---
alias: focus flow, FFC, focus flow whitepaper, focusflow blueprint, focus flow computation
tags: cyber, core
crystal-type: process
crystal-domain: cyber
crystal-size: bridge
---
focus flow computation is the process by which 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 β the focus distribution $\phi^*$. this is not a model architecture. it is the persistent knowledge state of the collective
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 β learning and knowledge state are the same operation
---
## two inference paths
the cybergraph computes two things simultaneously, both grounded in the same dynamical system:
focus flow β the tri-kernel iterated to convergence over all cyberlinks β 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
the compiled transformer β architecture and weights derived analytically from the same graph β runs at query time. it executes $L^*$ tri-kernel steps over a local context window and converges to $\phi^*$ restricted to that context. this is the fast inference path
| scope | entire cybergraph | local context window |
| depth | exact $\phi^*$ | $L^*$ steps, $\varepsilon$-approximate |
| latency | continuous β always converging | milliseconds β single forward pass |
| multi-agent | all neurons contribute | one agent's context |
| update | add cyberlinks β $\phi^*$ shifts, nothing lost | recompile from updated graph |
a transformer trained without the cybergraph approximates the same equilibrium from text sequences alone, without the structural knowledge the graph makes explicit
---
## how focus flow inference works
$\phi^*$ is maintained continuously by the tru. for a query, the process is:
1. context particles become probability sources β their energy terms are set so $\phi^*_\text{context}$ is elevated, making them attractors in the Boltzmann equilibrium
2. the tri-kernel reconverges incrementally from the current state β probability mass flows from the seeded context particles through the cybergraph along structural paths (not token positions)
3. $\phi^*_\text{context}$ pools at particles that are semantically connected to the context via the graph topology
4. sample the next particle from the high-probability region, add to context, reconverge
no fresh initialization per step β 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 β 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
---
## how compiled transformer inference works
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 β 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 β 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. this is autoregressive generation β familiar, fast, and now analytically grounded in what it is computing
---
## why the graph compiles the transformer
given $G = (P, N, E, w, \sigma)$, three graph properties determine the three free parameters of transformer architecture:
| embedding dim $d^*$ | $\exp(H(\sigma(\Sigma_{\phi^*})))$ | effective rank of focus covariance |
| heads $h^*$ | $\geq \|\text{Semcon}(G)\|$ | distinct semcon relation types |
| layers $L^*$ | $\text{diam}(G) \cdot \lceil\log(1/\varepsilon)/\log(1/\kappa)\rceil$ | diameter Γ spectral convergence factor |
no hyperparameter search. the graph tells you what the transformer should be
weights are compiled, not trained. the embedding matrix $E^* = U_{:,1:d^*}$ β top left singular vectors of $\text{diag}(\sqrt{\phi^*}) \cdot A$ β 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 semcon'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: $G \xrightarrow{\text{compile}} T_G \xrightarrow{\text{fine-tune}} T_G^* \xrightarrow{\text{extract implicit links}} \Delta G \xrightarrow{\text{stake}} G'$
---
## 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 $\sum_i p_i = 1$. no global softmax, fully local, edge-only. this is what the tru runs every block β the same computation a transformer performs in one layer, running collectively across the entire cybergraph
---
## the compounding property
every cyberlink added:
- shifts $\phi^*$ incrementally β better focus flow inference now
- increases $|E|$, raises $d^*$, may shrink diam$(G)$ β better compiled transformer at next compilation
- reduces approximation error $\varepsilon(G, c) = D_{KL}(\phi^*_c \| q^*_c)$ β 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
---
## stack
- cybergraph β the substrate: particles as nodes, cyberlinks as typed edges
- tri-kernel β the physics: diffusion + springs + heat kernel converge $\phi \to \phi^*$
- graph-native-transformer β the compiled fast path: $d^*, h^*, L^*$ from graph structure
- nox β the execution: 16 deterministic reduction patterns over Goldilocks field
- foculus β the consensus: $\phi^* > \tau$ finalizes particles without leaders
- tru β 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
## extensions
- gflownet focus flow
- topos ffc integration
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