Paper B1 — Emergence of Quantum State Representation

Abstract

This paper demonstrates that linear amplitude-based state representation is a representational necessity for cohesive substrates exhibiting mergeable divergent histories under finite tolerance $W$. We show that any representational calculus capable of faithfully tracking substrate evolution through branching constraint resolution paths must employ a linear composition rule. This result is derived from substrate mechanics (Paper A) and the formal mechanisms of cohesion, admissibility, and phase structure (Papers M1–M4), without importing quantum axioms.

Amplitudes emerge as bookkeeping objects for compatibility and provenance relationships between uncommitted substrate alternatives. Linearity is forced by consistency requirements: regrouping, associativity, and invariance under history reordering. The resulting amplitude space is the minimal structure required to represent substrate behaviour prior to any notions of entanglement, spectral discreteness, dynamics, or measurement.

This paper establishes what a quantum state is as a representational object. It does not yet explain why states evolve (B4), why spectra are discrete (B3), how composite systems factor (B2), or how outcomes are weighted (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 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 $(v \to s)$
• Commit semantics: configurations may diverge through admissible alternatives

From Paper M1 (Constructive Viability):

• Mismatch as structural degree of freedom, not violation
• Bounded tolerance $W$ enabling admissibility without rigidity
• Construction requires divergence and convergence capacity

From Paper M2 (Constraint Dynamics):

• Precedence-restricted admissibility: $\Delta s^* = \arg\min M(s + \Delta s)$
• Persistence via structural invariance
• Repair and reuse through constraint geometry

From Paper M3 (Modes):

• Modes as discrete basins in state space
• Finite stable configurations under precedence
• Mode invariance under admissible updates

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
• Decoherence as tolerance violation and provenance partition

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

• AX-REL: Relational evolution
• 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:

• Hilbert space structure (we derive amplitude spaces, not assume them)
• Wavefunctions as primitives (amplitudes are bookkeeping, not ontological)
• Superposition postulates (linearity is derived, not assumed)
• Probabilistic interpretation or Born rule (outcome weighting is B5)
• Operators or observables (representational tools come later)
• Time evolution or Hamiltonians (dynamics is B4)
• Measurement axioms (measurement is B5)
• Inner products or norms (metric structure is not yet required)

1.3 Explicitly Out of Scope

This paper does not address:

• Entanglement (B2) — composite systems and non-factorisability
• Spectral discreteness (B3) — quantisation and bound states
• Dynamics (B4) — evolution equations
• Measurement and probabilities (B5) — outcome weighting and Born rule

These are systematically deferred to later B-series papers.

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2. The Representational Problem

2.1 Substrate Evolution Produces Branching Histories

The substrate defined in Paper A permits divergent resolution paths. Given a configuration $X$ and an unresolved constraint, multiple local modifications may be equally admissible:

$$\mathcal{A}(v; X) = \{ s \in \Sigma \mid M(v; X^{(v \to s)}) \le M(v; X) \}$$

If $|\mathcal{A}(v; X)| > 1$, the substrate does not select a single outcome. By AX-PAR (partition axiom), all admissible alternatives are realized, not rejected.

This produces branching histories: distinct resolution trajectories that:

1. Originate from a common ancestor configuration
2. Remain individually admissible under the constraint system
3. May differ in local structure while satisfying global constraints

2.2 Mergeability of Compatible Histories

Two divergent histories $h_i$ and $h_j$ are compatible if their mutual mismatch remains within tolerance $W$ (from M4):

$$|\Delta \phi_{ij}| \le W_{\text{spin}}, \quad |\Delta_{\text{shape}}| \le W_{\text{shape}}, \quad |\Delta_{\text{clock}}| \le W_{\text{clock}}$$

Compatible histories may merge at a later constraint resolution. This is not a collapse or selection; it is joint satisfaction of a downstream constraint by multiple compatible histories.

Mergeability is central to coherent substrate behaviour. It enables:

• Interference effects (histories recombine)
• Persistent identity across divergence
• Recoverable alternatives
• Constructor reuse

2.3 The Representation Task

A representational calculus must track:

1. Identity of alternatives: Which distinct histories are present?
2. Compatibility relations: Which histories can merge?
3. Provenance structure: What phase/compatibility invariants characterize each history?
4. Merger outcomes: When histories recombine, what result?

The calculus must remain:

• Consistent: Same physical substrate state → same representation
• Regrouping-invariant: Merging (A+B) then C must equal merging A then (B+C)
• History-count independent: Representation should not depend arbitrarily on how we enumerate alternatives
• Extensible: Supports adding newly discovered compatible alternatives

Scope restriction: Incompatible alternatives are not jointly representable: they trigger provenance partition at commit (AX-PAR). The representational problem addressed here concerns only mutually compatible, uncommitted alternatives. Histories that violate tolerance $W$ belong to distinct provenance domains and are tracked separately.

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3. Logic Chain — Why Linearity Is Forced

This section provides a high-level argument outline before the detailed derivation. The reasoning proceeds as follows:

Step 1: Representation must distinguish alternatives

• If two histories are compatible but distinct, the representation must preserve their distinctness while allowing joint tracking.

Step 2: Merger requires composition

• When compatible histories merge at a constraint resolution, the representation must combine them into a single merged state.

Step 3: Non-linear composition violates consistency

• Any non-additive composition rule (e.g., $f(A, B) \neq A + B$) either:
  • Breaks associativity when merging three or more histories
  • Produces representation-dependent outcomes based on arbitrary history enumeration
  • Fails to preserve provenance/phase relationships

Step 4: Additive composition is forced

• The only composition rule satisfying associativity, commutativity, and history-count invariance is:

$$A_{\text{merged}} = \sum_i A_i$$

where $A_i$ represents the bookkeeping object for history $h_i$.

Step 5: Amplitudes are minimal bookkeeping objects

• These bookkeeping objects must carry:
  • Identity (which history)
  • Compatibility information (phase/provenance)
  • Compositional structure (additive)
• We call these objects amplitudes, though they are not yet complex numbers, vectors, or wavefunctions.

Step 6: Linear amplitude space is the result

• The set of all representational states forms a space closed under addition and scaling.
• This is a linear structure, not assumed, but forced by representational necessity.

Conclusion: Any faithful representation of mergeable divergent substrate histories must be (isomorphic to) a linear amplitude calculus.

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4. Derivation — Linearity from Representational Consistency

4.1 Notation and Setup

Let $\{h_i\}$ denote a set of divergent substrate histories originating from a common ancestor configuration.

Let $A_i$ denote the representational object associated with history $h_i$. At this stage, $A_i$ is abstract; we do not assume it is a number, vector, or function.

We require only:

• $A_i \neq A_j$ if $h_i \neq h_j$ (distinctness)
• $A_i$ carries compatibility/provenance information (phase from M4)

4.2 Merger as Composition

When two compatible histories $h_i$ and $h_j$ merge at a constraint resolution, the substrate produces a single outcome configuration. The representation must combine $A_i$ and $A_j$ into a merged representation.

Define a composition operation:

$$A_{\text{merged}} = A_i \oplus A_j$$

Question: What properties must $\oplus$ satisfy?

4.3 Constraint 1 — Associativity

Consider three compatible histories $h_1, h_2, h_3$ that merge simultaneously.

We can represent this merger as:

• First merge $h_1$ and $h_2$, then merge the result with $h_3$: $(A_1 \oplus A_2) \oplus A_3$
• First merge $h_2$ and $h_3$, then merge the result with $h_1$: $A_1 \oplus (A_2 \oplus A_3)$

Representational consistency demands:

$$(A_1 \oplus A_2) \oplus A_3 = A_1 \oplus (A_2 \oplus A_3)$$

If $\oplus$ is not associative, the representation depends on the order we process mergers, even though the substrate outcome is the same.

This forces associativity of composition.

4.4 Constraint 2 — Commutativity

The substrate does not distinguish an ordering among compatible histories. Merging $h_i$ then $h_j$ produces the same result as merging $h_j$ then $h_i$.

Representational consistency demands:

$$A_i \oplus A_j = A_j \oplus A_i$$

This forces commutativity of composition.

4.5 Constraint 3 — History-Count Invariance

Suppose we split history $h_1$ into two sub-histories $h_{1a}$ and $h_{1b}$ that are internally compatible but represent the same overall resolution path.

The physical outcome when merging $\{h_1, h_2\}$ should be the same as merging $\{h_{1a}, h_{1b}, h_2\}$, since $h_{1a}$ and $h_{1b}$ are just a finer description of $h_1$.

Representational consistency demands:

$$A_1 \oplus A_2 = (A_{1a} \oplus A_{1b}) \oplus A_2$$

If $A_{1a} \oplus A_{1b} = A_1$, this is automatically satisfied by associativity.

But what if we represent the “split” as $A_1 = \alpha A_{1a} + \beta A_{1b}$ for some weighting?

For consistency, we must have:

$$A_1 = A_{1a} + A_{1b} \quad \text{(when histories are equivalent splits)}$$

This forces additive regrouping.

4.6 Why Non-Linear Composition Fails

Consider a non-additive composition rule, such as:

$$A_i \oplus A_j = f(A_i, A_j)$$

where $f$ is some non-linear function (e.g., max, product, weighted average).

Example: Maximum rule Suppose $A_{\text{merged}} = \max(A_i, A_j)$.

While max is associative, it fails history-count invariance. If we split $h_1 = h_{1a} + h_{1b}$ (representing the same overall resolution path as two sub-paths), then:

$$\max(A_1, A_2) \neq \max(A_{1a}, A_{1b}, A_2)$$

The representation changes based on how we enumerate histories, not on physical substrate state.

For instance, if $A_1 = 5$, $A_2 = 3$, and we split $h_1$ into $h_{1a}$ with $A_{1a} = 2$ and $h_{1b}$ with $A_{1b} = 2$:

• Original: $\max(5, 3) = 5$
• Split: $\max(2, 2, 3) = 3$

The physical merger is the same, but the representation differs—violating representational consistency.

Example: Multiplicative rule Suppose $A_{\text{merged}} = A_i \cdot A_j$.

For three histories:

$$(A_1 \cdot A_2) \cdot A_3 = A_1 \cdot (A_2 \cdot A_3)$$

Associativity holds. But commutativity and history-count invariance still fail for scaled sub-divisions.

4.7 Additive Composition is Forced

The only composition rule satisfying:

• Associativity: $(A_i \oplus A_j) \oplus A_k = A_i \oplus (A_j \oplus A_k)$
• Commutativity: $A_i \oplus A_j = A_j \oplus A_i$
• History-count invariance: Splitting/regrouping preserves outcomes
• Extensibility: Adding new compatible alternatives does not retroactively change existing representations

is additive composition:

$$A_{\text{merged}} = \sum_i A_i$$

This is a mathematical necessity, not a choice.

4.8 Scalar Multiplication

If history $h_i$ admits $n$ identical realizations (e.g., $n$ equivalent paths through constraint space), we must represent this as:

$$A_{\text{total}} = n \cdot A_i$$

More generally, for any scalar $\alpha \in \mathbb{R}$ (or $\mathbb{C}$ once phase is incorporated), scaling is required:

$$\alpha A_i \text{ represents a weighted or phase-shifted variant}$$

This introduces scalar multiplication into the representational structure.

4.9 Linear Structure Emerges

We have derived:

• An additive composition operation: $A_i + A_j$
• Scalar multiplication: $\alpha A_i$

These operations satisfy:

• Closure: $A_i + A_j$ is a valid representation
• Associativity: $(A_i + A_j) + A_k = A_i + (A_j + A_k)$
• Commutativity: $A_i + A_j = A_j + A_i$
• Identity: A “no divergence” state acts as $0$
• Distributivity: $\alpha(A_i + A_j) = \alpha A_i + \alpha A_j$

This is a linear space (vector space structure).

The representation is not assumed to be linear. Linearity is forced by consistency requirements on representing mergeable divergent substrate histories.

Clarification on additivity: The additive structure derived here reflects the algebra required to track jointly admissible alternatives prior to commit; it does not imply physical overlap, wave-like superposition, or ontological mixing. Additive composition is a representational bookkeeping requirement, not a claim about physical superposition of substrate states.

Note on algebraic structure: Any representational algebra satisfying associativity, commutativity of compatible merger, and regrouping invariance is isomorphic to an additive linear space; more exotic algebras (e.g., non-commutative, non-associative, or tropical algebras) either violate regrouping invariance or collapse distinguishability of alternatives. The linear space is thus the unique minimal algebraic structure meeting our representational constraints.

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5. Minimal Amplitude Space Definition

5.1 Amplitudes as Bookkeeping Objects

We define amplitudes as the minimal mathematical objects satisfying the derived linear structure.

An amplitude $A_i$ represents:

1. Identity: Which substrate history it tracks
2. Provenance/Phase: Compatibility information (from M4 phase structure)
3. Compositional role: How it merges with other amplitudes

Amplitudes are not:

• Physical waves propagating in space
• Ontological entities separate from the substrate
• Probabilities (that comes in B5)
• Wavefunctions in the quantum sense (that requires Hilbert space structure, not yet established)

5.2 Basis Elements and History Identification

Each distinct admissible resolution path defines a basis element in the amplitude space.

If a constraint admits $n$ admissible resolutions $\{h_1, \ldots, h_n\}$, then the representational state is:

$$A = \sum_{i=1}^{n} \alpha_i A_i$$

where:

• $A_i$ is the basis amplitude for history $h_i$
• $\alpha_i$ encodes the phase/provenance of that history

5.3 Phase Structure and Complex Amplitudes

From M4, phase is a relational property of closure-cycle alignment. Histories that differ only in phase (within tolerance $W_{\text{spin}}$) remain compatible.

To represent phase, amplitudes must carry phase information. The minimal structure supporting phase is:

$$A_i = |A_i| e^{i\phi_i}$$

This introduces complex structure (or an equivalent two-parameter representation of phase and magnitude).

At this stage, we do not require a full complex Hilbert space. We require only:

• A magnitude (representing “how much” of this history is present)
• A phase (representing closure-cycle alignment from M4)

5.4 Amplitude Space is Not Yet Hilbert Space

The amplitude space derived here is:

• Linear: Closed under addition and scalar multiplication
• Phase-carrying: Amplitudes include provenance/phase information
• Basis-indexed: Each basis element corresponds to a substrate history

It is not yet:

• Equipped with an inner product (no notion of orthogonality or projection)
• Normalized (no constraint on total magnitude)
• Probabilistically interpreted (no Born rule)
• Operated upon by observables (no operators yet)
• Dynamically evolved (no Schrödinger equation)

Those structures are introduced in later B-series papers as representational necessities emerge.

5.5 States as Uncommitted Alternatives

Crucially, amplitude representations exist only over uncommitted substrate alternatives.

Once a constraint is resolved and committed to the substrate ledger, the associated amplitude collapses to a single basis element (or is removed from the superposition).

Amplitudes track unresolved divergence, not finalized substrate states.

This matches the substrate semantics from Paper A: configurations diverge during constraint resolution and reconverge (merge) when compatible alternatives jointly satisfy downstream constraints.

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6. Why This Is Not Yet Quantum Mechanics

The amplitude structure derived in this paper is necessary but not sufficient for quantum mechanics.

6.1 What We Have Established

• Linear amplitude spaces are forced by representational consistency
• Additive superposition is unavoidable for tracking mergeable histories
• Phase structure emerges from substrate phase/provenance (M4)
• Basis elements correspond to substrate resolution paths

6.2 What We Have NOT Established

We have not shown:

Composite systems and entanglement (B2):

• How do amplitude spaces for subsystems combine?
• Why do joint systems exhibit non-separable states?
• What is the tensor product structure?

Spectral discreteness (B3):

• Why do physical systems exhibit discrete spectra?
• How do bound states emerge?
• What selects allowed energy levels?

Quantum dynamics (B4):

• Why does the amplitude space evolve unitarily?
• What determines the Hamiltonian?
• Why is evolution governed by a Schrödinger-like equation?

Measurement and Born rule (B5):

• How are amplitudes related to outcome probabilities?
• Why is the weighting quadratic ($|\alpha|^2$)?
• What happens at commit/measurement?

6.3 Why Amplitudes Alone Are Insufficient

The amplitude space is a representational framework, not a physical theory.

To recover quantum mechanics, we must show:

• How dynamics arise (B4)
• How probabilities emerge (B5)
• How composite systems behave (B2)
• Why spectra are discrete (B3)

Each of these will be derived from substrate mechanics in subsequent papers.

6.4 No Quantum Axioms Imported

Critically, we have not assumed:

• Superposition as a postulate (we derived additive composition)
• Hilbert space structure (we have a linear space, not yet a complete inner product space)
• Probability interpretation (amplitudes are bookkeeping, not probabilities)
• Operators or observables (no measurement theory yet)
• Unitary evolution (dynamics is B4)

Everything derived here follows from:

• Substrate mechanics (Paper A)
• Mergeability of compatible histories (M4)
• Representational consistency (associativity, regrouping invariance)

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7. Implications for B2–B5

7.1 What B1 Enables

The linear amplitude structure established here provides the foundation for:

B2 (Entanglement):

• Composite systems will be represented as joint amplitude spaces
• Non-factorisability will emerge from joint admissibility constraints under tolerance $W$

B3 (Spectral Discreteness):

• Discrete modes (from M3) will correspond to discrete basis elements
• Closure stability will force quantization

B4 (Quantum Dynamics):

• Evolution of amplitude spaces will be derived from substrate closure dynamics
• Linearity of evolution will follow from linearity of representation

B5 (Measurement and Born Rule):

• Commit semantics will link amplitudes to outcome frequencies
• Quadratic weighting will emerge from preservation of interference structure

7.2 What B1 Does Not Prejudge

This paper does not presuppose:

• The form of dynamics (could be first-order, second-order, non-local, etc.)
• The inner product structure (not yet required)
• Normalization conventions (not yet meaningful)
• Basis choice (substrate-determined, not conventional)

These will be constrained by later derivations.

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8. Summary and Scope Boundaries

8.1 Central Result

We have shown:

Any representational calculus capable of faithfully tracking mergeable divergent substrate histories under finite tolerance $W$ must employ a linear amplitude space.

This is a necessity claim, not a modeling choice.

8.2 Derivation Path

1. Substrate evolution produces branching admissible histories
2. Compatible histories may merge at later constraint resolutions
3. Representation must track identity, compatibility, and provenance
4. Consistency demands associative, commutative, history-count-invariant composition
5. Only additive composition satisfies these constraints
6. Scalar multiplication follows from scaling and phase
7. Linear amplitude space emerges as representational necessity

8.3 What B1 Establishes

• Amplitude spaces are forced, not assumed
• Linearity is unavoidable, not conventional
• Phase structure comes from substrate (M4), not quantum postulates
• Superposition is representational merger, not ontological wave interference

8.4 What B1 Does Not Claim

• That amplitudes are ontologically fundamental (they represent substrate states)
• That quantum mechanics is fully derived (we have only established state representation)
• That this framework is empirically validated (empirical work is E-series)
• That amplitudes are probabilities (outcome weighting is B5)

8.5 Logical Position in B-Series

B1 provides:

• The representational framework for all subsequent B-papers
• The minimal structure required before addressing entanglement, dynamics, or measurement

B1 assumes:

• Substrate mechanics (A-series)
• Formal mechanisms (M1–M4)
• No quantum axioms

B1 enables:

• B2 (composite systems and entanglement)
• B3 (spectral discreteness)
• B4 (quantum dynamics)
• B5 (measurement and Born rule)

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9. Open Questions and Next Steps

9.1 Questions Deferred to Later Papers

For B2:

• How do amplitude spaces compose for subsystems?
• Why is the composition non-factorisable (entangled)?

For B3:

• How do modes (M3) map to quantum eigenstates?
• What selects the discrete spectrum?

For B4:

• What determines the evolution equation?
• Why is dynamics unitary?

For B5:

• How do amplitudes relate to probabilities?
• Why is the Born rule quadratic?

9.2 Potential Refinements to B1

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

• Tighten the necessity arguments for linearity
• Explore non-standard amplitude structures (quaternions, geometric algebra) and show why complex linear spaces are minimal
• Provide explicit substrate examples of mergeable histories
• Formalize the regrouping invariance constraint more rigorously

9.3 Relation to Empirical Work (E-series)

E-series papers empirically explore which substrate configurations produce constructor-capable continua. B1 shows that any such continuum must support linear amplitude representations if mergeable histories exist.

This creates a testable prediction: substrates lacking mergeability cannot exhibit quantum-like behaviour.

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10. Conclusion

We have established that linear amplitude-based state representation is not a quantum postulate, but a representational necessity for cohesive substrates with mergeable divergent histories.

Linearity is forced by:

• Associativity (consistent merger of multiple histories)
• Commutativity (history order irrelevance)
• History-count invariance (regrouping stability)
• Phase structure (from substrate provenance, M4)

Amplitudes are bookkeeping objects, not ontological entities. They track uncommitted substrate alternatives, not physical waves.

This paper provides the minimal representational framework required before addressing entanglement (B2), spectral discreteness (B3), dynamics (B4), and measurement (B5).

B1 establishes what a quantum state is. The remaining B-series papers will establish why states evolve as they do, why spectra are discrete, how systems compose, and how probabilities emerge.

The quantum formalism is not assumed—it is derived, step by step, from substrate mechanics.

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End of Paper B1