ricci_flow.007

-- SPDX-License-Identifier: MPL-2.0
-- CONFIDENTIAL — EXPERIMENTAL — NOT FOR DISTRIBUTION
--
-- Ricci Flow Equation
--
-- dg/dt = -2 Ric(g)
--
-- For a round 2-sphere of radius r:
--   Scalar curvature R = 2/r²
--   Ricci flow: dr/dt = -1/r (in 2D: dg/dt = -Rg = -2g/r²)
--   Solution: r(t) = sqrt(r₀² - 2t)
--   Singularity at t = r₀²/2

-- Initial sphere radius
@total data r0 = 5.0
@total data r0_squared = r0 * r0

-- Singularity time
@pure fn singularity_time(r0: Float) -> Float {
    return r0 * r0 / 2.0
}

-- Radius at time t under Ricci flow
@pure fn radius_at_time(r0: Float, t: Float) -> Float {
    let r2 = r0 * r0 - 2.0 * t
    return sqrt(r2)
}

-- Scalar curvature at radius r
@pure fn scalar_curvature(r: Float) -> Float {
    return 2.0 / (r * r)
}

-- Gaussian curvature at radius r (for 2-sphere: K = 1/r²)
@pure fn gaussian_curvature(r: Float) -> Float {
    return 1.0 / (r * r)
}

-- Compute flow at several time points
@total data t_sing = singularity_time(r0)

@total data r_at_0 = radius_at_time(5.0, 0.0)
@total data r_at_5 = radius_at_time(5.0, 5.0)
@total data r_at_10 = radius_at_time(5.0, 10.0)

@total data curvature_initial = scalar_curvature(5.0)
@total data curvature_at_5 = scalar_curvature(r_at_5)
@total data curvature_at_10 = scalar_curvature(r_at_10)

-- Euler characteristic: chi(S²) = 2
-- Gauss-Bonnet: integral of K dA = 2π χ = 4π
@total data euler_characteristic = 2
@total data gauss_bonnet_integral = 4.0 * 3.14159265358979