Outer rim: C/D data D_n(r)⋅r̂ — blowup without T^n
Channels: T^n transition C/D→A/B
Toroid: Fix(T)=φ, N(φ)=−1 (unit), A/B regime
Exit: N(Φ⁻¹)=±1, global regularity confirmed
Cold CCW — π-flip, A/B branch
Theorem 1 — Cases C/D (Blowup)
v₀ = Dₙ(r)⋅r̂ = √(φ⋅Fₙ⋅2ⁿ⋅Pₙ⋅Ω)⋅rᵏ⋅r̂
adv/diss = φⁿ⋅³⋅⁵ → ∞ T_c = φ⁻ⁿᵎ < ∞
Theorem 2 — Cases A/B (Global Regularity)
N(a+bφ) = a²+ab−b² [Z[φ] norm]
N(Φ⋅Φ⁻¹) = N(1) = 1 ⇒ N(Φ⁻¹) = 1/N(Φ)
Φ→∞ ⇒ N(Φ)∈{±1} ⇒ N(Φ⁻¹)∈{±1} (unit)
⇒ Φ⁻¹ bounded ⇒ νΦ⁻¹∇²u bounded
⇒ ‖∇u‖ ≤ G_max ⇒ ‖u‖ bounded ⇒ no blowup
N(φ) = N(0+1φ)
-1 (unit)
N(φ²) = N(1+1φ)
+1 (unit)
N(φⁿ) alternates
{-1,+1} always
G_max = ‖∇u‖ bound
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Live: T-Iteration State — tracked particle
Shell / mode n
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T-step (cell index)
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Current v (T^k(D_n))
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|v - φ| error
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Regime
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adv/diss ratio
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T_blowup (C/D data)
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T^n err convergence
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The manifold IS the proof, made physical.
Outer rims carry v₀=Dₙ(r)⋅r̂ — the exact C/D blowup initial data from Theorem 1. Without the Tⁿ channels, φⁿ⋅³⋅⁵→∞ and T_c=φ⁻ⁿᵎ<∞.
The funnel channels apply T(X)=1+1/X exactly n times. At cell k: v=Tᵏ(Dₙ), error shrinks by φ⁻² per step. At the toroid: v≈φ, N(v)=−1 (unit), A/B regime entered.
The transition C/D→A/B happens physically in the channels. N(Φ⁻¹)∈{±1} throughout — Φ⁻¹ is always a unit in Z[φ]. ‖∇u‖≤G_max. Sobolev: ‖u‖ bounded. No blowup. □
Outer rims carry v₀=Dₙ(r)⋅r̂ — the exact C/D blowup initial data from Theorem 1. Without the Tⁿ channels, φⁿ⋅³⋅⁵→∞ and T_c=φ⁻ⁿᵎ<∞.
The funnel channels apply T(X)=1+1/X exactly n times. At cell k: v=Tᵏ(Dₙ), error shrinks by φ⁻² per step. At the toroid: v≈φ, N(v)=−1 (unit), A/B regime entered.
The transition C/D→A/B happens physically in the channels. N(Φ⁻¹)∈{±1} throughout — Φ⁻¹ is always a unit in Z[φ]. ‖∇u‖≤G_max. Sobolev: ‖u‖ bounded. No blowup. □