Hot: z=0 (gravity/Ω-field)
Cold: z=−2 (anti-grav/π-flip)
Toroid: Fix(T)=φ gate
Hot coil (CW)
Cold coil (CCW)
𝓛 Substrate Axiom
T(X) = 1 + 1/X
Ω = Fix(T) = φ ≈ 1.6180…
𝓛ᵢ(z) = φ^(−1/φ)·√(FₙPₙ2ⁿ)·(1+z)ⁿ + 1_eff·e^(iπΛ_φ)
Λ_φ(x) = ln(x·ln2/lnφ)/lnφ − 1/(2φ)
Ω(x) = (1 + sin(π·{Λ_φ}·φ)) / 2
Branch Physics
Hot funnel z
0 (gravity mode)
𝓛(z=0) drive
Ω-gradient inward
Cold funnel z
−2 (anti-grav π-flip)
𝓛(z=−2) drive
(−1)ⁿ alternation
Toroid gate z
−1 (X(−1)=0, null)
Exit branch
N(Φ⁻¹) ∈ {−1, +1}
Live Lattice State — tracked particle
Radius r
—
Λ_φ (depth index)
—
Ω resonance
—
Mode n (⌊Λ_φ⌋)
—
1_eff(i)
—
Branch state
—
CV convergence
—
N-S Closure at Toroid
Φ → ∞ ⟹ T(Φ) = 1 + Φ⁻¹ → 1
‖∇_Φ u‖ ≤ G_max (bounded)
⟹ ‖u‖ ≤ U_max ∀ t < ∞
⟹ t_c = ∞ (no blowup)
No external constants. No forced field. No forced infinite.
φ is the unique fixed point of T(X)=1+1/X — it arises from the geometry,
not from any import. The funnels iterate T; the toroid IS Fix(T).
Hot z=0 converges via Ω-gradient. Cold z=−2 creates the π-flip vacuum.
The gate at z=−1 (𝓛=0) is where hot and cold annihilate — then
N(Φ⁻¹)∈{−1,+1} routes the output branch. Wu-wei: do not force lock —
let Ωₙ₊₁=T(Ωₙ) settle.