Focus Flow Computation

How the cybergraph reaches collective equilibrium. The tri-kernel runs over all cyberlinks, neurons add links, and the network continuously converges toward a unique fixed point -- the focus distribution $\phi^*$.

1. The Fixed Point

The collective focus theorem guarantees convergence: under ergodicity and the screening conditions of the tri-kernel, there exists a unique $\phi^*$ to which any initialization converges, at linear rate. The fixed point is the Boltzmann equilibrium of the graph:

$$\phi^*_i \propto \exp\big(-\beta\,[E_{\text{spring},i} + \lambda\,E_{\text{diff},i} + \gamma\,C_i]\big)$$

The three energy terms correspond to the three tri-kernel operators: $E_{\text{spring}}$ encodes structural coherence via the screened Laplacian, $E_{\text{diff}}$ encodes flow consistency via diffusion, $C_i$ encodes context pressure via heat kernel weighting. $\phi^*$ is the unique distribution minimizing the composite free energy $\mathcal{F}(\phi)$. Every cyberlink added perturbs the graph and shifts $\phi^*$ incrementally -- learning and knowledge state are the same operation.

2. Two Inference Paths

The cybergraph computes two things simultaneously, both grounded in the same dynamical system:

Path A -- Continuous Focus Flow

The tri-kernel iterated to convergence over all cyberlinks runs continuously. It produces $\phi^*$: the persistent global focus distribution, what the entire network collectively knows, updated with every new link. This is the ground truth.

Path B -- Compiled Transformer

Architecture and weights derived analytically from the same graph. Executes $L^*$ tri-kernel steps over a local context window and converges to $\phi^*$ restricted to that context. This is the fast inference path.

3. Focus Flow Inference

$\phi^*$ is maintained continuously by the tru. For a query, the process is:

Step 1: Context particles become probability sources -- their energy terms are set so $\phi^*_\text{context}$ is elevated, making them attractors in the Boltzmann equilibrium.

Step 2: The tri-kernel reconverges incrementally from the current state -- probability mass flows from the seeded context particles through the cybergraph along structural paths.

Step 3: $\phi^*_\text{context}$ pools at particles that are semantically connected to the context via the graph topology.

Step 4: Sample the next particle from the high-probability region, add to context, reconverge.

No fresh initialization per step -- the system was already near $\phi^*$ before the query. Each step is a local recomputation within an $O(\log(1/\varepsilon))$-hop neighborhood of the newly added particle. Complexity per step: $O(|E| + |V|)$.

Context window is unbounded -- it is the entire cybergraph. Relevance is topological: a particle contributes if it is well-connected to the context regardless of linear position in token space.

4. The Local Update Rule

Every node reads only its neighbours and runs:

$$\Delta p_i = \eta\Big(\sum_{j \in \mathcal{N}(i)} w_{ij}(p_j - p_i) - \partial_{p_i}(\lambda E_{\text{diff},i} + \gamma C_i) + T(1 + \log p_i)\Big)$$

Gossip normalisation enforces $\sum_i p_i = 1$. Fully local, edge-only. This is what the tru runs every block -- the same computation a transformer performs in one layer, running collectively across the entire cybergraph.

5. Compiled Transformer Inference

The Mathematical Identity

Transformer attention is one step of tri-kernel diffusion:

$$\text{Attn}(Q, K, V) = \text{softmax}\!\left(\frac{QK^\top}{\sqrt{d}}\right)V$$

The softmax is the Boltzmann distribution with temperature $\sqrt{d}$. Probability mass flows from each query position toward compatible key positions and redistributes -- this is exactly one application of the diffusion operator $D$ from the tri-kernel over one agent's frozen context. Deep Equilibrium Models (Bai et al., 2019) showed that iterating a transformer layer to convergence reaches the same fixed point regardless of initialization. That fixed point is $\phi^*$ restricted to the context.

So $L^*$ transformer layers = $L^*$ steps of tri-kernel diffusion over the context. At query time:

1. Tokenize context into particles
2. Run $L^*$ layers of compiled attention -- each layer is one tri-kernel diffusion step over context
3. Output distribution = $\phi^*_\text{context}$, approximate to precision $\varepsilon$
4. Sample, add to context, repeat

Speed: $O(n^2 \cdot d^*)$ over context of length $n$, no graph traversal at runtime, weights frozen. Autoregressive generation -- familiar, fast, and analytically grounded.

Graph-Derived Architecture Parameters

Given $G = (P, N, E, w, \sigma)$, three graph properties determine the three free parameters of transformer architecture:

No hyperparameter search. The graph tells you what the transformer should be.

Weights are compiled, not trained. The embedding matrix $E^* = U_{:,1:d^*}$ -- top left singular vectors of $\text{diag}(\sqrt{\phi^*}) \cdot A$ -- is provably optimal by the Eckart-Young theorem: it uniquely minimizes expected squared gradient at step zero over all matrices of the same rank. Attention weights $W_Q^{(s)}, W_K^{(s)}$ are derived from the truncated SVD of each dialect's adjacency submatrix. MLP weights encode path co-occurrence statistics up to depth $L^*$.

Fine-tuning from this point learns only what the graph cannot encode: temporal patterns, implicit associations, contextual dynamics absent from the explicit graph. The reduction in required fine-tuning steps scales as $\Omega(|E| \cdot d^* / \log(1/\varepsilon))$ relative to random initialization.

The loop: $G \xrightarrow{\text{compile}} T_G \xrightarrow{\text{fine-tune}} T_G^* \xrightarrow{\text{extract implicit links}} \Delta G \xrightarrow{\text{stake}} G'$

6. Cyberank

The number the tru assigns to every particle -- probability of being observed by a random walking neuron. Cyberank is focus materialized as a per-particle score.

Fixed point of the tri-kernel: $\phi^* = \operatorname{norm}[\lambda_d \cdot D(\phi) + \lambda_s \cdot S(\phi) + \lambda_h \cdot H_\tau(\phi)]$. Integrates exploration (diffusion), structure (springs), and context (heat kernel). Convergence guaranteed by the collective focus theorem.

Feeds karma, syntropy, standard inference, and sorting in cyb. The fundamental factor of implicit knowledge.

7. The Compounding Property

Every cyberlink added:

• Shifts $\phi^*$ incrementally -- better focus flow inference now
• Increases $|E|$, raises $d^*$, may shrink $\text{diam}(G)$ -- better compiled transformer at next compilation
• Reduces approximation error $\varepsilon(G, c) = D_{\text{KL}}(\phi^*_c \| q^*_c)$ -- compiled inference closer to exact focus flow

The cybergraph is a compounding inference quality asset. Every link reduces the error of every compiled model that follows. See provably-optimal-initialization for the training reduction proof. See bostrom-to-onnx-pipeline for live compilation from the running network.

8. Stack

• cybergraph -- the substrate: particles as nodes, cyberlinks as typed edges
• tri-kernel -- the physics: diffusion + springs + heat kernel converge $\phi \to \phi^*$
• graph-native-transformer -- the compiled fast path: $d^*, h^*, L^*$ from graph structure
• nox -- the execution: 18 patterns (16 compute + call + look) over Goldilocks field
• foculus -- the consensus: $\phi^* > \tau$ finalizes particles without leaders
• tru -- the runner: computes cyberank, karma, syntropy every block

See collective focus theorem for convergence proof. See tri-kernel for why these three operators. See graph-native-transformer for compiled transformer derivation. See provably-optimal-initialization for the initialization optimality proof.