Phase-Coherent Closure Dynamics and the Emergence of Interference

The Missing Phase

Paper XIV’s stochastic closure dynamics produces a probability distribution over admissible states — the squared-modulus rule emerges from the equilibrium measure. But the dynamics is entirely real-valued: path weights are positive, amplitudes are non-negative, and no phase variable appears anywhere in the formalism.

This is probabilistic but not quantum. Quantum mechanics requires complex amplitudes with a phase degree of freedom — without phase, there is no interference. Paper XIV recovers Born-rule statistics but cannot distinguish between a double slit and a single slit. The missing ingredient is structural.

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Phase-Augmented State Space

The state space is extended from S_adm to S_adm × S¹. Each admissible state ψ now carries an additional degree of freedom: a phase angle θ ∈ [0, 2π). The phase is not imposed externally — it is a structural variable on the closure landscape.

The amplitude assigned to each state is fixed by consistency with Paper XIV’s equilibrium:

The phase θ is a genuine degree of freedom — it is not determined by F alone. It evolves along paths on the augmented state space and accumulates according to the geometry of the closure landscape.

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The Complexified Path Integral

The central ansatz of this paper: the transition amplitude between states is computed by summing over all paths on the augmented state space, with each path contributing a complex weight:

The first factor is real and positive — it suppresses paths with large action, exactly as in Paper XIV. The second factor is oscillatory — it assigns a phase to each path proportional to the integrated closure functional along the trajectory. Together, they produce a complex amplitude for each path.

The total transition amplitude is the sum over all paths. This is the complexified closure path integral: structurally analogous to the Feynman path integral, but derived from the closure landscape rather than from a Lagrangian.

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Interference from Phase Accumulation

When multiple paths connect the same initial and final states, their complex amplitudes add. If two paths have similar real damping but different accumulated phases, the result depends on the phase difference:

• Constructive interference — paths with phase difference near 0 (mod 2π) reinforce each other. The total amplitude is larger than either individual contribution.
• Destructive interference — paths with phase difference near π cancel. The total amplitude can be much smaller than either individual contribution, or even zero.

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The Classical Limit

In the limit σ → 0, the real damping factor exp(−S/σ²) becomes sharply peaked around the path of minimum action. All other paths are exponentially suppressed. This is the stationary-phase approximation: only the classical path contributes.

In this limit, the complexified path integral reduces to single-path evaluation. No multi-path summation means no interference. The dynamics recovers Paper XII’s deterministic gradient flow on the closure landscape — classical, single-trajectory evolution.

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Phase-Coherent Crystallisation

Paper XIV’s crystallisation selected the realised state by maximising closure compatibility — the state with the deepest closure-functional minimum is selected. With the phase-augmented formalism, the selection criterion is extended: the realised state must maximise both closure compatibility and phase coherence.

A state that sits at a deep closure minimum but has incoherent phase across its contributing paths will be suppressed by destructive interference. Conversely, a state at a slightly shallower minimum but with coherent phase across all paths can dominate. Phase coherence acts as a second selection filter on top of closure depth.

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σ² and ℏ

The noise parameter σ² occupies part of ℏ’s structural role: it sets the scale at which multi-path effects become important, controls the width of the equilibrium distribution, and appears in the exponent of the path integral in the same position as ℏ.

This is a partial formal analogy, not a physical identification. The closure path integral retains a real damping factor exp(−S/σ²) that has no counterpart in standard quantum mechanics. The Feynman path integral is purely oscillatory — exp(iS/ℏ) — with no real exponential suppression. This structural difference means σ² cannot simply be identified with ℏ, and the closure dynamics is not yet unitary.

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Stated Limitations

• No full QM equivalence — the complexified path integral produces interference but the dynamics is not unitary
• Unitarity is absent — the real damping factor breaks probability conservation
• No Schrödinger equation — the evolution is defined by the path integral, not by a differential equation
• No Hilbert space — the state space is the augmented closure landscape, not a linear vector space
• σ² ≠ ℏ — structural analogy only, not physical identification
• The double-well interference example is schematic — illustrative, not a quantitative prediction

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