Gravitational Frequency Dragging: How Matter Interacts with the Chronostasis Field in the Fractal-Spectral Framework

If dark matter consists of temporal vibrational modes that are partially desynchronized from the √2 transport tower, how does it interact with ordinary matter? This paper derives the answer: through gravitational frequency dragging. Matter creates a local τ-gradient that shifts the frequencies of nearby temporal modes by δω/ω ~ GM/(rc²). This shift determines three distinct interaction regimes. At galactic scales (δω/ω ~ 10⁻⁶), the shift is negligible compared to the tower spacing (ln√2/2 ≈ 0.17): dark matter modes cannot synchronize with the √2 tower and pass through baryonic matter with virtually no coupling. This explains the Bullet Cluster observation — dark matter and baryonic matter separate during cluster collisions — without invoking exotic non-interacting particles. The dark matter is simply off-channel. At neutron star surfaces (δω/ω ~ 0.3), the gravitational frequency shift exceeds the tower spacing. Dark matter modes can be dragged into synchronization with the √2 tower, potentially converting into baryonic matter and releasing binding energy. This predicts anomalous heating of old neutron stars in dark-matter-rich environments — a specific, testable signature distinct from standard cooling curves. At black hole horizons (τ → 0), all temporal structure is erased: the tower itself collapses, the distinction between synchronized and desynchronized modes vanishes, and all information about the progenitor's dark matter content is lost. This provides a physical mechanism for the no-hair theorem within the fractal-temporal framework — the horizon is not merely a causal boundary but a frequency eraser. A key consequence is scale-dependent dark matter self-interaction. The effective cross-section σ/m depends on the local gravitational environment: < 0.1 cm²/g at cluster scales (consistent with Bullet Cluster constraints) but potentially 1–10 cm²/g in dwarf galaxy cores (where τ-gradients are stronger relative to the tower spacing). This naturally resolves the tension between cluster-scale constraints (requiring low self-interaction) and dwarf-scale observations (favoring higher self-interaction). Four falsifiable predictions are proposed: scale-dependent self-interaction cross-sections (testable through cluster merger lensing maps vs dwarf galaxy core profiles), neutron star dark matter heating (testable through temperature measurements of old neutron stars in high-DM environments), no-hair tower collapse (testable through gravitational wave ringdown spectroscopy), and anomalous baryon production near compact objects.