Toward Unitary Closure Dynamics: The Evolution Equation

The Retained Damping

Paper XV’s complexified path integral assigns each path an amplitude of the form exp(−S/σ²) · exp(iΘ). The first factor is real and positive — it damps paths with large action. The second factor is oscillatory — it produces interference.

The problem: quantum mechanics has only the oscillatory factor. The Feynman path integral is exp(iS/ℏ) — purely oscillatory, no real damping. The retained damping factor means the closure dynamics is dissipative, not unitary. Probability is not conserved. This paper asks: what differential equation does this path integral generate, and under what conditions does the damping become subdominant?

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The Closure Evolution Equation

Expanding the short-time propagator of Paper XV’s path integral to first order in Δt yields the closure evolution equation:

The first term (σ²/2)ΔΨ is a real diffusion — it spreads probability, exactly as in a Fokker–Planck equation. The second term −∇F · ∇Ψ is a drift toward closure minima. The third term (i/σ²)FΨ is an imaginary potential — it rotates phase, exactly as in a Schrödinger equation.

The equation inherits features from both classical stochastic dynamics and quantum mechanics, but is identical to neither.

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Structural Gap to Schrödinger

The closure evolution equation and the Schrödinger equation share the imaginary potential term but differ in the kinetic sector:

Term	Closure Evolution	Schrödinger
Kinetic	+(σ²/2)ΔΨ  (real, diffusive)	+(iℏ/2m)ΔΨ  (imaginary, oscillatory)
Drift	−∇F · ∇Ψ  (present)	absent
Potential	(i/σ²)FΨ  (imaginary)	−(i/ℏ)VΨ  (imaginary)
Probability	Dissipative — ‖Ψ‖² decays	Conserved — ‖Ψ‖² constant

The gap is in the kinetic term: real diffusion versus imaginary diffusion. The drift term is secondary — it can be absorbed by a change of variables. The kinetic-term difference is the structural obstruction to unitarity.

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Three Routes to Oscillatory Dynamics

Three candidate mechanisms for recovering oscillatory (unitary-like) dynamics from the dissipative closure equation:

(A) Measure absorption

Absorb the real damping into a redefinition of the path-integral measure. The modified measure produces a purely oscillatory propagator, but the redefinition is not unique and introduces measure-dependent ambiguities.

(B) WKB transport (σ → 0)

In the small-σ limit, the WKB approximation shows that the modulus of Ψ is locally preserved along characteristics while the phase evolves according to a Hamilton–Jacobi equation. The real diffusion becomes subdominant — the dynamics is approximately oscillatory without analytic continuation.

(C) Formal Wick rotation

Replace σ² → iσ² in the kinetic term. This converts real diffusion to imaginary diffusion and recovers a Schrödinger-type equation. But the rotation is a formal operation — it is not physically motivated within the closure framework.

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The Witten Connection

The drift term −∇F · ∇Ψ can be eliminated by a similarity transformation Ψ = e^−F/σ²Φ. After this change of variables, the evolution equation for Φ takes the form:

∂_tΦ = (σ²/2)ΔΦ − V_effΦ + (i/σ²)FΦ

where V_eff = (|∇F|² − σ²ΔF)/(2σ²). The real part is a Euclidean quantum mechanics Hamiltonian — recognisable as a Witten-type operator from supersymmetric quantum mechanics. The imaginary potential is an additional rotation. This connection suggests the closure framework naturally contains the algebraic structure of ground-state quantum mechanics.

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Stated Limitations

• No exact unitarity — the evolution equation is dissipative for all finite σ
• No Schrödinger equation — the kinetic term is real, not imaginary
• No Hilbert space — the state space remains the closure landscape, not a linear vector space
• The Wick rotation route is not physically justified within the framework
• The quadratic-potential example is schematic — illustrative, not a general proof

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