07

Shape Optimization

Interactive proof companion. Drag z in the complex plane to deform the pentagon and observe spectral coercivity, barrier containment, and the five-stage assembly of the inequality in real time.

Shape Coordinate z

Regular
r_max = 1/φr_loczImRe-0.50.5
z = 0.0000 + 0.0000i|z| = 0.0000arg = 0.0deg

Pentagon P(z)

v0v1v2v3v4
Area 1.000000
|z|/r_max 0.0%
F(z) 15.7000
Ξ_spec 0.000000
β0.30
IBarrier Lemmas & Compactness
IIComplex Shape Chart & D5 Normal Form

The DFT on ℤ/5ℤ identifies the local moduli space with z = δ̂h(2) ∈ ℂ. D₅ acts by r·z = ζ²z (rotation by 4π/5) and s·z = z̄. The normal form gives F(z) = Λ_reg + κ|z|² + o(|z|²).

delta-h perturbation
δh₀0.0000δh₁0.0000δh₂0.0000δh₃0.0000δh₄0.0000
Parseval: ‖δh‖² = (2/5)|z|²
‖δh‖² = 0.000000
(2/5)|z|² = 0.000000
δ̂h(2) = 0.0000 + 0.0000i
|δ̂h(2)| = 0.0000
D₅ orbit of z: ζ^0z = (0.000, 0.000); ζ^2z = (0.000, 0.000); ζ^4z = (0.000, 0.000); ζ^6z = (0.000, 0.000); ζ^8z = (0.000, 0.000)
IIISpectral Coercivity (κ > 0)
IVCertified Exclusion (Annulus)
VGlobal Assembly
ℚ(√5) Structure
φ = (1+√5)/2 = 1.618034 | r_max = 1/φ = 0.618034
cos(2π/5) = (√5−1)/4 = 0.309017
cos(4π/5) = −(1+√5)/4 = -0.809017
μ₁ = μ₄ = [(5−√5)+β(5+√5)]/10 = 0.493475
μ₂ = μ₃ = [(5+√5)+β(5−√5)]/10 = 0.806525