Paper B3 — Spectral Discreteness from Closure Stability

Abstract

This paper demonstrates that discrete spectra (quantisation) emerge as a representational necessity for cohesive substrates governed by commit-based closure and finite tolerance $W$. We show that only discrete mode configurations admit stable, repeatable closure, while continuous configurations generically fail closure and therefore cannot persist as representable states.

Building on the linear amplitude framework (B1) and composite system structure (B2), we derive that quantisation is not a boundary condition, external constraint, or empirical postulate—it is a selection effect imposed by closure stability. The derivation proceeds via substrate mechanics: closure requires global compatibility satisfaction, which is generically met only at isolated points in parameter space forming discrete families indexed by integers or equivalent labels.

Continuous spectra, when examined under commit semantics, exhibit structural instability. Small perturbations accumulate mismatch across commit cycles, preventing repeatable closure. Discrete modes, by contrast, correspond to configurations where closure conditions are exactly satisfied, enabling stable reconstruction and persistent identity.

This paper establishes the structural origin of quantisation without importing quantum axioms, eigenvalue problems, Hamiltonians, or energy minimisation principles. Discreteness is forced by tolerance-limited admissibility, not imposed by measurement or boundary conditions.

The result enables B4 (dynamics over discrete spectral bases) and B5 (outcome weighting over discrete eigenfamilies). It does not yet address how these discrete modes evolve or how probabilities emerge—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: $A_{\text{merged}} = \sum_i A_i$
• Phase structure emerges from substrate provenance (M4)
• Amplitudes track uncommitted alternatives, not finalized states

From Paper B2 (Entanglement):

• Non-factorisable composite states arise from joint admissibility constraints
• Tensor product structure emerges as representational bookkeeping
• Separability is conditional, not fundamental

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 diverge and reconverge through admissible resolution

From Paper M3 (Modes):

• Modes as discrete basins in constraint state space
• Finite stable configurations under precedence
• Mode invariance under admissible updates
• Modes are not particles or energy levels—they are structural attractors

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
• Closure as joint satisfaction of internal and external constraints

Substrate Capability Assumption:

This paper assumes a quantum-capable substrate (Derived Capability Class: DCC-QM), as defined in R-DCC. This capability class encompasses the relational and constraint structures required for quantum-like representational emergence, including finite tolerance, coherence, admissibility, and partition semantics.

Granular axioms are referenced explicitly only where they play a direct operational role in the derivation. Specific axioms cited in this paper include:

• AX-TOL (Finite tolerance window $W$) — central to closure stability analysis
• AX-PAR (Partition on tolerance violation) — governs failure modes
• AX-SEL (Precedence selection) — drives mode restoration dynamics

1.2 What This Paper Does NOT Assume

This paper does not import:

• Hamiltonians or energy operators (dynamics is B4)
• Eigenvalue problems as primitives (we derive spectral structure)
• Schrödinger equations or wave equations
• Boundary conditions as fundamental constraints
• Energy minimisation principles
• Canonical commutation relations
• Hilbert space completeness (inner product structure not yet required)
• Probabilistic interpretation or Born rule (outcome weighting is B5)
• Measurement postulates (measurement is B5)

1.3 Explicitly Out of Scope

This paper does not address:

• Quantum dynamics (B4) — how discrete modes evolve
• Measurement and probabilities (B5) — outcome weighting and Born rule
• Specific physical systems — hydrogen, harmonic oscillators (those are examples, not derivations)
• Continuous approximations — when discreteness becomes effectively continuous
• Spectral densities — measure-theoretic structure of eigenspaces

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

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2. The Representational Problem for Spectral Structure

2.1 Modes in Substrate Mechanics

From M3, we established that substrate evolution under constraint resolution produces discrete modes: stable basins in configuration space characterized by:

1. Finite admissible configurations satisfying local constraints
2. Precedence-selected updates minimizing mismatch
3. Structural invariance under admissible perturbations
4. Reusability — same mode can be re-entered from different histories

Modes are not energy levels, particles, or quantum states. They are structural features of constraint geometry.

2.2 The Question: Why Are Stable Modes Discrete?

While M3 established that modes exist as discrete basins, it did not explain why continuous families of stable modes are excluded.

In principle, one might expect:

• A continuous spectrum of admissible configurations
• Smooth variation in constraint satisfaction
• Arbitrary precision in mode specification

Instead, physical systems exhibit:

• Discrete energy levels (hydrogen: $E_n = -13.6/n^2$ eV)
• Quantised angular momentum ($L = n\hbar$)
• Discrete vibrational modes
• Integer-indexed quantum numbers

Central question: Why does substrate closure select discrete mode families rather than continuous spectra?

2.3 Closure as a Selection Mechanism

From M4 and substrate commit semantics (Paper A), a configuration persists as a representable state only if it admits closure: joint satisfaction of all constraints within tolerance $W$.

Closure is not a selection rule or optimization criterion. It is a consistency requirement:

A configuration closes if its internal constraint structure and external compatibility constraints are simultaneously satisfied to within tolerance $W$.

Configurations that fail closure cannot persist as repeatable, reconstructible states. They are not “rejected” or “forbidden”—they simply do not stabilize under repeated commit cycles.

Hypothesis (to be proven): Closure is satisfied only on a measure-zero subset of continuous parameter space, producing discrete families of stable modes.

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3. Logic Chain — Why Discreteness Is Forced

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

Step 1: Closure requires global compatibility

• A configuration closes when all constraints (internal and external) are jointly satisfied
• This imposes a global condition on configuration parameters
• Generic continuous configurations do not meet this condition

Step 2: Closure defect vanishes on isolated configurations

• Define a closure defect $\Delta(\phi)$ measuring deviation from exact closure
• For continuous parameter families, $\Delta(\phi) \neq 0$ generically
• $\Delta(\phi) = 0$ only at isolated points forming discrete sets

Step 3: Continuous spectra accumulate closure defects

• Configurations with $\Delta(\phi) > 0$ fail to close after finite cycles
• Perturbations grow across repeated commits
• Such states cannot be repeatably reconstructed

Step 4: Discrete modes admit exact closure

• Configurations where $\Delta(\phi) = 0$ satisfy closure exactly
• These form stable mode families indexed by integers $n \in \mathbb{Z}$
• They are repeatable, reconstructible, and identifiable

Step 5: Representable states are discrete

• Only configurations admitting stable closure can persist as representable amplitude states (from B1)
• Therefore, the representable spectrum is necessarily discrete

Conclusion: Quantisation is not a postulate—it is a selection effect imposed by commit-based closure under finite tolerance.

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4. Derivation — Closure Stability and Spectral Discreteness

Note on theorem language: Theorems in this paper are structural results within the defined substrate axioms; they assert necessity relative to those axioms, not absolute mathematical truth independent of the model.

4.1 Notation and Setup

Let $\phi$ denote a continuous parameter characterizing a family of substrate configurations. This could represent:

• Phase alignment (from M4)
• Internal mode structure
• Composite system compatibility parameters

Let $X(\phi)$ denote the configuration corresponding to parameter value $\phi$.

From B1, the amplitude representation for such a configuration is:

$$A(\phi) = \sum_i \alpha_i(\phi) A_i$$

where $\{\alpha_i(\phi)\}$ encode the compatibility and provenance structure.

4.2 Closure Condition

From M4, closure requires that the phase and compatibility structure satisfy:

$$|\Delta \phi_{\text{internal}}| \le W_{\text{spin}}$$

and that mismatch across one commit cycle returns to initial values:

$$M(X(\phi); \text{after cycle}) = M(X(\phi); \text{before cycle})$$

Definition (Closure Defect): The closure defect measures deviation from exact closure:

$$\Delta(\phi) = \left| M(X(\phi); \text{after cycle}) - M(X(\phi); \text{before cycle}) \right|$$

A configuration closes if and only if $\Delta(\phi) = 0$.

4.3 Generic Failure of Closure for Continuous Families

Theorem 4.3.1 (Closure is measure-zero): For a continuous parameter family $X(\phi)$ with $\phi \in \mathbb{R}$, the set of parameters satisfying $\Delta(\phi) = 0$ is generically measure zero.

Proof sketch:

1. Constraint satisfaction is overdetermined

   • Closure requires simultaneous satisfaction of multiple constraints: internal compatibility, external alignment, phase consistency
   • Each constraint imposes an independent condition on $\phi$
   • For $n$ independent constraints, exact satisfaction defines an $(n-1)$-dimensional submanifold or lower in parameter space, whose intersection in a one-dimensional parameter space is generically discrete

2. Intersection of constraint manifolds

   • Joint satisfaction requires intersection of multiple constraint manifolds
   • Generically, $n$ independent conditions in 1-dimensional parameter space intersect at isolated points
   • These points form a discrete set $\{\phi_k\}$ with $k \in \mathbb{Z}$ or equivalent discrete label

3. Continuous families have non-zero defect

   • For $\phi \notin \{\phi_k\}$, at least one constraint is not exactly satisfied
   • Mismatch accumulates: $\Delta(\phi) > 0$
   • After $N$ cycles, accumulated defect grows: $\Delta_N(\phi) \approx N \cdot \Delta(\phi)$

4. Tolerance violation

   • Once accumulated defect exceeds tolerance: $\Delta_N(\phi) > W$
   • Configuration violates AX-TOL and triggers partition (AX-PAR)
   • State cannot persist as coherent representation

Therefore, only the discrete set $\{\phi_k\}$ where $\Delta(\phi_k) = 0$ admits stable closure. $\square$

Clarification on measure-zero: “Measure zero” means the set of closure-satisfying parameters has zero Lebesgue measure in the continuous parameter space—it forms a countable and isolated discrete set, not a continuous interval or merely sparse subset. The closure-satisfying set is not dense: configurations arbitrarily close to $\phi_k$ but not exactly at $\phi_k$ accumulate closure defects and fail stability. This excludes dense-but-non-closing sets explicitly.

The strategic role of finite tolerance $W$: The finiteness and nonzero value of $W$ is critical for discrete spectral structure:

• If $W = \infty$ (infinite tolerance), closure defects never accumulate to violation → continuous spectra persist, no discreteness emerges.
• If $W = 0$ (zero tolerance), no configurations satisfy closure → no modes persist, no coherent states exist at all.
• Only finite, nonzero $W$ produces the regime where discrete, stable spectra emerge as the unique closure-admissible configurations.

This demonstrates that quantisation is not merely enabled by tolerance—it is forced by the specific constraint structure of finite $W$ under commit semantics.

4.4 Discrete Mode Families

Definition (Stable Mode): A stable mode is a configuration $X(\phi_k)$ where:

1. Closure defect vanishes: $\Delta(\phi_k) = 0$
2. Closure is stable under small perturbations: $\frac{d\Delta}{d\phi}\big|_{\phi=\phi_k} \neq 0$ (non-degenerate)
3. The configuration can be repeatably reconstructed

Proposition 4.4.1 (Integer indexing): Stable modes $\{\phi_k\}$ form a discrete family indexed by integers or equivalent discrete labels:

$$\phi_k, \quad k \in \mathbb{Z} \text{ or } k \in \mathbb{Z}^d \text{ for composite systems}$$

Proof: From topological properties of closure conditions:

• Closure defect $\Delta(\phi)$ is a continuous function
• Zeros of $\Delta(\phi)$ are isolated (non-degenerate case)
• Stable modes can be ordered and labeled sequentially
• In bounded regions, finite number of stable modes
• In unbounded parameter space, countably infinite modes

This produces integer or multi-integer indexing, which is the substrate origin of quantum numbers. $\square$

4.5 Why Continuous Spectra Are Unstable

Theorem 4.5.1 (Continuous spectra fail closure): Any continuous family of configurations $\{X(\phi) : \phi \in [a,b]\}$ with $b > a$ contains at most measure-zero subset satisfying closure.

Proof: From Theorem 4.3.1, closure requires $\Delta(\phi) = 0$.

For continuous interval $[a,b]$:

• If $\Delta(\phi) = 0$ for all $\phi \in [a,b]$, then $\Delta$ is identically zero
• This requires all constraint equations to be functionally dependent
• Generically, constraints are independent (no fine-tuning)
• Therefore, $\Delta(\phi) = 0$ only at isolated points, not intervals

Consequence: Continuous spectra cannot be stably represented. Any attempt to represent a continuous family results in:

• Accumulated closure defects across commit cycles
• Tolerance violation after finite time
• Loss of coherence and partition (AX-PAR)

Only discrete modes persist. $\square$

4.6 Stability Analysis

Why don’t small perturbations destroy discrete modes?

Discrete modes $\phi_k$ satisfying $\Delta(\phi_k) = 0$ are attractors in the following sense:

Proposition 4.6.1 (Basin of attraction): Each stable mode $\phi_k$ admits a neighborhood $B_k = [\phi_k - \epsilon, \phi_k + \epsilon]$ such that:

• Configurations $X(\phi)$ with $\phi \in B_k$ experience restoring dynamics
• Precedence selection (AX-SEL) drives configurations toward $\phi_k$
• Small perturbations $|\phi - \phi_k| < \epsilon$ relax back to $\phi_k$

Proof sketch: From M3 mode stability:

• Modes are basins in constraint state space
• Perturbations introduce mismatch
• Precedence-driven updates reduce mismatch
• Mismatch-minimizing path leads back to mode center $\phi_k$

This is why discrete modes are repeatable and reconstructible—they are stable under substrate dynamics. $\square$

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5. Representational Outcome — Discrete Spectral Structure

5.1 Amplitude Basis Indexed by Quantum Numbers

From B1, amplitude representations require basis elements corresponding to admissible substrate states.

From the derivation above, admissible stable states form discrete families $\{\phi_k\}$.

Therefore, the amplitude basis is necessarily discrete:

$$A = \sum_{k} \alpha_k A_k$$

where:

• $A_k$ represents the amplitude for stable mode $k$
• $k \in \mathbb{Z}$ (or $\mathbb{Z}^d$ for composite systems)
• Continuous superpositions over $k$ are allowed (linearity from B1)
• Continuous variation within a single $k$ is not admissible (fails closure)

This is the substrate origin of discrete quantum bases.

Two distinct notions of continuity: It is critical to distinguish:

1. Continuity of representation — superpositions over discrete modes with continuously varying coefficients $\alpha_k$ are allowed (this is linear amplitude composition from B1).
2. Continuity of admissible substrate configurations — continuous families of substrate states generically fail closure and are unstable (as proven in §4.3).

Quantum mechanics employs the first (continuous amplitude coefficients over discrete bases) while excluding the second (continuous mode spectra). This distinction is structurally forced, not conventional.

5.2 Quantum Numbers as Mode Labels

The discrete index $k$ is the substrate analogue of quantum numbers:

• In hydrogen: $k = (n, l, m)$ labeling energy, angular momentum, projection
• In harmonic oscillators: $k = n$ labeling vibrational quanta
• In spin systems: $k = \pm 1/2$ labeling spin orientation

These are not fundamental quantities. They are labels for discrete closure-admissible configurations.

5.3 Orthogonality via Incompatibility

Distinct stable modes $\phi_k$ and $\phi_{k'}$ with $k \neq k'$ are incompatible in the sense of M4:

$$|\Delta \phi_{kk'}| > W_{\text{spin}} \quad \Rightarrow \quad \text{modes } k, k' \text{ are non-mergeable}$$

This incompatibility is the substrate origin of orthogonality in quantum mechanics.

This is a representational orthogonality induced by incompatibility; inner product structure itself is derived later (B4), not assumed here. Modes that cannot merge cannot interfere. In the amplitude representation:

• Orthogonal modes: $\langle A_k | A_{k'} \rangle = 0$ (when inner product is introduced in B4)
• Non-orthogonal modes: compatible within tolerance, can merge

5.4 Continuous Approximations Are Derived, Not Fundamental

When discrete mode spacing becomes small relative to system resolution:

$$\Delta \phi_k = \phi_{k+1} - \phi_k \ll W_{\text{spin}}$$

the discrete spectrum appears continuous at accessible resolution.

But this is an approximation, not fundamental structure:

• True substrate states remain discrete
• Continuous descriptions are effective, not exact
• Discreteness re-emerges at sufficient precision

This explains why classical mechanics works (coarse-grained discreteness) while quantum mechanics is fundamental (exact discreteness).

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6. Why Quantisation Is Forced, Not Chosen

This section explicitly rules out alternative explanations for discrete spectra.

6.1 Not Boundary Conditions

Claim: Quantisation does not arise from spatial boundaries or confinement.

Argument:

• Closure stability applies to all substrate configurations, bounded or unbounded
• Discrete modes emerge from global compatibility constraints, not spatial edges
• Even in infinite parameter space, closure defect vanishes on discrete sets

Example: Free particles in quantum mechanics exhibit continuous momentum spectra. This is consistent: momentum corresponds to phase rate (from M4), which is a continuous parameter when no binding constraints exist. Discrete spectra arise only when closure constraints bind the system.

6.2 Not External Constraints

Claim: Quantisation is not imposed by measurement apparatus or external observers.

Argument:

• Closure conditions are internal to substrate (from Paper A)
• No external agent selects discrete values
• Discreteness arises from self-consistency of commit cycles

Measurement (B5) will later show how discrete outcomes are selected, but discreteness itself is prior to measurement.

6.3 Not Discretisation Artefacts

Claim: Quantisation is not a computational or simulator artifact.

Argument:

• Discrete substrate $\Sigma$ at the foundational level does not imply discrete modes
• Continuous parameter families $X(\phi)$ are representable in discrete substrate
• Spectral discreteness arises from closure stability, not alphabet granularity

The substrate is discrete, but mode structure could in principle be continuous. That it is not is a derived result, not an assumption.

6.4 Quantisation Is Necessary

Combining the above:

Discrete spectra are unavoidable in any cohesive substrate governed by commit-based closure and finite tolerance $W$.

This is not:

• A modeling choice
• An empirical postulate
• A boundary condition
• A measurement effect

It is a structural necessity following from substrate mechanics.

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7. Implications for B4 and B5

7.1 What B3 Enables

The discrete spectral structure established here provides the foundation for:

B4 (Quantum Dynamics):

• Evolution equations will act on discrete basis $\{A_k\}$
• Hamiltonian-like operators (if they emerge) will have discrete eigenvalues
• Transitions between modes will be governed by closure-preserving dynamics

B5 (Measurement and Born Rule):

• Outcome probabilities will be defined over discrete eigenfamilies
• Measurement will correspond to commit into one stable mode $k$
• Continuous measurement results will be shown to be coarse-grained discreteness

7.2 What B3 Does Not Prejudge

This paper does not presuppose:

• The form of dynamics (unitary, non-unitary, Schrödinger-like)
• Which observables have discrete spectra (position, momentum, energy)
• The relationship between discrete modes and physical quantities
• How transitions between discrete modes occur (B4)
• How discrete outcomes are weighted (B5)

These will be constrained by later derivations.

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8. Summary and Scope Boundaries

8.1 Central Result

We have shown:

In a substrate governed by commit-based closure and finite tolerance $W$, only discrete mode configurations admit stable, repeatable closure. Continuous spectra generically fail closure and cannot persist as representable states.

This is a necessity claim, not a modeling choice.

8.2 Derivation Path

1. Substrate evolution produces modes (M3)
2. Modes persist only if they admit closure (M4, Paper A)
3. Closure requires global compatibility satisfaction
4. Compatibility is satisfied only at isolated points in parameter space
5. Isolated points form discrete families $\{\phi_k\}$ indexed by integers
6. Continuous families accumulate closure defects and violate tolerance
7. Only discrete modes are stably representable
8. Discrete amplitude basis emerges as necessity

8.3 What B3 Establishes

• Discrete spectra are forced, not assumed
• Quantisation is a selection effect, not a postulate
• Continuous spectra are structurally unstable
• Quantum numbers are mode labels, not fundamental quantities
• Orthogonality arises from incompatibility, not inner product axioms

8.4 What B3 Does Not Claim

• That all observables have discrete spectra (position and momentum may differ)
• That discreteness is empirically validated (empirical work is E-series)
• That quantum mechanics is fully derived (dynamics and probabilities are B4, B5)
• That specific systems have been analyzed (hydrogen, etc., are examples)

8.5 Logical Position in B-Series

B3 assumes:

• Linear amplitude representation (B1)
• Composite system structure (B2)
• Substrate mechanics (A-series)
• Modes and phase structure (M3, M4)
• No quantum postulates

B3 provides:

• Discrete spectral structure
• Quantum number indexing
• Orthogonality from incompatibility
• Selection mechanism for quantisation

B3 enables:

• B4 (dynamics over discrete bases)
• B5 (outcome weighting over discrete states)

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9. Open Questions and Next Steps

9.1 Questions Deferred to Later Papers

For B4:

• How do discrete modes evolve dynamically?
• What determines transition amplitudes between modes?
• Why is evolution unitary (if it is)?

For B5:

• How are discrete outcomes weighted probabilistically?
• Why is the Born rule quadratic?
• How does measurement select one discrete mode?

9.2 Potential Refinements to B3

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

• Formalize closure defect $\Delta(\phi)$ more rigorously using constraint satisfaction theory
• Provide explicit substrate examples showing discrete mode emergence
• Analyze the transition from discrete to continuous approximations
• Develop quantitative criteria for mode stability
• Explore systems where continuous spectra arise (unbound states)

9.3 Relation to Empirical Work (E-series)

E-series papers (particularly E2/Q6) empirically explore which substrate configurations produce constructor-capable continua. B3 shows that any such continuum supporting closed modes must exhibit discrete spectra.

This creates a testable prediction: substrates lacking closure stability cannot exhibit quantisation.

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10. Conclusion

We have established that discrete spectra (quantisation) are not a quantum postulate, but a representational necessity for cohesive substrates with commit-based closure under finite tolerance $W$.

Discreteness is forced by:

• Closure requiring global compatibility (M4)
• Compatibility satisfied only at isolated parameter values
• Continuous families failing closure after finite commit cycles
• Stable modes forming discrete families indexed by integers

Quantisation is not imposed by:

• Boundary conditions
• Measurement apparatus
• Observer selection
• Computational artifacts

It is a selection effect: only configurations admitting exact closure persist as representable states.

This paper completes the spectral structure of quantum mechanics. The remaining B-series papers will establish how discrete modes evolve (B4) and how outcomes are weighted (B5).

B3 establishes why quantum spectra are discrete. B1 established what quantum states are. B2 established why composite systems entangle. Together, they form the core representational structure of quantum mechanics—derived, not assumed.

The quantum formalism emerges, step by step, from substrate mechanics.

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End of Paper B3