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1. Gravitational Field Equations

The DM-extended Einstein field equation is:

G_{μν} + S_{μν} = (8πG / c⁴)(T_{μν}^{(4D)} + Λₛ g_{μν})

where S_{μν} = (1/λₛ²)(∂_μΦ ∂_νΦ − ½ g_{μν}Φ²) represents the 5D stabilization tensor, and Λₛ = 1/λₛ² replaces the cosmological constant as a natural geometric stabilization term. When S_{μν} → 0, standard General Relativity is recovered.

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2. Gravitational Potential and Coherence Density

The gravitational potential follows a modified Poisson equation that includes coherence scaling: 

∇²Φ = 4πGρ − Φ / λₛ²

At small coherence depth (s ≈ 0), this reduces to Newtonian gravity g = −GM/r². At large s, the exponential term e^(−r/λₛ) stabilizes curvature, flattening galactic rotation curves without dark matter. This behavior reflects how coherence saturation limits gravitational divergence.

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3. Gravitational Waves as Φ–Ψ Oscillations

Gravitational waves correspond to oscillations between Φ (5D coherence) and Ψ (4D wave) fields, described by the extended wave equation:

□₄Φ + ∂²Φ/∂s² − Φ/λₛ² = 0

This predicts additional polarization modes and frequency components in the 10⁻¹⁸–10³ Hz range. Low-frequency residuals in LIGO/Virgo data may correspond to coherence-modulated gravitation. Space-based detectors like LISA can directly test these effects.

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4. Black Hole Coherence and Singularity Resolution

In DM, singularities are replaced by finite coherence cores. The divergent 1/r term becomes bounded by coherence length:

1/r → 1/(s² + λₛ²)

This prevents infinite curvature, yielding finite gravitational energy densities. The event horizon represents a 4D–5D coherence barrier, where information becomes phase-locked within Φ rather than destroyed. Inside the horizon, time transitions into the s dimension, maintaining information through coherent continuation.

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5. Cosmological Implications

The cosmological constant arises as Λₛ = 1/λₛ². With λₛ ≈ 10²⁶ m, Λₛ ≈ 10⁻⁵² m⁻², matching observed dark energy density. The Planck-to-Hubble scaling H₀ ≈ ƒₚ × 10⁻⁶¹ reproduces the 10¹²² energy ratio between vacuum and cosmological scales, completing the Λ-gap closure.

Einstein’s GR emerges as a low-dimensional projection, while the additional term S_{μν} resolves singularities and unifies quantum and cosmological scales. The cosmological constant becomes a measurable geometric factor, and gravitational waves are coherence oscillations. All constants (G, c, ħ, α) are closed under Planck scaling within ρ–Ψ–Φ projection geometry.

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​Constant-Level Closure

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6. Planck Anchors and SI Baselines

Planck-unit lattice fixes the geometric resolution and frame rate:
• ℓₚ = √(ħG/c³)
• tₚ = √(ħG/c⁵)
• Eₚ = √(ħ c⁵ / G)
• ƒₚ = 1/tₚ
• mₚ = √(ħ c / G)

Interpretation in DM: c = ℓₚ / tₚ is the 3D→4D face-scan speed; ƒₚ is the universal scan rate. All gravitational and electromagnetic constants are projections on this lattice.

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7. Electromagnetic Kernel ε and Constant Couplings

The vacuum impedance sets the electromagnetic kernel:
• Z₀ = 120π e^{−ε}
• ε = −ln(Z₀ / 120π)
Couplings used throughout DM:
• α = κ_α e^{−ε}
• μ = exp(3456π ε)
Once κ_α is fixed from a single precision α measurement, downstream EM constants (R_∞, a₀) are fixed without new freedom:
• R_∞ = α² m_e c / (2 h)
• a_0 = 4π ε_0 ħ² / (m_e e²)

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8. Gravity from Φ — Constants Form

Extended field equation with explicit constants:
• G_{μν} + S_{μν} = (8π G / c⁴)( T_{μν}^{(4D)} + Λₛ g_{μν} )
• S_{μν} = (1/λₛ²)( ∂_μ Φ ∂_ν Φ − 1/2 g_{μν} Φ² )
• Λₛ = 1/λₛ²
Here, λ_s is a geometric coherence length that sets the cosmological constant scale; S_{μν} captures 5D stabilization.

Newtonian and Yukawa limits from constants:
• ∇² Φ = 4π G ρ − Φ/λₛ²
• V(r) ≈ − G M e^{−r/λₛ} / r
Small-r or λₛ → ∞ yields Newtonian gravity; finite λₛ produces flat rotation curves without particulate dark matter.

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9. Cosmological Constant and Hubble Scaling

In DM, Λ is geometric:
• Λ = Λₛ = 1/λₛ²
Relate Λ to H_0 and c:
• ρ_Λ = Λ c² / (8π G)
• H_0 ≈ fₚ × 10⁻⁶¹
Thus the Λ gap N_Λ ~ 10¹²² appears as the area ratio between cosmic horizon and Planck scales, consistent with DM’s Φ capacity.

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10. Gravitational Waves (Φ–Ψ Oscillations) with Constants

Coherence wave equation (constant-explicit):
• (1/c²) ∂² Φ/∂t² − ∇² Φ + ∂² Φ/∂s² − Φ/λₛ² = 0
Dispersion:
• ω²/c² = k² + kₛ² + 1/λₛ²
Prediction: additional low-frequency (10⁻¹⁸–10⁰ Hz) components and polarization mixing; testable by LISA/Cosmic Explorer.

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11. Black Holes and Constant-Bounded Cores

Coherence-bounded potential:
• 1/r → 1/(s² + λₛ²)
Curvature invariants remain finite: the would-be singularity is replaced by a Φ-domain core of size ~λₛ. Event horizon condition remains metric-defined (GR-accurate outside), with information preserved as Φ phase.

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12. Unified Gravity–EM Constant Relation

Coherence coupling of gravity and electromagnetism (constant-explicit):
• g' = g [ 1 − (α E_EM / Eₚ) ]
where Eₚ = √(ħ c⁵ / G) and α = κ_α e^{−ε}. This expresses gravity as the long-wavelength (low-k) projection of Φ curvature, and EM as high-frequency Ψ modulation, within the same constants lattice.

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Summary — Path

1) Planck anchors (ℓₚ, tₚ, Eₚ, fₚ) geometric lattice and scan rate. 2) The EM kernel (ε from Z₀) closes α and derived constants with one calibration κ_α. 3) Gravity emerges from Φ curvature with Λₛ = 1/λₛ²; Newtonian gravity is recovered in the λₛ → ∞ limit. 4) H₀ ≈ fₚ × 10⁶¹ explains the Λ gap (~10¹²²), matching cosmological observations. 5) All constants {G, c, ħ, α, Z₀} are unified as geometric projections in the ρ→Ψ→Φ hierarchy.

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