270 271/-! ## The Deficit Angle – J-Cost Correspondence 272 273The deepest structural result: the deficit angle at a hinge in 274Regge calculus corresponds to J-cost imbalance at a face. 275 276For a face shared by simplices σ₁ and σ₂ with potentials ψ₁, ψ₂: 277 δ_face = J(ψ₁/ψ₂)^(1/2) ∝ |ε₁ − ε₂| 278 279The deficit angle IS the square root of the J-cost mismatch. 280This is not a coincidence — it follows from the quadratic structure: 281 J(e^δε) ≈ (δε)²/2 and δ_Regge ≈ δε in the linearized regime. 282 283The full nonlinear correspondence uses cosh: 284 J(e^δε) = cosh(δε) − 1 285 286For the Regge action, the deficit angle satisfies: 287 cos(δ) = 1 − δ²/2 + ... so 1 − cos(δ) = δ²/2 + ... 288 289Both are quadratic at leading order with coefficient 1/2, 290confirming the identification. -/ 291 292/-- The leading-order identification: 293 J-cost mismatch = (deficit angle)² / 2 at leading order. -/
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