Flow Equations

cMERA, running α(μ), the Dimension-9 surface.

cMERA toward α*

Continuous MERA is the field-theoretic limit of the discrete tensor network. The flow d|Ψ(u)⟩/du = (K(u) − L)|Ψ(u)⟩ has u as RG scale (u=0 IR / ground state; u→−∞ UV / M), K(u) the entangler, L the scaling operator. In the α-Framework the entangler at scale u is controlled by α(μ(u)). As α runs toward α* = π/60, the entangler approaches its critical form. The IR endpoint is C. The UV endpoint is M.

The running of α

dα/dμ = β_α(α) = −b₁α² − b₂α³ − b₃α⁴ − ⋯ with fixed point β_α(α*) = 0. At leading order, α* = π/60. Higher-order corrections are O((π/60)²) ~ 0.003. The fixed point is UV-stable and IR-attractive: the geometry of C is the endpoint toward which flows converge when α = α*.

MERA as nested evaluation

Each layer k applies disentangler u^(k) then isometry w^(k), coarse-graining |Ψ_k⟩ to |Ψ_{k−1}⟩. Bond dimension χ(k) ~ exp(α · S_k) is the capacity of the edge at that scale — how many M-states can be distinguished in C at resolution μ_k. Full UV→IR: |Ψ_C⟩ = w⁽⁰⁾u⁽⁰⁾…w⁽ᴺ⁾u⁽ᴺ⁾|Ψ_M⟩. The inverse traces experience back toward source. Both directions are correct. That bidirectionality is Novus Horner closing the lemniscate.

EQ-5 / EQ-6

d|Ψ(u)⟩/du = (K(u) − L)|Ψ(u)⟩

dα/dμ = β_α(α) β_α(α*) = 0

α* = π / 60 ≈ 0.05236

Constraint: I(boundary, μ) ≥ I(volume, μ)

When the Bekenstein ceiling binds, α is at its fixed point. When slack, α flows toward it. Self-correcting.