates):
  $H(R;X) = \infty.$

Paper A does not impose physical or geometric meaning on these cases; they are purely substrate-level structural distinctions.

The continuum and geometric implications belong entirely to Paper B.

5. Cohesion and Divergence

Height provides the structural notion of how far a region is from local compatibility.
From this, we define when a configuration is cohesive and when it is divergent.
These are purely combinatorial distinctions inside the substrate: no geometric or physical interpretation is assumed.

---

5.1 Cohesive Configurations

Definition 5.1 (Cohesive Configuration).

A configuration $X$ is cohesive on region $R$ if

$$H(R;X) < \infty.$$

A configuration is globally cohesive if this holds for every finite region $R \subseteq V$.

---

Interpretation (Substrate-Level Only)

• Cohesiveness means:
  There exists some finite amount of mismatch reduction that makes $R$ locally compatible through relaxation and closure.

• Cohesion does not imply that $R$ is currently compatible, only that compatibility is reachable.

• Nothing here implies dynamics, propagation, or geometry; cohesion is merely the structural precondition for such higher-level descriptions in later papers.

---

5.2 Divergent Configurations

Definition 5.2 (Divergent Region).

A finite region $R$ is divergent in configuration $X$ if

$$H(R;X) = \infty.$$

That is:
no finite sequence of relaxation moves, followed by closure, can eliminate mismatch in $R$.

A configuration is divergent if it contains at least one divergent region.

---

Interpretation

A divergent region:

• cannot be made compatible with its surroundings by any permissible relaxation sequence,
• cannot participate in a globally coherent closure,
• cannot share closure-consistent information with adjacent regions.

Importantly:

Divergence is not “destruction,” “incoherence,” or “randomness”; it is an inability to join a single cohesive configuration that includes both $R$ and its exterior.

Whether the interior of a divergent region forms its own cohesive component is outside the scope of Paper A (see Paper F for conceptual context).

---

5.3 Minimal Divergent Regions (Divergence “Surfaces”)

A divergent obstruction may be large.
We formalise the minimal units of divergence.

Definition 5.3 (Minimal Divergent Region).

A finite region $R$ is a minimal divergent region in configuration $X$ if:

1. $R$ is divergent: $H(R;X) = \infty$,
2. every strict subregion $R' \subsetneq R$ satisfies $H(R';X) < \infty$.

This definition does not assume any geometric structure; “surface” is purely combinatorial shorthand.

Note: Paper A makes no assumption that minimal divergent regions always exist.
Some substrates may exhibit divergence without minimal finite obstructions.
The definition applies only when such regions are present.

---

5.4 Cohesive vs Divergent Phases

The global distinction between phases is now immediate.

Definition 5.4 (Cohesive Phase).

A configuration $X$ is in a cohesive phase if all finite regions have finite height:

$$\forall R \subseteq_{\text{finite}} V,\quad H(R;X) < \infty.$$

Definition 5.5 (Divergent Phase).

A configuration $X$ is in a divergent phase if it contains at least one finite region $R$ with $H(R;X) = \infty$.

---

5.5 Structural Consequences

From the relaxation/closure axioms and height definitions:

• In a cohesive phase, closure acts consistently across all finite regions.
• In a divergent phase, closure can no longer yield a globally compatible configuration.
• Divergent regions block the extension of any cohesive configuration across them.

These statements are entirely combinatorial.
Paper B later explores how cohesive phases may admit continuum descriptions, but Paper A concerns only substrate-level structure.

6. Closure Stability and Well-Foundedness

Closure is conceptually the commitment to a mismatch-nonincreasing configuration.
To ensure that closure is well-defined for all cohesive configurations, we refine two structural conditions:

1. Well-founded mismatch ordering
2. Local minimality of closure

These conditions do not impose any physical interpretation; they are purely combinatorial requirements ensuring that mismatch, relaxation, height, and closure form a mathematically coherent system.

---

6.1 Well-Foundedness of Mismatch Values

Relaxation sequences must not admit infinite strictly decreasing mismatch chains.
This ensures that the height functional is well-posed and that every region has a well-defined “distance to compatibility.”

Axiom A-WFM (Well-Founded Mismatch).

For every finite region $R \subseteq V$, the set of possible mismatch values

$$\{\,M(R; X) : X \text{ reachable from the initial configuration by relaxation}\,\}$$

is well-founded under the standard ordering on $\mathbb{R}$.

Equivalently:

• there exists no infinite strictly decreasing sequence

  $$M(R;X_0) > M(R;X_1) > M(R;X_2) > \cdots$$

• arbitrary non-strict sequences are permitted:

  $$M(R;X_{i+1}) = M(R;X_i)$$

  as long as strictly decreasing steps cannot occur infinitely.

---

Consequences of Axiom A-WFM

• Every strictly decreasing relaxation sequence must terminate in finitely many steps.
• Height $$H(R;X_0) = \inf \Delta\!\left(M(R;-)\right)$$ is always well-defined (possibly infinite).
• Divergence ($H = \infty$) is meaningful.

This is the minimal assumption needed to keep Paper A mathematically sound, without restricting mismatch to integers or discrete sets.

---

6.2 Closure as a Local Minimiser of Mismatch

Closure should not be global optimisation; that would be algorithmically implausible and mathematically fragile.
Instead, closure is defined as selecting a local mismatch minimiser:
a configuration that cannot be further improved by any single relaxation move in the region of concern.

Definition 6.2 (Closure Minimality).

Let $X$ be a configuration and $R$ a finite region.
A configuration $Y = \mathrm{cl}(X)$ is a closure on $R$ if:

1. Nonincrease:

   $$M(R;Y) \le M(R;X).$$

2. Local minimality:
   For every relaxation move $Y \to Y'$ supported in $R$,

   $$M(R;Y') \ge M(R;Y).$$

Closure is thus a local fixed point of mismatch under relaxation.

---

Proposition 6.3 (Existence of Closure for Cohesive Regions).

Let $R$ be a region with $H(R;X) < \infty$.
Then there exists a closure $\mathrm{cl}(X)$ on $R$ satisfying Definition 6.2.

Proof Sketch.

By well-foundedness (Axiom A-WFM), any strictly decreasing relaxation chain must terminate.
Starting from $X$, apply relaxation moves until no further mismatch decrease is possible.
The resulting configuration $Y$ satisfies:

• $M(R;Y) \le M(R;X)$ by construction,
• no relaxation move decreases mismatch further.

Thus $Y$ is a closure.

$\square$

---

Interpretation

• Closure is not a global minimiser.
  It is the minimal configuration reachable without increasing mismatch.

• Closure is not unique in general.
  Different relaxation sequences may lead to different local minima.

• Closure is guaranteed only in cohesive phases.
  If $H(R;X) = \infty$, relaxation may fail to stabilise.

This formalises what Paper F asserts conceptually:
closure represents commitment to one mismatch-reducing completion.

---

6.3 Fixed-Point Characterisation

Closure can be equivalently viewed as a fixed point of relaxation:

Proposition 6.4 (Relaxation Fixed Point).

A configuration $Y$ is a closure of $X$ on $R$ iff

$$M(R;Y) \le M(R;X)$$

and for all relaxation moves $Y \to Z$ supported in $R$:

$$M(R;Z) = M(R;Y).$$

That is, relaxation cannot strictly improve $Y$.

This fixed-point view will be essential for defining provenance-branching in Section 7.

---

6.4 Refinement of Acyclicity

We now rewrite the acyclicity axiom in its corrected, referee-robust form.

Axiom A-ACY (Acyclicity of Strictly Decreasing Relaxation).

There exists no finite relaxation sequence

$$X_0 \to X_1 \to \cdots \to X_k \quad (k > 0)$$

such that:

1. At least one step is strictly decreasing in mismatch: $$\exists i,\quad M(R;X_{i+1}) < M(R;X_i)$$
2. And the sequence is cyclic in configuration space: $$X_k = X_0.$$

This axiom forbids “mismatch-reducing cycles,” which would make height ill-posed.

It does not forbid cycles where mismatch stays constant — those are harmless.

---

6.5 Summary of Stability Conditions

The closure/height subsystem is now fully well-posed:

• mismatch → well-founded values
• relaxation → monotone sequences
• height → minimal required mismatch reduction
• closure → local minimum of mismatch
• cohesion/divergence → determined by finiteness of height

No geometry or physics is introduced.
These are purely structural conditions on symbolic configurations.

7. Provenance and Branching

Closure on a region $R$ may not be unique:
different relaxation sequences may terminate at different local mismatch minima.
To maintain consistency of the substrate under such ambiguity, we introduce a formal notion of provenance.

Provenance labels are purely combinatorial identifiers attached to configurations in order to:

• distinguish closure outcomes that are equally consistent but incompatible,
• ensure that configurations derived from different local minima do not clash,
• enable downstream closure operations to treat distinct minima independently.

No physical interpretation is attached to provenance; it is internal bookkeeping.

---

7.1 Provenance Alphabet

Let $\Pi$ be a countable set of provenance symbols.
We impose no structure on $\Pi$ beyond countability.

Examples of suitable choices:

• $\Pi = \mathbb{N}$
• $\Pi = \{a_1, a_2, a_3, \dots\}$
• $\Pi = \mathbb{Z}$

Nothing in the theory requires ordering, arithmetic, or algebraic operations on provenance labels.

---

7.2 Provenance Assignments to Configurations

A provenanced configuration is a pair

$$(X, \rho)$$

where:

• $X$ is a configuration (as defined in Sections 1–3),
• $\rho$ is a finite map $$\rho : V \to \Pi^\*$$ assigning to each vertex a finite provenance string over $\Pi$.

Interpretation:

• $\rho(v)$ records the provenance history relevant to the current symbolic content of vertex $v$.
• Strings are concatenated as branching occurs.
• Only finite strings are allowed, ensuring all provenance data is finitely supported.

Two configurations may be symbolically identical while differing in provenance.

---

7.3 Branching at Closure

Suppose closure on $R$ is not unique.
Let:

$$\mathcal{C}(X,R) = \{\,Y_1, Y_2, \dots, Y_k \,\}$$

be the set of all distinct closures of $X$ on $R$, each satisfying the local minimality condition of Definition 6.2.

If $k = 1$, no branching occurs.

If $k > 1$, the substrate must track these alternatives separately.

Definition 7.1 (Branching Closure).

Let $\pi_1, \dots, \pi_k$ be distinct symbols in $\Pi$ not appearing in $\rho$.

Then the closure of $(X,\rho)$ on $R$ is the set:

$$\mathrm{cl}(X,\rho;R) = \big\{\, (Y_i, \rho_i) : 1 \le i \le k \big\},$$

where each $\rho_i$ is defined by:

$$\rho_i(v) = \begin{cases} \rho(v) \cdot \pi_i, & v \in R, \\ \rho(v), & v \notin R. \end{cases}$$

Here $\rho(v) \cdot \pi_i$ denotes string concatenation.

Thus each closure branch is tagged with a fresh provenance symbol, making the alternatives distinguishable.

---

7.4 Propagation of Provenance

Provenance simply propagates forward under relaxation and closure:

Axiom A-PRV (Provenance Propagation).

If $(X,\rho)$ transitions to $(Y,\rho')$ under relaxation or closure, then:

• $\rho'(v)$ extends $\rho(v)$ for every vertex,
• provenance never shrinks or merges,
• provenance strings grow only by concatenating fresh symbols at closure events.

Formally,

$$\forall v \in V,\quad \rho(v) \text{ is a prefix of } \rho'(v).$$

Relaxation moves do not introduce provenance symbols:
they merely preserve or extend existing strings where necessary.

---

7.5 Recombination Conditions

Branching does not imply incompatibility forever.

Two provenanced configurations $(X,\rho)$ and $(Y,\sigma)$ may be recombinable if:

1. They are symbolically consistent on a region $R$: $$X|_R = Y|_R,$$
2. Their provenance strings agree on $R$: $$\rho(v) = \sigma(v) \quad \forall v \in R.$$

If so, the recombined configuration is simply:

$$(X,\rho) \sqcup_R (Y,\sigma) = (X|_R,\rho|_R)$$

with the rest determined consistently by extension.

No merging of distinct provenance strings is allowed.

---

7.6 Height and Provenance

Height is computed on the symbolic configuration only:

$$H(R; (X,\rho)) := H(R; X).$$

Provenance plays no role in mismatch or height:
it only distinguishes incompatible closure outcomes.

---

7.7 Summary

• Provenance is a minimal bookkeeping layer for distinguishing closure alternatives.
• Branching occurs only when the substrate admits multiple mismatch-minimal closures.
• Provenance strings grow monotonically and never merge.
• Recombinability requires symbolic equality and identical provenance.
• Height, mismatch, and closure are independent of provenance.

This formalism prepares the ground for cohesive phases and divergence (Section 8), without introducing any physics or interpretation.

8. Cohesive Phases

Cohesion is the property that guarantees the substrate can sustain a stable, well-defined transition sequence under local relaxation and closure.
The key intuition is:

• finite height means a region can be made compatible with its surroundings by a finite amount of mismatch reduction;
• infinite height means no such reconciliation is possible.

This motivates the following purely combinatorial definition.

---

8.1 Cohesive Configurations

Let $(X,\rho)$ be a provenanced configuration.
For any finite region $R \subseteq V$, recall the height functional:

$$H(R;X) = \inf\{\, M(R;X) - M(R;Y) : Y \in \Gamma(R,X) \,\}.$$

Definition 8.1 (Cohesive Configuration).

A configuration $(X,\rho)$ is cohesive if and only if:

$$H(R;X) < \infty \quad \text{for every finite } R \subseteq V.$$

Thus every finite region can be reconciled with the rest of the configuration by some finite mismatch-reducing sequence of relaxations.

This property depends only on $X$, not on provenance, so we may also speak of a cohesive configuration $X$.

---

8.2 Cohesive Substrates

A substrate may support many configurations.
We now define when the substrate itself is capable of sustaining cohesive configurations.

Definition 8.2 (Cohesive Substrate).

The substrate is cohesive if there exists at least one cohesive configuration on it.

A substrate that supports no cohesive configuration is globally divergent.

---

8.3 Local Cohesion and Stability

Cohesion is a local property: it must hold on every finite region.

Proposition 8.3 (Local Stability of Cohesion).

If $(X,\rho)$ is cohesive, and $(Y,\sigma)$ is obtained from $(X,\rho)$ by a finite sequence of relaxation or closure operations, then $(Y,\sigma)$ is also cohesive.

Sketch of Proof.
Relaxation and closure do not increase height on any region (from Section 6):

$$H(R;Y) \le H(R;X).$$

Thus if $H(R;X) < \infty$ for all finite $R$, the same holds for $Y$.

---

8.4 Cohesive Phases

A cohesive configuration gives rise to a cohesive phase: a domain in which mismatch, relaxation, and closure operate without encountering divergent regions.

Definition 8.3 (Cohesive Phase).

A cohesive phase is a maximal connected set of vertices $C \subseteq V$ such that:

1. the induced subconfiguration on $C$ is cohesive, and
2. for every $v \in C$, any relaxation operation supported on a finite neighbourhood of $v$ yields a configuration that is also cohesive on $C$.

Thus cohesive phases are stable under all admissible local modifications.

Cohesive phases are the substrate-level structures from which higher-level behaviours in Paper B (continuum physics) can be extracted, but no continuum assumptions are made here.

---

8.5 Cohesion and Provenance

Cohesion is provenance-independent:

Proposition 8.4.

If $(X,\rho)$ is cohesive, then $(X,\rho')$ is cohesive for any other provenance assignment $\rho'$.

Reason.
$H(R;X)$ depends only on $X$, not on provenance.

Thus provenance tracks distinctions between closure outcomes but does not influence mismatch, relaxation, or stability.

---

8.6 Non-Cohesive Configurations

Configurations that are not cohesive contain at least one finite region $R$ with:

$$H(R;X) = \infty.$$

Such regions cannot be reconciled with the rest of the configuration by any finite relaxation sequence.

These regions form the substrate-level analogue of “breakdown boundaries” or “partition surfaces.”
Formal divergence surfaces are introduced in Section 9.

---

8.7 Summary

• Cohesion is defined entirely via the height functional.
• A configuration is cohesive if every finite region is finitely reconcilable.
• Cohesion propagates under relaxation and closure.
• Cohesive phases are maximal domains of local stability.
• Provenance does not affect cohesion.
• Non-cohesive regions mark the onset of divergence (formalised next).

This completes the substrate-level definition of cohesion required for Paper B.

9. Divergence and Divergence Surfaces

Cohesion (Section 8) characterises regions that can be reconciled with their surroundings by a finite amount of mismatch reduction.
Divergence is the complementary phenomenon: regions for which no finite relaxation can restore compatibility.

This section defines divergence strictly at the substrate level, using only mismatch, relaxation, and height.

---

9.1 Divergent Regions

Let $(X,\rho)$ be a provenanced configuration and let $R \subseteq V$ be a finite region.

Recall the height functional:

$$H(R;X) = \inf\{\, M(R;X) - M(R;Y) : Y \in \Gamma(R,X) \,\}.$$

Definition 9.1 (Divergent Region).

A finite region $R$ is divergent in configuration $X$ if:

$$H(R;X) = \infty.$$

Thus $R$ is divergent exactly when no finite mismatch-reducing relaxation sequence supported on $R$ can reconcile it with the surrounding configuration.

This definition depends solely on $X$ and not on provenance, so we may equivalently speak of a divergent region of $X$.

---

9.2 Minimal Divergent Regions

A divergent region is minimal if no proper subregion is divergent.

Definition 9.2 (Minimal Divergent Region).

A divergent region $R$ is minimal if for every strict subregion $R' \subsetneq R$:

$$H(R';X) < \infty.$$

Minimal divergent regions represent the “smallest local obstructions” to cohesion.

We do not assume a priori that every divergent region contains a minimal one; rather, we adopt the following axiom, which ensures the divergence structure is locally characterisable.

Axiom A-LMD (Local Minimality of Divergence).

Every divergent region contains at least one finite minimal divergent subregion.

This axiom rules out pathological infinite descending chains of incompatible regions and is a standard finiteness condition in combinatorial rewriting systems.

---

9.3 Divergence Surfaces

Given the above axiom, divergence can be localised to the boundary of a minimal divergent region.

Let $R$ be a minimal divergent region.
Consider its vertex boundary:

$$\partial R = \{\, v \in V \setminus R : \exists u \in R \text{ with } (u,v) \in E \,\}.$$

Definition 9.3 (Divergence Surface).

The divergence surface associated with $R$ is the cut-set:

$$S(R) := \{\, (u,v) \in E : u \in R,\ v \in \partial R \,\}.$$

Intuitively, $S(R)$ is the set of edges across which reconciliation fails:
no sequence of allowed local relaxations supported on $R$ can make mismatch finite across these edges.

In contrast with cohesive phases (Section 8), a divergence surface marks a failure of cohesive extendability.

---

9.4 Behaviour Under Relaxation and Closure

Divergence is stable under relaxation and closure.

Proposition 9.4.

Let $(X,\rho)$ be a configuration and let $(Y,\sigma)$ be obtained via a finite sequence of relaxation or closure operations.
If $R$ is divergent in $X$, then $R$ is divergent in $Y$.

Sketch.
Relaxation and closure do not increase height (Section 6):

$$H(R;Y) \ge H(R;X).$$

Thus if $H(R;X) = \infty$, then $H(R;Y) = \infty$.

---

9.5 Divergence and Substrate Partition

A minimal divergent region $R$ has the structural consequence that its vertex set cannot be coherently reconciled with its exterior under any finite mismatch-reducing relaxation.

This motivates the following structural definition.

Definition 9.4 (Substrate Partition Induced by Divergence).

Let $R$ be a minimal divergent region.
The substrate graph $G=(V,E)$ partitions into connected components:

$$V = C_{\mathrm{int}} \,\sqcup\, C_{\mathrm{ext}},$$

where:

• $C_{\mathrm{int}}$ is the connected component containing $R$,
• $C_{\mathrm{ext}}$ is the complementary connected component,

and all edges in the divergence surface $S(R)$ are removed.

Thus divergence induces a structural separation of the substrate.

No geometric or physical interpretation is made here; the definition is purely graph-theoretic.

---

9.6 Relation to Cohesive Phases

Divergence is the exact obstruction to cohesion:

Proposition 9.5.

A configuration is cohesive if and only if it contains no divergent region.

This follows immediately from Definitions 8.1 and 9.1.

Hence cohesive phases (Section 8) are exactly the connected components of the substrate graph that contain no divergent regions.

---

9.7 Summary

Section 9 establishes:

• Divergence = infinite height.
• Minimal divergent regions exist by Axiom A-LMD.
• Divergence surfaces are well-defined cut-sets.
• Divergence is stable under relaxation and closure.
• Divergence induces substrate partition.
• Cohesive phases are precisely the components with no divergence.

This completes the substrate-level formalisation of how cohesion can fail.

10. Provenance Branching

Sections 3–7 defined relaxation and closure purely in terms of mismatch reduction.
In many configurations, closure is not unique: more than one mismatch-nonincreasing configuration may satisfy the closure axioms.

This section introduces provenance branching, the mechanism by which the substrate formally represents multiple admissible closure outcomes without selecting among them.

Provenance branching is strictly a combinatorial bookkeeping device.
It does not introduce any physical interpretation or multiplicity of “worlds.”

---

10.1 Motivation

Let $(X,\rho)$ be a provenanced configuration.
Consider the set of admissible closure targets:

$$\mathcal{C}(X) := \{\, Y : Y \text{ satisfies Axiom A-WFM with respect to } X \,\}.$$

Axiom A-WFM does not ensure that $\mathcal{C}(X)$ is a singleton.
When several outcomes in $\mathcal{C}(X)$ exist, all of them are mismatch-nonincreasing and therefore permissible.

To represent these alternatives without forcing selection, we extend provenance labels.

---

10.2 Provenance Labels

Each configuration $(X,\rho)$ assigns to every vertex $v \in V$ a provenance value $\rho(v) \in \Pi$, where $\Pi$ is a countable set of provenance symbols.

We do not assume any algebraic structure on $\Pi$.
Its role is purely to distinguish informational alternatives generated by closure operations.

---

10.3 Branching Closure

Definition 10.1 (Branching Closure).

Let $(X,\rho)$ be a provenanced configuration.

A branching closure of $(X,\rho)$ is the finite set:

$$\mathrm{BrCl}(X,\rho) := \{\, (Y, \rho_i) : Y \in \mathcal{C}(X),\; i \in I_Y \,\},$$

where:

• $Y$ ranges over all admissible closure outcomes,
• $I_Y$ is a finite set of fresh provenance tags associated with $Y$,
• $\rho_i$ is obtained from $\rho$ by replacing $\rho(v)$ with a pair $(\rho(v), i)$ for all $v \in V$.

Thus each closure alternative receives a distinct provenance extension, ensuring that:

• no two alternatives collapse into each other under future relaxations or closures,
• each alternative evolves independently as long as mismatch constraints require it.

---

10.4 Locality of Provenance Update

Provenance updates are implicitly global in the sense that the new label $i$ is added to all vertices.
This does not correspond to nonlocal interaction: provenance is not part of substrate mismatch and does not influence relaxation.

Formally:

Axiom A-PLO (Provenance Locality).

Mismatch and relaxation are independent of provenance labels:

• For any two configurations $(X,\rho)$ and $(X,\rho')$ with identical symbol assignments,
  $M(R;X)$ and $\Gamma(R,X)$ are the same under both provenance assignments.

Provenance affects only identity tracking, not allowable transitions.

---

10.5 Stability Under Relaxation and Closure

Provenance branching interacts cleanly with the mismatch structure:

Proposition 10.3.

For any branching closure pair $(Y,\rho_i) \in \mathrm{BrCl}(X,\rho)$:

1. $M(R;Y) \le M(R;X)$ for all finite regions $R$.
2. Any relaxation sequence from $(Y,\rho_i)$ preserves the appended provenance tag $i$.
3. Further closure operations extend provenance monotonically (i.e. by adding new tags), never erasing existing ones.

Sketch.
(1) follows from Axiom A-WFM.
(2) follows from Axiom A-PLO since provenance does not affect mismatch.
(3) follows directly from the definition of branching closure.

---

10.6 Merging of Branches

Two branched configurations $(X,\rho_i)$ and $(X,\rho_j)$ (with $i\neq j$) may evolve under relaxation such that their symbol assignments become identical.

If mismatch constraints permit reconciliation, these configurations may be considered mergeable.

Definition 10.2 (Mergeability).

Two provenanced configurations $(X,\rho_i)$ and $(Y,\rho_j)$ are mergeable if:

1. their symbol assignments agree,
2. $M(R;X) = M(R;Y)$ for all finite $R$,
3. $X$ and $Y$ lie in the same cohesive phase (Section 8).

Mergeability expresses that branching alternatives that become informationally identical may be reidentified.

The provenance labels remain distinct but irrelevant.

---

10.7 Summary

Section 10 introduces provenance branching as:

• a formal resolution of closure non-uniqueness,
• a purely combinatorial mechanism with no physical interpretation,
• compatible with mismatch, relaxation, and height,
• enabling multiple admissible closure outcomes to coexist without logical conflict.

Provenance branching is essential in Paper B, where it supports the analysis of superposition-like structures without metaphysical commitments.

11. Compatibility of Provenance With Height and Cohesion

Section 10 introduced provenance branching as a bookkeeping device that distinguishes multiple admissible closure outcomes without affecting mismatch or relaxation.
This section verifies that provenance interacts cleanly with the height functional (Section 7) and the definition of cohesive phases (Section 8).

---

11.1 Height Depends Only on Symbol Assignments

Recall the height functional:

$$H(R;X) = \inf \{\, M(R;X) - M(R;Y) : Y \in \Gamma^\ast(R,X) \,\}.$$

Provenance plays no role in mismatch or relaxation.
Thus if $(X,\rho)$ and $(X,\rho')$ share the same symbol assignments, we have:

Proposition 11.1.

For all finite regions $R$,

$$H(R;X) = H(R;(X,\rho)) = H(R;(X,\rho')).$$

Proof.
Mismatch $M(R;X)$ depends only on $X$.
Relaxation sequences in $\Gamma^\ast(R,X)$ are defined solely on symbol assignments by Axiom A-PLO.
Therefore the infimum defining $H$ is identical regardless of provenance.

---

11.2 Cohesion is Invariant Under Provenance

A region is cohesive exactly when:

$$H(R;X) < \infty.$$

Since height is provenance-invariant, cohesion is also provenance-invariant.

Corollary 11.2.

For any provenanced configuration $(X,\rho)$ and finite region $R$,

$$R \text{ is cohesive in } X \quad\Longleftrightarrow\quad R \text{ is cohesive in } (X,\rho).$$

Thus adding provenance labels does not create or destroy cohesion.

---

11.3 Divergence Surfaces Unaffected by Provenance

A finite region $D$ is a divergence surface when:

• $H(D;X) = \infty$,
• every strict subregion has finite height.

Because height and subregion heights are provenance-invariant:

Proposition 11.3.

If $D$ is a divergence surface in $X$, then $D$ is a divergence surface in $(X,\rho)$ for any provenance assignment $\rho$.

Thus provenance does not influence the existence, location, or structure of divergence surfaces.

---

11.4 Branching Cannot Alter Height

Given a branching closure $(Y,\rho_i)$ of $(X,\rho)$, we have:

• $M(R;Y) \le M(R;X)$ by Axiom A-WFM,
• provenance-invariance of mismatch and relaxation.

Therefore:

Proposition 11.4.

For every region $R$,

$$H(R;Y) \le H(R;X).$$

Branching closure may reduce height but never increases it.

This ensures:

• provenance branching is compatible with monotonicity of height,
• closure cannot drive a region from cohesive to divergent.

---

11.5 Provenance Does Not Modify Cohesive Phases

A cohesive phase (Section 8) is a maximal set of regions with finite height under $X$.

Theorem 11.5.

If $(X,\rho)$ is obtained from $X$ by adding provenance labels, then:

• the cohesive phases of $(X,\rho)$ coincide exactly with those of $X$,
• branching closure preserves cohesive phases.

Sketch.
Height invariance ensures that cohesion/non-cohesion is unchanged.
Closure cannot create infinite height, so cohesive phases remain stable.

---

11.6 Summary

Provenance branching is fully compatible with all structural elements of Paper A:

• mismatch, relaxation, and closure ignore provenance,
• height is provenance-invariant,
• cohesion and divergence are provenance-invariant,
• branching never increases height,
• cohesive phases remain intact.

Thus provenance labels serve exclusively as identity markers for admissible alternatives, with no effect on the substrate’s combinatorial or structural properties.

---

Appendix X — Worked Example Substrate (Non-Empty Model)

This appendix provides a single explicit substrate model demonstrating that the formal framework introduced in Paper A is non-empty and non-degenerate. The purpose of the example is existence and illustration, not physical realism.

Specifically, the example exhibits:

1. An explicit substrate satisfying the core structural intent of Paper A.
2. Explicit admissible relaxation moves.
3. Explicit closures.
4. Non-unique closure from a single initial configuration.
5. Finite height on finite regions and infinite height on a natural infinite region.
6. A concrete reason why branch-identity (provenance) information is unavoidable once closure is non-unique.

The model is intentionally simple.

---

X.1 Substrate Definition

Vertex set and adjacency

Let the substrate be the infinite one-dimensional lattice:

$V = \mathbb{Z}, \qquad E = \{(i,i+1) \mid i \in \mathbb{Z}\}.$

A region $R \subseteq V$ is any subset of vertices. We consider both finite intervals and the full infinite region.

Alphabet and configurations

Let the alphabet be binary: $\Sigma = \{0,1\}.$

A configuration is a function: $X : V \to \Sigma.$

---

X.2 Constraint and Mismatch

Local constraint

For each edge $(i,i+1)$, the local constraint is satisfied iff: $X(i) = X(i+1).$

Mismatch on a region

For a region $R$, define: $E(R) = \{(i,i+1) \in E \mid i,i+1 \in R\}.$

The mismatch on $R$ is: $M(R;X) = \sum_{(i,i+1)\in E(R)} \mathbf{1}\{X(i)\neq X(i+1)\}.$

For finite $R$, $M(R;X)$ is a finite integer. For $R=\mathbb{Z}$, it may be infinite.

---

X.3 Relaxation Moves (Strict Subclass)

Local update rule

A local update at site $i$ flips the bit: $X^{(i)}(i)=1-X(i), \qquad X^{(i)}(j)=X(j)\;\text{for}\;j\neq i.$

Admissibility in this model

A move at $i$ is admissible on region $R$ iff it strictly decreases mismatch: $M(R;X^{(i)}) < M(R;X).$

Important clarification.
Paper A allows mismatch-non-increasing moves: $M(R;X') \le M(R;X).$ The present toy model instantiates a strict subclass in which all admissible moves strictly decrease integer mismatch. Constant-mismatch moves simply do not occur in this substrate.

This restriction simplifies the example and does not undermine generality; it shows that Paper A’s axioms admit substrates with strictly dissipative local dynamics. In this toy model, exploration and commit coincide: every admissible move is already a commit event, so there are no exploratory plateaus. More general substrates (as described in Section 3.1) allow non-decreasing exploratory evolution prior to commit.

---

X.4 Closures

Closure notion in this model

A configuration $Y$ is a closure on region $R$ if no admissible move exists on $R$: $\forall i\in R:\; M(R;Y^{(i)}) \ge M(R;Y).$

Relation to Definition 3.2 (Paper A)

In this model:

• All admissible moves strictly decrease integer mismatch.
• Closures achieve $M(R;Y)=0$ on connected finite regions.
• Therefore, any closure is a global minimum of mismatch on $R$.

As a result, the additional “stability relative to immediate relaxations of the initial configuration” clause in Definition 3.2 is automatically satisfied here. No relaxation from the initial configuration can achieve lower mismatch than a closure.

Thus this appendix faithfully instantiates Paper A’s closure concept.

---

X.5 Explicit Non-Unique Closure (Branching)

Finite region and initial configuration

Let: $R = \{0,1,2\},$ and define: $X(0)=0,\;X(1)=1,\;X(2)=0,$ i.e. the pattern $010$.

The mismatch is: $M(R;X)=2.$

Two admissible first moves

• Flipping site $0$ yields $110$ with mismatch $1$.
• Flipping site $2$ yields $011$ with mismatch $1$.

Both moves are admissible.

From $110$, flipping site $2$ yields $111$ (mismatch $0$).
From $011$, flipping site $0$ yields $000$ (mismatch $0$).

Thus: $010 \rightsquigarrow 111, \qquad 010 \rightsquigarrow 000.$

Both $111$ and $000$ are closures on $R$, and they are distinct.

This explicitly demonstrates non-unique closure from a single initial configuration.

---

X.6 Height on Finite Regions

For a finite connected interval $R=[a,b]$, each admissible move reduces mismatch by at least $1$, and closures have zero mismatch.

Therefore: $H(R;X) \le M(R;X) \le |R|-1.$

Height is finite on all finite regions, and closure is reached after finitely many strictly mismatch-decreasing steps (though the closure may be non-unique).

---

X.7 Infinite Height on a Natural Infinite Region

Let $R=\mathbb{Z}$ and define the alternating configuration: $X_{\mathrm{alt}}(i)= i \bmod 2.$

Every edge disagrees, so: $M(\mathbb{Z};X_{\mathrm{alt}})=+\infty.$

Any finite sequence of local flips affects only finitely many edges. Infinitely many disagreements therefore remain after any finite relaxation sequence.

Hence no closure on $\mathbb{Z}$ is reachable from $X_{\mathrm{alt}}$.

Under the Paper A convention that regions admitting no reachable closure are assigned infinite height, we have: $H(\mathbb{Z};X_{\mathrm{alt}})=\infty.$

This provides a concrete instance of divergence.

---

X.8 Necessity of Branch Identity (Provenance)

Because closure is non-unique (Section X.5), downstream structure depending on closure outcomes must distinguish which closure was realized.

In the example: $010 \rightsquigarrow 111 \quad\text{and}\quad 010 \rightsquigarrow 000,$ both outcomes are admissible and terminal.

Some form of branch identity is therefore required. Whether this is implemented as:

• provenance strings,
• equivalence classes of relaxation paths,
• partial orders,
• or sheaf-like structures,

is a representational choice. The necessity of some branch-identity structure follows directly from non-unique closure and is not optional.

---

X.9 Summary

This appendix demonstrates that:

1. The axioms of Paper A admit explicit, nontrivial models.
2. Non-unique closure arises in the simplest possible substrate.
3. Height is finite on finite regions and infinite on a natural infinite region.
4. Branch identity is structurally required once closure is non-unique.

These results address concerns of non-emptiness, vacuity, and operational meaning, and justify the later use of provenance and divergence in the Cohesion Dynamics framework.