Cohesion Dynamics: A Foundational Program for Emergent Physics (Paper F v1.4)

Author: Darrell Tunnell
Revision: v1.4
Series: F-series (Foundational Postulates and Priority Claims)
Epistemic Role: Ontological commitments and starting assumptions
Description: Priority claims and conceptual foundations for informational-constraint ontology

1. Introduction

Cohesion Dynamics (CD) proposes a substrate-independent conceptual framework describing the minimal informational structure required for a universe to exhibit:

coherent evolution,
stable classical regimes,
quantum-like branching with recombination,
emergent continuum behaviour.

This paper is not a mathematical formalisation (that is the role of Paper A), nor a derivation of continuum physics (the role of Paper B). Instead, this paper identifies the necessary conceptual primitives that any substrate must possess to produce a coherent, evolving world with quasi-classical behaviour.

These primitives are:

Mismatch tension — local informational inconsistency.
Tolerance — limits on resolvable gradients of mismatch.
Relaxation — ambiguity-resolving, mismatch-nonincreasing exploration processes.
Closure — mismatch-reducing commitments that define temporal succession.
Height — minimal mismatch deficit required to restore local compatibility.
Cohesive phases — regions where all finite subsets have finite height.
Provenance branching — necessary bookkeeping when multiple closure-compatible states exist.

Our aim is to show that these primitives are logically necessary for any constructive, evolving reality governed by informational consistency.

Scope Clarification

This paper establishes priority of conjecture and program, not completed derivations. Formal substrate mechanics, empirical emergence programs, and continuum-limit physics are developed in subsequent companion works. The present paper claims priority over: (i) the informational-constraint ontology, (ii) mismatch-tolerance-driven dynamics, and (iii) the hypothesis that quantum and spacetime structure emerge as stability-selected calculi within such a substrate.

2. Motivation for a Substrate-Independent Theory

Many foundational proposals attempt to describe the universe using a specific physical substrate: quantum fields, causal sets, spin networks, cellular automata, and so on. Any such proposal invites an immediate question:

Why that substrate, and not another equally permissible one?

Cohesion Dynamics avoids this pitfall by shifting perspective:

CD does not propose a particular substrate.
It proposes the minimal structural specification that any substrate must implement to support coherent physics.

This is analogous to the way:

Turing machines do not describe physical hardware,
thermodynamics does not depend on molecular details,
information theory does not depend on the storage medium.

CD plays a similar role for coherence, causality, and consistency. The primitives introduced here are not optional: they are the necessary logical components of any substrate capable of supporting meaningful change and stable constructive processes.

Priority Claims

The author claims priority for the following ideas, articulated here as conjectural but programmatically precise:

A physical ontology in which information under constraints is fundamental.
Dynamics driven by mismatch reduction under tolerance, not background time.
The emergence of constructors as stable, reusable patterns.
The emergence of interference, phase, and recombination from path-dependent histories.
The conjecture that quantum mechanics is the unique stable calculus of interference compatible with such a substrate.
The conjecture that spacetime geometry and gravitation emerge from large-scale cohesion structure.

Relationship to Supporting Papers

This priority paper is supported by a structured sequence of subsequent works:

M-series (M1–M5): Formal metaphysical and structural foundations (cohesion, admissibility, modes, phase, narrative synthesis).
E-series (E1, E2): Empirical and computational emergence programs establishing necessity and sufficiency of structural ingredients.
Paper A: Formal substrate mechanics (commit dynamics, constraint graphs, tolerance vectors).
Paper B: Continuum-limit physics (quantum mechanics, causal structure, and geometry as effective descriptions).

None of these are required to accept the priority claims made here; they exist to justify and elaborate them.

3. Conceptual Primitives

This section introduces the conceptual primitives of CD before any formalisation. Their formal counterparts appear in Paper A; continuum consequences appear in Paper B.

3.1 Mismatch Tension

Every informational system has constraints governing what counts as a consistent local configuration. When a configuration violates these constraints, we say it carries mismatch tension, denoted conceptually by a scalar

M \ge 0.

Interpretation:

$M = 0$ means all local constraints are satisfied.
$M > 0$ means local inconsistency; something “does not fit.”

Mismatch is not energy and not a physical potential. It is a consistency measure: the extent to which a state deviates from internally compatible structure.

Mismatch is the engine of all CD processes: relaxation attempts to neutralise it, closure commits to a reduced-mismatch configuration, and height measures how far a region sits from compatibility.

3.2 Tolerance

Not all mismatch gradients are resolvable. Some are too sharp or too large to be reconciled by local updates.

A region may tolerate mismatch up to some vector threshold

W = (W_\tau, W_\sigma, W_\theta).

This conceptual tolerance vector determines how much local change a substrate can absorb while remaining part of the same coherent component.

If mismatch gradients remain $\le W$ , the region stays coherent.
If mismatch gradients exceed $W$ , coherence fails and the region partitions informationally.

Tolerance is not a free parameter: it is a minimal structural requirement for a consistency-based substrate to avoid immediate fragmentation.

3.3 Relaxation

Relaxation is the process by which a system explores mismatch-nonincreasing alternatives outside the temporal sequence.

Relaxation:

does not define time,
explores allowed alternatives in parallel,
is not stochastic,
is not agent-driven,
is simply the substrate’s local attempt to avoid inconsistency.

Conceptually:

Relaxation defines the space of all locally compatible ways the system could resolve inconsistency, without yet committing to any one of them as the next moment.

Because mismatch never increases along relaxation paths, the space of relaxation alternatives forms a directed acyclic graph (DAG) in which edges correspond to allowed mismatch-nonincreasing moves.

This DAG becomes the precondition for closure.

3.4 Mapping Table (Patch F1)

To align conceptual primitives with the formal substrate defined in Paper A, we include:

Conceptual Primitive	Formal Object (Paper A)	Purpose
mismatch tension	$M(x;\sigma)$	Local incompatibility vs. constraints
tolerance	$W$	Bounds on resolvable mismatch gradients
relaxation	mismatch-nonincreasing local transitions	explores alternatives outside time
closure	mismatch-decreasing commit	defines time progression
height $H(R;X)$	height functional	minimal mismatch deficit to reach compatibility
cohesive phase	all finite regions have $H < \infty$	supports continuum-like behaviour
provenance	branching labels (vectors or equivalent schema)	bookkeeping for multi-solution closure and recombining

This makes Papers A and B logically downstream of Paper F:
Paper F explains why these objects must exist; Paper A defines them precisely; Paper B studies their continuum consequences.

4. Height: Why a Consistency-Based Substrate Must Include a Height Functional

A substrate built on informational consistency must include a way to distinguish recoverable tensions from irrecoverable disruptions.
This distinction cannot be made from mismatch $M$ alone, because:

$M$ is local and does not encode how far a region is from rejoining a coherent whole.
Two regions may have the same $M$ but differ drastically in whether they can be restored.
Local mismatch can be small while global incompatibility is catastrophic.

Thus we introduce the conceptual primitive height $H(R\,|\,X)$ :
the minimal mismatch reduction required (over all relaxation paths that do not increase mismatch) to bring region $R$ into consistency with its surrounding context $X$ .

Height encodes:

how far $R$ sits from any compatible configuration,
the amount of unavoidable corrective work required,
whether the inconsistency is locally resolvable or globally obstructed.

4.1 Why Height Must Exist in Any Consistency Substrate

If the substrate did not quantify distance to compatibility, then:

There would be no principled way to classify a region as repairable versus beyond repair.
There would be no concept of cohesion—everything would be either trivially consistent or undefined.
No boundary between coherent physics and incoherent substrate behaviour could be recognised.

Height is therefore not an optional axiom but a logical necessity for any substrate that distinguishes:

“cohesive physics” (finite height),
from “divergence” (infinite height).

Height expresses a global structural property even though defined through local relaxation rules.

4.2 Finite vs Infinite Height

Finite height ( $H < \infty$ )
The region is salvageable using mismatch-nonincreasing relaxation steps.
Infinite height ( $H = \infty$ )
No sequence of allowable relaxations can restore consistency.
The region is informationally unreachable from the outside context.

This distinction is the conceptual source of horizons in Cohesion Dynamics:
they arise not from geometry but from structural reachability in the substrate.

4.3 Minimality of the Height Functional

Among all conceivable ways to quantify “distance from compatibility,” the height functional is uniquely minimal. Any consistency-distance measure must satisfy three conditions:

Monotonicity under relaxation — distance must never increase when local mismatch is reduced.
Local definability — the distance of a region must depend only on that region and its immediate constraint-neighbourhood.
Minimal deficit principle — the measure must be the smallest deficit such that some mismatch-nonincreasing relaxation sequence can reach a compatible state.

Any functional lacking (1) fails to guide relaxation; lacking (2) breaks substrate locality; lacking (3) artificially enlarges the class of divergent regions. Height is the only primitive satisfying all three conditions without introducing additional structure. For this reason, it is not merely one option but the canonical consistency-distance notion required by a mismatch-based substrate.

5. Closure: Why Time Is the Commitment to Reduced Mismatch

Relaxation explores all admissible mismatch–nonincreasing changes to a region.
But if a substrate were only relaxation, it would:

never stabilise,
never construct persistent structures,
never produce a directed history.

Thus closure is required as a second primitive.

Closure is the irreversible commitment to one compatible configuration among the relaxation-accessible alternatives.

5.1 Why Closure Must Exist

Without closure:

no constructor (biological, chemical, or informational) could persist,
no stable record or memory could form,
no causal chain could be well-defined,
no notion of before and after could emerge.

Closure is therefore the substrate-level definition of a coherence cycle.

A closure step:

selects one mismatch-nondecreasing terminal configuration,
resolves local tension,
produces a new consistent substrate state.

Each closure step is irreversible in informational terms: once the mismatch-reducing choice is made, the substrate designates that choice as the next committed configuration.

5.2 Closure and the Emergence of Time

Time in CD is not a fundamental parameter.
It is the ordering induced by closure steps:

S_0 \rightarrow S_1 \rightarrow S_2 \rightarrow \cdots,

where each $S_{n+1}$ is the result of a closure following relaxation from $S_n$ .

Thus time exists because closure happens.
Relaxation provides the branching structure; closure provides the directed edge through it.

6. Cohesive Phases and Why They Are Necessary for Physics

A cohesive phase is a domain of the substrate in which every finite region has finite height:

\forall R \subset \text{finite},\quad H(R\,|\,X) < \infty.

Cohesive phases represent the parts of the substrate where:

mismatch is low enough,
relaxation is effective enough,
closure always finds a locally compatible successor,
no region is cut off by infinite inconsistency.

6.1 Why Cohesive Phases Are the Arena of Physics

Physics requires:

stable signals,
predictable causal influence,
well-defined locality,
repeatability of processes,
persistence of structures.

These are only possible if relaxation can always repair small disruptions and closure never fails.

If a region had infinite height, then:

it could not be reintegrated into the surrounding substrate,
causal propagation would fail,
the region would fragment into incompatible informational components,
no continuum-like behaviour could emerge from it.

Thus cohesive phases are not optional — they are the only regions in which:

effective fields can be defined (Paper B),
classical structures can persist,
probabilistic branching recombines (where $W$ allows),
constructive processes (life, cognition) can exist.

6.2 Why Physics Cannot Exist Outside Cohesive Phases

Outside cohesive phases:

relaxation cannot heal inconsistencies,
closure may have no viable consistent successor,
provenance becomes undefined,
locality collapses,
the informational substrate fails to support any continuous description.

Therefore the continuum physics derived in Paper B is valid only inside cohesive phases.

Regions of divergence (infinite height) are not “places in spacetime” but structural disconnections in the substrate, where the continuum description breaks down.

7. Divergence, Partition, and Structural Horizons

Height allows us to distinguish cohesive regions (all finite height) from regions of divergence (infinite height).
When height diverges across some cut in the substrate, something stronger happens than “large tension”:

the substrate partitions.

A partition is not a region of high mismatch.
It is a region where no relaxation path exists that preserves mismatch nonincrease and leads to compatibility with its surroundings.

Formally:

H(R\,|\,X) = \infty \quad\Longrightarrow\quad \text{region } R \text{ is structurally unreachable from } X.

This yields a substrate-level concept of horizon:

A horizon is not a geometric surface.
It is not a causal boundary in the continuum sense (Paper B).
It is the boundary of a substrate component beyond which the outside cohesive phase cannot be extended.

A horizon is therefore a failure of structural reachability, not a physical object.

7.1 Why Divergence Creates Partition

If $H = \infty$ , then no mismatch-reducing sequence can restore local consistency.
Because only mismatch-nonincresing relaxations are allowed, any attempt to “pull” the region into alignment would violate the substrate rules.

Thus the substrate has only one option:

drop the problematic edges from the admissible neighbourhood graph,
thereby creating two or more disconnected cohesive components.

This is analogous to cutting stressed connections in a network to preserve global consistency.

Each resulting component may remain internally cohesive, with its own closure cycles and provenance evolution, but no consistent adjacency relation survives across the cut.

This explains why divergence surfaces are not places where physics “goes wild” — they are simply boundaries where that cohesive phase ends.

7.2 Structural Horizons vs Physical Horizons

A structural horizon is more primitive than a physical (continuum) horizon:

It does not require spacetime.
It does not involve curvature.
It is defined entirely at the consistency level.

Physical horizons (e.g., Paper B) arise when cohesive phases themselves admit a continuum limit with finite propagation speed; divergence partitions then become causal boundaries.

Thus structural horizons are the substrate origin of physical horizons.

7.3 Why Substrate Independence Requires Horizons

Any substrate-independent informational theory must include:

a notion of local incompatibility,
a boundary where that incompatibility cannot be resolved,
which prevents the extension of global structure across it.

Otherwise:

the system could not sustain constructive processes,
inconsistencies would propagate uncontrollably,
no layered emergence would be possible.

Horizons are therefore a necessary feature of any substrate whose coherence defines physics at larger scales.

8. Provenance: Why Closure Creates Irreducible Causal Structure

Relaxation explores many alternative mismatch-nonincreasing configurations.
Closure selects one.
But closure alone is insufficient to encode:

which alternatives were available,
what incompatibilities existed,
why two closures cannot recombine later.

Therefore each closure step carries provenance: a minimal record of which relaxation alternatives were available prior to the commit.

Provenance is not history.
It is a structural discriminator that ensures:

incompatible branches do not recombine,
compatible branches can recombine,
information flow respects substrate constraints.

8.1 Why Provenance Must Exist

Without provenance:

two states that arose from different unresolved incompatibilities could appear identical,
yet their later interactions would require mutually inconsistent relaxations,
causing unresolvable conflicts in closure.

Thus provenance is the substrate’s minimal bookkeeping to ensure:

structural correctness of future relaxations,
correct routing of information,
correct determination of whether two regions can be jointly closed.

Provenance is not optional — it is required for consistency.

8.2 Provenance as a Minimal Structural Record

Each closure step appends a provenance value $p$ to the local region.
Provenance values are:

local,
minimal,
incomparable unless structurally compatible,
merged only when all contributing relaxations permit it.

Two regions may interact only if their provenance values do not imply contradictory commitments.

This is how the substrate ensures that:

classicality emerges whenever provenance differences become widespread and nonmergeable,
quantum-like behaviour emerges when provenance remains compatible and mergeable.

8.3 Provenance Is Not Physical Memory

It is important to emphasise:

provenance is not recorded in spacetime,
provenance is not a particle label,
provenance is not an observer-dependent object.

Rather:

provenance ensures that closure cannot accidentally recombine incompatible alternatives,
provenance enforces local structural correctness of closure.

It is thus a structural guardrail, not a physical entity.

8.4 Provenance as an Equivalence Relation

The substrate does not store tuples, vectors, or any explicit branching data structure. Provenance is conceptually an equivalence relation on configuration histories: two regions share provenance if they arise from closure operations that are compatible with the same mismatch-reducing sequence. Representing provenance by a finite tag, list, or label in this paper is a modelling convenience. The foundational claim is only that the substrate must preserve distinctions between incompatible closure histories; it is agnostic about how those distinctions are implemented.

9. Branching: Why a Consistency Substrate Must Allow Multiple Alternatives

Because relaxation explores all mismatch-nonincreasing alternatives, and closure is a local commit, a consistency-driven substrate must allow branching whenever:

more than one relaxation terminal state exists,
and none of them can be ruled out by mismatch constraints.

Branching is therefore not a stochastic rule, not a choice, and not multiplicity of worlds.
It is a logical consequence of the substrate’s two-phase structure.

9.1 Why Branching Is Unavoidable

Branching is necessary because:

Relaxation must explore multiple possibilities in parallel to maintain coherence.
Closure must commit locally without access to a global selector.
Some alternatives are equally compatible with all constraints.
Eliminating one possibility arbitrarily would violate substrate consistency.

Thus branching is what happens when:

the substrate cannot eliminate an alternative by mismatch arguments,
and closure cannot represent a superposed state (closure must commit).

9.2 Branches Are Not Worlds

A branch is:

a causal lineage distinguished by provenance,
not a separate reality,
not a parallel universe,
not metaphysical duplication.

All branches coexist only as potential commitments inside a single evolving substrate,
and only until coherence constraints force one of two outcomes:

merge: if provenance and mismatch permit recombination,
separate: if $W$ prohibits reconciliation.

9.3 Why Branching Does Not Lead to Everettian Multiplicity

Everett’s many-worlds posits:

parallel universes,
independent dynamics,
ontic duplication of states.

CD posits none of these.

In CD:

all alternatives exist only as internal tension structures during relaxation,
closure selects one at each region,
provenance persists only as long as required for structural correctness,
all branches inhabit a single underlying substrate.

Thus branching is:

algebraic, not metaphysical,
structural, not ontological,
consistency-preserving, not world-creating.

9.4 Branching as the Basis of Decoherence

When provenance differences propagate:

many regions become incompatible for recombination,
mismatch constraints cause relaxation paths to diverge,
large-scale branch distinctions emerge.

This produces a natural decoherence-like regime:

large incompatible alternatives become structurally nonmergeable,
closure aligns local regions along their respective provenance paths,
classicality emerges as provenance partitions become stable.

This is not a quantum-mechanical claim; it is a structural statement about how CD enforces consistency across large cohesive regions.

9.5 Why Branching Cannot Be Eliminated

One might imagine strengthening the substrate postulates so that every region has a unique mismatch-minimising relaxation trajectory, thereby avoiding branching altogether. Such a rule is inadmissible. Enforcing uniqueness requires a new primitive not present in the postulated structure—namely, a global selector that breaks ties between equally mismatch-reducing options. This selector would violate locality, introduce an external ordering principle, and compromise the minimality of the axioms. Because mismatch constraints alone do not guarantee unique minimisers, branching is not optional: it is a structural consequence of the substrate and not an additional assumption.

10. Cohesive Phases: The Threshold for Meaningful Large-Scale Physics

Up to this point, the substrate has been described in terms of:

mismatch $M$ ,
tolerance $W$ ,
height $H$ ,
relaxation and closure,
provenance and branching,
divergence and horizons.

These primitives are entirely local.
They make no reference to geometry, dimension, metric, or physical spacetime.

Yet large-scale physics requires something more:

stable extended regions,
locally compatible updates,
nondivergent constraint propagation,
consistent closure cycles across many substrate nodes.

This substrate condition is what we call a cohesive phase.

A cohesive phase is not a region of low mismatch, nor of static stability.
It is a region where every finite neighbourhood has finite height, meaning:

H(R\,|\,X) < \infty \quad \text{for all finite regions } R.

This guarantees that:

relaxation can always find a mismatch-nonincresing path,
closure can always succeed,
divergence cannot spontaneously appear internally,
provenance differences do not explode uncontrollably,
structural horizons cannot form inside the region.

Cohesion is thus a phase property, not a point property.

10.1 Why Cohesion Is Necessary for Any Higher-Level Physics

In a noncohesive region:

height diverges,
closure cannot propagate consistently,
provenance becomes incompatible,
edges of the underlying constraint graph become effectively unreliable,
and no extended structure can survive.

In such regions, the substrate cannot maintain:

stable signals,
causal order,
consistent long-range relations.

Therefore no physics can emerge from a noncohesive phase.

All observable physics—classical, quantum, and relativistic—must arise from cohesive phases only.

10.2 Why Cohesion Does Not Assume Geometry

Cohesion assumes:

H < \infty \ \text{locally} \quad \text{and} \quad \text{closure cycles succeed everywhere}.

It does not assume:

metric distances,
neighbourhood embeddings,
dimensionality,
continuity,
Lorentzian behaviour.

Those belong to continuum-limit eligibility, introduced later.

Cohesion is the purely structural condition that a region of the substrate behaves like a single computationally consistent medium, not necessarily a spacetime.

10.3 Cohesion as the Precondition for Constructive Processes

Constructors—systems that transform information into new information reliably—require:

consistent local updates,
stable provenance,
bounded mismatch,
closure success,
and no divergence.

Thus constructors can exist only inside cohesive phases.

This includes:

atoms,
molecules,
organisms,
computers,
and knowledge-producing systems.

Cohesive phases are therefore the arena in which structured, persistent computation occurs—not the output of such computation.

10.4 Cohesion vs Stability

Cohesion does not imply that:

mismatch is small,
dynamics are slow,
systems are near equilibrium,
nothing changes.

On the contrary, cohesive regions can host vigorous, nonlinear, rapidly changing structure.

Cohesion merely means that:

mismatch remains manageable via relaxation,
closure is always possible,
alternatives remain bounded in provenance,
and divergence does not spontaneously occur.

A turbulent plasma can be cohesive; a quiescent region near a divergence boundary may not be.

10.5 Cohesive Phases and the “Lightning / Tension” Picture

The intuitive picture where:

mismatch acts as tension,
relaxation acts as “background lightning,” searching for consistent arrangements,
closure acts as the stitching of consistent layers (“zipping”),
provenance tracks which zips are compatible,

becomes precise in cohesive phases:

mismatch cannot build beyond what $W$ permits locally,
height is finite everywhere,
relaxations always find a non-divergent path,
closure cycles progress smoothly throughout the region.

This is the substrate analogue of “no tears” in a physical fabric.

10.6 Why the Continuum Limit Requires Cohesion (But Not Vice Versa)

Later (in Paper B), the continuum limit imposes additional conditions:

compatibility with embedding,
controlled variation of mismatch density,
refinement stability,
effective locality of propagation,
existence of coarse fields.

But these rely on cohesion.

Cohesion does not guarantee a continuum limit.
But without cohesion, continuum physics is impossible.

Thus:

Cohesion = the prerequisite.
Continuum = an additional emergent regime.

This distinction protects the theory from circularity:
we do not assume spacetime; we show when spacetime-like behaviour becomes possible.

11. Divergence, Partition, and Horizons

The previous section described why cohesive structure is necessary for meaningful physical evolution. We now contrast this with regions where cohesion fails and where the height functional diverges, clarifying what CD does and does not assert about divergent regimes.

In a consistency-based substrate, not every region can necessarily be reconciled with its surrounding context. The conceptual tool that captures this limit is the height functional $H(R \mid X)$ , introduced earlier: it measures the minimal mismatch reduction required for a region $R$ to become compatible with its neighbourhood when the surrounding configuration is held fixed at $X$ .

11.1 When Height Diverges

A region has divergent height when

H(R \mid X) = +\infty,

meaning:
there exists no finite sequence of allowed relaxation steps that can reconcile $R$ with the surrounding configuration.
This does not mean the interior is inconsistent or dynamically inert; only that the exterior cannot coherently extend its own state into $R$ .

This situation arises conceptually when:

local mismatch exceeds the tolerance structure $W$ , and
all mismatch-reducing relaxation sequences are blocked from the outside.

The failure is not dynamical but structural:
the constraints defining coherence cannot be satisfied across that boundary.

11.2 Substrate Partition

When $H = \infty$ for some boundary region, the substrate undergoes a partition:

the global constraint graph separates into components,
the exterior component cannot impose coherence conditions across the boundary,
the interior component may still form a cohesive phase.

Partition is therefore a topological failure of extendability, not a destruction of information or substrate activity.

11.3 Conceptual Horizon

A horizon in CD is defined as:

a boundary across which a given cohesive component of the substrate cannot be coherently extended.

Formally, if $\partial R$ is a boundary region such that

H(\partial R \mid X_{\text{ext}}) = +\infty,

then the exterior cohesive phase cannot continue across $\partial R$ .
From the viewpoint of that component:

time cannot be extended across the boundary,
the continuum interpretation collapses at $\partial R$ ,
the interior is “cut off”.

No geometric interpretation is made here; that belongs to Paper B.
Conceptually, a horizon is an obstruction to the continuation of coherent updates, not a geometric singularity.

11.4 Interior Cohesion Remains Possible

Nothing in the definition of $H$ requires that the interior region lacks coherence.
It may independently satisfy all cohesion conditions:

finite height internally,
consistent closure operations,
emergent temporal ordering.

The exterior simply cannot synchronise updates with it.
Thus horizons represent limits of one component’s coherence, not global breakdowns.

Analysis of interior cohesive phases—if present—is deferred to later papers on gravitational and cosmological structure.

12. Provenance Branching: Why a Consistency-Driven Substrate Cannot Always Choose a Single Path

The substrate defined by Cohesion Dynamics is deterministic in its rules, but not necessarily in its outcomes.
When several locally consistent closures are available that each reduce mismatch without violating any tolerance $W$ ,
the substrate cannot single-out one of them a priori without an external selector.
This section explains why such branching is both inevitable and physically benign.

12.1 The Source of Branching

Consider a region $R$ and its local mismatch field $M(x; X)$ .
Suppose relaxation identifies multiple sequences of local updates,
each yielding a configuration $\sigma_i$ such that:

M(R; \sigma_i) < M(R; X),

and for all $i, j$ , neither $\sigma_i$ nor $\sigma_j$ violates local tolerances $W$ .
Each represents a locally admissible closure.

Nothing in the substrate rules distinguishes which should be “chosen.”
The system’s consistency principle allows all mismatch-decreasing closures simultaneously as informational possibilities.
Therefore, rather than forcing an arbitrary selection, CD records provenance branches —
distinct, internally consistent closure outcomes that all remain valid under the same substrate rules.

12.2 Provenance as Consistency Bookkeeping

Each closure carries a record of which combination of local constraints participated in its construction.
This record — the provenance — is represented abstractly by a tuple of identifiers tracing the closure’s ancestry in the constraint graph.

When multiple closures are admissible, their provenances diverge,
forming a branching structure in the informational history of the substrate.
This is not a new dynamical rule; it is bookkeeping required to prevent contradictions.
Without it, two equally valid but distinct closures would overwrite each other’s constraints,
breaking the invariant that mismatch must never increase under closure.

12.3 Branching Without Stochasticity

Branching in CD does not depend on randomness.
It follows logically from the combination of:

mismatch-nonincreasing closure,
finite tolerance $W$ ,
absence of a unique global minimiser of $M$ .

If multiple closure sequences achieve equal mismatch reduction,
the substrate must retain them all as informationally real until further constraints collapse or merge them.

No stochastic noise or probabilistic sampling is introduced;
the multiplicity of admissible closures is purely structural.

12.4 When Branches Can Recombine

Two provenance branches can recombine if their local configurations differ only in mismatch-neutral ways
— meaning that rejoining them does not increase $M$ and does not violate any component of $W$ .

Recombination restores a single cohesive record of state evolution.
When recombination fails (because $W$ would be exceeded),
the branches persist as independent cohesive components of the same substrate.

Thus branching is reversible in principle, subject to the same tolerance structure that governs divergence and cohesion.

12.5 No Multiplication of Worlds

All provenance branches exist within one substrate,
sharing the same underlying constraint network.
They represent alternative consistent histories of update — not parallel universes.

Branching does not imply ontological duplication;
it is the informational expression of a system that must maintain consistency
in the presence of multiple valid mismatch-reducing closures.

12.6 Summary

Branching arises from the coexistence of multiple admissible closures.
Provenance tracks which constraints each closure satisfied.
No stochasticity or “many-worlds” ontology is required.
Branches may later recombine if mismatch and tolerance allow.
Provenance branching is therefore a structural necessity,
not an interpretational embellishment.

13. Why Provenance Branching Is Not Many-Worlds

Cohesion Dynamics permits branching when multiple mismatch-decreasing closures are locally admissible.
This section clarifies why such branching does not commit CD to a Many-Worlds ontology.

13.1 One Substrate, One Ontology

All provenance branches coexist inside the same substrate configuration space.
They do not generate new substrates, new manifolds, or new “copies” of the universe.
They are alternative internally consistent closures of the same constraint graph.

Nothing in the theory multiplies ontological entities.
The branching mechanism is purely representational, not metaphysical.

13.2 A Branch Is an Informational Alternative, Not a World

A provenance branch records a chain of mismatch-reducing commitments.
It is analogous to:

a version history in a distributed database,
a consistent proof-path in constructive logic,
or a compatible sequence of local rewrites in a rewriting system.

These are not “worlds” in any physical sense;
they are informational alternatives that remain viable until
tolerance (W) or further constraints differentiate or recombine them.

13.3 No Ontology of Duplication

In Everettian quantum mechanics, each branch corresponds to a physically realised universe.
In CD, only one coherent configuration exists at each closure step;
branching applies to possible successor closures, not to entire physical histories persisting independently.

The substrate does not replicate matter, geometry, or observers.
It simply tracks which constraints can be satisfied without increasing mismatch.

13.4 Recombination Undermines Many-Worlds Interpretation

Branches can recombine when their differences are mismatch-neutral and compatible with tolerance (W).
Recombination contradicts the standard many-worlds picture, where branches never merge.

In CD, provenance branching is inherently non-terminal.
It expresses temporary divergence in local constraint satisfaction,
not permanent world-splitting.

13.5 No Global Incompatibility

All branches share the same:

underlying constraint graph,
informational substrate,
tolerance vector (W),
global mismatch budget.

There is no sense in which branches are causally sealed from each other by ontology;
they are separated only when mismatch-neutral recombination becomes impossible
(i.e., when merging would violate a component of (W)).

13.6 Summary

CD branching arises from local non-uniqueness of closure, not universal duplication.
All branches reside in one substrate.
Provenance is bookkeeping, not multiverse ontology.
Recombination is possible and expected.
Therefore CD is fundamentally single-world:
branching describes informational alternatives,
not parallel universes.

14. Why CD Predicts Emergent Geometry but Does Not Define It

Cohesion Dynamics is not a geometric theory.
It does not assume manifolds, coordinates, metrics, or causal cones.
Nevertheless, once a substrate enters a cohesive phase, a form of geometry becomes unavoidable.

This section explains why emergent geometry is a structural consequence of cohesion,
not an additional postulate.

14.1 Cohesion Forces Stable Local Neighbourhood Structure

In a cohesive phase, every finite region has finite height:

H(R) < \infty \quad \text{for all finite } R.

This ensures:

local mismatch is bounded,
local relaxations succeed,
closure can propagate consistently,
local updates remain synchronised.

When these conditions hold, each region can reliably track:

which neighbouring regions influence it,
which constraints they share,
how small changes propagate.

This induces something equivalent to a local adjacency structure.
CD does not assert that this structure is geometric, but that it behaves as if it were.

14.2 Overlapping Cohesion Leads to Topological Order

Cohesion implies that local neighbourhoods remain compatible under closure.
Given regions (R) and (R’) whose closures remain jointly coherent, their compatibility defines:

an overlap relation (“can be consistently combined”),
a refinement relation (“can be jointly replaced by smaller regions”),
and a separation relation (“cannot extend cohesion beyond this surface”).

These three relations—overlap, refinement, separation—are exactly the relations needed to define a topological space.

CD does not introduce topology as a primitive;
it shows that cohesion necessarily induces one.

14.3 Why Localised Influence Implies Coordinate-Like Behaviour

Because mismatch cannot propagate arbitrarily in one closure step, and because tolerance (W) restricts how sharply constraints can vary, each region has an effective influence radius.

This means:

influence is local,
influence is approximately symmetric under small perturbations,
influence decays with closure distance.

Any substrate in which information has:

local influence,
controlled spread,
bounded mismatch gradients,

will necessarily admit something functionally similar to coordinates.

CD does not specify what these coordinates are;
Paper B shows how one can be constructed in cohesive phases.

14.4 Geometry Emerges as a Description, Not as a Primitive

Once local adjacency, overlap, and influence relations are stable under closure, it becomes mathematically convenient—though not required—to represent them using:

neighbourhoods as open sets,
refinement as shrinking coordinate patches,
mismatch gradients as geometric derivatives.

In this sense, emergent geometry is a representation theorem:

Cohesive substrates can always be modelled as if they were manifolds
because their consistency conditions enforce the structural relations
needed to interpret them as such.

Nothing in CD posits:

a metric,
a dimension,
a signature,
or curvature.

Those arise only in Paper B, and only after additional continuity assumptions.

14.5 Why CD Does Not Attempt to Define Geometry

The goal of CD is to explain why geometry appears, not to prescribe it.

Geometry is a description of cohesive behaviour, not the behaviour itself.
The substrate never “contains” distances, speeds, or angles.
What it contains is:

mismatch,
tolerance,
closure propagations,
compatibility relations.

These form the combinatorial skeleton from which geometry can be reconstructed.

14.6 The Boundary of CD’s Conceptual Claim

We emphasise the boundary clearly:

CD predicts that cohesive phases admit a geometric description.
CD does not specify what that geometry ultimately looks like.
CD does not derive metric formulas or physical field dynamics here.
CD does not assume spacetime exists a priori.

Everything beyond this point belongs to Paper B, which formalises the emergence of continuum physics from cohesive phases.

14.7 Summary

Cohesion ensures stable neighbourhood interactions.
Stable overlaps induce topology.
Local influence constraints induce coordinate-like behaviour.
Geometry is a representational tool for describing consistency patterns.
CD predicts geometry must emerge, but does not define the geometry itself.

This preserves the conceptual boundary between Papers F, A, and B,
and makes clear that CD’s core claim is structural:
cohesion implies geometry, but CD itself is not a geometric theory.

14.8 Scope Clarification

This section concerns only the emergence of topological and coordinate structure. Nothing stated here asserts that a metric, curvature tensor, Lorentzian signature, or Einsteinian dynamics must appear. Those are addressed only in Paper B under additional regularity and continuum-limit assumptions. The present section establishes only that cohesive phases necessarily admit geometric representation, not that any particular geometric content is entailed.

15. What This Paper Does Not Claim

Because this is a conceptual–foundational paper, not a physical or mathematical derivation, it is important to state explicitly what Cohesion Dynamics (CD) does not assert.
These boundary statements are essential for referee clarity and for maintaining the conceptual integrity of the project.

15.1 CD Does Not Provide a Physical Substrate Model

CD does not claim:

that the real universe is literally a graph,
that physical constituents correspond to nodes,
that mismatch or tolerance are material quantities.

Rather, CD proposes the minimal structure that any substrate must implement if it is to support:

consistent information,
constructive update cycles,
emergent temporal order,
and cohesive phases.

What actual substrate implements CD (if any) lies outside the scope of this paper.

15.2 CD Does Not Derive Quantum Mechanics or Classical Mechanics

CD supplies conceptual primitives—mismatch, tolerance, height, relaxation, closure, and provenance—that explain:

why superposition-like alternatives appear,
why closure produces definite outcomes,
why branching is necessary but not metaphysical,
why recombination is permitted unless prevented by divergence.

CD does not claim to derive:

Schrödinger dynamics,
Born amplitudes,
Hilbert spaces,
canonical quantisation,
classical particle trajectories.

These belong to later model-specific work.

15.3 CD Does Not Derive Spacetime Geometry or Field Equations

CD predicts that cohesive phases admit a geometric description, but does not provide:

a metric,
a signature,
a dimensionality,
curvature tensors,
geodesic equations,
Einstein equations.

These appear only in Paper B, where additional assumptions about continuity and refinement stability are introduced.

15.4 CD Does Not Claim Cosmological or Gravitational Results

CD does not assert:

an explanation of inflation,
dark matter,
black hole interiors,
cosmic horizons,
singularity resolution.

Divergence, horizons, and partitioning are defined at the level of informational structure, not astrophysical models.

Interpretation in physical terms requires additional assumptions and belongs to future work.

15.5 CD Does Not Provide a Selection Rule Beyond Consistency

CD does not introduce:

randomness,
stochastic postulates,
collapse rules,
measurement axioms,
selection operators.

Closure is defined purely in terms of consistency:

mismatch must be reduced or kept minimal,
the resulting configuration must satisfy local constraints,
provenance branching occurs when multiple consistent closures exist.

Nothing in CD resembles, or replaces, the probabilistic postulates of QM.

15.6 CD Does Not Assert That Branching Creates Multiple Worlds

CD explicitly denies that provenance branching represents:

multiple universes,
separate ontological worlds,
spatially disjoint manifolds,
metaphysical duplication.

Branches are informational alternatives that coexist inside a single evolving substrate.
Their separation is a bookkeeping necessity, not an ontological claim.

15.7 CD Does Not Claim Completeness

This paper does not pretend to fully describe:

the ultimate nature of information,
the ontology of physical reality,
the origin of constraints,
why the universe exists,
or any metaphysical foundation.

It provides only what is needed for Papers A and B:
a minimal consistency-based conceptual ontology.

15.8 Summary

This paper does not introduce physics;
it introduces the conceptual structure that physics will be built on in later work.

It does not derive geometry, quantum dynamics, or cosmology;
it explains why these must emerge from any substrate that maintains coherent, consistency-preserving constructive cycles.

It does not specify a substrate;
it specifies the constraints that any physically viable substrate must satisfy.

The purpose of this paper is simply:

to articulate the minimal conceptual framework that makes
Papers A (substrate mechanics) and B (continuum physics) meaningful.

16. Program Roadmap

This paper provides the conceptual foundations of Cohesion Dynamics (CD).
It defines the minimal informational primitives—mismatch, relaxation, closure, tolerance, height, provenance, cohesion, and divergence—that any substrate must satisfy in order for coherent temporal evolution, constructive processes, and emergent continuity to be possible.

The technical development of CD proceeds in subsequent papers as follows.

16.1 Paper A — Substrate Mechanics (Formalisation of Dynamics)

Paper A provides the formal discrete mechanics of the substrate implied by the conceptual structure here.
Its goals are:

To formalise:
- mismatch (M),
- tolerance (W),
- height (H),
- relaxation transitions,
- closure transitions,
- provenance branching and recombination,
- divergence and partition.
To define precisely:
- the allowed local moves,
- the structure of relaxation DAGs,
- the mathematical conditions for closure existence,
- the substrate conditions for cohesion.
To prove:
- that cohesive phases exist for nontrivial classes of substrates,
- that closure-induced sequences are consistent,
- that divergence surfaces arise when and only when (H = \infty).

Paper A develops the discrete mathematics of CD and introduces no continuum assumptions.

16.2 Paper B — Continuum Physics (Emergent Geometry and Effective Fields)

Paper B assumes:

a cohesive phase,
stable local refinement structure,
bounded-height regions,
and the conceptual primitives defined here.

It then develops the continuum limit, showing how:

coarse regions and overlaps induce a topological domain,
coherent refinement induces a coordinate structure,
smoothness emerges from bounded height and tolerance,
effective fields arise from coarse mismatch,
hyperbolic propagation emerges in cohesive phases,
causal constraints appear as information-propagation boundaries.

Paper B does not derive general relativity, quantum mechanics, or specific field equations.
It merely shows why geometry and continuum dynamics must arise generically from CD in cohesive regimes.

16.3 Later Papers (C, D, …): Physics Proper

After Papers A and B, later work will examine:

matter models,
gauge-like constraint families,
quantum amplitudes and branch weights,
decoherence and classicality,
gravitational phenomena,
cosmological regimes,
black-hole interiors and partitions,
horizon mechanics derived from (H) and (W),
possible unification strategies.

These are model-building papers:
they explore specific ways an actual physical substrate could instantiate the abstract CD framework.

16.4 Purpose of the Conceptual Foundations

This paper (Paper F) occupies a specific role:

It is not a derivation.
It is not a model.
It is not a physical prediction.

It is the conceptual ontology without which Papers A and B cannot be understood or evaluated.

Its purpose is to show:

why mismatch must exist,
why tolerance must limit admissible change,
why height must distinguish coherent and divergent regimes,
why relaxation and closure are necessary dual processes,
why provenance branching is unavoidable,
why coherence is rare and precious,
and why cohesive phases naturally support emergent geometry.

Everything beyond this—substrate mechanics, continuum physics, and real-world phenomenology—is developed in later papers.

16.5 Closing Summary

Cohesion Dynamics proposes that coherence is not a property of physical systems, but a property of informational consistency that any viable substrate must satisfy.

From this perspective:

time is closure,
change is relaxation,
space is adjacency in cohesive phases,
geometry is emergent refinement structure,
probability is branch-provenance bookkeeping,
horizons are infinite height,
and physics is the effective behaviour of cohesive subsystems.

This conceptual layer is not the end of the story but the beginning.

Paper F provides the minimal postulates needed to motivate and justify the technical development of Papers A and B, ensuring the entire research program rests on explicit, defensible foundations.

Author’s Note (2025)

Since the initial circulation of this paper, substantial progress has been made on formal and empirical programs supporting these conjectures, including derivations of linear interference composition and constraint-based decoherence. These results are reported separately and are not required for the priority claims asserted here.