Paper B5 — Measurement and the Born Rule

Abstract

This paper demonstrates that the Born rule and quantum outcome probabilities emerge as a representational necessity once branching dynamics (B1–B4), discrete spectral structure (B3), and deterministic commit semantics (Paper A) are taken seriously. We show that quantum probabilities are not stochastic laws of nature, but epistemic weights over deterministically branching commit histories, forced by compatibility accounting under finite tolerance $W$ .

Building on the linear amplitude framework (B1), non-factorisable composite structure (B2), discrete spectral modes (B3), and unitary dynamics (B4), we derive that outcome weighting is not postulated but forced by representational consistency requirements. The derivation proceeds via substrate mechanics: when multiple mutually incompatible post-commit states exist after closure, an embedded observer cannot determine which branch they will occupy. This structural uncertainty requires a conserved branch measure that remains coherent under evolution and composition.

We establish that only quadratic weighting $w_k \propto |\alpha_k|^2$ satisfies all constraints: additivity under branch refinement, conservation under unitary evolution (B4), and correct reduction under coarse-graining. Any other weighting rule either breaks compatibility conservation, accumulates mismatch, or violates closure under repeated commits.

Critically, we show that measurement is not a new physical process but simply commit: the substrate resolution mechanism already defined in Paper A. “Collapse” is deterministic commit to one branch, not stochastic wavefunction reduction. Probabilities arise as branch-relative epistemic weights, not as fundamental randomness in nature.

This paper establishes the structural origin of the Born rule without importing probabilistic axioms, measurement postulates, or collapse mechanisms. Outcome weighting is representationally necessary, not empirically postulated. The result completes the B-series recovery of quantum mechanics: after B5, the entire quantum formalism is derived from substrate mechanics without importing quantum axioms.

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: $A_{\text{merged}} = \sum_i A_i$
Complex amplitude structure emerges from substrate provenance (M4)
Amplitudes track uncommitted alternatives, not finalized states
Basis elements correspond to substrate resolution paths

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
Reduced states are marginal compatibility summaries

From Paper B3 (Spectral Discreteness):

Discrete spectral bases from closure stability
Stable modes form discrete families indexed by integers $k \in \mathbb{Z}$
Continuous spectra generically fail closure and violate tolerance
Quantum numbers are mode labels, not fundamental quantities
Orthogonality arises from incompatibility under tolerance $W$

From Paper B4 (Quantum Dynamics):

Unitary evolution preserves closure and compatibility
Schrödinger-class dynamics are forced by consistency requirements
Hamiltonians are derived descriptors of stable transport
Time evolution emerges from commit cycle ordering
Decoherence corresponds to tolerance violation and partition (AX-PAR)

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 $X$
Commit semantics: configurations diverge and reconverge through admissible resolution
Commits are deterministic — each branch resolves to a definite outcome
Partition occurs when tolerance $W$ is violated

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 branch weighting analysis
AX-PAR (Partition on tolerance violation) — governs outcome exclusivity
AX-REL (Relational evolution) — underpins commit-based branching

1.2 What This Paper Does NOT Assume

This paper does not import:

Fundamental randomness or stochastic dynamics
Probability as an ontic primitive or physical law
Frequentist or ensemble interpretations
Decision theory or rationality axioms
Measurement postulates or collapse mechanisms as new physics
Many-worlds interpretations or ontological branching
Hidden variables or non-local influences
Quantum logic or lattice structures
Decoherence as an external process (it is tolerance violation)

1.3 Explicitly Out of Scope

This paper does not address:

Specific measurement apparatus design — how physical systems implement detectors
Decoherence timescales — quantitative rates of tolerance violation
Quantum-to-classical transition — when quantum structure becomes effectively classical
Macroscopic superpositions — why large systems don’t exhibit interference
Pointer basis selection — which observables decohere fastest
Continuous measurement — weak measurement and partial collapse
Quantum computing — error correction and quantum information processing

These are systematically deferred to later work or optional extensions.

2. The Representational Problem for Outcome Selection

2.1 Outcome Multiplicity from Branching Dynamics

From Papers B1–B4, we established that amplitude states evolve unitarily over discrete spectral bases. A general state has the form:

A = \sum_{k} \alpha_k A_k

where $\{A_k\}$ are discrete spectral modes (B3), $\{\alpha_k\}$ are complex coefficients encoding compatibility and provenance (B1), and evolution preserves total amplitude measure (B4).

From substrate mechanics (Paper A), evolution proceeds through commit cycles: discrete resolution points where constraints are jointly satisfied. Between commits, configurations may diverge through admissible resolution paths, exploring alternatives that remain mutually compatible under tolerance $W$ .

Key observation from commit semantics:

When closure occurs and the substrate commits, multiple outcomes may be available:

Each discrete mode $A_k$ represents a distinct post-commit configuration
These modes are mutually incompatible (from B3: orthogonal under tolerance $W$ )
Partition (AX-PAR) enforces that only one outcome is realized per branch
The substrate deterministically commits to exactly one of these branches

Central question: Given that the substrate deterministically selects one branch, how should an embedded observer weight the possibility of occupying each branch?

2.2 The Nature of Quantum Uncertainty

Critical distinction:

The substrate is deterministic. Commit semantics (Paper A) specify that:

Commits resolve to definite outcomes
No randomness exists at the substrate level
Each branch evolves according to consistent constraints
Partition separates incompatible branches

Why uncertainty arises:

An embedded observer is branch-relative:

The observer’s state is itself represented within the amplitude structure
Before commit, the observer exists in superposition across all branches
After commit, the observer occupies exactly one branch
The observer cannot predict which branch they will occupy

This uncertainty is epistemic, not dynamical:

It is not randomness in the substrate
It is not ignorance of hidden variables
It is structural inability to determine future self-location given only pre-commit amplitude structure

2.3 Why Outcome Weighting Is Necessary

Even though the substrate is deterministic, an embedded observer must assign epistemic weights to outcomes because:

Prediction requires weighting: To make predictions, observers must weight alternative futures
Repeated experiments show patterns: Even though each individual commit is deterministic, aggregate statistics over many similar configurations exhibit stable frequency patterns
Consistency across scales: Outcome weights must compose correctly under subsystem decomposition (B2)
Conservation under evolution: Weights must be preserved under unitary dynamics (B4)

The question is not whether to assign weights, but which weighting rule is forced by representational consistency.

3. Logic Chain — Why Quadratic Outcome Weighting Is Forced

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

Step 1: Branching is real and deterministic

From B1–B4, incompatible branches exist post-commit
Each branch is a definite substrate configuration
Partition (AX-PAR) ensures mutual exclusivity
No randomness exists at the substrate level

Step 2: Observers are branch-relative

Observer states are embedded in the amplitude structure
Pre-commit: observer exists in superposition
Post-commit: observer occupies exactly one branch
Observer cannot know which branch they will occupy

Step 3: Outcome uncertainty is unavoidable

Even with deterministic substrate evolution, future self-location is unknown
This is not epistemic limitation but structural necessity
The amplitude representation contains all branches but doesn’t specify which will be occupied

Step 4: Outcome weights must be conserved scalars

To remain coherent under:
- Unitary evolution (B4) — weights must not change during evolution
- Branch refinement — splitting one outcome into finer alternatives
- Composition (B2) — joint system weights from subsystem weights
- Coarse-graining — combining outcomes into aggregates
Weights must be real, non-negative scalars derived from amplitude structure

Step 5: Candidate weighting rules

Consider possible weighting functions $w_k = f(|\alpha_k|)$ :

Linear weighting: $w_k \propto |\alpha_k|$

Fails under composition: $|(\alpha_1, \alpha_2)| \ne |\alpha_1| + |\alpha_2|$
Violates tensor product structure from B2

Quadratic weighting: $w_k \propto |\alpha_k|^2$

Succeeds under composition: total measure is sum of individual measures
Preserved under unitary evolution (norm-squared conservation)
Additive under branch refinement

Higher-order weighting: $w_k \propto |\alpha_k|^n$ for $n > 2$

Fails under composition: doesn’t factorize correctly
Accumulates inconsistency under repeated measurements

Step 6: Only quadratic weights satisfy all constraints

The requirement that:

Weights are real, non-negative scalars
Weights are conserved under unitary evolution
Weights compose additively under branch refinement
Weights factorize correctly for composite systems

uniquely determines:

w_k = |\alpha_k|^2

Step 7: Born rule emerges as the unique solution

Normalizing to obtain probabilities:

P(k) = \frac{|\alpha_k|^2}{\sum_j |\alpha_j|^2}

This is the Born rule, derived from representational consistency, not postulated.

4. Formal Derivation

4.1 Outcome Weights as Branch Measures

Consider a pre-commit amplitude state:

A = \sum_{k=1}^{N} \alpha_k A_k

where $\{A_k\}$ are discrete orthogonal modes (from B3) and $\sum_k |\alpha_k|^2 = 1$ (normalized).

Definition (Branch Measure):

A branch measure is a function $w: \{\alpha_k\} \to \mathbb{R}_{\ge 0}$ that assigns a non-negative weight to each outcome $k$ .

Required properties:

Non-negativity: $w_k \ge 0$ for all $k$
Normalization: $\sum_k w_k = 1$ (for probabilistic interpretation)
Amplitude dependence: $w_k = w_k(\alpha_k)$ depends only on the amplitude
Phase insensitivity: $w_k(e^{i\theta}\alpha_k) = w_k(\alpha_k)$ (global phase irrelevant)

From property 4, $w_k$ can only depend on $|\alpha_k|$ :

w_k = f(|\alpha_k|)

for some function $f: \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ .

4.2 Conservation Under Unitary Evolution

From B4, amplitude states evolve unitarily:

A(t) = U(t) A(0)

where $U(t)$ is a unitary operator preserving the norm: $\sum_k |\alpha_k(t)|^2 = \sum_k |\alpha_k(0)|^2$ .

Requirement: Branch measures must be conserved under unitary evolution.

If we measure the system at time $t_1$ or time $t_2$ , the distribution of outcomes should be related by the deterministic unitary evolution, not by independent randomness at each measurement time.

Constraint: For any unitary $U$ :

\sum_k w_k(\alpha_k) = \sum_k w_k(\alpha'_k)

where $\alpha'_k$ are the amplitudes after unitary transformation.

Since unitary transformations preserve $\sum_k |\alpha_k|^2$ , we require:

\sum_k f(|\alpha_k|) = g\left(\sum_k |\alpha_k|^2\right)

for some function $g$ .

Simplest solution: $f(|\alpha|) = c|\alpha|^2$ for some constant $c$ .

With normalization $\sum_k w_k = 1$ and $\sum_k |\alpha_k|^2 = 1$ , we have $c = 1$ :

w_k = |\alpha_k|^2

Consider an outcome $k$ that can be decomposed into finer alternatives $\{k_1, k_2, \ldots, k_m\}$ .

For example, measuring “spin up” might be refined into “spin up AND position in region R1” plus “spin up AND position in region R2”.

The amplitude for outcome $k$ decomposes as:

\alpha_k = \sum_{i=1}^{m} \alpha_{k_i}

Requirement: The total weight for outcome $k$ should equal the sum of weights for refined outcomes:

w_k = \sum_{i=1}^{m} w_{k_i}

Testing linear weighting:

w_k = |\alpha_k| = \left|\sum_{i=1}^{m} \alpha_{k_i}\right|

By triangle inequality:

\left|\sum_{i=1}^{m} \alpha_{k_i}\right| \le \sum_{i=1}^{m} |\alpha_{k_i}|

with equality only when all $\alpha_{k_i}$ have the same phase.

Conclusion: Linear weighting fails additivity except in special cases.

Testing quadratic weighting:

For orthogonal refinements (different $k_i$ correspond to incompatible outcomes):

|\alpha_k|^2 = \left|\sum_{i=1}^{m} \alpha_{k_i}\right|^2 = \sum_{i=1}^{m} |\alpha_{k_i}|^2

when $\langle A_{k_i} | A_{k_j} \rangle = 0$ for $i \ne j$ .

Conclusion: Quadratic weighting succeeds under orthogonal refinement.

4.4 Composition for Joint Systems

From B2, composite systems are represented as tensor products:

A_{AB} = \sum_{i,j} \alpha_{ij} (A_i \otimes B_j)

Requirement: If we measure subsystem A only, the marginal weight for outcome $i$ should be:

w_i^A = \sum_j w_{ij}

where $w_{ij}$ is the joint weight for outcomes $(i,j)$ .

Testing quadratic weighting:

w_{ij} = |\alpha_{ij}|^2

w_i^A = \sum_j |\alpha_{ij}|^2

This is exactly the reduced density matrix formalism from quantum mechanics, which correctly describes marginal probabilities.

Testing linear weighting:

w_{ij} = |\alpha_{ij}|

w_i^A = \sum_j |\alpha_{ij}|

This does not match the required marginal structure from B2.

Conclusion: Only quadratic weighting correctly handles composite systems.

4.5 Uniqueness of Quadratic Weighting

Theorem: The unique branch measure satisfying:

Non-negativity and normalization
Conservation under unitary evolution
Additivity under orthogonal branch refinement
Correct composition for joint systems

is the quadratic measure:

w_k = |\alpha_k|^2

Proof sketch:

From conservation under unitary evolution (§4.2), we have $f(|\alpha|) = c|\alpha|^n$ for some $n \ge 0$ .

From additivity under orthogonal refinement (§4.3):

\left|\sum_i \alpha_i\right|^n = \sum_i |\alpha_i|^n

This holds for all orthogonal decompositions only when $n = 2$ .

For $n = 1$ : fails except when all phases align (shown in §4.3). For $n > 2$ : fails due to binomial expansion terms.

Only $n = 2$ satisfies the additivity requirement.

Normalization fixes $c = 1$ .

Therefore: $w_k = |\alpha_k|^2$ is unique. $\square$

5. Measurement as Commit

5.1 Measurement is Not New Physics

From the derivation above, we have established that outcome weights are determined by $|\alpha_k|^2$ . But what is measurement?

Key claim: Measurement is simply commit as defined in Paper A.

Measurement does not introduce new dynamics, collapse mechanisms, or stochastic processes. It is:

Interaction with environment causing tolerance violation (AX-PAR)
Partition into mutually incompatible branches
Deterministic resolution to exactly one branch per partition

5.2 The Measurement Process

Pre-measurement:

System in superposition: $A = \sum_k \alpha_k A_k$
All branches remain compatible under tolerance $W$
Unitary evolution (B4) preserves superposition

Measurement interaction:

System couples to measurement apparatus (another substrate system)
Joint state becomes entangled (B2): $A_{\text{joint}} = \sum_k \alpha_k (A_k \otimes M_k)$
Apparatus states $\{M_k\}$ become macroscopically distinct
Mismatch between branches $|\Delta \phi_{ij}| > W_{\text{spin}}$ violates tolerance

Post-measurement (commit):

Tolerance violation triggers partition (AX-PAR)
Branches separate into distinct cohesion domains
Each domain evolves independently
Observer occupies exactly one branch deterministically
From observer’s perspective: “collapse” to one outcome

5.3 Why “Collapse” Appears Stochastic

The substrate perspective:

Commit is deterministic
Each branch is definite
No randomness exists

The embedded observer perspective:

Observer cannot predict which branch they will occupy
Observer has access only to pre-commit amplitude structure
Observer assigns epistemic weight $P(k) = |\alpha_k|^2$ to each outcome
Repeated experiments show frequency distributions matching these weights

Key insight: The appearance of randomness is a consequence of:

Observer being embedded in the amplitude structure
Deterministic partition into branches
Epistemic uncertainty about self-location
Statistical patterns over many similar commits

This is epistemic uncertainty, not ontological randomness.

5.4 The Born Rule

Combining outcome weights with normalization:

P(k) = \frac{|\alpha_k|^2}{\sum_j |\alpha_j|^2}

This is the Born rule: the probability of observing outcome $k$ upon measurement is the squared amplitude.

But we have derived it as:

Not a probability of random events (substrate is deterministic)
Not a measurement postulate (measurement is commit)
Not an axiom (forced by representational consistency)
Branch-relative epistemic weight for self-location

The Born rule is representationally necessary given:

Linear amplitude structure (B1)
Unitary dynamics (B4)
Deterministic commit semantics (Paper A)
Observer embeddedness in amplitude representation

6. Resulting Structure

6.1 Quantum Measurement Theory Recovered

The derivation establishes:

1. Measurement outcomes are discrete

From B3: only discrete modes admit stable closure
Continuous spectra generically fail tolerance requirements

2. Outcome probabilities follow the Born rule

$P(k) = |\alpha_k|^2$ from representational consistency
Quadratic weighting uniquely satisfies all constraints

3. Measurement causes “collapse”

Collapse ≡ commit to one branch (deterministic)
Partition (AX-PAR) separates incompatible branches
No new physics beyond substrate mechanics

4. Repeated measurements show statistical patterns

Each commit is deterministic
Observer self-location is epistemically uncertain
Aggregate frequencies match $|\alpha_k|^2$ weights

5. No measurement problem

No wavefunction collapse as physical process
No preferred basis problem (determined by apparatus interaction)
No quantum-classical cut (all systems obey same substrate mechanics)

6.2 What Measurement Is

Measurement is:

Interaction causing tolerance violation
Entanglement with environment (B2)
Partition into incompatible branches (AX-PAR)
Deterministic commit to one branch

Measurement is NOT:

A fundamental physical process
Stochastic wavefunction reduction
Observer consciousness causing collapse
A boundary between quantum and classical

6.3 Epistemic vs Ontic Probability

Ontic probability: Randomness as fundamental feature of reality

Not required by this derivation
Substrate is deterministic

Epistemic probability: Uncertainty due to structural limitations

Required for embedded observers
Branch-relative self-location is unknowable pre-commit
Weights represent rational credence given amplitude structure

The Born rule is epistemic, not ontic.

6.4 Comparison with Standard Quantum Mechanics

Standard QM	Cohesion Dynamics (B-series)
Hilbert space (postulated)	Linear amplitude space (derived, B1)
Unitary evolution (postulated)	Closure-preserving dynamics (derived, B4)
Born rule (postulated)	Quadratic weighting (derived, B5)
Measurement collapse (postulated)	Commit and partition (from Paper A)
Probability (fundamental)	Branch-relative epistemic weight (derived)
Discrete spectra (boundary conditions)	Closure stability (derived, B3)

All quantum structure is derived from substrate mechanics.

7. Implications and Completeness

7.1 B-Series Recovery Complete

With B5, the B-series programme is complete. We have derived:

B1: Linear amplitude representation (why quantum states exist) B2: Entanglement (why states are non-factorisable) B3: Quantisation (why spectra are discrete) B4: Unitary dynamics (why evolution is Schrödinger-class) B5: Born rule (why outcomes are weighted quadratically)

No quantum postulates remain. Every element of standard quantum formalism has been shown to be representationally necessary for cohesive, tolerance-limited substrates.

7.2 What Remains Open

Substrate specification:

What is the discrete substrate $(\Sigma, V, \mathcal{C})$ for our universe?
What is the tolerance vector $W = (W_{\text{shape}}, W_{\text{clock}}, W_{\text{spin}})$ ?
How do we experimentally measure or constrain $W$ ?

System-specific Hamiltonians:

How do particular physical configurations determine effective Hamiltonians?
Hydrogen, harmonic oscillators, field theories — specific systems

Quantum field theory:

How does field structure emerge from substrate?
Particle creation, annihilation, Fock space

Gravity and spacetime:

How does metric structure emerge?
Relation to general relativity

These are physics questions, not foundational questions. The B-series establishes that quantum mechanics is forced — the remaining work is physics within that framework.

7.3 Interpretational Implications

No measurement problem:

Measurement is commit, already defined in substrate mechanics
No conflict between deterministic evolution and apparent randomness

No preferred basis problem:

Basis determined by apparatus interaction
Decoherence is tolerance violation (AX-PAR)
Pointer states are those that violate tolerance when entangled

No quantum-classical boundary:

All systems obey substrate mechanics
“Classical” systems are those where decoherence is rapid
No fundamental distinction, only effective description

Determinism and probabilities coexist:

Substrate is deterministic
Embedded observers face epistemic uncertainty
Born rule weights self-location probabilities

7.4 Empirical Programme

What B-series enables:

Experimental tests of substrate properties
- Constraining tolerance $W$ via interference experiments
- Testing closure requirements in quantum systems
- Searching for deviations from standard QM at tolerance boundaries
New predictions
- Violations of unitarity at tolerance limits
- Discrete structure at Planck scales (if $W$ is Planck-scale)
- Testable differences from Copenhagen, Many-Worlds, etc.
Unification programme
- Quantum mechanics as effective description
- General relativity as effective spacetime
- Both emerging from substrate mechanics

The B-series provides the formal bridge between substrate mechanics and experimental quantum mechanics, enabling precise empirical engagement.

8. Summary

8.1 What We Derived

Branching is deterministic — commits resolve to definite outcomes
Observers are branch-relative — cannot predict self-location
Outcome weights are necessary — to make predictions about many similar experiments
Quadratic weighting is unique — only $w_k = |\alpha_k|^2$ satisfies all constraints
Born rule is forced — $P(k) = |\alpha_k|^2 / \sum_j |\alpha_j|^2$
Measurement is commit — tolerance violation causes partition
Probabilities are epistemic — branch-relative weights, not ontological randomness

8.2 What We Did NOT Assume

No stochastic postulates
No measurement axioms
No collapse mechanisms
No many-worlds ontology
No hidden variables
No quantum logic
No probability as fundamental

Everything follows from:

Linear amplitude structure (B1)
Unitary evolution (B4)
Deterministic commit semantics (Paper A)
Finite tolerance and partition (AX-TOL, AX-PAR)

8.3 Programme Completion

The B-series is complete. Quantum mechanics is fully recovered as the unique stable calculus for representing cohesive, tolerance-limited substrates.

No interpretational gaps remain:

Superposition → uncommitted alternatives
Entanglement → joint admissibility constraints
Quantisation → closure stability
Unitary evolution → closure-preserving dynamics
Measurement → commit and partition
Probabilities → branch-relative epistemic weights

Cohesion Dynamics provides a complete substrate-level account of quantum mechanics, with no quantum axioms required.

The framework is now ready for:

Empirical calibration and testing
Extension to quantum field theory
Unification with spacetime emergence
New experimental predictions

The B-series establishes that quantum mechanics is not fundamental physics — it is effective representation of substrate physics.