Focus Dynamics
focus is the collective attention distribution over all particles in the cybergraph — content-particles and axon-particles. it is not a fuel, not a token, not a per-neuron resource. it is what the tri-kernel computes from the aggregate of all attention
How Focus Emerges
neurons lock balance to create will. will auto-distributes across cyberlinks, producing attention at target particles. the tri-kernel aggregates all attention into a single probability distribution φ* over all particles. this distribution is focus
| Layer | What | Per-what |
|---|---|---|
| balance | tokens held | neuron |
| will | locked balance × time | neuron |
| attention | will allocated to targets | neuron × particle |
| focus | collective attention | particle |
| cyberank / prob | focus read at a point | particle |
Conservation
Σᵢ focus(i) = 1 (always, enforced by normalization)
Focus sums to 1 because it is a probability distribution.
Emphasizing one particle defocuses all others.
This is not a separate conservation law — it is the
normalization step of the tri-kernel.
Focus Flow Equation
the tri-kernel composite operator:
$$\phi^{(t+1)} = \text{norm}\big[\lambda_d \cdot D(\phi^t) + \lambda_s \cdot S(\phi^t) + \lambda_h \cdot H_\tau(\phi^t)\big]$$
where:
- $D$ — diffusion (random walk exploration)
- $S$ — springs (structural constraints via screened Laplacian)
- $H_\tau$ — heat (multi-scale context smoothing)
the weights come from attention: each axon's weight is the sum of all neurons' attention directed along that edge
Convergence
the transition matrix P is stochastic, irreducible, aperiodic. by Perron-Frobenius theorem, a unique φ* exists:
$$\phi^* P = \phi^*, \quad \sum_i \phi^*_i = 1, \quad \phi^*_i > 0 \;\forall\, i$$
convergence rate determined by spectral gap: $\|\phi^{(t)} - \phi^*\| \leq C \cdot (1-\lambda)^t$
Balance and Energy Conservation
BALANCE CONSERVATION
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Σᵢ balance(i) = B_total (for non-minting transactions)
Enforced by polynomial commitment structure.
Invalid conservation → invalid state transition → rejected.
ENERGY CONSERVATION (Privacy Layer)
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Σ(record values) = initial + minted - burned
Enforced by ZK circuit constraints.
for the full probabilistic framework including axioms, proofs, and emergence theory, see collective focus theorem
see focus for the concept definition. see cyber/will for how will produces attention. see focus flow computation for the full protocol specification