Dark Matter as Topological Phase Persistence of the Existence Equation

Evidence Paper V of the Existence Equation series In the Existence Equation 1, the deviation field Ψ = AeiΦ carries two degrees of freedom. The gradient energy decomposes exactly: |∇Ψ|² = |∇A|² + A²|∇Φ|². The first term is the amplitude-gradient energy. The second is the phase-gradient energy. The two are independent contributions to the stress-energy tensor — both gravitate equally — but they couple differently to electromagnetic probes. The amplitude |Ψ| is gauge-invariant and determines all observable emission, absorption, and scattering. The phase Φ is gauge-dependent; electromagnetic probes do not see it. Now add topology. When the phase winds by 2πn around a closed loop, the winding number n is an integer. Integers cannot relax continuously. Under damped evolution toward the vacuum, the amplitude can smooth — its localized core structure dissolves, the gradient cost |∇A|² is minimized — but the phase winding is stuck. The result is a field configuration in which the amplitude is uniform and featureless (electromagnetically silent) while the phase gradient persists (gravitationally active). This paper demonstrates that separation numerically, in three experiments on a 2048² finite-difference grid solving Eq. (1) of 1 directly. Experiment Configuration Key measurement Result 1. Single vortex n = +1, 6000 steps, Γ = 0.02 ⟨vθ·r⟩ 1.0000 ± 0.0001 2a. Opposite pair (+1, −1), 6000 steps Winding at r = 10 0 → 0 (annihilates) 2b. Same-sign pair (+1, +1), 6000 steps ⟨vθ·r⟩ far-field 2.0000 ± 0.0003 3. N = 10, σ ∝ 1/r Inverse-radial distribution Flatness ℱ3, 9 0.82 (flat rotation curve) 3. N = 10, uniform Uniform disk distribution Flatness ℱ3, 9 0.63 (rising-then-declining) Experiment 2b is the central result. Two same-sign vortices are placed at (±2, 0) and evolved under damped dynamics. The local amplitude cores dissolve completely — |Ψ| returns to the vacuum value vvac = √(λ/α) throughout the formerly defect-bearing region. But the winding on any loop enclosing both original positions remains exactly +2. The far-field rotation curve vθ(r) = 2/r persists to four significant digits, driven entirely by the phase gradient ∇Φ, with no amplitude structure remaining to support it. The amplitude has smoothed. The phase has not. The circulation endures. The structural identification follows directly: Amplitude A Phase Φ Electromagnetic couples does not couple Gravitational couples couples Topologically can relax cannot relax The amplitude channel is what standard terminology calls baryonic matter: it gravitates and it radiates. The phase channel has precisely the two properties that define dark matter: it gravitates and it does not radiate. The configuration produced by damped relaxation of a topologically nontrivial initial condition — smooth amplitude, persistent winding — is, by construction, a gravitationally active structure that leaves no electromagnetic signature. Dark matter ≡ topological phase persistence of Ψ. No new field. No new particle. No new parameter. No modification to the equation. The rotation curve vθ(r) = Nenc(r)/r is determined entirely by the spatial distribution of topological charge, and is insensitive to the amplitude profile. A distribution σ(r) ∝ 1/r produces a flat rotation curve as a direct kinematic consequence. The amplitude tells you where the visible matter is. The phase tells you how the galaxy rotates. What this paper has not done is equally important: no value in physical units is computed, no galactic rotation curve is fit, no cosmological observable ΩDM is derived. The claim is structural, and for that reason strong: the Existence Equation, as already written, contains the configuration that dark matter is operationally defined to be. References 1 J.-A. Shin, "The Existence Equation: The Grammar of Persistence," Zenodo (2026). doi: 10.5281/zenodo.18639316 All simulation code and raw data are publicly available at https://github.com/Galileo-leo/existence-equation.