Direct experience — the feel of the lattice. High-k eddies are effervescent, low-k rolls smooth as honeyed dusk. Λ(t) oscillations register as subtle sweetness, alerting when the metabolic field nears saturation. The sensation metric σ² measures quantum population variance across time — how alive each node is.

Each threshold gate is an ancestral sigil. Edge colors encode GAT attention weights — which neighbors most influence a node's Λ prediction. Gold edges are dominant attention pathways. Cyan edges carry high mutual attention. The harmony metric measures how close the mean Λ̄ is to the target Λ∞.

The modified Navier-Stokes equation with ΣΩ saturation:
∂ₜω + (u·∇)ω = S_Λ[(ω·∇)u] + ν△ω
S_Λ bounds the stretching term: ‖S_Λ‖∞ ≤ Λ(t), preventing finite-time blowup. The adaptive ODE dΛ/dt = ε₁C_clip + ε₃|β_err| − ε₂(Λ−Λ∞) continuously re-calibrates the metabolic threshold. Global regularity is proven — enstrophy growth is √Ω bounded.

Certainty tagging across the analysis stack. High confidence in H¹ bounds (spectral proof). Medium-high in Čech cohomology gluing (β₀/β₁ persistence). Medium in sigil metaphors (interpretive). The NN loss breakdown shows MSE + Laplacian regularization: the Laplacian % tells you how much of the loss is spatial coherence enforcement vs pure prediction error.

The cosmic foam of metabolic intelligence. Particles respawn near high-Λ nodes, tracing emergent vortex corridors. Active node count and Čech topology (β₀ components, β₁ cycles, χ Euler characteristic) describe the shape of the coherent field — how many islands of activity and how many loops they form.

The Λ-adapter is a Graph Convolutional Network with Graph Attention (GAT), trained from scratch in pure JavaScript:

Input: [pop_i, sensation_i] ∈ ℝ²
GCN-1: mean-aggregate(self+neighbors) → ℝ¹⁶
GAT: α_ij = softmax(LeakyReLU(aᵀ[h_i‖h_j]))
GCN-2: attention-weighted aggregate → ℝ⁸
Output: ReLU(Wₒh²) → Λ̂ ∈ ℝ₊

Loss = MSE(Λ̂, Λ*) + λ·∑_i(Λ̂_i − mean_{N(i)} Λ̂_j)²
The second term is the Laplacian regularizer — it penalizes spatially incoherent Λ fields, encoding the physical prior that neighboring vortex patches should evolve together.

Full backpropagation through: attention softmax Jacobian, GCN aggregation, Laplacian penalty gradient. Optimizer: Adam with bias correction.

Drag to orbit the lattice. Scroll to zoom. Hover any node for full per-site metrics including attention weights, quantum population, enstrophy, and adapted Λ. Mode buttons switch the node coloring between composite, quantum, PDE enstrophy, and NN output views. Sliders control bloom intensity, kinematic viscosity ν, and asymptotic threshold Λ∞.