Focus Dynamics

focus is a single conserved quantity in CORE that serves three roles simultaneously. This unifies what other systems split into separate mechanisms—gas, stake, priority, reputation.

The mathematical foundation is formalized in the cft. This page specifies the engineering: conservation laws, flow equation, and convergence properties as implemented in the CORE substrate.

Conservation Laws

FOCUS CONSERVATION
──────────────────
Σᵢ focus(i) = 1   (always, enforced structurally)

Focus can:
  ✓ Flow between neurons (transfer)
  ✓ Be consumed (computation)
  ✓ Regenerate (from pool, proportional to balance)

Focus cannot:
  ✗ Be created from nothing
  ✗ Be destroyed (only redistributed)
  ✗ Exceed 1 in total


BALANCE CONSERVATION
────────────────────
Σᵢ balance(i) = B_total   (for non-minting transactions)

Enforced by polynomial commitment structure.
Invalid conservation → invalid state transition → rejected.


ENERGY CONSERVATION (Privacy Layer)
───────────────────────────────────
Σ(record values) = initial + minted - burned

Enforced by ZK circuit constraints.

Focus Flow Equation

CONTINUOUS FORM (for analysis)
──────────────────────────────
∂f/∂t = -∇·(D∇f) + R - C

  f = focus distribution vector
  D = diffusion tensor (derived from weights × balances)
  R = regeneration rate
  C = consumption rate

DISCRETE FORM (for implementation)
──────────────────────────────────
f'ᵢ = Σⱼ Pᵢⱼ · fⱼ + rᵢ - cᵢ

  Pᵢⱼ = wᵢⱼ · bⱼ / Σₖ wₖⱼ · bₖ

  wᵢⱼ = edge weight from i to j
  bⱼ  = balance of neuron j
  rᵢ  = regeneration for neuron i
  cᵢ  = consumption by neuron i

Convergence Theorem

Theorem: Focus dynamics converge to a unique stationary distribution π.

Proof: The transition matrix P is:

• Stochastic (rows sum to 1)
• Irreducible (graph is strongly connected by assumption)
• Aperiodic (self-loops or odd cycles exist)

By Perron-Frobenius theorem, there exists a unique π where: πP = π Σᵢ πᵢ = 1 πᵢ > 0 for all i

All initial distributions converge to π geometrically fast. ∎

Convergence rate: Determined by spectral gap λ = 1 - |λ₂| ‖f^(t) - π‖ ≤ C · (1-λ)^t

For the full probabilistic framework including axioms, proofs, and emergence theory, see cft.