Paper B2 — Non-Factorisable Composition and Entanglement

Abstract

This paper demonstrates that non-factorisable composite states (entanglement) are a representational necessity for cohesive substrates when multiple subsystems are subject to joint admissibility constraints under finite tolerance $W$. We show that any representational calculus capable of faithfully tracking composite substrate systems with joint constraint evaluation must allow non-separable states.

Building on the linear amplitude framework established in Paper B1, we derive that subsystem separability is conditional, not fundamental. The derivation proceeds via a five-step necessity chain: (1) conditional separability holds only when joint constraints decompose additively, (2) joint admissibility generically produces non-decomposable cross-terms in the compatibility load, (3) factorised representations cannot encode these cross-terms without identifying distinct substrate states, (4) faithful representation forces non-factorisable coefficients, and (5) this is precisely the structure quantum mechanics calls entanglement.

Any representational scheme that remains linear (per B1) and faithful to joint admissibility must either (a) encode non-factorisable coefficients or (b) discard admissible substrate distinctions. The former is entanglement; the latter is information loss. There is no third option.

Critically, we show that entanglement is not a physical interaction, but a representational consequence of joint constraint bookkeeping. No quantum postulates, non-locality assumptions, or measurement axioms are imported. The derivation proceeds purely from substrate mechanics (Paper A), finite tolerance (AX-TOL), joint admissibility (AX-ADM), and the linear amplitude structure proven necessary in B1.

This paper establishes why composite quantum systems exhibit entanglement as an unavoidable representational feature. It does not yet address spectral discreteness (B3), dynamics (B4), or outcome probabilities (B5). Those derivations follow in subsequent B-series papers.

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1. Scope and Dependencies

1.1 Assumed Results

This paper assumes without re-derivation:

From Paper B1 (Quantum State Representation):

• Linear amplitude spaces are representationally necessary for mergeable divergent histories
• Additive composition: $A_{\text{merged}} = \sum_i A_i$
• Scalar multiplication and phase structure
• Amplitudes as bookkeeping objects for compatibility and provenance
• Basis elements correspond to substrate resolution paths

From Paper A (Substrate Mechanics):

• Discrete substrate with finite alphabet $\Sigma$ and locations $V$
• Local constraint system $\mathcal{C}$ defining admissibility
• Mismatch measure $M(v;X)$ for configurations $X$
• Local modification operations $(v \to s)$
• Relaxation moves: mismatch-non-increasing updates

From Paper M2 (Constraint Dynamics):

• Precedence-restricted admissibility: $\Delta s^* = \arg\min M(s + \Delta s)$
• Admissible move sets $\mathcal{A}(s)$
• Joint constraint resolution
• Persistence and repair mechanisms

From Paper M4 (Phase and Coherence):

• Phase $\phi$ as closure-cycle alignment
• Tolerance vector $W = (W_{\text{shape}}, W_{\text{clock}}, W_{\text{spin}})$
• Compatibility: $|\Delta \phi_{ij}| \le W_{\text{spin}}$
• Coherence as finite tolerance-bounded regime

From Axioms (R-series and C-series): We reference axioms by their codes:

• AX-TOL: Finite tolerance window $W$
• AX-COH: Cohesive informational units (CIUs)
• AX-PAR: Partition on tolerance violation
• AX-ADM: Admissible moves exist
• AX-SEL: Precedence selection

1.2 What This Paper Does NOT Assume

This paper does not import:

• Tensor product structure as a primitive (we derive it as representational necessity)
• Hilbert space direct products (structure emerges, not assumed)
• Non-locality or spacelike correlations (no spacetime structure yet)
• Bell inequalities or hidden variable assumptions
• Measurement postulates or collapse (measurement is B5)
• Density matrices or mixed states (not yet needed)
• Von Neumann entropy or entanglement measures (not yet defined)
• Probabilistic interpretation or Born rule (outcome weighting is B5)

1.3 Explicitly Out of Scope

This paper does not address:

• Bell inequalities and non-locality — correlation structure without spacetime assumptions
• Measurement and decoherence (B5) — outcome selection and Born rule
• Dynamics of entanglement (B4) — how entanglement evolves
• Quantification of entanglement — entropy measures and entanglement witnesses
• Spectral discreteness (B3) — bound states and quantisation

These are systematically deferred to later B-series papers or optional extensions.

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2. The Representational Problem for Composite Systems

2.1 Single Systems vs. Composite Systems

In Paper B1, we established that single substrate systems with mergeable divergent histories require linear amplitude representation:

$$A = \sum_i \alpha_i A_i$$

where each basis element $A_i$ represents a distinct admissible resolution path.

Question: How do we represent systems composed of multiple subsystems?

2.2 Subsystem Decomposition

Consider a substrate configuration $X$ that can be decomposed into two spatially or structurally distinct regions:

• Subsystem $A$: locations $V_A \subset V$
• Subsystem $B$: locations $V_B \subset V$
• Joint system: $V_{AB} = V_A \cup V_B$ (with $V_A \cap V_B = \varnothing$ for simplicity)

Each subsystem, considered in isolation, would have its own amplitude representation:

$$A_A = \sum_i \alpha_i A_i^{(A)}, \quad A_B = \sum_j \beta_j A_j^{(B)}$$

where:

• $A_i^{(A)}$ represents admissible resolution paths for subsystem $A$
• $A_j^{(B)}$ represents admissible resolution paths for subsystem $B$

2.3 Naive Factorisation Assumption

A natural initial assumption would be that the joint system’s representation factorises:

$$A_{AB} = A_A \otimes A_B = \left( \sum_i \alpha_i A_i^{(A)} \right) \otimes \left( \sum_j \beta_j A_j^{(B)} \right)$$

This would mean:

$$A_{AB} = \sum_{i,j} \alpha_i \beta_j \left( A_i^{(A)} \otimes A_j^{(B)} \right)$$

Critical question: Is this factorised form always sufficient?

2.4 Joint Constraint Evaluation

The substrate does not evaluate constraints on subsystems independently. From Paper A, mismatch is defined globally:

$$M(V_{AB}; X) = \sum_{v \in V_{AB}} M(v; X)$$

where each $M(v; X)$ includes contributions from all constraints whose neighbourhoods include $v$.

Crucially, there may exist constraints whose neighbourhoods span both subsystems:

$$N(i) \cap V_A \neq \varnothing \quad \text{and} \quad N(i) \cap V_B \neq \varnothing$$

Such constraints couple the subsystems. Their satisfaction depends on the joint configuration $(s_A, s_B)$, not independently on $s_A$ and $s_B$.

2.5 The Core Problem

When joint constraints exist, the compatibility and admissibility of the composite system cannot generally be decomposed into independent subsystem evaluations.

This means:

• A state admissible for subsystem $A$ alone may become inadmissible when paired with subsystem $B$
• The compatibility load $L$ (from M4) is not additive: $L_{AB}^2 \neq L_A^2 + L_B^2$
• Factorised representations cannot faithfully track joint admissibility

The question is: What representational structure is forced when joint constraints are present?

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3. Logic Chain — Why Non-Factorisability Is Forced

This section provides a high-level argument outline before the detailed derivation.

Step 1: Conditional Separability

• Subsystem separability (factorisation) holds only when:
  • No joint constraints exist, or
  • Joint constraints decompose additively into independent subsystem constraints
• This is a special case, not the generic situation

Step 2: Joint Admissibility Generates Cross-Terms

• When constraints couple subsystems, the compatibility load takes the form: $$L_{AB}^2 = L_A^2 + L_B^2 + L_{\text{cross}}^2$$
• Where $L_{\text{cross}}$ encodes mismatch arising from joint constraint evaluation
• Generically, $L_{\text{cross}} \neq 0$

Step 3: Factorised Representations Cannot Encode Cross-Terms

• Any representation assuming: $$A_{AB} = A_A \otimes A_B$$ can only encode independent subsystem information
• Cross-coupling information ($L_{\text{cross}}$) has no representation in factorised form

Step 4: Faithful Representation Requires Non-Factorisable States

• To preserve compatibility and admissibility bookkeeping, the representation must allow: $$A_{AB} = \sum_{i,j} \alpha_{ij} \left( A_i^{(A)} \otimes A_j^{(B)} \right)$$ where $\alpha_{ij}$ are not factorisable as $\alpha_{ij} = \alpha_i \beta_j$

Step 5: This Is Entanglement

• Non-factorisable amplitude states are precisely what quantum mechanics calls entangled states
• The structure is forced by joint admissibility, not imported as a postulate

Conclusion: Any faithful representation of composite substrate systems with joint constraints must allow non-separable states. Entanglement is representationally necessary, not mysterious.

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4. Derivation — When Separability Holds and When It Fails

4.1 Notation for Composite Systems

Let:

• $s_A$ denote the state of subsystem $A$
• $s_B$ denote the state of subsystem $B$
• $(s_A, s_B)$ denote the joint configuration

Let:

• $\mathcal{A}_A(s_A)$ denote admissible moves for subsystem $A$ in isolation
• $\mathcal{A}_B(s_B)$ denote admissible moves for subsystem $B$ in isolation
• $\mathcal{A}_{AB}(s_A, s_B)$ denote admissible moves for the joint system

4.2 Separable Case — No Joint Constraints

Definition: Subsystems $A$ and $B$ have no joint constraints if every constraint $C_i$ satisfies:

$$N(i) \subseteq V_A \quad \text{or} \quad N(i) \subseteq V_B$$

That is, no constraint neighbourhood spans both subsystems.

Proposition 4.2.1 (Separability under independence): If subsystems $A$ and $B$ have no joint constraints, then:

$$\mathcal{A}_{AB}(s_A, s_B) = \mathcal{A}_A(s_A) \times \mathcal{A}_B(s_B)$$

and the mismatch decomposes additively:

$$M_{AB}(s_A, s_B) = M_A(s_A) + M_B(s_B)$$

Proof sketch: Without joint constraints, each subsystem’s mismatch is independent. Admissibility is evaluated separately. Composition is independent. $\square$

Corollary: In this case, the amplitude representation factorises:

$$A_{AB} = A_A \otimes A_B$$

and subsystems evolve independently.

Interpretation: Separability is conditional on the absence of joint constraints. This is a special case.

4.3 Non-Separable Case — Joint Constraints Present

Now consider the generic case where constraints couple subsystems.

Definition: A joint constraint is one whose neighbourhood spans both subsystems:

$$N(i) \cap V_A \neq \varnothing \quad \text{and} \quad N(i) \cap V_B \neq \varnothing$$

Such constraints contribute to mismatch based on the joint configuration $(s_A, s_B)$.

Let:

$$M_{\text{joint}}(s_A, s_B) = \sum_{i : N(i) \cap V_A \neq \varnothing, N(i) \cap V_B \neq \varnothing} C_i(X|_{N(i)})$$

Then total mismatch is:

$$M_{AB}(s_A, s_B) = M_A(s_A) + M_B(s_B) + M_{\text{joint}}(s_A, s_B)$$

Key observation: $M_{\text{joint}}(s_A, s_B)$ is not generally decomposable as $f(s_A) + g(s_B)$.

It depends on the correlation between $s_A$ and $s_B$.

4.4 Compatibility Load Decomposition

From Paper M4, compatibility is governed by tolerance $W$. For composite systems, the compatibility load takes the form:

$$L_{AB}^2 = L_A^2 + L_B^2 + L_{\text{cross}}^2$$

where:

• $L_A^2$ encodes subsystem $A$‘s internal mismatch
• $L_B^2$ encodes subsystem $B$‘s internal mismatch
• $L_{\text{cross}}^2$ encodes mismatch from joint constraint evaluation

Clarification on $L_{\text{cross}}$: $L_{\text{cross}}^2$ is not an additional load or interaction term; it is the component of the existing compatibility load that cannot be assigned to either subsystem independently. It arises purely from the non-decomposability of joint constraint satisfaction.

Proposition 4.4.1 (Cross-term necessity): If joint constraints exist and are not trivially satisfied, then generically:

$$L_{\text{cross}}^2 > 0$$

Proof sketch: Joint constraints produce mismatch contributions that depend on $(s_A, s_B)$ in a non-additive way. The cross-term $L_{\text{cross}}^2$ arises from the covariance structure of joint constraint violations. $\square$

4.5 Failure of Factorised Representations

Theorem 4.5.1 (Factorised representations are insufficient): When $L_{\text{cross}} \neq 0$, any representation of the form:

$$A_{AB} = A_A \otimes A_B = \left( \sum_i \alpha_i A_i^{(A)} \right) \otimes \left( \sum_j \beta_j A_j^{(B)} \right)$$

cannot faithfully encode the joint admissibility constraints.

Proof: A factorised representation can only encode:

• Subsystem $A$ amplitudes: $\{\alpha_i\}$
• Subsystem $B$ amplitudes: $\{\beta_j\}$

The joint amplitudes are constrained to be:

$$\alpha_{ij} = \alpha_i \beta_j$$

This means the representation assumes statistical independence of subsystem states.

However, joint constraints create correlations between admissible $(s_A, s_B)$ pairs:

• Some $(s_A, s_B)$ pairs may be individually admissible but jointly inadmissible
• The joint admissibility depends on $L_{\text{cross}}$, which has no representation in factorised form

Critically: Factorised representations fail because they identify distinct admissible substrate states. Two distinct joint substrate configurations that differ only in their joint constraint satisfaction are mapped to the same factorised representation $\alpha_i \beta_j$. This is not “quantum weirdness”—it is lossy compression of admissibility structure.

Therefore, factorised representations lose information about joint constraint satisfaction.

A faithful representation must allow:

$$\alpha_{ij} \neq \alpha_i \beta_j$$

This is the definition of non-factorisability. $\square$

4.6 Forced Non-Factorisable State Space

Theorem 4.6.1 (Necessity of non-separable states): To faithfully represent composite substrate systems with joint constraints, the amplitude space must allow states of the form:

$$A_{AB} = \sum_{i,j} \alpha_{ij} \left( A_i^{(A)} \otimes A_j^{(B)} \right)$$

where the coefficients $\{\alpha_{ij}\}$ are not constrained to factorise.

Proof: From Theorem 4.5.1, factorised representations fail when joint constraints exist.

The minimal extension is to allow arbitrary coefficients $\alpha_{ij}$, subject only to:

• Linear composition (from B1)
• Compatibility constraints (from M4)
• Normalization (not yet required, deferred to B5)

This produces the general form above, which includes both:

• Separable states (when $\alpha_{ij} = \alpha_i \beta_j$)
• Non-separable (entangled) states (when no such factorisation exists)

No additional structure is required. $\square$

Interpretation: Non-separable states are not an added feature. They are the minimal representational extension required when joint constraints couple subsystems.

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5. Tensor Product Structure as Representation

5.1 Emergence of Tensor Product

The notation $A_i^{(A)} \otimes A_j^{(B)}$ suggests a tensor product structure.

This structure is not assumed. It emerges as follows:

Basis construction:

• Subsystem $A$ has basis elements $\{A_i^{(A)}\}$ (from B1)
• Subsystem $B$ has basis elements $\{A_j^{(B)}\}$ (from B1)
• Joint system basis: pairs $(A_i^{(A)}, A_j^{(B)})$

Linearity: From B1, amplitude spaces are linear. The joint space must be linear over pairs.

Tensor product is the minimal such structure: The space of bilinear maps on $(A_i^{(A)}, A_j^{(B)})$ is the tensor product $\mathcal{H}_A \otimes \mathcal{H}_B$. Any faithful bilinear extension of linear subsystem spaces is isomorphic (up to representation) to a tensor product—there is no alternative structure that preserves linearity and composition.

Interpretation: Tensor product structure is representational bookkeeping, not ontological combination.

5.2 Reduced States as Marginal Summaries

Given a joint amplitude:

$$A_{AB} = \sum_{i,j} \alpha_{ij} \left( A_i^{(A)} \otimes A_j^{(B)} \right)$$

What is the “state” of subsystem $A$ alone?

Definition (Reduced amplitude): The reduced amplitude for subsystem $A$ is obtained by marginalizing over subsystem $B$:

$$A_A^{\text{reduced}} = \sum_i \left( \sum_j |\alpha_{ij}|^2 \right)^{1/2} A_i^{(A)}$$

(The appearance of squared magnitudes here anticipates B5; at this stage, reduced states should be understood only as marginal compatibility summaries, not probabilistic objects. The precise form depends on normalization, to be addressed in B5.)

Interpretation: Reduced states are marginal compatibility summaries, not independent subsystem states.

When the joint state is non-factorisable, the reduced state loses information about joint constraints. This is why entangled states exhibit correlations not present in subsystem marginals.

5.3 Correlations as Shared Admissibility Constraints

Entanglement manifests as correlations in measurement outcomes (to be formalized in B5).

From the substrate perspective:

• Correlations reflect joint admissibility constraints
• They are bookkeeping of which $(s_A, s_B)$ pairs satisfy joint constraints
• They are not causal influences or non-local signals

Example: If a joint constraint requires $s_A = s_B$ (same local state), then:

• Admissible joint configurations: $\{(s_1, s_1), (s_2, s_2), \ldots\}$
• This produces perfect correlation in subsystem outcomes
• The correlation arises from constraint geometry, not interaction

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6. Why Entanglement Is Not a Physical Interaction

This section addresses a common misconception about entanglement.

6.1 Entanglement Is Representational, Not Ontological

Entanglement is often described as a “mysterious connection” or “spooky action at a distance.”

From the substrate perspective:

• Entanglement is bookkeeping of joint admissibility constraints
• It reflects which composite configurations are admissible, not how subsystems “interact”
• It is representational structure, not physical mechanism

Analogy: Consider a constraint: “If $A$ is red, then $B$ must be blue.”

This constraint correlates $A$ and $B$, but it does not mean:

• $A$ sends a signal to $B$
• $B$ is influenced by $A$
• There is a physical connection between $A$ and $B$

The correlation is constraint geometry, not causation.

6.2 No Non-Locality Required

Entanglement is often associated with “non-local correlations” violating Bell inequalities.

From the substrate perspective:

• Joint constraints are local (their neighbourhoods $N(i)$ are finite)
• Constraint evaluation is local (mismatch is computed locally)
• No non-local coordination is required (AX-LOC axiom)

The appearance of non-locality arises because:

• The representational amplitude space summarizes joint constraint structure
• Marginal subsystem states lose information about joint constraints
• Correlation appears “mysterious” only if subsystems are assumed independent

Clarification: Bell inequalities (to be addressed in optional extension B-BELL) reflect constraints on factorisable representations, not on substrate structure.

Violation of Bell inequalities is evidence against separability, consistent with the necessity of non-factorisable states derived here.

Scope guard: This paper makes no claims about spacetime locality or relativistic causality; it only establishes the representational origin of non-separability prior to any spacetime structure. Questions of spacelike correlations, causal structure, and Bell violations are deferred to later work where spacetime emergence is addressed.

6.3 No Action, No Influence, Just Bookkeeping

Key insight: When we say subsystems $A$ and $B$ are entangled, we mean:

The joint system’s amplitude representation cannot be factorised into independent subsystem representations because joint constraints couple their admissibility.

This is not a claim that:

• $A$ influences $B$
• Changing $A$ affects $B$
• Information travels between $A$ and $B$

It is a claim about representational structure, not physical dynamics.

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7. Explicitly Out of Scope and What Is Deferred

7.1 What B2 Establishes

• Non-factorisable composite states are representationally necessary
• Separability is conditional, not fundamental
• Tensor product structure emerges as representational bookkeeping
• Entanglement arises from joint admissibility constraints under finite tolerance $W$

7.2 What B2 Does NOT Explain

Measurement and collapse (B5):

• How are entangled states “measured”?
• What happens at commit when subsystems are separated?
• How do outcome probabilities arise?

Dynamics of entanglement (B4):

• How does entanglement evolve in time?
• What determines entangling vs. disentangling dynamics?

Bell inequalities and non-locality (B-BELL extension):

• How do Bell violations arise?
• What constraints do they place on substrate structure?

Quantification of entanglement:

• Entropy measures (von Neumann, entanglement entropy)
• Entanglement witnesses and separability criteria

Practical entanglement:

• Quantum gates and entangling operations
• Decoherence and entanglement degradation

These are deferred to later work.

7.3 No Quantum Postulates Imported

Critically, we have not assumed:

• Tensor product structure as a primitive (we derived it as representational necessity)
• Non-locality or spacelike correlations (no spacetime structure invoked)
• Measurement postulates (deferred to B5)
• Probabilistic interpretation (deferred to B5)
• Unitary evolution (deferred to B4)

Everything derived here follows from:

• Substrate mechanics (Paper A)
• Joint constraint evaluation
• Finite tolerance (AX-TOL)
• Linear amplitude structure (B1)

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8. Implications for B3–B5

8.1 What B2 Enables

The non-factorisable amplitude structure established here provides the foundation for:

B3 (Spectral Discreteness):

• Discrete modes (from M3) will map to composite system eigenspaces
• Joint bound states will exhibit entangled character

B4 (Quantum Dynamics):

• Evolution of composite systems will preserve or alter entanglement
• Unitary evolution will act on the full tensor product space

B5 (Measurement and Born Rule):

• Commit semantics on entangled states will produce correlated outcomes
• Joint measurement statistics will reflect joint admissibility structure

8.2 What B2 Does Not Prejudge

This paper does not presuppose:

• How entanglement evolves (B4)
• How measurement affects entanglement (B5)
• Whether entanglement is preserved or lost in physical processes
• The role of entanglement in emergent spacetime structure

These will be constrained by later derivations.

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9. Summary and Scope Boundaries

9.1 Central Result

We have shown:

Any representational calculus capable of faithfully tracking composite substrate systems with joint admissibility constraints under finite tolerance $W$ must allow non-separable (entangled) states.

This is a necessity claim, not a modeling choice.

9.2 Derivation Path

1. Single subsystems require linear amplitude representation (B1)
2. Composite systems admit joint constraints spanning subsystems
3. Joint constraints generate non-additive compatibility load: $L_{AB}^2 = L_A^2 + L_B^2 + L_{\text{cross}}^2$
4. Factorised representations cannot encode $L_{\text{cross}}$
5. Faithful representation requires non-factorisable coefficients $\alpha_{ij}$
6. Tensor product structure emerges as minimal representational extension
7. Entanglement is forced representational necessity

9.3 What B2 Establishes

• Non-separable states are forced, not assumed
• Separability is conditional, not fundamental
• Separability is a property of representations, not substrates — the substrate has joint constraints; representations may or may not preserve this structure
• Tensor product structure is representational, not ontological
• Entanglement is bookkeeping, not physical interaction
• Correlations reflect constraint geometry, not non-local influence

9.4 What B2 Does Not Claim

• That entanglement is ontologically fundamental (it represents substrate joint constraints)
• That quantum mechanics is fully derived (we have only established composite state structure)
• That this framework is empirically validated (empirical work is E-series)
• That entanglement violates locality (no spacetime structure invoked)

9.5 Logical Position in B-Series

B2 assumes:

• Linear amplitude representation (B1)
• Substrate mechanics (A-series)
• Formal mechanisms (M1–M4)
• No quantum postulates

B2 provides:

• Representational structure for composite systems
• Explanation of why entanglement is unavoidable
• Foundation for multi-particle quantum mechanics

B2 enables:

• B3 (spectral discreteness of composite systems)
• B4 (dynamics of entangled states)
• B5 (measurement outcomes on entangled states)

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10. Open Questions and Next Steps

10.1 Questions Deferred to Later Papers

For B3:

• How do joint bound states exhibit entanglement?
• What role does entanglement play in spectral structure?

For B4:

• How does entanglement evolve dynamically?
• What determines entangling vs. disentangling processes?

For B5:

• How do measurement outcomes on entangled states arise?
• Why do correlations violate classical bounds (Bell inequalities)?

10.2 Potential Refinements to B2

This is a first-draft derivation. Future refinements may:

• Formalize the cross-term structure $L_{\text{cross}}$ more rigorously
• Provide explicit substrate examples of joint constraints
• Explore when separability is approximately valid (weak coupling regime)
• Develop quantitative entanglement measures from substrate compatibility structure

10.3 Relation to Empirical Work (E-series)

E-series papers empirically explore which substrate configurations produce constructor-capable continua. B2 shows that any such continuum with composite systems must exhibit entanglement if joint constraints exist.

This creates a testable prediction: substrates lacking joint constraints cannot exhibit entangled behaviour.

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11. Conclusion

We have established that non-factorisable composite state representation (entanglement) is not a quantum postulate, but a representational necessity for cohesive substrates with joint admissibility constraints under finite tolerance $W$.

Non-separability is forced by:

• Joint constraint evaluation (substrate mechanics)
• Finite tolerance $W$ (AX-TOL axiom)
• Linear amplitude structure (B1)
• Faithful representational bookkeeping

Entanglement is not a physical interaction. It is bookkeeping of joint admissibility constraints.

This paper completes the representational framework for composite systems. The remaining B-series papers will establish spectral structure (B3), dynamics (B4), and measurement (B5).

B2 establishes why composite quantum systems are entangled. B1 established what quantum states are. Together, they form the representational core of quantum mechanics—derived, not assumed.

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End of Paper B2