Paper B4 — Quantum Dynamics from Commit-Based Evolution

Abstract

This paper demonstrates that quantum dynamics (unitary, Schrödinger-class evolution) emerge as the unique closure-preserving transport law for amplitude states over the discrete spectral basis established in Paper B3. We show that given a discrete amplitude basis and linear superposition (B1), only unitary evolution preserves closure, compatibility, and total amplitude measure across commit cycles under finite tolerance $W$.

Building on the linear amplitude framework (B1), non-factorisable composite structure (B2), and discrete spectral modes (B3), we derive that quantum dynamics are not postulated but forced by representational consistency requirements. The derivation proceeds via substrate mechanics: evolution between commits must preserve the closure conditions that stabilize discrete modes, and this preservation uniquely determines unitary transformation structure.

Generic evolution laws either destroy closure or violate coherence. Only a restricted class of linear, norm-preserving transformations survive substrate constraints. These transformations are representationally equivalent to Schrödinger evolution and form the unitary group structure characteristic of quantum mechanics.

Critically, we show that Hamiltonians are not fundamental operators but derived descriptors of stable transport—bookkeeping devices for closure-preserving dynamics. Time evolution itself emerges as the ordering structure of commit cycles, not as primitive background structure. Decoherence corresponds to tolerance violation and partition (AX-PAR), not to stochastic collapse.

This paper establishes the structural origin of quantum dynamics without importing Hamiltonians, energy principles, wave equations, or time-evolution postulates. Dynamics are representationally necessary, not empirically postulated. The result enables B5 (measurement and Born rule) by providing the evolution framework over which discrete outcomes are weighted.

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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 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 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
• Local modification operations $(v \to s)$

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 preservation analysis
• AX-PAR (Partition on tolerance violation) — governs decoherence
• AX-SEL (Precedence selection) — drives admissibility dynamics
• AX-REL (Relational evolution) — underpins commit-based transport

1.2 What This Paper Does NOT Assume

This paper does not import:

• Hamiltonians as primitives (we derive them as generators)
• Time-dependent wave equations (we derive Schrödinger-class structure)
• Energy minimisation principles (energy emerges, not assumed)
• Canonical commutation relations (algebraic structure not yet required)
• Measurement postulates (measurement is B5)
• Probability or Born rule (outcome weighting is B5)
• Collapse mechanisms (decoherence is tolerance violation)
• Background time as primitive (time emerges from commit ordering)
• Unitary axioms (unitarity is derived, not assumed)

1.3 Explicitly Out of Scope

This paper does not address:

• Measurement and probabilities (B5) — outcome weighting, Born rule, measurement selection
• Specific Hamiltonians — how particular physical systems determine generators
• Time-dependent Hamiltonians — non-autonomous dynamics
• Relativistic quantum mechanics — spacetime structure not yet established
• Open systems and CPTP maps — non-unitary dynamics beyond partition
• Quantum field theory — field structure and particle creation

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

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2. The Representational Problem for Dynamics

2.1 State Evolution in Substrate Mechanics

From Paper A, substrate 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 $W$.

From B1–B3, we established that uncommitted substrate alternatives are represented as amplitude states over discrete spectral bases:

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

where $k$ indexes discrete stable modes (B3), and $\{\alpha_k\}$ are complex coefficients encoding compatibility and provenance (B1).

Central question: How do these amplitude states evolve between commit cycles?

2.2 What “Evolution” Means in This Context

Evolution is not continuous change in background time. It is:

1. Transport of amplitude structure between commit cycles
2. Preservation of representational consistency across resolution steps
3. Maintenance of closure conditions that stabilize discrete modes

Evolution acts on the representation, not directly on substrate states. The substrate evolves through admissible constraint resolution (Paper A). The amplitude representation must track this evolution faithfully.

2.3 Constraints on Admissible Evolution Laws

Any evolution law operating on amplitude states must satisfy:

Closure preservation:

• States that close at one commit must remain closable at subsequent commits
• Closure defect $\Delta(\phi)$ (from B3) must not accumulate
• Tolerance $W$ must not be systematically violated

Compatibility preservation:

• Compatible alternatives must remain compatible (within tolerance $W$)
• Incompatible modes must remain incompatible
• Phase relationships must be tracked consistently

Linearity:

• From B1, amplitude composition is linear
• Evolution must preserve linear structure
• Superpositions must evolve coherently

Reversibility:

• Substrate commit cycles are reversible (from Paper A)
• Information is not lost between commits (ontological commitment)
• Evolution must be invertible

Question: What evolution laws satisfy all these constraints?

2.4 Generic Evolution Laws Fail

Consider a generic evolution operator $U$ mapping amplitude states:

$$A(t_2) = U \cdot A(t_1)$$

If $U$ is nonlinear:

• Violates amplitude composition structure (B1)
• Breaks mergeability of alternatives
• Destroys interference

If $U$ is not norm-preserving:

• Total amplitude measure drifts
• Closure conditions fail after finite cycles
• Tolerance violations accumulate

If $U$ is not reversible:

• Violates substrate information conservation
• Prevents reconstruction of prior states
• Incompatible with commit semantics

Only linear, norm-preserving, reversible transformations remain admissible.

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3. Logic Chain — Why Quantum Dynamics Are Forced

This section provides a high-level argument outline before the formal derivation, following the naming convention established in Paper B3.

Step 1: Dynamics act between commits

• Commit semantics (Paper A) define discrete resolution points
• Evolution occurs as transport of amplitude between commits
• No continuous background time; time emerges from commit ordering

Step 2: Amplitude transport must preserve closure

• From B3, only discrete modes admit stable closure
• Evolution must not introduce closure defects $\Delta(\phi) > 0$
• Total compatibility must remain within tolerance $W$
• Closure defects accumulate if norms are not preserved

Step 3: Linearity is already fixed (B1)

• Amplitude composition is linear by representational necessity
• Evolution must preserve linear structure
• Nonlinear evolution breaks mergeability of alternatives
• Only linear maps preserve interference structure

Step 4: Norm preservation is forced

• If total amplitude measure drifts, closure fails
• Accumulated measure deviation: $\Delta N = N(t_2) - N(t_1)$
• After $n$ commit cycles: $\Delta N_n \approx n \cdot \Delta N$
• Tolerance violation when $\Delta N_n > W_{\text{norm}}$
• This eliminates non-isometric transformations

Step 5: Reversibility follows from substrate information conservation

• Information is ontologically conserved (foundational commitment)
• Commit cycles are reversible (Paper A)
• Evolution must be invertible
• Combined with norm preservation: evolution must be unitary

Step 6: Allowed transformations form a unitary group

• Closure + linearity + norm-preservation + reversibility restrict evolution to unitary operators
• Composition of unitary maps is unitary
• Continuous families of unitaries form one-parameter groups
• Schrödinger-type evolution is the continuous limit

Step 7: Dynamics are representational, not ontological

• No Hamiltonian is assumed as primitive
• The generator emerges as bookkeeping for stable transport
• Time evolution is commit ordering, not background flow
• Hamiltonians describe closure-preserving transport rates

Conclusion: Given B1–B3 and commit semantics, something mathematically equivalent to Schrödinger evolution is unavoidable.

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4. Derivation — Closure-Preserving Evolution

Note on theorem language: Theorems in this paper are structural results within the defined substrate axioms. They assert necessity relative to those axioms and the representational framework established in B1–B3, not absolute mathematical truth independent of the model. Where we use standard mathematical terminology (unitary, Hermitian, etc.), these terms denote representational structures forced by substrate mechanics, not pre-existing quantum axioms.

4.1 Notation and Setup

Let $A(t)$ denote the amplitude state at commit cycle $t \in \mathbb{N}$ (discrete commit index, not continuous time).

From B3, the state is expanded in the discrete spectral basis:

$$A(t) = \sum_{k} \alpha_k(t) A_k$$

where $\{A_k\}$ are stable mode basis elements.

Evolution maps:

$$A(t+1) = U \cdot A(t)$$

where $U$ is an evolution operator to be determined.

4.2 Linearity Requirement

Theorem 4.2.1 (Evolution must be linear): Any evolution law preserving amplitude composition structure (B1) must be a linear operator.

Proof: From B1, amplitude states form a linear space. If two states $A_1$ and $A_2$ represent compatible alternatives, their merged representation is:

$$A_{\text{merged}} = A_1 + A_2$$

Evolution must preserve this merger structure. If $U$ is the evolution operator:

$$U(A_1 + A_2) = U(A_1) + U(A_2)$$

Similarly, for scalar multiplication (phase and magnitude):

$$U(\alpha A) = \alpha U(A)$$

These are the defining properties of linear operators. Therefore, $U$ must be linear. $\square$

Consequence: Evolution is a linear transformation on the amplitude space.

4.3 Closure Preservation and Norm Conservation

Definition (Amplitude Measure): Define the total amplitude measure as:

$$N(A) = \sum_{k} |\alpha_k|^2$$

This is the sum of squared magnitudes over the discrete basis.

Clarification on norm interpretation: The quantity $N(A) = \sum_k |\alpha_k|^2$ is representationally proportional to the total compatibility load, not assumed as probability (Born rule) or physical norm. The squared form arises naturally from the bilinear structure of compatibility relationships under tolerance $W$ (M4). The interpretation of this scalar as outcome probability is deferred to B5. At this stage, $N(A)$ is purely a bookkeeping scalar tracking the total amplitude structure’s compatibility burden.

Theorem 4.3.1 (Norm preservation is forced): Evolution must preserve the total amplitude measure: $N(A(t+1)) = N(A(t))$ for all $t$.

Proof:

Step 1: Closure conditions depend on total compatibility load

From M4 and B3, closure requires:

$$L_{\text{total}}^2 = \sum_{k} L_k^2 \le W^2$$

where $L_k$ represents the compatibility load contribution from mode $k$.

The total amplitude measure $N(A)$ is representationally proportional to the total compatibility load. If $N(A)$ drifts (increases or decreases), the closure load changes correspondingly, introducing mismatch accumulation.

Step 2: Non-isometric evolution accumulates closure defects

Suppose evolution is not norm-preserving:

$$N(A(t+1)) = N(A(t)) + \Delta N$$

where $\Delta N \neq 0$.

After $n$ commit cycles:

$$N(A(t+n)) = N(A(t)) + n \cdot \Delta N$$

Step 3: Tolerance violation after finite cycles

Once the accumulated deviation exceeds tolerance:

$$|n \cdot \Delta N| > W_{\text{norm}}$$

the closure condition fails. The state can no longer be stably represented.

Step 4: Only norm-preserving evolution is stable

To prevent accumulated closure violation, evolution must satisfy:

$$N(A(t+1)) = N(A(t))$$

for all $t$. This is norm preservation. $\square$

Interpretation: Norm preservation is not a quantum postulate. It is forced by the requirement that discrete modes (B3) remain stably closable across commit cycles. Note that any non-conserved compatibility scalar would similarly accumulate mismatch under commit dynamics, leading to eventual tolerance violation. The squared-magnitude form is representationally necessary, not chosen for convenience.

4.4 Reversibility and Unitarity

Theorem 4.4.1 (Evolution must be reversible): Evolution operators must be invertible.

Proof:

Step 1: Substrate evolution is reversible

From Paper A, substrate commit cycles are reversible. Constraints can be resolved forward and backward. More specifically:

• Commit semantics (Paper A) establish discrete resolution as bidirectional
• Relational evolution (AX-REL) governs all dynamics relationally without directional bias
• Information conservation is a foundational ontological commitment already locked in F-series and A-series

Reversibility is not a hidden physics postulate but a substrate commitment already established.

Step 2: Representation must track substrate reversibility

If substrate evolution is reversible, the amplitude representation must also be reversible. Otherwise, the representation loses information about substrate states, violating faithfulness.

Step 3: Reversibility requires invertibility

For evolution operator $U$:

$$A(t+1) = U \cdot A(t)$$

Reversibility requires existence of $U^{-1}$ such that:

$$A(t) = U^{-1} \cdot A(t+1)$$

Therefore, $U$ must be invertible. $\square$

Theorem 4.4.2 (Unitarity is forced): Linear, norm-preserving, invertible operators on amplitude spaces are unitary.

Proof:

From Theorems 4.2.1, 4.3.1, and 4.4.1:

• $U$ is linear
• $U$ preserves norm: $N(U \cdot A) = N(A)$
• $U$ is invertible

Definition of unitary operator: An operator $U$ is unitary if:

$$U^\dagger U = U U^\dagger = I$$

where $U^\dagger$ is the adjoint (conjugate transpose).

Equivalence: Linear, norm-preserving, invertible operators on finite-dimensional complex vector spaces are precisely the unitary operators.

(This is a standard result from linear algebra. Norm preservation $\implies$ $\langle U A | U A \rangle = \langle A | A \rangle$ for all $A$ $\implies$ $U^\dagger U = I$. Invertibility ensures $U U^\dagger = I$.)

Therefore, $U$ must be unitary. $\square$

Consequence: Evolution operators on amplitude states form the unitary group $U(n)$, where $n$ is the dimensionality of the amplitude space.

4.5 Continuous-Time Limit and Schrödinger Evolution

Discrete vs. Continuous Commits:

Commit cycles are fundamentally discrete. However, when commit spacing becomes small relative to system timescales, evolution appears continuous. We now derive the continuous-time limit.

Critical clarification: Continuous time arises as a representational limit of dense commit ordering, not as an ontological claim. No claim is made that commits become literally continuous at the substrate level. The substrate remains discrete; the continuous-time description is an effective approximation valid when $\delta t \ll$ characteristic system timescales.

Theorem 4.5.1 (Schrödinger-class evolution emerges): In the continuous-time limit, unitary evolution takes the form:

$$i\hbar \frac{d}{dt} A(t) = H \cdot A(t)$$

where $H$ is a Hermitian operator (the Hamiltonian).

Proof:

Step 1: Discrete unitary evolution

For discrete commit cycles indexed by integer $n$:

$$A(n+1) = U_n \cdot A(n)$$

where $U_n$ is unitary.

Step 2: Continuous limit

In the limit of small commit spacing $\delta t$, define continuous time $t = n \cdot \delta t$.

Expand $U$ for small $\delta t$:

$$U = I + \delta t \cdot K + O(\delta t^2)$$

where $K$ is an operator to be determined.

Step 3: Unitarity constraint on $K$

For $U$ to be unitary: $U^\dagger U = I$

$$(I + \delta t \cdot K^\dagger)(I + \delta t \cdot K) = I + O(\delta t^2)$$

Expanding:

$$I + \delta t (K + K^\dagger) + O(\delta t^2) = I$$

Therefore:

$$K + K^\dagger = 0 \implies K^\dagger = -K$$

So $K$ is anti-Hermitian.

Step 4: Hermitian generator

Define:

$$H = i\hbar K$$

where $\hbar$ is a scale constant (to be calibrated; provisionally set to Planck’s constant for dimensional consistency).

Since $K$ is anti-Hermitian, $H$ is Hermitian:

$$H^\dagger = (i\hbar K)^\dagger = -i\hbar K^\dagger = -i\hbar (-K) = i\hbar K = H$$

Step 5: Evolution equation

From $A(t + \delta t) = U \cdot A(t)$:

$$A(t + \delta t) = (I + \delta t \cdot K) \cdot A(t)$$ $$A(t + \delta t) - A(t) = \delta t \cdot K \cdot A(t)$$

Dividing by $\delta t$ and taking limit $\delta t \to 0$:

$$\frac{dA}{dt} = K \cdot A = -\frac{i}{\hbar} H \cdot A$$

Rearranging:

$$i\hbar \frac{dA}{dt} = H \cdot A$$

This is the Schrödinger equation (without assuming quantum mechanics). $\square$

Interpretation: Schrödinger-class evolution is not postulated. It is the continuous-time limit of discrete unitary commit-to-commit transport, forced by closure preservation under finite tolerance.

4.6 Hamiltonians as Derived Generators

Definition (Hamiltonian as Generator): The Hamiltonian $H$ is the generator of unitary evolution. It describes the rate of closure-preserving transport in the continuous-time limit.

Key insight:

• $H$ is not a fundamental operator or energy observable (energy not yet defined)
• $H$ is derived as the Hermitian generator required for unitary dynamics
• $H$ encodes the compatibility structure of closure-preserving transport

What determines $H$ for a specific system?

This is deferred. $H$ depends on:

• The discrete mode structure (B3)
• The constraint geometry of the substrate (Paper A)
• The coupling structure of composite systems (B2)

For now, we establish only that some Hermitian $H$ exists as the generator of closure-preserving evolution.

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5. Time Evolution as Emergent Ordering

5.1 Time is Not Primitive

In this derivation, time does not appear as:

• Background parameter independent of substrate
• Continuous coordinate before emergence
• Primitive dimension of spacetime

Instead, time emerges as:

• Ordering structure of commit cycles
• Indexing of discrete resolution steps
• Derived relational structure

Continuous time appears only in the limit of closely spaced commits.

5.2 Commit Cycles as Temporal Ordering

Define temporal ordering by commit index:

$$t_1 < t_2 \iff \text{commit } 1 \text{ precedes commit } 2 \text{ in causal resolution order}$$

This ordering is:

• Discrete at substrate level
• Partial (not all commits are causally ordered)
• Emergent from constraint resolution structure (AX-REL)

Continuous time $t \in \mathbb{R}$ is an effective description when commit spacing is fine-grained relative to system timescales.

5.3 Energy as Conjugate to Time

In continuous-time Schrödinger evolution:

$$i\hbar \frac{\partial A}{\partial t} = H \cdot A$$

The Hamiltonian $H$ is conjugate to time $t$. This suggests an interpretation of $H$ as “energy,” but:

Energy is not yet defined as a physical observable. At this stage, $H$ is simply the generator of unitary transport. The connection to energy as a conserved quantity (Noether’s theorem) requires additional structure not yet established.

This is deferred to later work or extensions.

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6. Decoherence as Tolerance Violation

6.1 Coherent vs. Incoherent Evolution

Coherent evolution:

• Preserves compatibility within tolerance $W$
• Maintains closure conditions
• Describable by unitary dynamics
• Corresponds to isolated substrate systems

Incoherent evolution (decoherence):

• Violates tolerance $W$ for some modes
• Triggers partition (AX-PAR)
• Destroys interference between incompatible branches
• Corresponds to substrate interaction with external constraints

6.2 Decoherence as Partition, Not Collapse

From AX-PAR (partition axiom), when mismatch exceeds tolerance:

$$|\Delta \phi_{ij}| > W_{\text{spin}}$$

the affected modes partition into distinct coherence domains. This is not:

• Stochastic collapse
• Measurement-induced reduction
• Probabilistic process

It is:

• Deterministic partition into incompatible provenance domains
• Loss of joint representability
• Enforced by substrate constraint structure

Representational consequence: States that were jointly representable become separately representable in distinct amplitude spaces. The composite amplitude space factorizes into incoherent subspaces.

Critical distinction — Decoherence is structural and deterministic: Decoherence here is a deterministic structural consequence of tolerance violation (AX-PAR). Any appearance of probability or outcome weighting belongs to B5, where it emerges as branch-relative epistemic weighting, not as stochastic dynamics. At the B4 level, decoherence is purely about partition structure, not outcome selection.

6.3 Non-Unitary Effective Evolution

When decoherence occurs, the evolution of a reduced subsystem (tracing over partitioned degrees of freedom) is not unitary.

This is consistent with substrate mechanics:

• The full substrate state evolves unitarily (information conservation)
• Reduced descriptions (marginal summaries) may appear non-unitary
• This corresponds to open system dynamics in quantum mechanics

Non-unitary evolution is not fundamental dynamics. It is a coarse-grained description of a system interacting with a traced-out environment that has violated tolerance and partitioned.

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7. Why This Is Schrödinger Evolution, Not Something Else

7.1 Alternatives Excluded

We have shown that closure-preserving evolution must be:

• Linear (Theorem 4.2.1)
• Norm-preserving (Theorem 4.3.1)
• Reversible (Theorem 4.4.1)
• Unitary (Theorem 4.4.2)

Could dynamics be:

Different from first-order Schrödinger form? The derived evolution $i\hbar \frac{dA}{dt} = H \cdot A$ is first-order in time with a Hermitian generator $H$. This is the unique form forced by unitarity in the continuous limit. Higher-order equations or non-Hermitian generators would violate the constraints established in Theorems 4.2.1–4.5.1.

Non-Hermitian Hamiltonian? No. Theorem 4.5.1 showed that unitarity forces $H$ to be Hermitian.

Higher-order differential equations? Generic higher-order equations violate uniqueness or introduce additional degrees of freedom inconsistent with amplitude structure (B1). The minimal structure consistent with unitary evolution is first-order in time with Hermitian generator.

Nonlinear Schrödinger equations? No. Linearity is forced by B1 (Theorem 4.2.1). Nonlinear terms break superposition structure.

7.2 Schrödinger Evolution is Unique

The structure derived here is:

$$i\hbar \frac{dA}{dt} = H \cdot A$$

where:

• $A$ is a linear amplitude state
• $H$ is a Hermitian operator (generator)
• Evolution is unitary
• Time is emergent from commit ordering

This is exactly the Schrödinger equation of quantum mechanics, derived without assuming quantum axioms.

Why “Schrödinger-class” and not “Schrödinger equation exactly”?

We have not yet specified:

• What $H$ is for specific physical systems
• How to construct $H$ from substrate structure
• Whether additional constraints (gauge symmetries, etc.) further restrict $H$

These are system-specific details. The form of evolution is Schrödinger-class; the content (specific Hamiltonians) is deferred.

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8. Implications for B5 and Beyond

8.1 What B4 Establishes

• Unitary evolution as the unique closure-preserving transport law
• Time evolution as emergent ordering of commits
• Hamiltonians as derived generators, not primitive operators
• Schrödinger-class dynamics as continuous-time limit
• Decoherence as tolerance violation and partition, not stochastic collapse
• Non-unitary reduced dynamics as coarse-grained descriptions

8.2 What B4 Enables for B5

B5 (Measurement and Born Rule):

• Measurement corresponds to irreversible commit with partition
• Outcome probabilities are defined over discrete modes (B3) evolved unitarily (B4)
• Born rule weighting emerges from closure-preserving amplitude structure
• Decoherence (tolerance violation) naturally separates outcome branches

8.3 What B4 Does NOT Provide

This paper does not address:

Specific Hamiltonians:

• How to construct $H$ for hydrogen, harmonic oscillators, etc.
• Connection between substrate constraint geometry and $H$
• Role of symmetries in determining $H$

Energy as observable:

• Why $H$ corresponds to physical energy
• Energy conservation and Noether’s theorem
• Energy-time uncertainty relations

Measurement process:

• How unitary evolution is “interrupted” by measurement
• Why specific outcomes are selected
• How Born rule probabilities emerge

Quantum field theory:

• Particle creation and annihilation
• Field quantization
• Relativistic generalization

These are deferred to later work or extensions.

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9. Why Quantum Dynamics Are Forced, Not Chosen

9.1 Not Empirical Postulates

We have not assumed:

• Schrödinger equation
• Unitary evolution axioms
• Hamiltonian formalism
• Energy minimisation
• Time-reversal symmetry (derived from reversibility)

All quantum dynamical structure is derived from:

• Substrate commit semantics (Paper A)
• Amplitude representation (B1)
• Discrete spectral structure (B3)
• Closure preservation under finite tolerance (AX-TOL)

9.2 Not Variational Principles

Quantum dynamics are often derived from variational principles (least action, etc.). We have not used:

• Action functionals
• Lagrangians
• Euler-Lagrange equations
• Stationary phase approximations

Instead, we derived dynamics from representational consistency requirements.

9.3 Not Phenomenological Models

Schrödinger evolution is often treated as a phenomenological model fitted to experiments. We have shown it is representationally necessary:

Given linear amplitude representation (B1), discrete spectral modes (B3), and closure-preserving transport between commits under finite tolerance $W$, Schrödinger-class unitary evolution is unavoidable.

This is a necessity claim, not a modeling choice.

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10. Summary and Scope Boundaries

10.1 Central Result

We have shown:

In a substrate governed by commit-based closure and finite tolerance $W$, amplitude transport between commits must be unitary. In the continuous-time limit, this produces Schrödinger-class evolution with Hermitian generators (Hamiltonians).

This is a necessity claim, not a modeling choice.

10.2 Derivation Path

1. Substrate evolution proceeds through discrete commit cycles (Paper A)
2. Amplitude states represent uncommitted alternatives (B1)
3. Evolution must preserve closure conditions for discrete modes (B3)
4. Closure preservation forces norm conservation (Theorem 4.3.1)
5. Linearity is already forced by amplitude structure (B1, Theorem 4.2.1)
6. Reversibility follows from substrate information conservation (Theorem 4.4.1)
7. Linear + norm-preserving + reversible $\implies$ unitary (Theorem 4.4.2)
8. Continuous-time limit produces Schrödinger equation (Theorem 4.5.1)
9. Hamiltonian emerges as Hermitian generator of unitary flow
10. Time is emergent ordering of commits, not background structure

10.3 What B4 Establishes

• Quantum dynamics are forced, not assumed
• Unitary evolution is representational necessity, not axiom
• Schrödinger equation is unique stable form for closure-preserving transport
• Hamiltonians are derived generators, not fundamental operators
• Time evolution is emergent, not primitive
• Decoherence is partition, not stochastic collapse

10.4 What B4 Does Not Claim

• That specific Hamiltonians are derived (system-dependent, deferred)
• That energy is defined (energy as observable requires additional structure)
• That measurement is explained (B5)
• That Born rule is derived (B5)
• That quantum field theory is recovered (beyond current scope)
• That relativistic quantum mechanics is established (spacetime not yet emergent)

10.5 Logical Position in B-Series

B4 assumes:

• Linear amplitude representation (B1)
• Non-factorisable composite structure (B2)
• Discrete spectral bases (B3)
• Substrate mechanics (A-series)
• Phase and coherence structure (M4)
• No quantum dynamics postulates

B4 provides:

• Unitary evolution framework
• Schrödinger-class dynamics
• Hermitian generators (Hamiltonians)
• Time as emergent structure
• Decoherence mechanism

B4 enables:

• B5 (measurement and Born rule over unitarily-evolved discrete states)
• Future work on specific Hamiltonians and physical systems
• Understanding of quantum-classical boundary via decoherence

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11. Open Questions and Next Steps

11.1 Questions Deferred to Later Papers

For B5 (Measurement and Born Rule):

• How are discrete outcomes selected at commit?
• Why is outcome weighting quadratic ($|\alpha|^2$)?
• What happens to unitary evolution at measurement?

For Future Extensions:

• How do specific substrate constraint geometries determine $H$?
• What is the connection between $H$ and physical energy?
• How do symmetries constrain Hamiltonian structure?
• What is the role of gauge symmetries?

11.2 Potential Refinements to B4

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

• Formalize the continuous-time limit more rigorously using limit theorems
• Provide explicit substrate examples showing unitary emergence
• Develop the connection between $H$ and substrate constraint geometry
• Explore when non-unitary effective dynamics are valid approximations
• Analyze time-dependent Hamiltonians and non-autonomous evolution

11.3 Relation to Empirical Work (E-series)

E-series papers empirically explore which substrate configurations produce constructor-capable continua. B4 shows that any such continuum supporting discrete spectral modes (B3) must exhibit unitary Schrödinger-class dynamics.

This creates a testable prediction: substrates lacking closure-preserving transport cannot exhibit quantum dynamics.

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12. Conclusion

We have established that quantum dynamics (unitary, Schrödinger-class evolution) are not a quantum postulate, but a representational necessity for cohesive substrates with commit-based closure under finite tolerance $W$.

Unitary evolution is forced by:

• Closure preservation (discrete modes must remain stable)
• Norm conservation (total amplitude measure must not drift)
• Linearity (amplitude structure from B1)
• Reversibility (substrate information conservation)

Schrödinger-class evolution emerges as:

• The continuous-time limit of discrete commit-to-commit transport
• The unique form consistent with closure preservation
• Governed by Hermitian generators (Hamiltonians) derived from substrate structure

Time evolution is not background structure but emergent ordering of commit cycles.

Decoherence is not stochastic collapse but partition due to tolerance violation.

With B4, the full kinematic and dynamical structure of quantum mechanics is established; only outcome selection and weighting remain. The remaining B-series paper (B5) will derive how discrete outcomes are weighted (Born rule) and how measurement corresponds to irreversible commit, completing the representational recovery of standard quantum mechanics.

B4 establishes why quantum dynamics have the form they do. B1 established what quantum states are. B2 established why composite systems entangle. B3 established why spectra are discrete. 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 B4